Subgroups — #30873626

Order 17152.

744 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562613	C₂	1	1	yes	maximal, characteristic
2	#22560926	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560924	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560772	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560768	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560767	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560192	C₂	1	1	yes	maximal, characteristic
8	#22560190	C₂	1	1	yes	maximal, characteristic
9	#22560188	C₂	1	1	yes	maximal, characteristic
10	#22560077	C₂	1	1	yes	maximal, characteristic, complemented
11	#22560070	C₂	1	1	yes	maximal, characteristic, complemented
12	#22559978	C₂	1	1	yes	maximal, characteristic
13	#22559974	C₂	1	1	yes	maximal, characteristic
14	#22559966	C₂	1	1	yes	maximal, characteristic
15	C₆₇×#1917	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14524000	C₂²	1	1	yes	characteristic
17	C₂×#14524000	C₂²	1	1	yes	characteristic
18	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×(C₂×D₆₇)⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×#14523953	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
33	#16863265	C₄	1	1	yes	characteristic
34	#16863265	C₄	1	1	yes	characteristic
35	#16863260	C₄	1	1	yes	characteristic
36	#16863260	C₄	1	1	yes	characteristic
37	#16863255	C₄	1	1	yes	characteristic
38	#16863255	C₄	1	1	yes	characteristic
39	#16863253	C₄	1	1	yes	characteristic
40	#16863253	C₄	1	1	yes	characteristic
41	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
42	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
44	#16863151	C₂²	1	1	yes	characteristic
45	#16863151	C₂²	1	1	yes	characteristic
46	#16863151	C₂²	1	1	yes	characteristic
47	#16863151	C₂²	1	1	yes	characteristic
48	#16863150	C₂²	1	1	yes	characteristic
49	#16863150	C₂²	1	1	yes	characteristic
50	#16863150	C₂²	1	1	yes	characteristic
51	#16863150	C₂²	1	1	yes	characteristic
52	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
54	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
55	#336×C₆₇	C₂²	1	1	yes	characteristic
56	#336×C₆₇	C₂²	1	1	yes	characteristic
57	#336×C₆₇	C₂²	1	1	yes	characteristic
58	#336×C₆₇	C₂²	1	1	yes	characteristic
59	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
63	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
64	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
66	#14524000	—	4	1	no	—
67	#14524000	—	4	1	no	—
68	#14524000	—	4	1	no	—
69	#14524000	—	4	1	no	—
70	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
77	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₂×Dic₆₇⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
89	C₄²×D₆₇	—	2	2	no	—
90	C₄²×D₆₇	—	2	2	no	—
91	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
92	Dic₆₇⋊C₈	—	2	2	no	—
