Subgroups — #30874087

Order 17152.

868 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11761612	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863468	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863444	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14523978	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560984	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560982	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560414	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560414	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560413	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560413	C₂	1	1	yes	maximal, characteristic, complemented
11	#22560383	C₂	1	1	yes	maximal, characteristic, complemented
12	#22560249	C₂	1	1	yes	maximal, characteristic, complemented
13	#22560245	C₂	1	1	yes	maximal, characteristic, complemented
14	#22560116	C₂	1	1	yes	maximal, characteristic, complemented
15	D₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
16	C₂²×#11761612	C₂²	1	1	yes	characteristic
17	C₂²×#11761612	C₂²	1	1	yes	characteristic
18	C₂²×#11761612	C₂²	1	1	yes	characteristic
19	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	#16863474	—	2	1	no	—
21	#16863468	—	2	1	no	—
22	#16863468	—	2	1	no	—
23	#16863468	C₄	1	2	yes	—
24	#16863468	C₄	1	2	yes	—
25	#16863468	C₄	1	2	yes	—
26	#16863468	C₄	1	2	yes	—
27	D₄×C₆₇⋊C₈	—	2	1	no	—
28	#16863444	—	2	4	no	—
29	C₂×#14523978	—	2	4	no	—
30	C₂×#14523978	C₂²	1	1	yes	characteristic
31	C₂×#14523978	C₂²	1	1	yes	characteristic
32	C₂×#14523978	C₂²	1	1	yes	characteristic
33	C₂×#14523978	C₂²	1	1	yes	characteristic
34	C₂×#14523978	C₂²	1	1	yes	characteristic
35	C₂×#14523978	C₂²	1	1	yes	characteristic
36	C₂×#14523978	C₂²	1	1	yes	characteristic
37	C₂×#14523978	C₂²	1	1	yes	characteristic
38	C₂×#14523978	C₂²	1	1	yes	characteristic
39	C₂×#14523978	C₂²	1	1	yes	characteristic
40	#16863420	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
42	#16863360	C₂²	1	1	yes	characteristic
43	#16863360	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
46	C₄×#11761612	C₂²	1	1	yes	characteristic
47	C₂×#14523954	C₂²	1	1	yes	characteristic
48	#16863160	C₂²	1	1	yes	characteristic
49	#16863160	C₂²	1	1	yes	characteristic
50	#16863160	C₂²	1	1	yes	characteristic
51	#16863160	C₂²	1	1	yes	characteristic
52	#16863160	C₂²	1	1	yes	characteristic
53	#16863160	C₂²	1	1	yes	characteristic
54	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
55	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
56	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
57	#436×C₆₇	—	2	1	no	—
58	#435×C₆₇	—	2	1	no	—
59	#435×C₆₇	—	2	1	no	—
60	C₈×D₄×C₆₇	—	2	1	no	—
61	#423×C₆₇	C₂²	1	1	yes	characteristic
62	#407×C₆₇	C₂²	1	1	yes	characteristic
63	#407×C₆₇	C₂²	1	1	yes	characteristic
64	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
65	C₂×#11761612	—	2	2	no	—
66	C₂×#11761612	—	2	2	no	—
67	C₂×#11761612	—	2	2	no	—
68	C₂×#11761612	—	2	2	no	—
69	C₂×#11761612	—	2	1	no	—
70	C₂×#11761612	—	2	1	no	—
71	C₂×#11761612	—	4	1	no	—
72	C₂×#11761612	—	4	1	no	—
73	C₂×#11761612	C₂×C₄	1	2	yes	—
74	C₂×#11761612	C₂×C₄	1	2	yes	—
75	C₂×#11761612	C₂³	1	1	yes	characteristic
76	C₂×#11761612	C₂³	1	1	yes	characteristic
77	C₂×#11761612	C₂³	1	1	yes	characteristic
78	C₂×#11761612	C₂³	1	1	yes	characteristic
79	C₂²×C₆₇⋊C₈	—	2	2	no	—
80	C₂²×C₆₇⋊C₈	—	2	2	no	—
81	C₂²×C₆₇⋊C₈	—	2	1	no	—
82	C₂²×C₆₇⋊C₈	—	2	1	no	—
83	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
