Subgroups — #30875280

Order 17152.

740 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	#22568871	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568775	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560923	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560917	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560809	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560807	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560538	C₂	1	1	yes	maximal, characteristic
9	#22560345	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560227	C₂	1	1	yes	maximal, characteristic, complemented
11	#22560227	C₂	1	1	yes	maximal, characteristic, complemented
12	#22560075	C₂	1	1	yes	maximal, characteristic, complemented
13	#22560054	C₂	1	1	yes	maximal, characteristic, complemented
14	#22560051	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1876	C₂	1	1	yes	maximal, characteristic
16	C₂×#14524050	C₂²	1	1	yes	characteristic
17	#41×Dic₆₇	C₂²	1	1	yes	characteristic
18	#41×Dic₆₇	C₂²	1	1	yes	characteristic
19	Q₁₆×Dic₆₇	C₄	1	1	yes	characteristic, complemented
20	Q₁₆×Dic₆₇	C₄	1	1	yes	characteristic, complemented
21	SD₁₆×Dic₆₇	C₄	1	1	yes	characteristic, complemented
22	SD₁₆×Dic₆₇	C₄	1	1	yes	characteristic, complemented
23	SD₁₆×Dic₆₇	C₄	1	1	yes	characteristic, complemented
24	SD₁₆×Dic₆₇	C₄	1	1	yes	characteristic, complemented
25	D₈×Dic₆₇	C₄	1	1	yes	characteristic, complemented
26	D₈×Dic₆₇	C₄	1	1	yes	characteristic, complemented
27	#16863454	—	2	1	no	—
28	#16863454	—	2	1	no	—
29	#16863450	—	2	1	no	—
30	#16863446	—	2	1	no	—
31	#16863434	C₂²	1	1	yes	characteristic
32	#16863424	—	2	1	no	—
33	#16863424	C₂²	1	1	yes	characteristic
34	#16863422	—	2	1	no	—
35	#16863422	C₂²	1	1	yes	characteristic
36	#16863419	—	2	1	no	—
37	#16863419	C₂²	1	1	yes	characteristic
38	#16863419	C₂²	1	1	yes	characteristic
39	C₂×Dic₆₇⋊C₈	—	2	1	no	—
40	C₂×Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×#14523970	C₂²	1	1	yes	characteristic
42	C₂×#14523970	C₂²	1	1	yes	characteristic
43	#16863392	C₂²	1	1	yes	characteristic
44	#16863392	C₂²	1	1	yes	characteristic
45	#16863363	C₂²	1	1	yes	characteristic
46	#16863354	C₂²	1	1	yes	characteristic
47	#16863351	C₂²	1	1	yes	characteristic
48	C₂×#14523954	C₂²	1	1	yes	characteristic
49	C₂×#14523953	C₂²	1	1	yes	characteristic
50	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
51	C₄×Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
52	#16863161	C₂²	1	1	yes	characteristic
53	#16863160	C₂²	1	1	yes	characteristic
54	#16863160	C₂²	1	1	yes	characteristic
55	#16863157	C₂²	1	1	yes	characteristic
56	#16863157	C₂²	1	1	yes	characteristic
57	#16863156	C₂²	1	1	yes	characteristic
58	#16863156	C₂²	1	1	yes	characteristic
59	#16863150	C₂²	1	1	yes	characteristic
60	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
61	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
