Subgroups — C₆₇×#22454 #30701705

Order 17152.

1380 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#443×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2880	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1974	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1876	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1789	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	2	yes	—
10	C₂×#135×C₆₇	—	2	8	no	—
11	C₂×#135×C₆₇	C₄	1	1	yes	characteristic, complemented
12	C₂×#135×C₆₇	C₄	1	2	yes	complemented
13	C₂×#135×C₆₇	C₄	1	1	yes	characteristic, complemented
14	C₂×#135×C₆₇	C₂²	1	2	yes	—
15	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic, complemented
18	C₂²×SD₁₆×C₆₇	C₄	1	2	yes	complemented
19	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic, complemented
20	C₂×#131×C₆₇	C₂²	1	4	yes	complemented
21	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#443×C₆₇	—	2	8	no	—
23	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#104×C₆₇	C₂²	1	4	yes	complemented
25	C₂×#104×C₆₇	C₂²	1	4	yes	complemented
26	#408×C₆₇	—	2	4	no	—
27	#408×C₆₇	C₂²	1	4	yes	complemented
28	#407×C₆₇	—	2	2	no	—
29	#407×C₆₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
31	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
33	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	#341×C₆₇	C₂²	1	2	yes	complemented
35	#336×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#41×C₆₇	—	2	2	no	—
37	C₂×#41×C₆₇	—	2	2	no	—
38	C₂×#41×C₆₇	—	2	8	no	—
39	C₂×#41×C₆₇	—	4	4	no	—
40	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×#41×C₆₇	C₂³	1	2	yes	—
43	C₂×#41×C₆₇	C₂³	1	2	yes	—
44	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂³×C₄×C₆₇	—	2	2	no	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	#135×C₆₇	—	2	8	no	—
50	#135×C₆₇	—	2	4	no	—
51	#135×C₆₇	—	2	8	no	—
52	#135×C₆₇	—	2	4	no	—
53	#135×C₆₇	—	2	4	no	—
54	#135×C₆₇	D₄	1	8	yes	—
55	C₂×Q₁₆×C₆₇	—	2	4	no	—
56	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×SD₁₆×C₆₇	—	2	8	no	—
60	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×D₈×C₆₇	—	2	4	no	—
65	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	#131×C₆₇	—	2	8	no	—
69	#131×C₆₇	—	4	4	no	—
70	C₂×M₁₆×C₆₇	—	2	4	no	—
71	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
72	C₂²×C₈×C₆₇	—	2	4	no	—
73	C₂²×C₈×C₆₇	—	2	4	no	—
74	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
80	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
83	#108×C₆₇	—	2	4	no	—
84	#104×C₆₇	—	2	8	no	—
85	#104×C₆₇	—	4	4	no	—
86	C₂²⋊C₈×C₆₇	—	2	2	no	—
87	C₂²⋊C₈×C₆₇	—	2	2	no	—
88	C₂²⋊C₈×C₆₇	—	2	4	no	—
89	C₂²⋊C₈×C₆₇	—	2	4	no	—
90	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
91	C₂²⋊C₈×C₆₇	D₄	1	1	yes	characteristic
92	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
93	C₄×C₈×C₆₇	—	2	4	no	—
94	#95×C₆₇	—	2	1	no	—
95	C₂⁴×C₆₇	—	2	2	no	—
96	C₂⁴×C₆₇	#41	1	1	yes	characteristic
97	#41×C₆₇	—	2	4	no	—
98	#41×C₆₇	—	2	4	no	—
99	#41×C₆₇	—	2	4	no	—
100	#41×C₆₇	—	2	4	no	—
101	#41×C₆₇	—	2	4	no	—
102	#41×C₆₇	—	4	8	no	—
103	#41×C₆₇	—	4	2	no	—
104	#41×C₆₇	—	4	2	no	—
105	#41×C₆₇	—	4	8	no	—
106	#41×C₆₇	C₂×D₄	1	8	yes	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	2	4	no	—
109	C₂×Q₈×C₆₇	—	4	2	no	—
110	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	—	2	4	no	—
114	C₂×D₄×C₆₇	—	2	2	no	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	2	2	no	—
117	C₂×D₄×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	—	4	2	no	—
119	C₂×D₄×C₆₇	—	4	8	no	—
120	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
121	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	4	8	no	—
131	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	Q₁₆×C₆₇	—	2	4	no	—
138	Q₁₆×C₆₇	—	2	2	no	—
139	Q₁₆×C₆₇	C₂²⋊C₄	1	4	yes	complemented
140	SD₁₆×C₆₇	—	2	4	no	—
141	SD₁₆×C₆₇	—	2	4	no	—
142	SD₁₆×C₆₇	—	2	4	no	—
143	SD₁₆×C₆₇	C₂²⋊C₄	1	8	yes	complemented
144	D₈×C₆₇	—	2	4	no	—
145	D₈×C₆₇	—	2	2	no	—
146	D₈×C₆₇	C₂²⋊C₄	1	4	yes	complemented