93	Dic₆₇⋊C₈	—	2	2	no	—
94	Dic₆₇⋊C₈	—	4	1	no	—
95	Dic₆₇⋊C₈	—	4	1	no	—
96	(C₂×D₆₇)⋊C₈	—	2	2	no	—
97	(C₂×D₆₇)⋊C₈	—	2	2	no	—
98	(C₂×D₆₇)⋊C₈	—	2	2	no	—
99	(C₂×D₆₇)⋊C₈	Q₈	1	4	yes	—
100	#14523953	—	4	1	no	—
101	C₄×C₆₇⋊C₈	—	4	1	no	—
102	C₈×Dic₆₇	—	2	2	no	—
103	C₈×Dic₆₇	—	2	2	no	—
104	C₈×Dic₆₇	—	4	1	no	—
105	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
106	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
107	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
108	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
109	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
110	C₆₇⋊#95	C₂×C₄	1	1	yes	characteristic
111	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
112	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
113	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	C₄⋊C₈×C₆₇	—	4	1	no	—
121	#97×C₆₇	—	4	1	no	—
122	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
123	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
124	C₂³×D₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄×D₆₇	—	2	1	no	—
133	C₂×C₄×D₆₇	—	2	1	no	—
134	C₂×C₄×D₆₇	—	2	1	no	—
135	C₂×C₄×D₆₇	—	2	1	no	—
136	C₂×C₄×D₆₇	—	2	2	no	—
137	C₂×C₄×D₆₇	—	2	2	no	—
138	C₂×C₄×D₆₇	—	2	2	no	—
139	C₂×C₄×D₆₇	—	2	2	no	—
140	C₂×C₄×D₆₇	—	2	2	no	—
141	C₂×C₄×D₆₇	—	4	1	no	—
142	C₂×C₄×D₆₇	C₂×Q₈	1	2	yes	—
143	C₂×C₄×D₆₇	#41	1	2	yes	—
144	C₂×C₆₇⋊C₈	—	2	1	no	—
145	C₂×C₆₇⋊C₈	—	2	1	no	—
146	C₂×C₆₇⋊C₈	—	2	1	no	—
147	C₂×C₆₇⋊C₈	—	2	1	no	—
148	C₂×C₆₇⋊C₈	—	2	1	no	—
149	C₂×C₆₇⋊C₈	—	2	1	no	—
150	C₂×C₆₇⋊C₈	—	2	1	no	—
151	C₂×C₆₇⋊C₈	—	2	1	no	—
152	C₂×C₆₇⋊C₈	—	2	1	no	—
153	C₂×C₆₇⋊C₈	—	2	1	no	—
154	C₂×C₆₇⋊C₈	—	2	1	no	—
155	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
156	C₄⋊Dic₆₇	—	2	2	no	—
157	C₄⋊Dic₆₇	—	4	1	no	—
158	Dic₆₇⋊C₄	—	2	2	no	—
159	Dic₆₇⋊C₄	—	2	2	no	—
160	Dic₆₇⋊C₄	—	2	2	no	—
161	Dic₆₇⋊C₄	—	2	2	no	—
162	Dic₆₇⋊C₄	—	4	1	no	—
163	Dic₆₇⋊C₄	—	4	1	no	—
164	Dic₆₇⋊C₄	—	4	1	no	—
165	Dic₆₇⋊C₄	—	4	1	no	—
166	(C₂×D₆₇)⋊C₄	—	4	1	no	—
167	(C₂×D₆₇)⋊C₄	—	4	1	no	—
168	(C₂×D₆₇)⋊C₄	—	4	1	no	—
169	(C₂×D₆₇)⋊C₄	—	4	1	no	—
170	C₄×Dic₆₇	—	2	1	no	—
171	C₄×Dic₆₇	—	2	1	no	—
172	C₄×Dic₆₇	—	2	1	no	—
173	C₄×Dic₆₇	—	2	1	no	—
174	C₄×Dic₆₇	—	2	2	no	—
175	C₄×Dic₆₇	—	2	2	no	—
176	C₄×Dic₆₇	#41	1	2	yes	—
177	C₄×Dic₆₇	#41	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂×C₈×C₆₇	—	2	1	no	—
186	C₂×C₈×C₆₇	—	2	1	no	—
187	C₂×C₈×C₆₇	—	2	1	no	—
188	C₂×C₈×C₆₇	—	2	1	no	—
189	C₂×C₈×C₆₇	—	2	1	no	—
190	C₂×C₈×C₆₇	—	2	1	no	—
191	C₂×C₈×C₆₇	—	2	1	no	—
192	C₂×C₈×C₆₇	—	2	1	no	—
193	C₂×C₈×C₆₇	—	2	1	no	—
194	C₂×C₈×C₆₇	—	2	1	no	—
195	C₂×C₈×C₆₇	—	2	1	no	—
196	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
197	C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₄⋊C₄×C₆₇	—	4	1	no	—
199	C₄²×C₆₇	—	2	1	no	—
200	C₄²×C₆₇	—	2	1	no	—
201	C₄²×C₆₇	—	2	1	no	—
202	C₄²×C₆₇	—	2	1	no	—
203	C₄²×C₆₇	—	2	1	no	—