84	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
85	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
86	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
87	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
88	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
89	#14523978	—	2	2	no	—
90	#14523978	—	2	2	no	—
91	#14523978	—	2	2	no	—
92	#14523978	—	2	2	no	—
93	#14523978	—	2	2	no	—
94	#14523978	—	2	2	no	—
95	#14523978	—	2	2	no	—
96	#14523978	—	4	1	no	—
97	#14523978	—	4	1	no	—
98	#14523978	D₄	1	4	yes	complemented
99	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
100	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
101	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
102	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
103	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
104	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
105	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
106	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
107	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
108	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
109	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
110	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
111	#14523954	—	2	1	no	—
112	#14523954	C₂×C₄	1	2	yes	—
113	#14523954	C₂×C₄	1	2	yes	—
114	C₄×C₆₇⋊C₈	—	2	1	no	—
115	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
116	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
117	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×M₁₆×C₆₇	—	2	1	no	—
119	C₂×M₁₆×C₆₇	—	2	1	no	—
120	C₂×M₁₆×C₆₇	—	2	2	no	—
121	C₂×M₁₆×C₆₇	—	2	2	no	—
122	C₂×M₁₆×C₆₇	—	4	1	no	—
123	C₂×M₁₆×C₆₇	—	4	1	no	—
124	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
125	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
126	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
128	C₂²×C₈×C₆₇	—	2	1	no	—
129	C₂²×C₈×C₆₇	—	2	1	no	—
130	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
131	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
132	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
133	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
134	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
135	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
136	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
137	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
138	C₂×C₄⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
139	C₂×C₂²⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
140	C₂×C₂²⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
141	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
142	C₄⋊C₈×C₆₇	—	2	1	no	—
143	C₄⋊C₈×C₆₇	—	2	1	no	—
144	C₂²⋊C₈×C₆₇	—	2	1	no	—
145	C₂²⋊C₈×C₆₇	—	2	1	no	—
146	C₂²⋊C₈×C₆₇	—	2	1	no	—
147	C₂²⋊C₈×C₆₇	—	2	1	no	—
148	#97×C₆₇	—	2	1	no	—
149	C₄×C₈×C₆₇	—	2	1	no	—
150	#11761612	—	2	1	no	—
151	#11761612	—	2	4	no	—
152	#11761612	—	2	4	no	—
153	#11761612	—	2	1	no	—
154	#11761612	—	2	1	no	—
155	#11761612	—	2	1	no	—
156	#11761612	—	2	1	no	—
157	#11761612	—	2	1	no	—
158	#11761612	—	2	1	no	—
159	#11761612	—	4	2	no	—
160	#11761612	—	4	2	no	—
161	#11761612	C₂×D₄	1	2	yes	—
162	C₂×C₆₇⋊C₈	—	2	1	no	—
163	C₂×C₆₇⋊C₈	—	2	1	no	—
164	C₂×C₆₇⋊C₈	—	2	1	no	—
165	C₂×C₆₇⋊C₈	—	2	1	no	—
166	C₂×C₆₇⋊C₈	—	2	1	no	—
167	C₂×C₆₇⋊C₈	—	2	1	no	—
168	C₂×C₆₇⋊C₈	—	2	1	no	—
169	C₂×C₆₇⋊C₈	—	2	4	no	—
170	C₂×C₆₇⋊C₈	—	2	4	no	—
171	C₂×C₆₇⋊C₈	—	2	1	no	—
172	C₂×C₆₇⋊C₈	—	2	1	no	—