147	M₁₆×C₆₇	—	2	4	no	—
148	M₁₆×C₆₇	—	2	4	no	—
149	M₁₆×C₆₇	—	4	8	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	4	no	—
158	C₂×C₈×C₆₇	—	4	8	no	—
159	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
164	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
166	C₂²⋊C₄×C₆₇	—	2	4	no	—
167	C₄²×C₆₇	—	2	2	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₂³×C₆₇	—	2	2	no	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	—	2	2	no	—
172	C₂³×C₆₇	—	2	2	no	—
173	C₂³×C₆₇	—	4	1	no	—
174	C₂³×C₆₇	—	4	4	no	—
175	C₂³×C₆₇	—	4	4	no	—
176	C₂³×C₆₇	—	8	2	no	—
177	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
178	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
179	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
180	Q₈×C₆₇	—	2	2	no	—
181	Q₈×C₆₇	—	2	2	no	—
182	Q₈×C₆₇	—	2	2	no	—
183	Q₈×C₆₇	—	4	4	no	—
184	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
185	D₄×C₆₇	—	2	8	no	—
186	D₄×C₆₇	—	2	2	no	—
187	D₄×C₆₇	—	2	2	no	—
188	D₄×C₆₇	—	2	2	no	—
189	D₄×C₆₇	—	4	4	no	—
190	D₄×C₆₇	—	4	4	no	—
191	D₄×C₆₇	—	4	4	no	—
192	D₄×C₆₇	—	4	4	no	—
193	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	1	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	8	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	4	4	no	—
206	C₂×C₄×C₆₇	—	4	4	no	—
207	C₂×C₄×C₆₇	—	4	4	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	—	4	4	no	—
211	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
217	C₈×C₆₇	—	2	1	no	—
218	C₈×C₆₇	—	2	1	no	—
219	C₈×C₆₇	—	2	2	no	—
220	C₈×C₆₇	—	2	2	no	—
221	C₈×C₆₇	—	4	4	no	—
222	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
223	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	—	2	1	no	—
229	C₂²×C₆₇	—	2	4	no	—
230	C₂²×C₆₇	—	2	1	no	—
231	C₂²×C₆₇	—	2	1	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	2	no	—
234	C₂²×C₆₇	—	4	2	no	—
235	C₂²×C₆₇	—	4	2	no	—
236	C₂²×C₆₇	—	4	2	no	—
237	C₂²×C₆₇	—	4	2	no	—
238	C₂²×C₆₇	—	8	8	no	—
239	C₂²×C₆₇	#443	1	2	yes	complemented
240	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
241	C₂²×C₆₇	#546	1	2	yes	—
242	C₄×C₆₇	—	2	1	no	—
243	C₄×C₆₇	—	2	1	no	—
244	C₄×C₆₇	—	2	4	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₄×C₆₇	—	4	2	no	—
247	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
248	C₄×C₆₇	#518	1	1	yes	characteristic
249	C₄×C₆₇	D₄²	1	2	yes	—
250	#22454	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂×C₆₇	—	2	2	no	—
252	C₂×C₆₇	—	2	2	no	—
253	C₂×C₆₇	—	4	1	no	—
254	C₂×C₆₇	—	4	4	no	—
255	C₂×C₆₇	—	8	2	no	—
256	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
258	C₂×C₆₇	#2957	1	1	yes	characteristic
259	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
260	C₂×#443	C₂×C₆₇	1	2	yes	complemented
261	#2880	C₂×C₆₇	1	2	yes	complemented
262	#1974	C₂×C₆₇	1	2	yes	complemented
263	#1876	C₂×C₆₇	1	4	yes	complemented
264	#1810	C₂×C₆₇	1	1	yes	characteristic, complemented
265	#1789	C₂×C₆₇	1	2	yes	complemented
266	C₈×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₆₇	#22454	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²×#41	C₂²×C₆₇	1	2	yes	—
269	C₂×#135	—	2	8	no	—
270	C₂×#135	C₄×C₆₇	1	1	yes	characteristic, complemented
271	C₂×#135	C₄×C₆₇	1	2	yes	complemented
272	C₂×#135	C₄×C₆₇	1	1	yes	characteristic, complemented
273	C₂×#135	C₂²×C₆₇	1	2	yes	—
274	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
276	C₂²×Q₁₆	C₄×C₆₇	1	1	yes	characteristic, complemented
277	C₂²×SD₁₆	C₄×C₆₇	1	2	yes	complemented
278	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic, complemented
279	C₂×#131	C₂²×C₆₇	1	4	yes	complemented
280	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
281	#443	—	2	8	no	—
282	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
283	C₂×#104	C₂²×C₆₇	1	4	yes	complemented
284	C₂×#104	C₂²×C₆₇	1	4	yes	complemented
285	#408	—	2	4	no	—
286	#408	C₂²×C₆₇	1	4	yes	complemented
287	#407	—	2	2	no	—
288	#407	C₂²×C₆₇	1	2	yes	—
289	C₂×C₂²⋊C₈	—	2	2	no	—
290	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
