Subgroups — #30865836

Order 17152.

1242 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561154	C₂	1	2	yes	maximal, complemented
4	#22561129	C₂	1	2	yes	maximal, complemented
5	#22560653	C₂	1	4	yes	maximal, complemented
6	#22560229	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3444	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
9	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
10	#16863453	—	2	1	no	—
11	#16863453	—	2	1	no	—
12	#16863449	—	2	1	no	—
13	#16863449	—	2	1	no	—
14	#16863420	C₂²	1	2	yes	—
15	C₂×#14523970	C₂²	1	8	yes	—
16	#16863360	C₂²	1	2	yes	—
17	C₄×#11761612	C₂²	1	2	yes	—
18	C₄×C₄⋊Dic₆₇	—	2	1	no	—
19	C₄×C₄⋊Dic₆₇	—	2	1	no	—
20	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
22	C₂×C₆₇⋊#95	—	2	1	no	—
23	#16863155	C₂²	1	8	yes	—
24	#533×C₆₇	C₄	1	1	yes	characteristic, complemented
25	#533×C₆₇	C₄	1	1	yes	characteristic, complemented
26	#533×C₆₇	C₂²	1	1	yes	characteristic
27	#532×C₆₇	C₄	1	1	yes	characteristic, complemented
28	#532×C₆₇	C₄	1	1	yes	characteristic, complemented
29	#532×C₆₇	C₂²	1	1	yes	characteristic
30	#529×C₆₇	—	2	1	no	—
31	#529×C₆₇	—	2	1	no	—
32	#529×C₆₇	—	2	1	no	—
33	#529×C₆₇	—	2	1	no	—
34	#529×C₆₇	C₄	1	2	yes	complemented
35	#529×C₆₇	C₄	1	2	yes	complemented
36	#525×C₆₇	—	2	1	no	—
37	#525×C₆₇	—	2	1	no	—
38	#525×C₆₇	C₂²	1	2	yes	—
39	C₄×#41×C₆₇	—	2	1	no	—
40	C₄×#41×C₆₇	—	2	1	no	—
41	C₄×#41×C₆₇	C₂²	1	2	yes	—
42	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
43	C₂³×Dic₆₇	—	2	2	no	—
44	C₂×#11761612	C₂³	1	4	yes	—
45	C₂×C₄⋊Dic₆₇	—	2	8	no	—
46	C₂×C₄⋊Dic₆₇	—	2	2	no	—
47	C₂×C₄⋊Dic₆₇	—	2	2	no	—
48	C₂×C₄×Dic₆₇	—	2	2	no	—
49	C₂×C₄×Dic₆₇	—	4	4	no	—
50	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
51	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
52	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
53	#14523970	—	4	8	no	—
54	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
55	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
56	#14523954	—	2	2	no	—
57	C₄×C₆₇⋊C₈	—	2	2	no	—
58	C₆₇⋊#95	—	2	2	no	—
59	C₆₇⋊#95	—	2	2	no	—
60	C₂×#41×C₆₇	—	2	1	no	—
61	C₂×#41×C₆₇	—	2	1	no	—
62	C₂×#41×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
67	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
68	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
69	#127×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
70	#126×C₆₇	—	2	2	no	—
71	#126×C₆₇	—	2	2	no	—
72	#125×C₆₇	—	2	1	no	—
73	#125×C₆₇	—	2	1	no	—
74	#125×C₆₇	C₂×C₄	1	2	yes	complemented
75	#124×C₆₇	—	2	1	no	—
76	#124×C₆₇	—	2	1	no	—
77	#124×C₆₇	C₂×C₄	1	2	yes	complemented
78	#123×C₆₇	—	2	2	no	—
79	#123×C₆₇	—	2	2	no	—
80	#123×C₆₇	—	2	2	no	—
81	C₂²⋊Q₈×C₆₇	—	2	1	no	—
82	C₂²⋊Q₈×C₆₇	—	2	1	no	—
83	C₂²⋊Q₈×C₆₇	—	2	1	no	—
84	C₂²⋊Q₈×C₆₇	—	2	1	no	—
85	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
86	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
87	#121×C₆₇	—	2	1	no	—
88	#121×C₆₇	—	2	1	no	—
89	#121×C₆₇	—	2	1	no	—
90	#121×C₆₇	—	2	1	no	—
91	#121×C₆₇	C₂×C₄	1	2	yes	—
92	#121×C₆₇	C₂×C₄	1	2	yes	—
93	#120×C₆₇	—	2	2	no	—
94	C₄×Q₈×C₆₇	—	2	1	no	—
95	C₄×Q₈×C₆₇	—	2	1	no	—