173	C₂×C₆₇⋊C₈	—	2	1	no	—
174	C₂×C₆₇⋊C₈	—	4	2	no	—
175	C₂×C₆₇⋊C₈	—	4	2	no	—
176	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
177	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
178	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
179	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
180	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
181	C₂×C₆₇⋊C₈	#41	1	2	yes	—
182	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
183	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
184	C₂×D₄×C₆₇	—	4	2	no	—
185	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
186	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
187	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
188	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	—	2	1	no	—
190	C₂²×C₄×C₆₇	—	2	1	no	—
191	C₂²×C₄×C₆₇	—	2	1	no	—
192	C₂²×C₄×C₆₇	—	2	1	no	—
193	C₂²×C₄×C₆₇	—	2	1	no	—
194	C₂²×C₄×C₆₇	—	2	1	no	—
195	C₂²×C₄×C₆₇	—	2	1	no	—
196	C₂²×C₄×C₆₇	—	2	1	no	—
197	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
198	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
199	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
200	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
205	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
206	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
207	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
208	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
209	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
210	M₁₆×C₆₇	—	2	1	no	—
211	M₁₆×C₆₇	—	2	1	no	—
212	M₁₆×C₆₇	—	2	1	no	—
213	M₁₆×C₆₇	—	2	1	no	—
214	M₁₆×C₆₇	—	2	1	no	—
215	M₁₆×C₆₇	—	2	1	no	—
216	M₁₆×C₆₇	—	2	1	no	—
217	M₁₆×C₆₇	—	4	2	no	—
218	M₁₆×C₆₇	—	4	2	no	—
219	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
220	C₂×C₈×C₆₇	—	2	1	no	—
221	C₂×C₈×C₆₇	—	2	1	no	—
222	C₂×C₈×C₆₇	—	2	1	no	—
223	C₂×C₈×C₆₇	—	2	1	no	—
224	C₂×C₈×C₆₇	—	4	2	no	—
225	C₂×C₈×C₆₇	—	4	2	no	—
226	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
227	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
228	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
229	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
230	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
231	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
232	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
233	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
234	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
235	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
236	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
237	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
238	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
239	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
240	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
241	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
242	C₂³×C₆₇	—	2	1	no	—
243	C₂³×C₆₇	—	2	1	no	—
244	C₂³×C₆₇	—	2	2	no	—
245	C₂³×C₆₇	—	2	1	no	—
246	C₂³×C₆₇	—	2	1	no	—
247	C₂³×C₆₇	—	4	1	no	—
248	C₂³×C₆₇	—	4	1	no	—
249	C₂³×C₆₇	—	4	1	no	—
250	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
251	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
252	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
253	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
254	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
255	D₄×C₆₇	—	2	2	no	—
256	D₄×C₆₇	—	2	2	no	—
257	D₄×C₆₇	—	4	1	no	—
258	D₄×C₆₇	—	4	1	no	—