292	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
293	#341	C₂²×C₆₇	1	2	yes	complemented
294	#336	C₂²×C₆₇	1	2	yes	complemented
295	C₂×#41	—	2	2	no	—
296	C₂×#41	—	2	2	no	—
297	C₂×#41	—	2	8	no	—
298	C₂×#41	—	4	4	no	—
299	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
300	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
301	C₂×#41	C₂³×C₆₇	1	2	yes	—
302	C₂×#41	C₂³×C₆₇	1	2	yes	—
303	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
304	C₂²×D₄	—	2	2	no	—
305	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
306	C₂³×C₄	—	2	2	no	—
307	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
308	#135	—	2	8	no	—
309	#135	—	2	4	no	—
310	#135	—	2	8	no	—
311	#135	—	2	4	no	—
312	#135	—	2	4	no	—
313	#135	D₄×C₆₇	1	8	yes	—
314	C₂×Q₁₆	—	2	4	no	—
315	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
316	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
317	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
318	C₂×SD₁₆	—	2	8	no	—
319	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
320	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
321	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
322	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
323	C₂×D₈	—	2	4	no	—
324	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
325	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
326	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
327	#131	—	2	8	no	—
328	#131	—	4	4	no	—
329	C₂×M₁₆	—	2	4	no	—
330	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
331	C₂²×C₈	—	2	4	no	—
332	C₂²×C₈	—	2	4	no	—
333	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
334	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
335	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
336	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
339	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
340	C₂×C₂²⋊C₄	—	2	1	no	—
341	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
342	#108	—	2	4	no	—
343	#104	—	2	8	no	—
344	#104	—	4	4	no	—
345	C₂²⋊C₈	—	2	2	no	—
346	C₂²⋊C₈	—	2	2	no	—
347	C₂²⋊C₈	—	2	4	no	—
348	C₂²⋊C₈	—	2	4	no	—
349	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
350	C₂²⋊C₈	D₄×C₆₇	1	1	yes	characteristic
351	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
352	C₄×C₈	—	2	4	no	—
353	#95	—	2	1	no	—
354	C₂⁴	—	2	2	no	—
355	C₂⁴	#41×C₆₇	1	1	yes	characteristic
356	#41	—	2	4	no	—
357	#41	—	2	4	no	—
358	#41	—	2	4	no	—
359	#41	—	2	4	no	—
360	#41	—	2	4	no	—
361	#41	—	4	8	no	—
362	#41	—	4	2	no	—
363	#41	—	4	2	no	—
364	#41	—	4	8	no	—
365	#41	C₂×D₄×C₆₇	1	8	yes	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	4	no	—
368	C₂×Q₈	—	4	2	no	—
369	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
370	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	8	no	—
379	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
380	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	4	8	no	—
390	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
391	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
394	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
396	Q₁₆	—	2	4	no	—
397	Q₁₆	—	2	2	no	—
398	Q₁₆	C₂²⋊C₄×C₆₇	1	4	yes	complemented
399	SD₁₆	—	2	4	no	—
400	SD₁₆	—	2	4	no	—
401	SD₁₆	—	2	4	no	—
402	SD₁₆	C₂²⋊C₄×C₆₇	1	8	yes	complemented
403	D₈	—	2	4	no	—
404	D₈	—	2	2	no	—
405	D₈	C₂²⋊C₄×C₆₇	1	4	yes	complemented
406	M₁₆	—	2	4	no	—
407	M₁₆	—	2	4	no	—
408	M₁₆	—	4	8	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	4	no	—
417	C₂×C₈	—	4	8	no	—
418	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
419	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
420	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
421	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
422	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
423	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
424	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
425	C₂²⋊C₄	—	2	4	no	—
426	C₄²	—	2	2	no	—