204	C₄²×C₆₇	—	2	1	no	—
205	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
206	C₂²×D₆₇	—	2	1	no	—
207	C₂²×D₆₇	—	2	1	no	—
208	C₂²×D₆₇	—	2	1	no	—
209	C₂²×D₆₇	—	2	1	no	—
210	C₂²×D₆₇	—	2	1	no	—
211	C₂²×D₆₇	—	2	1	no	—
212	C₂²×D₆₇	C₄×Q₈	1	2	yes	—
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	#131	1	2	yes	—
231	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
235	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
237	C₂×Dic₆₇	—	2	1	no	—
238	C₂×Dic₆₇	—	2	1	no	—
239	C₂×Dic₆₇	—	2	1	no	—
240	C₂×Dic₆₇	—	2	1	no	—
241	C₂×Dic₆₇	—	2	1	no	—
242	C₂×Dic₆₇	—	2	1	no	—
243	C₂×Dic₆₇	—	2	1	no	—
244	C₂×Dic₆₇	—	2	1	no	—
245	C₂×Dic₆₇	—	2	1	no	—
246	C₂×Dic₆₇	—	2	1	no	—
247	C₂×Dic₆₇	—	2	1	no	—
248	C₂×Dic₆₇	—	2	1	no	—
249	C₂×Dic₆₇	—	2	1	no	—
250	C₂×Dic₆₇	—	2	1	no	—
251	C₂×Dic₆₇	—	2	1	no	—
252	C₂×Dic₆₇	—	2	1	no	—
253	C₂×Dic₆₇	#117	1	2	yes	—
254	C₂×Dic₆₇	#117	1	2	yes	—
255	C₂×Dic₆₇	C₄×Q₈	1	2	yes	—
256	C₂×Dic₆₇	#131	1	2	yes	—
257	C₂×Dic₆₇	#131	1	2	yes	—
258	C₄×D₆₇	—	2	2	no	—
259	C₄×D₆₇	—	2	2	no	—
260	C₄×D₆₇	—	2	2	no	—
261	C₄×D₆₇	—	2	2	no	—
262	C₄×D₆₇	—	4	4	no	—
263	C₄×D₆₇	—	4	4	no	—
264	C₈×C₆₇	—	2	2	no	—
265	C₈×C₆₇	—	4	1	no	—
266	C₈×C₆₇	—	4	1	no	—
267	C₈×C₆₇	—	4	1	no	—
268	C₆₇⋊C₈	—	2	2	no	—
269	C₆₇⋊C₈	—	4	1	no	—
270	C₆₇⋊C₈	—	4	1	no	—
271	C₆₇⋊C₈	—	4	1	no	—
272	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
273	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
274	C₂²×C₆₇	#518	1	1	yes	characteristic
275	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
276	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
277	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
278	C₂²×C₆₇	#568	1	1	yes	characteristic
279	C₂×D₆₇	—	2	2	no	—
280	C₂×D₆₇	—	2	2	no	—
281	C₂×D₆₇	—	2	2	no	—
282	C₂×D₆₇	—	4	1	no	—
283	C₂×D₆₇	—	4	1	no	—
284	C₂×D₆₇	—	4	1	no	—
285	C₂×D₆₇	—	4	1	no	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	4	1	no	—
288	C₄×C₆₇	—	4	1	no	—
289	C₄×C₆₇	—	4	1	no	—
290	C₄×C₆₇	—	4	1	no	—
291	C₄×C₆₇	—	4	1	no	—
292	C₄×C₆₇	#553	1	1	yes	characteristic
293	C₄×C₆₇	#553	1	1	yes	characteristic
294	C₄×C₆₇	#554	1	1	yes	characteristic
295	C₄×C₆₇	#554	1	1	yes	characteristic
296	Dic₆₇	—	2	2	no	—
297	Dic₆₇	—	2	2	no	—
298	Dic₆₇	—	4	1	no	—
299	Dic₆₇	—	4	1	no	—
300	Dic₆₇	—	4	1	no	—
301	Dic₆₇	—	4	1	no	—
302	Dic₆₇	#433	1	4	yes	—
303	#22626	—	67	1	no	maximal, Sylow
304	C₂×C₆₇	#2345	1	1	yes	characteristic
305	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
306	C₂×C₆₇	#2924	1	1	yes	characteristic
307	C₂×C₆₇	#2960	1	1	yes	characteristic
308	C₂×C₆₇	#2962	1	1	yes	characteristic
309	C₂×C₆₇	#2979	1	1	yes	characteristic
310	C₂×C₆₇	#2981	1	1	yes	characteristic
311	D₆₇	—	4	2	no	—
312	#2285	—	67	1	no	—
313	#1939	—	67	1	no	—
314	#1939	—	67	1	no	—
315	#1917	—	67	1	no	—
316	#1917	D₆₇	1	1	yes	characteristic, complemented