259	C₂×C₄×C₆₇	—	2	1	no	—
260	C₂×C₄×C₆₇	—	2	1	no	—
261	C₂×C₄×C₆₇	—	2	1	no	—
262	C₂×C₄×C₆₇	—	2	1	no	—
263	C₂×C₄×C₆₇	—	2	1	no	—
264	C₂×C₄×C₆₇	—	2	1	no	—
265	C₂×C₄×C₆₇	—	2	1	no	—
266	C₂×C₄×C₆₇	—	2	1	no	—
267	C₂×C₄×C₆₇	—	2	1	no	—
268	C₂×C₄×C₆₇	—	2	1	no	—
269	C₂×C₄×C₆₇	—	2	1	no	—
270	C₂×C₄×C₆₇	—	2	1	no	—
271	C₂×C₄×C₆₇	—	2	1	no	—
272	C₂×C₄×C₆₇	—	2	1	no	—
273	C₂×C₄×C₆₇	—	2	1	no	—
274	C₂×C₄×C₆₇	—	2	1	no	—
275	C₂×C₄×C₆₇	—	2	1	no	—
276	C₂×C₄×C₆₇	—	2	1	no	—
277	C₂×C₄×C₆₇	—	2	1	no	—
278	C₂×C₄×C₆₇	—	2	1	no	—
279	C₂×C₄×C₆₇	—	4	1	no	—
280	C₂×C₄×C₆₇	—	4	1	no	—
281	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
282	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
283	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
284	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
285	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
286	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
287	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
288	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
289	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
290	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
293	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
294	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
295	C₈×C₆₇	—	2	2	no	—
296	C₈×C₆₇	—	4	1	no	—
297	C₈×C₆₇	—	4	1	no	—
298	C₈×C₆₇	—	4	1	no	—
299	C₆₇⋊C₈	—	2	2	no	—
300	C₆₇⋊C₈	—	2	2	no	—
301	C₆₇⋊C₈	—	2	2	no	—
302	C₆₇⋊C₈	—	2	2	no	—
303	C₆₇⋊C₈	—	4	1	no	—
304	C₆₇⋊C₈	—	4	1	no	—
305	C₆₇⋊C₈	—	4	1	no	—
306	C₆₇⋊C₈	C₄×D₄	1	4	yes	—
307	C₂²×C₆₇	—	2	1	no	—
308	C₂²×C₆₇	—	2	1	no	—
309	C₂²×C₆₇	—	2	1	no	—
310	C₂²×C₆₇	—	2	1	no	—
311	C₂²×C₆₇	—	2	1	no	—
312	C₂²×C₆₇	—	2	1	no	—
313	C₂²×C₆₇	—	2	1	no	—
314	C₂²×C₆₇	—	2	1	no	—
315	C₂²×C₆₇	—	2	1	no	—
316	C₂²×C₆₇	—	2	1	no	—
317	C₂²×C₆₇	—	2	1	no	—
318	C₂²×C₆₇	—	2	1	no	—
319	C₂²×C₆₇	—	2	1	no	—
320	C₂²×C₆₇	—	4	2	no	—
321	C₂²×C₆₇	—	4	2	no	—
322	C₂²×C₆₇	—	8	1	no	—
323	C₂²×C₆₇	—	8	1	no	—
324	C₂²×C₆₇	#430	1	2	yes	complemented
325	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
326	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
327	C₂²×C₆₇	#554	1	1	yes	characteristic
328	C₄×C₆₇	—	2	1	no	—
329	C₄×C₆₇	—	2	1	no	—
330	C₄×C₆₇	—	2	1	no	—
331	C₄×C₆₇	—	2	1	no	—
332	C₄×C₆₇	—	2	1	no	—
333	C₄×C₆₇	—	2	1	no	—
334	C₄×C₆₇	—	2	1	no	—
335	C₄×C₆₇	—	2	1	no	—
336	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
337	C₄×C₆₇	#520	1	1	yes	characteristic
338	C₄×C₆₇	D₄²	1	1	yes	characteristic
339	C₄×C₆₇	#547	1	1	yes	characteristic
340	#22668	—	67	1	no	maximal, Sylow
341	C₂×C₆₇	—	2	1	no	—
342	C₂×C₆₇	—	2	1	no	—
343	C₂×C₆₇	—	2	2	no	—
344	C₂×C₆₇	—	4	1	no	—
345	C₂×C₆₇	—	4	1	no	—
346	C₂×C₆₇	—	4	1	no	—
347	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
348	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
349	C₂×C₆₇	#2916	1	1	yes	characteristic
350	D₄×M₁₆	—	67	1	no	—
351	D₄×M₁₆	D₆₇	1	1	yes	characteristic
352	C₂×#435	—	67	1	no	—
353	C₂×#430	—	67	1	no	—
354	C₂²×#108	—	67	1	no	—
355	#1950	—	67	1	no	—
356	#1950	—	67	1	no	—
357	#1855	—	67	1	no	—
358	#1855	—	67	1	no	—
359	#1855	—	67	1	no	—
360	#1855	—	67	1	no	—
361	#1849	—	67	1	no	—
362	#1811	—	67	1	no	—
363	#1810	—	67	1	no	—
364	#1778	—	67	1	no	—
365	C₆₇	#22668	1	1	yes	characteristic, complemented, Sylow
366	C₂²×M₁₆	—	67	1	no	—
367	C₂²×M₁₆	—	67	1	no	—
368	C₂²×M₁₆	—	67	1	no	—
369	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
370	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
371	C₂³×C₈	—	67	1	no	—