427	C₄²	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	4	1	no	—
433	C₂³	—	4	4	no	—
434	C₂³	—	4	4	no	—
435	C₂³	—	8	2	no	—
436	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
437	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
438	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
439	Q₈	—	2	2	no	—
440	Q₈	—	2	2	no	—
441	Q₈	—	2	2	no	—
442	Q₈	—	4	4	no	—
443	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
444	D₄	—	2	8	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	2	no	—
448	D₄	—	4	4	no	—
449	D₄	—	4	4	no	—
450	D₄	—	4	4	no	—
451	D₄	—	4	4	no	—
452	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	8	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	4	no	—
466	C₂×C₄	—	4	4	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
471	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
472	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
473	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
474	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
475	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
476	C₈	—	2	1	no	—
477	C₈	—	2	1	no	—
478	C₈	—	2	2	no	—
479	C₈	—	2	2	no	—
480	C₈	—	4	4	no	—
481	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
482	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	2	1	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	2	1	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	8	8	no	—
498	C₂²	#443×C₆₇	1	2	yes	complemented
499	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
500	C₂²	#546×C₆₇	1	2	yes	—
501	C₄	—	2	1	no	—
502	C₄	—	2	1	no	—
503	C₄	—	2	4	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	2	no	—
506	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
507	C₄	#518×C₆₇	1	1	yes	characteristic
508	C₄	D₄²×C₆₇	1	2	yes	—
509	C₂	—	2	2	no	—
510	C₂	—	2	2	no	—
511	C₂	—	4	1	no	—
512	C₂	—	4	4	no	—
513	C₂	—	8	2	no	—
514	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
515	C₂	C₂×#443×C₆₇	1	1	yes	characteristic
516	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₂²⋊C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#443×C₆₇	C₂	semidirect
3. C₆₇×#2880	C₂	semidirect
4. C₆₇×#1974	C₂	semidirect
5. C₆₇×#1876	C₂	semidirect
6. C₆₇×#1810	C₂	semidirect
7. C₆₇×#1789	C₂	semidirect
8. C₈×C₂²⋊C₄×C₆₇	C₂	semidirect
11. C₂×#135×C₆₇	C₄	semidirect
12. C₂×#135×C₆₇	C₄	semidirect
13. C₂×#135×C₆₇	C₄	semidirect
17. C₂²×Q₁₆×C₆₇	C₄	semidirect
18. C₂²×SD₁₆×C₆₇	C₄	semidirect
19. C₂²×D₈×C₆₇	C₄	semidirect
20. C₂×#131×C₆₇	C₂²	semidirect
23. C₂×#108×C₆₇	C₂²	semidirect
24. C₂×#104×C₆₇	C₂²	semidirect
25. C₂×#104×C₆₇	C₂²	semidirect
27. #408×C₆₇	C₂²	semidirect
32. C₂×C₄×C₈×C₆₇	C₂²	semidirect
34. #341×C₆₇	C₂²	semidirect
35. #336×C₆₇	C₂²	semidirect
139. Q₁₆×C₆₇	C₂²⋊C₄	semidirect
143. SD₁₆×C₆₇	C₂²⋊C₄	semidirect
146. D₈×C₆₇	C₂²⋊C₄	semidirect
239. C₂²×C₆₇	#443	semidirect
250. #22454	C₆₇	direct (N × H)
260. C₂×#443	C₂×C₆₇	semidirect
261. #2880	C₂×C₆₇	semidirect
262. #1974	C₂×C₆₇	semidirect
263. #1876	C₂×C₆₇	semidirect
264. #1810	C₂×C₆₇	semidirect
265. #1789	C₂×C₆₇	semidirect
266. C₈×C₂²⋊C₄	C₂×C₆₇	semidirect
267. C₆₇	#22454	direct (N × H)
270. C₂×#135	C₄×C₆₇	semidirect
271. C₂×#135	C₄×C₆₇	semidirect
272. C₂×#135	C₄×C₆₇	semidirect
276. C₂²×Q₁₆	C₄×C₆₇	semidirect
277. C₂²×SD₁₆	C₄×C₆₇	semidirect
278. C₂²×D₈	C₄×C₆₇	semidirect
279. C₂×#131	C₂²×C₆₇	semidirect
282. C₂×#108	C₂²×C₆₇	semidirect
283. C₂×#104	C₂²×C₆₇	semidirect
284. C₂×#104	C₂²×C₆₇	semidirect
286. #408	C₂²×C₆₇	semidirect
291. C₂×C₄×C₈	C₂²×C₆₇	semidirect
293. #341	C₂²×C₆₇	semidirect
294. #336	C₂²×C₆₇	semidirect
398. Q₁₆	C₂²⋊C₄×C₆₇	semidirect
402. SD₁₆	C₂²⋊C₄×C₆₇	semidirect
405. D₈	C₂²⋊C₄×C₆₇	semidirect
498. C₂²	#443×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₇	2
C₂×#443×C₆₇	2
C₆₇×#2880	2
C₆₇×#1974	2
C₆₇×#1876	2
C₆₇×#1810	2
C₆₇×#1789	2
C₈×C₂²⋊C₄×C₆₇	2
#22454	67

Subgroups — C67×#22454 #30701705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22454 #30701705