62	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
63	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
64	#408×C₆₇	C₂²	1	1	yes	characteristic
65	#341×C₆₇	C₂²	1	1	yes	characteristic
66	#336×C₆₇	C₂²	1	1	yes	characteristic
67	#14524050	—	2	2	no	—
68	#14524050	—	4	1	no	—
69	C₂×#11761612	—	2	1	no	—
70	C₂×#11761612	C₂³	1	1	yes	characteristic
71	C₂×#11761612	C₂³	1	1	yes	characteristic
72	C₂²×C₆₇⋊C₈	—	2	1	no	—
73	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
75	Q₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
76	Q₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
77	Q₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
78	Q₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
79	D₄×Dic₆₇	—	2	1	no	—
80	D₄×Dic₆₇	—	2	1	no	—
81	D₄×Dic₆₇	—	2	1	no	—
82	D₄×Dic₆₇	—	2	1	no	—
83	D₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	D₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
85	D₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
86	D₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
87	#14524002	—	2	1	no	—
88	#14524002	—	2	1	no	—
89	#14524002	C₂³	1	1	yes	characteristic
90	#14524002	C₂³	1	1	yes	characteristic
91	C₂×Dic₆₇⋊C₄	—	2	1	no	—
92	C₂×C₄×Dic₆₇	—	2	1	no	—
93	C₂×C₄×Dic₆₇	—	2	1	no	—
94	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
96	#14523977	C₂×C₄	1	1	yes	characteristic
97	#14523977	C₂×C₄	1	1	yes	characteristic
98	#14523975	C₂×C₄	1	1	yes	characteristic
99	#14523975	C₂×C₄	1	1	yes	characteristic
100	Dic₆₇⋊C₈	—	2	1	no	—
101	Dic₆₇⋊C₈	—	2	1	no	—
102	Dic₆₇⋊C₈	—	2	1	no	—
103	Dic₆₇⋊C₈	—	2	1	no	—
104	Dic₆₇⋊C₈	—	2	1	no	—
105	Dic₆₇⋊C₈	—	2	1	no	—
106	Dic₆₇⋊C₈	—	2	1	no	—
107	Dic₆₇⋊C₈	—	2	1	no	—
108	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
109	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
110	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
111	Dic₆₇⋊C₈	D₄	1	1	yes	characteristic
112	#14523970	—	2	2	no	—
113	#14523970	—	4	1	no	—
114	#14523967	C₂×C₄	1	1	yes	characteristic
115	#14523967	C₂×C₄	1	1	yes	characteristic
116	#14523967	C₂×C₄	1	1	yes	characteristic
117	#14523967	C₂×C₄	1	1	yes	characteristic
118	#14523964	C₂×C₄	1	1	yes	characteristic
119	#14523964	C₂×C₄	1	1	yes	characteristic
120	#14523964	C₂×C₄	1	1	yes	characteristic
121	#14523964	C₂×C₄	1	1	yes	characteristic
122	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
123	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
124	#14523954	—	4	1	no	—
125	#14523953	—	2	1	no	—
126	#14523953	—	2	1	no	—
127	#14523953	—	4	1	no	—
128	C₈×Dic₆₇	—	2	1	no	—
129	C₈×Dic₆₇	—	2	1	no	—
130	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
131	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
132	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
133	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
134	C₆₇⋊#95	—	2	1	no	—
135	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