372	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
373	#436	—	2	1	no	—
374	#436	—	134	1	no	—
375	#435	—	2	1	no	—
376	#435	—	2	1	no	—
377	#435	—	67	2	no	—
378	#435	—	67	2	no	—
379	#435	—	67	2	no	—
380	#435	—	67	2	no	—
381	#435	—	134	1	no	—
382	#435	—	134	1	no	—
383	C₈×D₄	—	2	1	no	—
384	C₈×D₄	—	134	1	no	—
385	#430	—	134	4	no	—
386	C₂×#108	—	67	1	no	—
387	C₂×#108	—	67	1	no	—
388	C₂×#108	—	67	1	no	—
389	C₂×#108	—	67	1	no	—
390	C₂×#108	—	67	1	no	—
391	C₂×#108	—	67	1	no	—
392	C₂×#108	—	67	1	no	—
393	C₂×#108	—	67	1	no	—
394	C₂×#108	—	67	1	no	—
395	C₂×#108	—	67	1	no	—
396	C₂×#108	—	134	4	no	—
397	#423	—	67	1	no	—
398	#423	C₂×D₆₇	1	1	yes	characteristic
399	C₂×C₄⋊C₈	—	67	1	no	—
400	#407	—	67	1	no	—
401	#407	—	67	1	no	—
402	#407	C₂×D₆₇	1	1	yes	characteristic
403	#407	C₂×D₆₇	1	1	yes	characteristic
404	C₂×C₂²⋊C₈	—	67	1	no	—
405	C₂×C₂²⋊C₈	—	67	1	no	—
406	C₄×M₁₆	—	67	1	no	—
407	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
408	C₂×#97	—	67	1	no	—
409	#341	—	67	1	no	—
410	#341	—	67	1	no	—
411	#341	—	67	1	no	—
412	#341	—	67	1	no	—
413	#341	—	67	1	no	—
414	#341	—	67	1	no	—
415	C₂²×D₄	Dic₁₃₄	1	1	yes	characteristic
416	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
417	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
418	C₂×M₁₆	—	2	1	no	—
419	C₂×M₁₆	—	2	1	no	—
420	C₂×M₁₆	—	2	2	no	—
421	C₂×M₁₆	—	2	2	no	—
422	C₂×M₁₆	—	4	1	no	—
423	C₂×M₁₆	—	4	1	no	—
424	C₂×M₁₆	—	268	1	no	—
425	C₂×M₁₆	—	268	1	no	—
426	C₂×M₁₆	—	67	2	no	—
427	C₂×M₁₆	—	67	2	no	—
428	C₂×M₁₆	—	67	1	no	—
429	C₂×M₁₆	—	67	1	no	—
430	C₂×M₁₆	—	67	1	no	—
431	C₂×M₁₆	—	67	1	no	—
432	C₂×M₁₆	—	134	2	no	—
433	C₂×M₁₆	—	134	2	no	—
434	C₂×M₁₆	—	134	2	no	—
435	C₂×M₁₆	—	134	2	no	—
436	C₂×M₁₆	—	134	1	no	—
437	C₂×M₁₆	—	134	1	no	—
438	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
439	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
440	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
441	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
442	C₂²×C₈	—	2	1	no	—
443	C₂²×C₈	—	2	1	no	—
444	C₂²×C₈	—	67	1	no	—
445	C₂²×C₈	—	67	2	no	—
446	C₂²×C₈	—	67	2	no	—
447	C₂²×C₈	—	67	1	no	—
448	C₂²×C₈	—	67	1	no	—
449	C₂²×C₈	—	67	1	no	—
450	C₂²×C₈	—	134	1	no	—
451	C₂²×C₈	—	134	1	no	—
452	C₂²×C₈	—	134	2	no	—
453	C₂²×C₈	—	134	2	no	—
454	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
455	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
456	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
457	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
458	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
459	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
460	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
461	C₄×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
462	C₂×C₄⋊C₄	Dic₁₃₄	1	1	yes	characteristic
463	C₂×C₂²⋊C₄	Dic₁₃₄	1	1	yes	characteristic
464	C₂×C₂²⋊C₄	Dic₁₃₄	1	1	yes	characteristic
465	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
466	#108	—	268	1	no	—
467	#108	—	268	1	no	—
468	#108	—	67	4	no	—
469	#108	—	134	2	no	—
470	#108	—	134	2	no	—
471	#108	—	134	2	no	—
472	#108	—	134	2	no	—
473	#108	—	134	2	no	—
474	#108	—	134	2	no	—
475	#108	—	134	2	no	—
476	C₄⋊C₈	—	2	1	no	—
477	C₄⋊C₈	—	2	1	no	—
478	C₄⋊C₈	—	67	2	no	—
479	C₄⋊C₈	—	67	2	no	—
480	C₄⋊C₈	—	134	1	no	—
481	C₄⋊C₈	—	134	1	no	—
482	C₂²⋊C₈	—	2	1	no	—
483	C₂²⋊C₈	—	2	1	no	—
484	C₂²⋊C₈	—	2	1	no	—
485	C₂²⋊C₈	—	2	1	no	—
486	C₂²⋊C₈	—	67	2	no	—
487	C₂²⋊C₈	—	67	2	no	—
488	C₂²⋊C₈	—	67	2	no	—
489	C₂²⋊C₈	—	67	2	no	—
