Subgroups — #30822535

Order 17152.

1026 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
3	#22558664	C₂	1	4	yes	maximal, complemented
4	#22558638	C₂	1	4	yes	maximal, complemented
5	#22558612	C₂	1	4	yes	maximal, complemented
6	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	—	2	1	no	—
8	#16863449	—	2	1	no	—
9	#16863449	—	2	1	no	—
10	#16863449	C₂²	1	2	yes	—
11	#16863449	C₂²	1	2	yes	—
12	#16863420	—	2	1	no	—
13	#16863420	—	2	1	no	—
14	#16863420	C₂²	1	2	yes	—
15	#16863420	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
17	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
19	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
20	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
21	Q₈⋊(C₆₇⋊C₈)	C₂²	1	16	yes	complemented
22	#533×C₆₇	—	2	1	no	—
23	#533×C₆₇	—	2	1	no	—
24	#533×C₆₇	—	2	1	no	—
25	#533×C₆₇	—	2	1	no	—
26	#533×C₆₇	C₄	1	2	yes	—
27	#533×C₆₇	C₄	1	2	yes	—
28	#533×C₆₇	C₂²	1	2	yes	—
29	#533×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
31	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
32	C₂×#125×C₆₇	—	2	1	no	—
33	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
34	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
35	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
36	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
37	C₂×#117×C₆₇	—	2	1	no	—
38	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#11761612	—	2	2	no	—
40	C₂×#11761612	—	2	2	no	—
41	C₂×#11761612	—	4	4	no	—
42	C₂²×C₆₇⋊C₈	—	2	2	no	—
43	C₂²×C₆₇⋊C₈	—	2	2	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
45	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
48	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
49	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
50	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
51	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
52	C₄×C₆₇⋊C₈	—	2	2	no	—
53	C₄×C₆₇⋊C₈	C₂³	1	4	yes	—
54	C₂²×Q₈×C₆₇	—	2	1	no	—
55	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
56	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₇	—	2	1	no	—
60	C₄⋊Q₈×C₆₇	—	2	1	no	—
61	C₄⋊Q₈×C₆₇	—	2	1	no	—
62	C₄⋊Q₈×C₆₇	—	2	1	no	—
63	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
64	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
65	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
66	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
67	#125×C₆₇	—	2	2	no	—
68	#125×C₆₇	—	2	2	no	—
69	#125×C₆₇	—	2	2	no	—
70	#125×C₆₇	—	2	2	no	—
71	C₂²⋊Q₈×C₆₇	—	2	2	no	—
72	C₂²⋊Q₈×C₆₇	—	2	2	no	—
73	C₂²⋊Q₈×C₆₇	—	2	2	no	—
74	C₂²⋊Q₈×C₆₇	—	2	2	no	—
75	C₂²⋊Q₈×C₆₇	—	2	2	no	—
76	C₂²⋊Q₈×C₆₇	—	2	2	no	—
77	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
78	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
79	C₄×Q₈×C₆₇	—	2	2	no	—
80	C₄×Q₈×C₆₇	—	2	2	no	—
81	C₄×Q₈×C₆₇	—	2	2	no	—
82	C₄×Q₈×C₆₇	—	2	2	no	—
83	C₄×Q₈×C₆₇	—	2	2	no	—
84	C₄×Q₈×C₆₇	—	2	2	no	—
85	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
86	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
87	#117×C₆₇	—	2	2	no	—
88	#117×C₆₇	—	2	2	no	—
89	#117×C₆₇	—	2	2	no	—
90	#117×C₆₇	—	2	2	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₄²×C₆₇	—	2	2	no	—
99	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
108	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
111	#11761612	—	4	4	no	—
112	#11761612	—	4	4	no	—
113	C₂×C₆₇⋊C₈	—	2	4	no	—
114	C₂×C₆₇⋊C₈	—	2	4	no	—
115	C₂×C₆₇⋊C₈	—	4	4	no	—
116	C₂×C₆₇⋊C₈	—	4	4	no	—
117	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	—	2	2	no	—
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	—	4	4	no	—
122	C₂×Q₈×C₆₇	—	4	4	no	—
123	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	4	2	no	—
127	C₂²×C₄×C₆₇	—	4	2	no	—
128	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