317	#1916	—	67	1	no	—
318	#1916	—	67	1	no	—
319	#1798	—	67	1	no	—
320	#1798	—	67	1	no	—
321	#1797	—	67	1	no	—
322	#1770	—	67	1	no	—
323	C₈×C₄⋊C₄	—	67	1	no	—
324	#1750	—	67	1	no	—
325	#1750	—	67	1	no	—
326	C₄×C₂²⋊C₈	—	67	1	no	—
327	C₆₇	#22626	1	1	yes	characteristic, complemented, Sylow
328	C₂×#117	—	67	1	no	—
329	C₂×#117	—	67	1	no	—
330	C₂²×C₄²	—	67	1	no	—
331	C₂×C₄⋊C₈	—	67	1	no	—
332	C₂×C₄⋊C₈	—	67	1	no	—
333	C₂×C₄⋊C₈	—	67	1	no	—
334	C₂×C₄⋊C₈	—	67	1	no	—
335	C₂×C₄⋊C₈	—	67	1	no	—
336	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
337	C₂×C₂²⋊C₈	—	67	1	no	—
338	C₂×C₂²⋊C₈	—	67	1	no	—
339	C₂×C₂²⋊C₈	—	67	1	no	—
340	C₂×C₂²⋊C₈	—	67	1	no	—
341	C₂×#97	—	67	1	no	—
342	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
343	C₂×C₄×C₈	—	67	1	no	—
344	C₂×C₄×C₈	—	67	1	no	—
345	C₂×C₄×C₈	—	67	1	no	—
346	C₂×C₄×C₈	—	67	1	no	—
347	#384	—	67	1	no	—
348	#384	—	67	1	no	—
349	#382	—	67	1	no	—
350	#382	—	67	1	no	—
351	#380	—	67	1	no	—
352	#380	—	67	1	no	—
353	#380	—	67	1	no	—
354	#380	—	67	1	no	—
355	C₄×C₄⋊C₄	—	67	1	no	—
356	C₄×C₄⋊C₄	—	67	1	no	—
357	C₄×C₄⋊C₄	—	67	1	no	—
358	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
359	#336	—	67	1	no	—
360	#336	—	67	1	no	—
361	#336	—	67	1	no	—
362	#336	—	67	1	no	—
363	#336	—	67	1	no	—
364	#336	—	67	1	no	—
365	#336	—	67	1	no	—
366	#336	—	67	1	no	—
367	#336	C₂×D₆₇	1	1	yes	characteristic
368	#336	C₂×D₆₇	1	1	yes	characteristic
369	#336	C₂×D₆₇	1	1	yes	characteristic
370	#336	C₂×D₆₇	1	1	yes	characteristic
371	C₂³×C₄	—	67	1	no	—
372	C₂³×C₄	—	67	1	no	—
373	C₂³×C₄	—	67	1	no	—
374	C₂²×C₈	—	67	1	no	—
375	C₂²×C₈	—	67	1	no	—
376	C₂²×C₈	—	67	1	no	—
377	C₂²×C₈	—	67	1	no	—
378	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
379	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
380	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
381	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
382	#117	—	268	1	no	—
383	#117	—	268	1	no	—
384	#117	—	268	1	no	—
385	#117	—	268	1	no	—
386	C₂×C₄⋊C₄	—	67	1	no	—
387	C₂×C₄⋊C₄	—	67	1	no	—
388	C₂×C₄⋊C₄	—	67	1	no	—
389	C₂×C₄⋊C₄	—	67	1	no	—
390	C₂×C₄⋊C₄	—	67	1	no	—
391	C₂×C₄⋊C₄	—	67	1	no	—
392	C₂×C₄⋊C₄	—	67	1	no	—
393	C₂×C₄⋊C₄	—	67	1	no	—
394	C₂×C₄⋊C₄	—	67	1	no	—
395	C₂×C₄⋊C₄	—	67	1	no	—
396	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	—	67	1	no	—
399	C₂×C₂²⋊C₄	—	67	1	no	—
400	C₂×C₂²⋊C₄	—	67	1	no	—
401	C₂×C₂²⋊C₄	—	67	1	no	—
402	C₂×C₄²	—	67	1	no	—
403	C₂×C₄²	—	67	1	no	—
404	C₂×C₄²	—	67	1	no	—
405	C₂×C₄²	—	67	1	no	—
406	C₂×C₄²	—	67	1	no	—
407	C₂×C₄²	—	134	2	no	—
408	C₂×C₄²	—	134	2	no	—
409	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
410	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
411	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
412	C₄⋊C₈	—	4	1	no	—
413	C₄⋊C₈	—	268	1	no	—
414	C₄⋊C₈	—	268	1	no	—
415	C₄⋊C₈	—	268	1	no	—