490	C₂²⋊C₈	—	134	1	no	—
491	C₂²⋊C₈	—	134	1	no	—
492	C₂²⋊C₈	—	134	1	no	—
493	C₂²⋊C₈	—	134	1	no	—
494	#97	—	2	1	no	—
495	#97	—	67	2	no	—
496	#97	—	67	2	no	—
497	#97	—	134	1	no	—
498	C₄×C₈	—	2	1	no	—
499	C₄×C₈	—	134	1	no	—
500	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
501	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
502	C₂×D₄	—	4	2	no	—
503	C₂×D₄	Dic₆₇⋊C₄	1	1	yes	characteristic
504	C₂×D₄	Dic₆₇⋊C₄	1	1	yes	characteristic
505	C₂×D₄	Dic₆₇⋊C₄	1	1	yes	characteristic
506	C₂×D₄	Dic₆₇⋊C₄	1	1	yes	characteristic
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
516	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
517	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
518	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
519	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
520	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
521	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
522	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
523	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
524	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
525	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
526	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
527	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
528	M₁₆	—	2	1	no	—
529	M₁₆	—	2	1	no	—
530	M₁₆	—	2	1	no	—
531	M₁₆	—	2	1	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	—	2	1	no	—
534	M₁₆	—	2	1	no	—
535	M₁₆	—	4	2	no	—
536	M₁₆	—	4	2	no	—
537	M₁₆	—	268	2	no	—
538	M₁₆	—	268	2	no	—
539	M₁₆	—	67	2	no	—
540	M₁₆	—	134	1	no	—
541	M₁₆	—	134	1	no	—
542	M₁₆	—	134	1	no	—
543	M₁₆	—	134	1	no	—
544	M₁₆	—	134	1	no	—
545	M₁₆	—	134	1	no	—
546	M₁₆	—	134	4	no	—
547	M₁₆	—	134	4	no	—
548	M₁₆	—	134	1	no	—
549	M₁₆	D₄×D₆₇	1	2	yes	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	4	2	no	—
555	C₂×C₈	—	4	2	no	—
556	C₂×C₈	—	268	2	no	—
557	C₂×C₈	—	268	2	no	—
558	C₂×C₈	—	67	2	no	—
559	C₂×C₈	—	67	2	no	—
560	C₂×C₈	—	67	2	no	—
561	C₂×C₈	—	67	2	no	—
562	C₂×C₈	—	67	2	no	—
563	C₂×C₈	—	67	2	no	—
564	C₂×C₈	—	134	1	no	—
565	C₂×C₈	—	134	1	no	—
566	C₂×C₈	—	134	1	no	—
567	C₂×C₈	—	134	4	no	—
568	C₂×C₈	—	134	4	no	—
569	C₂×C₈	—	134	1	no	—
570	C₂×C₈	—	134	1	no	—
571	C₂×C₈	—	134	1	no	—
572	C₂×C₈	—	134	1	no	—
573	C₂×C₈	—	134	1	no	—
574	C₂×C₈	—	134	1	no	—
575	C₂×C₈	—	134	1	no	—
576	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
577	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
578	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
579	C₄⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
580	C₂²⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
581	C₂²⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
582	C₂²⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
583	C₂²⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
584	C₂²⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
585	C₂²⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
586	C₂²⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
587	C₂²⋊C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
588	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
589	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
590	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
591	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
592	C₂³	—	2	1	no	—
593	C₂³	—	2	1	no	—
594	C₂³	—	2	2	no	—
595	C₂³	—	2	1	no	—
596	C₂³	—	2	1	no	—
597	C₂³	—	4	1	no	—
598	C₂³	—	4	1	no	—
599	C₂³	—	4	1	no	—