136	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
137	#135×C₆₇	—	2	2	no	—
138	#135×C₆₇	—	2	2	no	—
139	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
140	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
141	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
142	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
143	#131×C₆₇	—	2	2	no	—
144	#131×C₆₇	—	4	1	no	—
145	C₂×M₁₆×C₆₇	—	2	1	no	—
146	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
147	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
148	C₂²×C₈×C₆₇	—	2	1	no	—
149	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
150	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
151	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
152	#104×C₆₇	—	2	2	no	—
153	#104×C₆₇	—	4	1	no	—
154	C₂²⋊C₈×C₆₇	—	2	1	no	—
155	C₂²⋊C₈×C₆₇	—	2	1	no	—
156	C₂²×Dic₆₇	—	2	1	no	—
157	C₂²×Dic₆₇	—	2	1	no	—
158	C₂²×Dic₆₇	—	2	1	no	—
159	C₂²×Dic₆₇	—	2	1	no	—
160	C₂²×Dic₆₇	—	2	1	no	—
161	C₂²×Dic₆₇	#41	1	1	yes	characteristic
162	C₂²×Dic₆₇	#41	1	1	yes	characteristic
163	#11761612	—	2	1	no	—
164	#11761612	—	2	1	no	—
165	#11761612	—	2	1	no	—
166	#11761612	—	2	1	no	—
167	#11761612	—	4	2	no	—
168	C₂×C₆₇⋊C₈	—	2	1	no	—
169	C₂×C₆₇⋊C₈	—	2	1	no	—
170	C₂×C₆₇⋊C₈	—	2	1	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	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
178	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
179	C₄⋊Dic₆₇	—	2	1	no	—
180	C₄⋊Dic₆₇	—	2	1	no	—
181	C₄⋊Dic₆₇	—	2	1	no	—
182	C₄⋊Dic₆₇	—	2	1	no	—
183	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
184	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
185	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
186	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
187	Dic₆₇⋊C₄	—	4	2	no	—
188	C₂²⋊Dic₆₇	—	2	1	no	—
189	C₂²⋊Dic₆₇	—	2	1	no	—
190	C₂²⋊Dic₆₇	—	2	1	no	—
191	C₂²⋊Dic₆₇	—	2	1	no	—
192	C₂²⋊Dic₆₇	—	2	1	no	—
193	C₂²⋊Dic₆₇	—	2	1	no	—
194	C₄×Dic₆₇	—	2	1	no	—
195	C₄×Dic₆₇	—	2	1	no	—
196	C₄×Dic₆₇	—	2	1	no	—
197	C₄×Dic₆₇	—	2	1	no	—
198	C₄×Dic₆₇	—	2	1	no	—
199	C₄×Dic₆₇	—	2	1	no	—
200	C₄×Dic₆₇	—	2	1	no	—
201	C₄×Dic₆₇	—	2	1	no	—
202	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
203	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
204	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
205	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
206	#41×C₆₇	—	2	1	no	—
207	#41×C₆₇	—	2	1	no	—
208	#41×C₆₇	—	2	1	no	—
209	#41×C₆₇	—	2	1	no	—
210	#41×C₆₇	—	2	1	no	—
211	#41×C₆₇	—	2	1	no	—
212	#41×C₆₇	#41	1	2	yes	—
213	#41×C₆₇	#41	1	2	yes	—
214	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
215	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
216	C₂×D₄×C₆₇	—	2	1	no	—
217	C₂×D₄×C₆₇	—	2	1	no	—
218	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
219	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
220	C₂²×C₄×C₆₇	—	2	1	no	—
221	C₂²×C₄×C₆₇	—	2	1	no	—