416	C₄⋊C₈	—	134	2	no	—
417	C₄⋊C₈	—	134	2	no	—
418	C₂²⋊C₈	—	67	4	no	—
419	C₂²⋊C₈	—	134	2	no	—
420	C₂²⋊C₈	—	134	2	no	—
421	C₂²⋊C₈	—	134	2	no	—
422	#97	—	4	1	no	—
423	#97	—	268	1	no	—
424	C₄×C₈	—	268	1	no	—
425	C₄×C₈	—	268	1	no	—
426	C₄×C₈	—	134	2	no	—
427	C₄×C₈	—	134	2	no	—
428	#95	—	67	1	no	—
429	#95	—	67	1	no	—
430	#95	—	67	1	no	—
431	#95	—	67	1	no	—
432	#95	—	67	1	no	—
433	#95	—	67	1	no	—
434	#95	C₄×D₆₇	1	1	yes	characteristic
435	#95	C₄×D₆₇	1	1	yes	characteristic
436	C₂⁴	—	67	1	no	—
437	C₂²×C₄	—	268	1	no	—
438	C₂²×C₄	—	67	1	no	—
439	C₂²×C₄	—	67	1	no	—
440	C₂²×C₄	—	67	1	no	—
441	C₂²×C₄	—	67	2	no	—
442	C₂²×C₄	—	67	2	no	—
443	C₂²×C₄	—	67	1	no	—
444	C₂²×C₄	—	67	1	no	—
445	C₂²×C₄	—	67	1	no	—
446	C₂²×C₄	—	67	1	no	—
447	C₂²×C₄	—	134	2	no	—
448	C₂²×C₄	—	134	2	no	—
449	C₂²×C₄	—	134	2	no	—
450	C₂²×C₄	—	134	2	no	—
451	C₂²×C₄	—	134	2	no	—
452	C₂²×C₄	—	134	1	no	—
453	C₂²×C₄	—	134	1	no	—
454	C₂²×C₄	—	134	1	no	—
455	C₂²×C₄	—	134	1	no	—
456	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
457	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
458	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
459	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
460	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
461	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
462	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	67	2	no	—
475	C₂×C₈	—	134	1	no	—
476	C₂×C₈	—	134	1	no	—
477	C₂×C₈	—	134	1	no	—
478	C₂×C₈	—	134	1	no	—
479	C₂×C₈	—	134	1	no	—
480	C₂×C₈	—	134	1	no	—
481	C₂×C₈	—	134	1	no	—
482	C₂×C₈	—	134	1	no	—
483	C₂×C₈	—	134	1	no	—
484	C₂×C₈	—	134	1	no	—
485	C₂×C₈	—	134	1	no	—
486	C₂×C₈	Q₈×D₆₇	1	2	yes	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	4	1	no	—
489	C₄⋊C₄	—	268	1	no	—
490	C₄⋊C₄	—	268	1	no	—
491	C₄⋊C₄	—	268	1	no	—
492	C₄⋊C₄	—	268	1	no	—
493	C₄⋊C₄	—	268	1	no	—
494	C₄⋊C₄	—	134	2	no	—
495	C₄⋊C₄	—	134	2	no	—
496	C₄⋊C₄	—	134	2	no	—
497	C₄⋊C₄	—	134	2	no	—
498	C₄⋊C₄	—	134	2	no	—
499	C₂²⋊C₄	—	268	1	no	—
500	C₂²⋊C₄	—	268	1	no	—
501	C₂²⋊C₄	—	268	1	no	—
502	C₂²⋊C₄	—	268	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	67	2	no	—
510	C₄²	—	67	2	no	—
511	C₄²	—	134	2	no	—
512	C₄²	—	134	2	no	—
513	C₄²	—	134	1	no	—
514	C₄²	—	134	1	no	—
515	C₄²	—	134	1	no	—
516	C₄²	—	134	1	no	—
517	C₂³	—	67	2	no	—
518	C₂³	—	134	1	no	—
519	C₂³	—	134	1	no	—
520	C₂³	—	134	1	no	—
521	C₂³	—	134	1	no	—
522	C₂³	—	134	1	no	—
523	C₂³	—	134	1	no	—
524	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	268	4	no	—
543	C₂×C₄	—	268	4	no	—
544	C₂×C₄	—	67	2	no	—
545	C₂×C₄	—	67	2	no	—
546	C₂×C₄	—	67	2	no	—
547	C₂×C₄	—	67	2	no	—
548	C₂×C₄	—	67	2	no	—
549	C₂×C₄	—	134	1	no	—
550	C₂×C₄	—	134	1	no	—
551	C₂×C₄	—	134	1	no	—
552	C₂×C₄	—	134	1	no	—
553	C₂×C₄	—	134	1	no	—
554	C₂×C₄	—	134	1	no	—
555	C₂×C₄	—	134	1	no	—
556	C₂×C₄	—	134	1	no	—
557	C₂×C₄	—	134	1	no	—