600	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
601	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
602	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
603	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
604	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
605	D₄	—	2	2	no	—
606	D₄	—	2	2	no	—
607	D₄	—	4	1	no	—
608	D₄	—	4	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
632	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
633	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
634	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
635	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
636	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
637	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
638	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
639	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
640	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
641	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
642	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
643	C₂×C₄	#14524088	1	1	yes	characteristic
644	C₂×C₄	#14524090	1	1	yes	characteristic
645	C₈	—	2	2	no	—
646	C₈	—	4	1	no	—
647	C₈	—	4	1	no	—
648	C₈	—	4	1	no	—
649	C₈	—	268	1	no	—
650	C₈	—	268	1	no	—
651	C₈	—	268	1	no	—
652	C₈	—	67	4	no	—
653	C₈	—	134	2	no	—
654	C₈	—	134	2	no	—
655	C₈	—	134	2	no	—
656	C₈	—	134	2	no	—
657	C₂²	—	2	1	no	—
658	C₂²	—	2	1	no	—
659	C₂²	—	2	1	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	1	no	—
663	C₂²	—	2	1	no	—
664	C₂²	—	2	1	no	—
665	C₂²	—	2	1	no	—
666	C₂²	—	2	1	no	—
667	C₂²	—	2	1	no	—
668	C₂²	—	2	1	no	—
669	C₂²	—	2	1	no	—
670	C₂²	—	4	2	no	—
671	C₂²	—	4	2	no	—
672	C₂²	—	8	1	no	—
673	C₂²	—	8	1	no	—
674	C₂²	#16863444	1	2	yes	complemented
675	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
676	C₂²	D₄×Dic₁₃₄	1	1	yes	characteristic
677	C₂²	#16864105	1	1	yes	characteristic
678	C₄	—	2	1	no	—
679	C₄	—	2	1	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	1	no	—
682	C₄	—	2	1	no	—
683	C₄	—	2	1	no	—
684	C₄	—	2	1	no	—
685	C₄	—	2	1	no	—
686	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
687	C₄	#16863875	1	1	yes	characteristic
688	C₄	D₄×C₂²⋊D₆₇	1	1	yes	characteristic
689	C₄	#16864045	1	1	yes	characteristic
690	C₂	—	2	1	no	—
691	C₂	—	2	1	no	—
692	C₂	—	2	2	no	—
693	C₂	—	4	1	no	—
694	C₂	—	4	1	no	—
695	C₂	—	4	1	no	—
696	C₂	D₄×Dic₆₇⋊C₄	1	1	yes	characteristic
697	C₂	C₂×#16863444	1	1	yes	characteristic
698	C₂	#22568818	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	D₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₆₇	Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	D₄×M₁₆	D₆₇
hypocenter	C₆₇	#22668
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ D₄×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×#16863444	C₂	semidirect
4. C₂²×#14523978	C₂	semidirect
5. #22560984	C₂	semidirect
6. #22560982	C₂	semidirect
7. #22560414	C₂	semidirect
8. #22560414	C₂	semidirect
9. #22560413	C₂	semidirect
10. #22560413	C₂	semidirect
11. #22560383	C₂	semidirect
12. #22560249	C₂	semidirect
13. #22560245	C₂	semidirect
14. #22560116	C₂	semidirect
98. #14523978	D₄	semidirect
324. C₂²×C₆₇	#430	semidirect
365. C₆₇	#22668	semidirect
674. C₂²	#16863444	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₄×#11761612	2
C₂×#16863468	2
C₂×#16863444	2
C₂²×#14523978	2
#22560984	2
#22560982	2
#22560414	2
#22560413	2
#22560383	2
#22560249	2
#22560245	2
#22560116	2
D₄×M₁₆×C₆₇	2
#22668	67