96	C₄×Q₈×C₆₇	—	2	1	no	—
97	C₄×Q₈×C₆₇	—	2	1	no	—
98	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	complemented
99	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	complemented
100	C₄×D₄×C₆₇	—	2	2	no	—
101	C₄×D₄×C₆₇	—	2	2	no	—
102	C₄×D₄×C₆₇	—	2	2	no	—
103	C₄×D₄×C₆₇	—	2	2	no	—
104	C₄×D₄×C₆₇	—	2	2	no	—
105	C₄×D₄×C₆₇	—	2	2	no	—
106	C₄×D₄×C₆₇	—	2	1	no	—
107	C₄×D₄×C₆₇	—	2	1	no	—
108	C₄×D₄×C₆₇	—	2	1	no	—
109	C₄×D₄×C₆₇	—	2	1	no	—
110	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	complemented
111	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	complemented
112	#117×C₆₇	—	2	1	no	—
113	#117×C₆₇	—	2	1	no	—
114	#117×C₆₇	—	2	2	no	—
115	#117×C₆₇	—	2	2	no	—
116	#117×C₆₇	—	2	2	no	—
117	#117×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄²×C₆₇	—	2	2	no	—
119	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
120	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
121	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
122	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
131	C₂²×Dic₆₇	—	2	8	no	—
132	C₂²×Dic₆₇	—	2	4	no	—
133	C₂²×Dic₆₇	—	4	4	no	—
134	C₂²×Dic₆₇	—	4	4	no	—
135	#11761612	—	2	4	no	—
136	#11761612	—	2	4	no	—
137	C₂×C₆₇⋊C₈	—	2	4	no	—
138	C₄⋊Dic₆₇	—	2	8	no	—
139	C₄⋊Dic₆₇	—	2	8	no	—
140	C₄×Dic₆₇	—	2	8	no	—
141	C₄×Dic₆₇	—	4	4	no	—
142	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
143	#41×C₆₇	—	2	4	no	—
144	#41×C₆₇	—	2	4	no	—
145	#41×C₆₇	—	4	4	no	—
146	#41×C₆₇	—	4	4	no	—
147	C₂×Q₈×C₆₇	—	2	1	no	—
148	C₂×Q₈×C₆₇	—	2	1	no	—
149	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂×D₄×C₆₇	—	2	2	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	—	2	1	no	—
154	C₂×D₄×C₆₇	—	2	1	no	—
155	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
156	C₂²×C₄×C₆₇	—	2	4	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	4	2	no	—
163	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
181	C₄⋊C₄×C₆₇	—	2	1	no	—
182	C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₄⋊C₄×C₆₇	—	2	1	no	—
184	C₄⋊C₄×C₆₇	—	2	1	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
192	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
193	C₂²⋊C₄×C₆₇	—	2	2	no	—
194	C₂²⋊C₄×C₆₇	—	2	2	no	—
195	C₂²⋊C₄×C₆₇	—	2	2	no	—
196	C₂²⋊C₄×C₆₇	—	2	2	no	—
197	C₂²⋊C₄×C₆₇	—	2	2	no	—
198	C₂²⋊C₄×C₆₇	—	2	2	no	—
199	C₂²⋊C₄×C₆₇	—	2	2	no	—
200	C₂²⋊C₄×C₆₇	—	2	2	no	—
201	C₂²⋊C₄×C₆₇	—	2	2	no	—
202	C₄²×C₆₇	—	2	2	no	—
203	C₄²×C₆₇	—	2	2	no	—
204	C₄²×C₆₇	—	2	2	no	—
205	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
209	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
210	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
211	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
212	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
213	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
214	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
215	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
216	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
217	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
221	C₂³×C₆₇	—	2	1	no	—