558	C₂×C₄	—	134	1	no	—
559	C₂×C₄	—	134	1	no	—
560	C₂×C₄	—	134	1	no	—
561	C₂×C₄	—	134	1	no	—
562	C₂×C₄	—	134	1	no	—
563	C₂×C₄	—	134	1	no	—
564	C₂×C₄	—	134	1	no	—
565	C₂×C₄	—	134	2	no	—
566	C₂×C₄	—	134	2	no	—
567	C₂×C₄	—	134	2	no	—
568	C₂×C₄	—	134	2	no	—
569	C₂×C₄	#14524051	1	2	yes	—
570	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
571	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
572	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
573	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
574	C₂×C₄	#14524088	1	1	yes	characteristic
575	C₂×C₄	#14524089	1	1	yes	characteristic
576	C₈	—	2	2	no	—
577	C₈	—	4	1	no	—
578	C₈	—	4	1	no	—
579	C₈	—	4	1	no	—
580	C₈	—	268	1	no	—
581	C₈	—	268	1	no	—
582	C₈	—	268	1	no	—
583	C₈	—	134	2	no	—
584	C₂²	—	268	1	no	—
585	C₂²	—	268	1	no	—
586	C₂²	—	268	1	no	—
587	C₂²	—	268	1	no	—
588	C₂²	—	134	2	no	—
589	C₂²	—	134	2	no	—
590	C₂²	—	134	2	no	—
591	C₂²	#117×D₆₇	1	1	yes	characteristic
592	C₂²	C₄×Q₈×D₆₇	1	1	yes	characteristic
593	C₂²	#16863868	1	1	yes	characteristic
594	C₂²	#131×D₆₇	1	1	yes	characteristic
595	C₂²	#131×D₆₇	1	1	yes	characteristic
596	C₂²	C₂×#14524051	1	1	yes	characteristic
597	C₂²	#16864162	1	1	yes	characteristic
598	C₄	—	2	2	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	1	no	—
601	C₄	—	4	1	no	—
602	C₄	—	4	1	no	—
603	C₄	—	4	1	no	—
604	C₄	—	268	1	no	—
605	C₄	—	268	1	no	—
606	C₄	—	268	1	no	—
607	C₄	—	268	1	no	—
608	C₄	—	67	4	no	—
609	C₄	—	134	2	no	—
610	C₄	—	134	2	no	—
611	C₄	#16864091	1	1	yes	characteristic
612	C₄	#16864097	1	1	yes	characteristic
613	C₄	#16864100	1	1	yes	characteristic
614	C₄	#16864102	1	1	yes	characteristic
615	C₂	—	268	2	no	—
616	C₂	#22563083	1	1	yes	characteristic
617	C₂	#433×D₆₇	1	1	yes	characteristic
618	C₂	#22568868	1	1	yes	characteristic
619	C₂	#22569140	1	1	yes	characteristic
620	C₂	#22569154	1	1	yes	characteristic
621	C₂	#22569274	1	1	yes	characteristic
622	C₂	#22569281	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#1917	C₂
prime radical	C₂³×D₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1917	D₆₇
hypocenter	C₆₇	#22626
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1917

Complements

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

N	H ≅ G/N	type
2. #22560926	C₂	semidirect
3. #22560924	C₂	semidirect
4. #22560772	C₂	semidirect
5. #22560768	C₂	semidirect
6. #22560767	C₂	semidirect
10. #22560077	C₂	semidirect
11. #22560070	C₂	semidirect
15. C₆₇×#1917	C₂	semidirect
316. #1917	D₆₇	semidirect
327. C₆₇	#22626	semidirect

Maximal subgroups

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

subgroup	index
#22562613	2
#22560926	2
#22560924	2
#22560772	2
#22560768	2
#22560767	2
#22560192	2
#22560190	2
#22560188	2
#22560077	2
#22560070	2
#22559978	2
#22559974	2
#22559966	2
C₆₇×#1917	2
#22626	67