138	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	2	2	no	—
151	C₄⋊C₄×C₆₇	—	2	2	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	4	no	—
155	C₄⋊C₄×C₆₇	—	2	4	no	—
156	C₄⋊C₄×C₆₇	—	2	4	no	—
157	C₄⋊C₄×C₆₇	—	2	2	no	—
158	C₄⋊C₄×C₆₇	—	2	4	no	—
159	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
160	C₂²⋊C₄×C₆₇	—	2	4	no	—
161	C₂²⋊C₄×C₆₇	—	2	4	no	—
162	C₄²×C₆₇	—	2	4	no	—
163	C₄²×C₆₇	—	2	4	no	—
164	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
165	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
166	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
167	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
168	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
169	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
172	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂³×C₆₇	—	4	1	no	—
181	C₂³×C₆₇	—	4	1	no	—
182	C₂³×C₆₇	#103	1	4	yes	—
183	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
184	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	Q₈×C₆₇	—	2	8	no	—
187	Q₈×C₆₇	—	4	4	no	—
188	Q₈×C₆₇	—	4	4	no	—
189	Q₈×C₆₇	—	4	4	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	4	1	no	—
199	C₂×C₄×C₆₇	—	4	1	no	—
200	C₂×C₄×C₆₇	—	4	1	no	—
201	C₂×C₄×C₆₇	—	4	1	no	—
202	C₂×C₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	4	1	no	—
204	C₂×C₄×C₆₇	—	4	4	no	—
205	C₂×C₄×C₆₇	—	4	4	no	—
206	C₂×C₄×C₆₇	#103	1	4	yes	—
207	C₂×C₄×C₆₇	#103	1	4	yes	—
208	C₂×C₄×C₆₇	#103	1	4	yes	—
209	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
219	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
220	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
221	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
222	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
223	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
224	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₆₇⋊C₈	—	4	4	no	—
228	C₆₇⋊C₈	—	4	4	no	—
229	C₂²×C₆₇	—	2	1	no	—
230	C₂²×C₆₇	—	2	1	no	—
231	C₂²×C₆₇	—	2	1	no	—
232	C₂²×C₆₇	—	2	1	no	—
233	C₂²×C₆₇	—	2	1	no	—
234	C₂²×C₆₇	—	2	1	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	C₂×#103	1	2	yes	—
237	C₂²×C₆₇	C₂×#103	1	2	yes	—
238	C₂²×C₆₇	C₂×#103	1	2	yes	—
239	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
240	C₄×C₆₇	—	2	1	no	—
241	C₄×C₆₇	—	2	1	no	—
242	C₄×C₆₇	—	2	1	no	—
243	C₄×C₆₇	—	2	1	no	—
244	C₄×C₆₇	—	2	1	no	—
245	C₄×C₆₇	—	2	1	no	—
246	C₄×C₆₇	—	4	4	no	—
247	C₄×C₆₇	—	4	4	no	—
248	C₄×C₆₇	C₂×#103	1	2	yes	—
249	C₄×C₆₇	C₂×#103	1	2	yes	—
250	C₄×C₆₇	C₂×#103	1	2	yes	—
251	C₄×C₆₇	#421	1	2	yes	—
252	C₄×C₆₇	#568	1	2	yes	—
253	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
254	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
255	#17486	—	67	1	no	maximal, Sylow
256	C₂×C₆₇	—	2	2	no	—
257	C₂×C₆₇	—	2	2	no	—
258	C₂×C₆₇	—	2	2	no	—
259	C₂×C₆₇	#2879	1	1	yes	characteristic
260	C₂×C₆₇	C₂²×#103	1	1	yes	characteristic
261	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
262	C₂×#533	D₆₇	1	1	yes	characteristic
263	C₂×#431	—	67	1	no	—
264	C₂×#423	—	67	1	no	—
265	#1493	—	67	4	no	—
266	#1487	—	67	4	no	—
267	#1481	—	67	4	no	—
268	C₆₇	#17486	1	1	yes	characteristic, complemented, Sylow
269	C₂²×M₁₆	—	134	1	no	—
270	#533	—	2	1	no	—
271	#533	—	2	1	no	—
272	#533	—	2	1	no	—
273	#533	—	2	1	no	—
274	#533	Dic₆₇	1	2	yes	—
275	#533	Dic₆₇	1	2	yes	—
276	#533	C₂×D₆₇	1	2	yes	—
277	#533	C₂×D₆₇	1	2	yes	—
278	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
279	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
280	C₂×#125	—	2	1	no	—
281	C₂×C₂²⋊Q₈	—	2	1	no	—
282	C₂×C₂²⋊Q₈	Dic₆₇	1	2	yes	—
283	C₂×C₄×Q₈	—	2	1	no	—
284	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
285	C₂×#117	—	2	1	no	—
286	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
287	#431	—	67	2	no	—
288	#431	—	67	2	no	—
289	#431	—	134	1	no	—