222	C₂³×C₆₇	—	2	1	no	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	2	2	no	—
225	C₂³×C₆₇	—	2	2	no	—
226	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
229	Q₈×C₆₇	—	2	4	no	—
230	Q₈×C₆₇	—	4	2	no	—
231	Q₈×C₆₇	—	4	2	no	—
232	D₄×C₆₇	—	2	4	no	—
233	D₄×C₆₇	—	4	2	no	—
234	D₄×C₆₇	—	4	2	no	—
235	D₄×C₆₇	—	4	2	no	—
236	D₄×C₆₇	—	4	4	no	—
237	D₄×C₆₇	—	4	4	no	—
238	D₄×C₆₇	—	4	2	no	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	4	1	no	—
256	C₂×C₄×C₆₇	—	4	1	no	—
257	C₂×C₄×C₆₇	—	4	1	no	—
258	C₂×C₄×C₆₇	—	4	1	no	—
259	C₂×C₄×C₆₇	—	4	4	no	—
260	C₂×C₄×C₆₇	—	4	4	no	—
261	C₂×C₄×C₆₇	—	4	4	no	—
262	C₂×C₄×C₆₇	#104	1	4	yes	complemented
263	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
274	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
276	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
277	C₂×Dic₆₇	—	2	8	no	—
278	C₂×Dic₆₇	—	4	16	no	—
279	C₂×Dic₆₇	—	4	4	no	—
280	C₂×Dic₆₇	—	4	4	no	—
281	C₆₇⋊C₈	—	4	4	no	—
282	C₂²×C₆₇	—	2	4	no	—
283	C₂²×C₆₇	—	2	4	no	—
284	C₂²×C₆₇	—	2	1	no	—
285	C₂²×C₆₇	—	2	1	no	—
286	C₂²×C₆₇	—	2	1	no	—
287	C₂²×C₆₇	—	2	1	no	—
288	C₂²×C₆₇	—	2	1	no	—
289	C₂²×C₆₇	—	2	1	no	—
290	C₂²×C₆₇	—	4	2	no	—
291	C₂²×C₆₇	—	4	2	no	—
292	C₂²×C₆₇	—	4	2	no	—
293	C₂²×C₆₇	C₂×#104	1	2	yes	—
294	C₂²×C₆₇	#421	1	2	yes	—
295	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
296	C₂²×C₆₇	#535	1	1	yes	characteristic
297	C₂²×C₆₇	#536	1	1	yes	characteristic
298	C₄×C₆₇	—	2	1	no	—
299	C₄×C₆₇	—	2	1	no	—
300	C₄×C₆₇	—	2	1	no	—
301	C₄×C₆₇	—	2	1	no	—
302	C₄×C₆₇	—	2	1	no	—
303	C₄×C₆₇	—	2	1	no	—
304	C₄×C₆₇	—	2	1	no	—
305	C₄×C₆₇	—	2	1	no	—
306	C₄×C₆₇	—	4	2	no	—
307	C₄×C₆₇	—	4	2	no	—
308	C₄×C₆₇	—	4	2	no	—
309	C₄×C₆₇	C₂×#104	1	2	yes	—
310	C₄×C₆₇	C₂×#104	1	2	yes	—
311	C₄×C₆₇	#520	1	2	yes	—
312	C₄×C₆₇	#537	1	2	yes	—
313	Dic₆₇	—	4	8	no	—
314	#21899	—	67	1	no	maximal, Sylow
315	C₂×C₆₇	—	2	1	no	—
316	C₂×C₆₇	—	2	1	no	—
317	C₂×C₆₇	—	2	2	no	—
318	C₂×C₆₇	—	2	2	no	—
319	C₂×C₆₇	—	8	2	no	—
320	C₂×C₆₇	#2311	1	1	yes	characteristic
321	C₂×C₆₇	C₂²×#104	1	1	yes	characteristic
322	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
323	#3444	D₆₇	1	1	yes	characteristic
324	#2920	—	67	1	no	—
325	C₂×C₄×C₄⋊C₄	—	67	1	no	—
326	#1986	—	67	2	no	—
327	#1980	—	67	2	no	—
328	#1896	—	67	4	no	—
329	#1806	—	67	4	no	—
330	C₆₇	#21899	1	1	yes	characteristic, complemented, Sylow
331	#533	Dic₆₇	1	1	yes	characteristic, complemented
332	#533	Dic₆₇	1	1	yes	characteristic, complemented
333	#533	C₂×D₆₇	1	1	yes	characteristic
334	#532	Dic₆₇	1	1	yes	characteristic, complemented
335	#532	Dic₆₇	1	1	yes	characteristic, complemented
336	#532	C₂×D₆₇	1	1	yes	characteristic
337	#529	—	2	1	no	—
338	#529	—	2	1	no	—
339	#529	—	2	1	no	—
340	#529	—	2	1	no	—
341	#529	Dic₆₇	1	2	yes	complemented
342	#529	Dic₆₇	1	2	yes	complemented
343	#525	—	2	1	no	—
344	#525	—	2	1	no	—
345	#525	C₂×D₆₇	1	2	yes	—
346	C₄×#41	—	2	1	no	—
347	C₄×#41	—	2	1	no	—
348	C₄×#41	C₂×D₆₇	1	2	yes	—
349	C₂²×C₄⋊C₄	—	134	1	no	—
350	C₂²×C₄²	—	67	2	no	—
351	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
352	#432	—	134	1	no	—
353	#432	—	134	1	no	—
354	#431	—	134	1	no	—
355	#431	—	134	1	no	—
356	#423	—	67	2	no	—