222	C₂²×C₄×C₆₇	—	2	1	no	—
223	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
224	Q₁₆×C₆₇	—	2	2	no	—
225	SD₁₆×C₆₇	—	2	2	no	—
226	SD₁₆×C₆₇	—	2	2	no	—
227	D₈×C₆₇	—	2	2	no	—
228	M₁₆×C₆₇	—	2	1	no	—
229	M₁₆×C₆₇	—	2	1	no	—
230	M₁₆×C₆₇	—	2	1	no	—
231	M₁₆×C₆₇	—	2	1	no	—
232	M₁₆×C₆₇	—	4	2	no	—
233	C₂×C₈×C₆₇	—	2	1	no	—
234	C₂×C₈×C₆₇	—	2	1	no	—
235	C₂×C₈×C₆₇	—	2	1	no	—
236	C₂×C₈×C₆₇	—	2	1	no	—
237	C₂×C₈×C₆₇	—	2	1	no	—
238	C₂×C₈×C₆₇	—	2	1	no	—
239	C₂×C₈×C₆₇	—	4	2	no	—
240	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
241	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
242	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
243	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
244	C₄²×C₆₇	—	2	1	no	—
245	C₄²×C₆₇	—	2	1	no	—
246	C₂³×C₆₇	—	2	1	no	—
247	C₂³×C₆₇	—	2	1	no	—
248	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
249	Q₈×C₆₇	—	2	1	no	—
250	Q₈×C₆₇	—	2	1	no	—
251	Q₈×C₆₇	—	2	1	no	—
252	Q₈×C₆₇	#117	1	2	yes	—
253	D₄×C₆₇	—	2	2	no	—
254	D₄×C₆₇	—	2	2	no	—
255	D₄×C₆₇	—	2	1	no	—
256	D₄×C₆₇	—	2	1	no	—
257	D₄×C₆₇	—	2	1	no	—
258	D₄×C₆₇	—	4	1	no	—
259	D₄×C₆₇	—	4	1	no	—
260	D₄×C₆₇	#117	1	2	yes	—
261	C₂×C₄×C₆₇	—	2	1	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	1	no	—
265	C₂×C₄×C₆₇	—	4	1	no	—
266	C₂×C₄×C₆₇	—	4	1	no	—
267	C₂×C₄×C₆₇	—	4	1	no	—
268	C₂×C₄×C₆₇	—	4	1	no	—
269	C₂×C₄×C₆₇	—	4	1	no	—
270	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
271	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
275	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
276	C₂×Dic₆₇	—	2	2	no	—
277	C₂×Dic₆₇	—	2	1	no	—
278	C₂×Dic₆₇	—	2	1	no	—
279	C₂×Dic₆₇	—	2	1	no	—
280	C₂×Dic₆₇	—	2	1	no	—
281	C₂×Dic₆₇	—	4	1	no	—
282	C₂×Dic₆₇	—	4	2	no	—
283	C₂×Dic₆₇	—	4	2	no	—
284	C₂×Dic₆₇	—	4	1	no	—
285	C₂×Dic₆₇	—	4	1	no	—
286	C₂×Dic₆₇	—	4	1	no	—
287	C₂×Dic₆₇	—	8	1	no	—
288	C₂×Dic₆₇	—	8	1	no	—
289	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
290	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
291	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
292	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
293	C₈×C₆₇	—	2	1	no	—
294	C₈×C₆₇	—	2	1	no	—
295	C₈×C₆₇	—	2	1	no	—
296	C₈×C₆₇	—	2	1	no	—
297	C₈×C₆₇	—	4	1	no	—
298	C₈×C₆₇	—	4	1	no	—
299	C₆₇⋊C₈	—	2	1	no	—
300	C₆₇⋊C₈	—	2	1	no	—
301	C₆₇⋊C₈	—	2	1	no	—
302	C₆₇⋊C₈	—	2	1	no	—
303	C₆₇⋊C₈	—	4	1	no	—
304	C₆₇⋊C₈	—	4	1	no	—
305	C₂²×C₆₇	—	2	1	no	—
306	C₂²×C₆₇	—	2	1	no	—
307	C₂²×C₆₇	—	2	2	no	—
308	C₂²×C₆₇	—	4	1	no	—
309	C₂²×C₆₇	—	4	1	no	—
310	C₂²×C₆₇	—	4	1	no	—
311	C₂²×C₆₇	—	4	1	no	—
312	C₂²×C₆₇	—	4	1	no	—
313	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
314	C₂²×C₆₇	#552	1	1	yes	characteristic
315	C₂²×C₆₇	#553	1	1	yes	characteristic
316	C₄×C₆₇	—	2	2	no	—
317	C₄×C₆₇	—	4	1	no	—
318	C₄×C₆₇	—	8	1	no	—
319	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
320	C₄×C₆₇	#520	1	1	yes	characteristic
321	C₄×C₆₇	#546	1	1	yes	characteristic