290	#431	—	134	1	no	—
291	#423	—	67	2	no	—
292	#423	—	67	2	no	—
293	#423	—	134	1	no	—
294	#423	—	134	1	no	—
295	C₂×C₄⋊C₈	—	67	2	no	—
296	C₂×C₄⋊C₈	—	134	1	no	—
297	C₂×C₄⋊C₈	—	134	1	no	—
298	C₂×C₂²⋊C₈	—	134	1	no	—
299	C₂×C₄×C₈	—	67	2	no	—
300	Q₈⋊C₈	—	67	16	no	—
301	C₂²×Q₈	—	2	1	no	—
302	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
303	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
304	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
305	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
306	C₂×M₁₆	—	268	4	no	—
307	C₂×M₁₆	—	134	2	no	—
308	C₂×M₁₆	—	134	2	no	—
309	C₂²×C₈	—	134	2	no	—
310	C₂²×C₈	—	134	2	no	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	—	2	1	no	—
313	C₄⋊Q₈	—	2	1	no	—
314	C₄⋊Q₈	—	2	1	no	—
315	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
316	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
317	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
318	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
319	#125	—	2	2	no	—
320	#125	—	2	2	no	—
321	#125	—	2	2	no	—
322	#125	—	2	2	no	—
323	C₂²⋊Q₈	—	2	2	no	—
324	C₂²⋊Q₈	—	2	2	no	—
325	C₂²⋊Q₈	—	2	2	no	—
326	C₂²⋊Q₈	—	2	2	no	—
327	C₂²⋊Q₈	—	2	2	no	—
328	C₂²⋊Q₈	—	2	2	no	—
329	C₂²⋊Q₈	C₂×Dic₆₇	1	4	yes	—
330	C₂²⋊Q₈	C₂×Dic₆₇	1	4	yes	—
331	C₄×Q₈	—	2	2	no	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×Q₈	—	2	2	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	—	2	2	no	—
336	C₄×Q₈	—	2	2	no	—
337	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
338	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
339	#117	—	2	2	no	—
340	#117	—	2	2	no	—
341	#117	—	2	2	no	—
342	#117	—	2	2	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	2	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₄²	—	2	2	no	—
351	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
352	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
353	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
354	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
355	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
356	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
357	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
358	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
359	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
360	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
361	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
362	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
363	C₄⋊C₈	—	67	4	no	—
364	C₄⋊C₈	—	134	2	no	—
365	C₄⋊C₈	—	134	2	no	—
366	C₄⋊C₈	—	134	2	no	—
367	C₄⋊C₈	—	134	2	no	—
368	C₄⋊C₈	—	134	2	no	—
369	C₂²⋊C₈	—	134	2	no	—
370	C₂²⋊C₈	—	134	2	no	—
371	C₄×C₈	—	67	4	no	—
372	C₄×C₈	—	134	2	no	—
373	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	4	4	no	—
378	C₂×Q₈	—	4	4	no	—
379	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	4	2	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
385	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
386	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
387	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
388	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
389	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
390	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
391	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
393	C₂²×C₄	#11761615	1	4	yes	—
394	C₂²×C₄	#11761618	1	4	yes	—
395	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
399	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
400	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
401	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
404	M₁₆	—	268	4	no	—
405	M₁₆	—	268	4	no	—
406	C₂×C₈	—	268	4	no	—
407	C₂×C₈	—	268	4	no	—
408	C₂×C₈	—	134	4	no	—