357	C₂×#104	—	67	8	no	—
358	#407	—	67	2	no	—
359	C₄×M₁₆	—	67	2	no	—
360	C₄×C₄⋊C₄	—	67	2	no	—
361	C₄×C₄⋊C₄	—	67	2	no	—
362	C₄×C₄⋊C₄	—	134	1	no	—
363	C₄×C₄⋊C₄	—	134	1	no	—
364	C₂×#95	—	134	1	no	—
365	#339	—	67	8	no	—
366	C₂×#41	—	2	1	no	—
367	C₂×#41	—	2	1	no	—
368	C₂×#41	C₂²×D₆₇	1	2	yes	—
369	C₂³×C₄	—	134	2	no	—
370	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
371	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
372	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
373	C₂×M₁₆	—	67	4	no	—
374	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic, complemented
375	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic, complemented
376	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic, complemented
377	#127	C₂×Dic₆₇	1	1	yes	characteristic, complemented
378	#126	—	2	2	no	—
379	#126	—	2	2	no	—
380	#125	—	2	1	no	—
381	#125	—	2	1	no	—
382	#125	C₂×Dic₆₇	1	2	yes	complemented
383	#124	—	2	1	no	—
384	#124	—	2	1	no	—
385	#124	C₂×Dic₆₇	1	2	yes	complemented
386	#123	—	2	2	no	—
387	#123	—	2	2	no	—
388	#123	—	2	2	no	—
389	C₂²⋊Q₈	—	2	1	no	—
390	C₂²⋊Q₈	—	2	1	no	—
391	C₂²⋊Q₈	—	2	1	no	—
392	C₂²⋊Q₈	—	2	1	no	—
393	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
394	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
395	#121	—	2	1	no	—
396	#121	—	2	1	no	—
397	#121	—	2	1	no	—
398	#121	—	2	1	no	—
399	#121	C₂×Dic₆₇	1	2	yes	—
400	#121	C₂×Dic₆₇	1	2	yes	—
401	#120	—	2	2	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	—	2	1	no	—
404	C₄×Q₈	—	2	1	no	—
405	C₄×Q₈	—	2	1	no	—
406	C₄×Q₈	C₂×Dic₆₇	1	2	yes	complemented
407	C₄×Q₈	C₂×Dic₆₇	1	2	yes	complemented
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	1	no	—
418	C₄×D₄	C₂×Dic₆₇	1	2	yes	complemented
419	C₄×D₄	C₂×Dic₆₇	1	2	yes	complemented
420	#117	—	2	1	no	—
421	#117	—	2	1	no	—
422	#117	—	2	2	no	—
423	#117	—	2	2	no	—
424	#117	—	2	2	no	—
425	#117	C₂²×D₆₇	1	2	yes	—
426	C₂×C₄⋊C₄	—	134	8	no	—
427	C₂×C₄⋊C₄	—	134	2	no	—
428	C₂×C₄⋊C₄	—	134	2	no	—
429	C₂×C₄²	—	2	2	no	—
430	C₂×C₄²	—	268	4	no	—
431	C₂×C₄²	—	67	4	no	—
432	C₂×C₄²	—	134	2	no	—
433	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
434	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
435	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
436	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
437	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
438	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
439	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
440	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
441	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
442	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
443	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
444	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
445	C₄⋊C₈	—	134	2	no	—
446	C₄⋊C₈	—	134	2	no	—
447	#104	—	268	8	no	—
448	C₂²⋊C₈	—	134	2	no	—
449	C₂²⋊C₈	—	134	2	no	—
450	#97	—	134	2	no	—
451	C₄×C₈	—	134	2	no	—
452	#95	—	134	2	no	—
453	#95	—	134	2	no	—
454	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
455	#41	—	2	4	no	—
456	#41	—	2	4	no	—