322	C₄×C₆₇	#548	1	1	yes	characteristic
323	Dic₆₇	—	2	1	no	—
324	Dic₆₇	—	2	1	no	—
325	Dic₆₇	—	2	1	no	—
326	Dic₆₇	—	4	1	no	—
327	Dic₆₇	—	4	1	no	—
328	Dic₆₇	—	4	1	no	—
329	Dic₆₇	—	4	1	no	—
330	Dic₆₇	—	8	1	no	—
331	Dic₆₇	#444	1	2	yes	—
332	#22782	—	67	1	no	maximal, Sylow
333	C₂×C₆₇	—	2	1	no	—
334	C₂×C₆₇	—	2	1	no	—
335	C₂×C₆₇	—	4	2	no	—
336	C₂×C₆₇	—	8	1	no	—
337	C₂×C₆₇	#2352	1	1	yes	characteristic
338	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
339	C₂×C₆₇	#2957	1	1	yes	characteristic
340	C₄×#135	—	67	1	no	—
341	#2925	—	67	1	no	—
342	#2906	—	67	1	no	—
343	#1938	—	67	1	no	—
344	#1938	—	67	1	no	—
345	#1922	—	67	1	no	—
346	#1922	—	67	1	no	—
347	#1876	—	67	1	no	—
348	#1876	D₆₇	1	1	yes	characteristic
349	#1837	—	67	1	no	—
350	#1806	—	67	1	no	—
351	#1806	—	67	1	no	—
352	#1770	—	67	1	no	—
353	#1765	—	67	1	no	—
354	#1765	—	67	1	no	—
355	C₆₇	#22782	1	1	yes	characteristic, complemented, Sylow
356	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
357	C₂×#131	—	67	1	no	—
358	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
359	C₄×#41	—	67	1	no	—
360	C₄×#41	—	67	1	no	—
361	C₄×Q₁₆	—	67	1	no	—
362	C₄×Q₁₆	—	67	1	no	—
363	C₄×SD₁₆	—	67	1	no	—
364	C₄×SD₁₆	—	67	1	no	—
365	C₄×SD₁₆	—	67	1	no	—
366	C₄×SD₁₆	—	67	1	no	—
367	C₄×D₈	—	67	1	no	—
368	C₄×D₈	—	67	1	no	—
369	#433	—	134	1	no	—
370	#433	—	134	1	no	—
371	#432	—	134	1	no	—
372	#431	—	134	1	no	—
373	#427	—	67	1	no	—
374	#424	—	67	1	no	—
375	#424	—	67	1	no	—
376	#424	—	134	1	no	—
377	#424	—	134	1	no	—
378	#423	—	67	1	no	—
379	#423	—	67	1	no	—
380	#423	—	134	1	no	—
381	C₂×C₄⋊C₈	—	67	1	no	—
382	C₂×C₄⋊C₈	—	134	1	no	—
383	C₂×#104	—	67	1	no	—
384	C₂×#104	—	67	1	no	—
385	C₂×#104	C₂×D₆₇	1	1	yes	characteristic
386	C₂×#104	C₂×D₆₇	1	1	yes	characteristic
387	#416	—	67	1	no	—
388	#416	—	67	1	no	—
389	#408	—	67	1	no	—
390	#408	C₂×D₆₇	1	1	yes	characteristic
391	#405	—	67	1	no	—
392	#405	—	67	1	no	—
393	C₂×#97	—	67	1	no	—
394	C₂×#97	—	67	1	no	—
395	C₂×C₄×C₈	—	67	1	no	—
396	C₄×C₄⋊C₄	—	67	1	no	—
397	#341	—	67	1	no	—
398	#341	—	67	1	no	—
399	#341	—	67	1	no	—
400	#341	C₂×D₆₇	1	1	yes	characteristic
401	#339	—	67	1	no	—
402	#339	—	67	1	no	—
403	#339	—	67	1	no	—
404	#339	—	67	1	no	—
405	#336	—	67	1	no	—
406	#336	C₂×D₆₇	1	1	yes	characteristic
407	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
408	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
409	#135	—	2	2	no	—
410	#135	—	2	2	no	—
411	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
412	C₂×SD₁₆	C₄×D₆₇	1	1	yes	characteristic
413	C₂×SD₁₆	C₄×D₆₇	1	1	yes	characteristic
414	C₂×D₈	C₄×D₆₇	1	1	yes	characteristic
415	#131	—	2	2	no	—
416	#131	—	4	1	no	—
417	#131	—	268	1	no	—
418	#131	—	134	2	no	—
419	C₂×M₁₆	—	2	1	no	—
420	C₂×M₁₆	—	67	1	no	—
421	C₂×M₁₆	—	67	1	no	—
422	C₂×M₁₆	—	134	1	no	—
423	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
424	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
425	C₂²×C₈	—	2	1	no	—