409	C₂×C₈	—	134	4	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₄²	—	2	4	no	—
425	C₄²	—	2	4	no	—
426	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
427	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
428	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
429	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
430	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
431	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
432	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
433	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
434	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
435	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
436	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
437	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
438	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
439	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
440	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
441	C₄²	C₂³×D₆₇	1	1	yes	characteristic
442	C₂³	—	4	1	no	—
443	C₂³	—	4	1	no	—
444	C₂³	#14523967	1	4	yes	—
445	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
446	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
447	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
448	Q₈	—	2	8	no	—
449	Q₈	—	4	4	no	—
450	Q₈	—	4	4	no	—
451	Q₈	—	4	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	4	no	—
467	C₂×C₄	—	4	4	no	—
468	C₂×C₄	#14523967	1	4	yes	—
469	C₂×C₄	#14523967	1	4	yes	—
470	C₂×C₄	#14523967	1	4	yes	—
471	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
472	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
473	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
474	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
475	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
476	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
477	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
478	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
479	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
480	C₂×C₄	C₂×#11761615	1	2	yes	—
481	C₂×C₄	C₂×#11761615	1	2	yes	—
482	C₂×C₄	C₂×#11761615	1	2	yes	—
483	C₂×C₄	C₂×#11761618	1	2	yes	—
484	C₂×C₄	C₂×#11761618	1	2	yes	—
485	C₂×C₄	C₂×#11761618	1	2	yes	—
486	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
487	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
488	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
489	C₈	—	268	4	no	—
490	C₈	—	268	4	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	1	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	4	4	no	—
498	C₂²	C₂×#14523967	1	2	yes	—
499	C₂²	C₂×#14523967	1	2	yes	—
500	C₂²	C₂×#14523967	1	2	yes	—
501	C₂²	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
502	C₄	—	2	1	no	—
503	C₄	—	2	1	no	—
504	C₄	—	2	1	no	—
505	C₄	—	2	1	no	—
506	C₄	—	2	1	no	—
507	C₄	—	2	1	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	4	no	—
510	C₄	C₂×#14523967	1	2	yes	—
511	C₄	C₂×#14523967	1	2	yes	—
512	C₄	C₂×#14523967	1	2	yes	—
513	C₄	#16863413	1	2	yes	—
514	C₄	#16864163	1	2	yes	—
515	C₄	C₂²×#11761615	1	1	yes	characteristic
516	C₄	C₂²×#11761618	1	1	yes	characteristic
517	C₂	—	2	2	no	—
518	C₂	—	2	2	no	—
519	C₂	—	2	2	no	—
520	C₂	#22568654	1	1	yes	characteristic
521	C₂	C₂²×#14523967	1	1	yes	characteristic
522	C₂	C₂×#16863413	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#533×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#533	D₆₇
hypocenter	C₆₇	#17486
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. #22558664	C₂	semidirect
4. #22558638	C₂	semidirect
5. #22558612	C₂	semidirect
21. Q₈⋊(C₆₇⋊C₈)	C₂²	semidirect
268. C₆₇	#17486	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863449	2
C₂×#16863420	2
#22558664	2
#22558638	2
#22558612	2
C₂×#533×C₆₇	2
#17486	67