457	#41	—	4	4	no	—
458	#41	—	4	4	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	268	4	no	—
476	C₂²×C₄	—	268	4	no	—
477	C₂²×C₄	—	134	8	no	—
478	C₂²×C₄	—	134	4	no	—
479	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
482	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
484	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
485	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
487	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
488	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
489	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
490	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
492	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
493	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
494	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
497	M₁₆	—	134	4	no	—
498	M₁₆	—	134	4	no	—
499	C₂×C₈	—	134	4	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	134	8	no	—
511	C₄⋊C₄	—	134	8	no	—
512	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
513	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	2	no	—
525	C₄²	—	2	2	no	—
526	C₄²	—	268	4	no	—
527	C₄²	—	134	8	no	—
528	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
529	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
530	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
531	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
532	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
533	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
534	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
535	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
536	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
537	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
538	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
539	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
540	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
541	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
542	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
543	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	1	no	—
547	C₂³	—	2	1	no	—
548	C₂³	—	2	2	no	—
549	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
550	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
551	C₂³	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
552	Q₈	—	2	4	no	—
553	Q₈	—	4	2	no	—
554	Q₈	—	4	2	no	—
555	D₄	—	2	4	no	—
556	D₄	—	4	2	no	—
557	D₄	—	4	4	no	—
558	D₄	—	4	4	no	—
559	D₄	—	4	2	no	—
560	D₄	—	4	2	no	—
561	D₄	—	4	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	—	4	4	no	—
585	C₂×C₄	—	268	16	no	—
586	C₂×C₄	—	268	4	no	—
587	C₂×C₄	—	268	4	no	—
588	C₂×C₄	—	134	8	no	—
589	C₂×C₄	#14523970	1	4	yes	complemented
590	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
592	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
593	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
594	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
595	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
596	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
597	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
598	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
599	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
600	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
601	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
602	C₂×C₄	#14524089	1	2	yes	—
603	C₂×C₄	#14524090	1	2	yes	—
604	C₈	—	268	4	no	—
605	C₂²	—	2	4	no	—
606	C₂²	—	2	4	no	—
607	C₂²	—	2	1	no	—
608	C₂²	—	2	1	no	—
609	C₂²	—	2	1	no	—
610	C₂²	—	2	1	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	2	1	no	—
613	C₂²	—	4	2	no	—
614	C₂²	—	4	2	no	—
615	C₂²	—	4	2	no	—
616	C₂²	C₂×#14523970	1	2	yes	—
617	C₂²	#16863413	1	2	yes	—
618	C₂²	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
619	C₂²	#16863960	1	1	yes	characteristic
620	C₂²	#16863967	1	1	yes	characteristic
621	C₄	—	2	1	no	—
622	C₄	—	2	1	no	—
623	C₄	—	2	1	no	—
624	C₄	—	2	1	no	—
625	C₄	—	2	1	no	—
626	C₄	—	2	1	no	—
627	C₄	—	2	1	no	—
628	C₄	—	2	1	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	2	no	—
632	C₄	—	268	8	no	—
633	C₄	C₂×#14523970	1	2	yes	—
634	C₄	C₂×#14523970	1	2	yes	—
635	C₄	#16863879	1	2	yes	—
636	C₄	#16863974	1	2	yes	—
637	C₂	—	2	1	no	—
638	C₂	—	2	2	no	—
639	C₂	—	2	2	no	—
640	C₂	—	2	1	no	—
641	C₂	—	8	2	no	—
642	C₂	#22562801	1	1	yes	characteristic
643	C₂	C₂²×#14523970	1	1	yes	characteristic
644	C₂	C₂×#16863413	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, prime radical	C₆₇×#3444	C₂
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3444	D₆₇
hypocenter	C₆₇	#21899
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #3444

Complements

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

N	H ≅ G/N	type
1. #22568842	C₂	semidirect
2. C₂×C₄×C₄⋊Dic₆₇	C₂	semidirect
3. #22561154	C₂	semidirect
4. #22561129	C₂	semidirect
5. #22560653	C₂	semidirect
6. #22560229	C₂	semidirect
24. #533×C₆₇	C₄	semidirect
25. #533×C₆₇	C₄	semidirect
27. #532×C₆₇	C₄	semidirect
28. #532×C₆₇	C₄	semidirect
34. #529×C₆₇	C₄	semidirect
35. #529×C₆₇	C₄	semidirect
66. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
67. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
68. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
69. #127×C₆₇	C₂×C₄	semidirect
74. #125×C₆₇	C₂×C₄	semidirect
77. #124×C₆₇	C₂×C₄	semidirect
98. C₄×Q₈×C₆₇	C₂×C₄	semidirect
99. C₄×Q₈×C₆₇	C₂×C₄	semidirect
110. C₄×D₄×C₆₇	C₂×C₄	semidirect
111. C₄×D₄×C₆₇	C₂×C₄	semidirect
262. C₂×C₄×C₆₇	#104	semidirect
330. C₆₇	#21899	semidirect
331. #533	Dic₆₇	semidirect
332. #533	Dic₆₇	semidirect
334. #532	Dic₆₇	semidirect
335. #532	Dic₆₇	semidirect
341. #529	Dic₆₇	semidirect
342. #529	Dic₆₇	semidirect
374. C₄⋊Q₈	C₂×Dic₆₇	semidirect
375. C₄⋊Q₈	C₂×Dic₆₇	semidirect
376. C₄⋊Q₈	C₂×Dic₆₇	semidirect
377. #127	C₂×Dic₆₇	semidirect
382. #125	C₂×Dic₆₇	semidirect
385. #124	C₂×Dic₆₇	semidirect
406. C₄×Q₈	C₂×Dic₆₇	semidirect
407. C₄×Q₈	C₂×Dic₆₇	semidirect
418. C₄×D₄	C₂×Dic₆₇	semidirect
419. C₄×D₄	C₂×Dic₆₇	semidirect
589. C₂×C₄	#14523970	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
C₂×C₄×C₄⋊Dic₆₇	2
#22561154	2
#22561129	2
#22560653	2
#22560229	2
C₆₇×#3444	2
#21899	67