426	C₂²×C₈	—	67	1	no	—
427	C₂²×C₈	—	67	1	no	—
428	C₂²×C₈	—	134	1	no	—
429	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
430	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
431	C₄×Q₈	—	67	1	no	—
432	C₄×Q₈	—	67	1	no	—
433	C₄×Q₈	—	67	1	no	—
434	C₄×Q₈	—	67	1	no	—
435	C₄×D₄	—	67	1	no	—
436	C₄×D₄	—	67	1	no	—
437	C₄×D₄	—	67	1	no	—
438	C₄×D₄	—	67	1	no	—
439	C₄×D₄	—	134	1	no	—
440	C₄×D₄	—	134	1	no	—
441	C₄×D₄	—	134	1	no	—
442	C₄×D₄	—	134	1	no	—
443	#117	—	67	1	no	—
444	#117	—	67	1	no	—
445	#117	—	134	1	no	—
446	#117	—	134	1	no	—
447	C₂×C₄⋊C₄	—	134	1	no	—
448	C₂×C₄²	—	67	1	no	—
449	C₂×C₄²	—	67	1	no	—
450	C₂×C₄²	—	134	1	no	—
451	C₂×C₄²	—	134	1	no	—
452	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
453	#107	—	67	1	no	—
454	#107	—	67	1	no	—
455	#106	—	67	1	no	—
456	#106	—	67	1	no	—
457	C₄⋊C₈	—	67	1	no	—
458	C₄⋊C₈	—	67	1	no	—
459	C₄⋊C₈	—	67	1	no	—
460	C₄⋊C₈	—	67	1	no	—
461	C₄⋊C₈	—	134	1	no	—
462	C₄⋊C₈	—	134	1	no	—
463	C₄⋊C₈	—	134	1	no	—
464	C₄⋊C₈	—	134	1	no	—
465	C₄⋊C₈	—	134	1	no	—
466	C₄⋊C₈	—	134	1	no	—
467	C₄⋊C₈	—	134	1	no	—
468	C₄⋊C₈	—	134	1	no	—
469	#104	—	2	2	no	—
470	#104	—	4	1	no	—
471	#104	—	268	1	no	—
472	#104	—	134	2	no	—
473	#103	—	67	1	no	—
474	#103	—	67	1	no	—
475	#103	—	67	1	no	—
476	#103	—	67	1	no	—
477	#102	—	67	1	no	—
478	#102	—	67	1	no	—
479	#102	—	67	1	no	—
480	#102	—	67	1	no	—
481	C₂²⋊C₈	—	2	1	no	—
482	C₂²⋊C₈	—	2	1	no	—
483	C₂²⋊C₈	—	134	1	no	—
484	C₂²⋊C₈	—	134	1	no	—
485	#97	—	268	1	no	—
486	#97	—	268	1	no	—
487	#97	—	134	1	no	—
488	#97	—	134	1	no	—
489	C₄×C₈	—	67	1	no	—
490	C₄×C₈	—	67	1	no	—
491	C₄×C₈	—	67	1	no	—
492	C₄×C₈	—	67	1	no	—
493	C₄×C₈	—	134	1	no	—
494	C₄×C₈	—	134	1	no	—
495	#95	—	134	1	no	—
496	#41	—	2	1	no	—
497	#41	—	2	1	no	—
498	#41	—	2	1	no	—
499	#41	—	2	1	no	—
500	#41	—	2	1	no	—
501	#41	—	2	1	no	—
502	#41	#11761627	1	2	yes	—
503	#41	#11761627	1	2	yes	—
504	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
505	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
506	C₂×D₄	—	2	1	no	—
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
509	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	67	1	no	—
514	C₂²×C₄	—	67	1	no	—
515	C₂²×C₄	—	134	1	no	—
516	C₂²×C₄	—	134	1	no	—
517	C₂²×C₄	—	134	1	no	—
518	C₂²×C₄	—	134	1	no	—
519	C₂²×C₄	—	134	1	no	—
520	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
521	Q₁₆	—	2	2	no	—
522	SD₁₆	—	2	2	no	—
523	SD₁₆	—	2	2	no	—
524	D₈	—	2	2	no	—
525	M₁₆	—	2	1	no	—
526	M₁₆	—	2	1	no	—
527	M₁₆	—	2	1	no	—
528	M₁₆	—	2	1	no	—
529	M₁₆	—	4	2	no	—
530	M₁₆	—	268	2	no	—
531	M₁₆	—	134	1	no	—
532	M₁₆	—	134	1	no	—
533	M₁₆	—	134	1	no	—
534	M₁₆	—	134	1	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	1	no	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	4	2	no	—
542	C₂×C₈	—	268	2	no	—
543	C₂×C₈	—	67	1	no	—
544	C₂×C₈	—	67	1	no	—
545	C₂×C₈	—	67	1	no	—
546	C₂×C₈	—	67	1	no	—
547	C₂×C₈	—	134	1	no	—
548	C₂×C₈	—	134	1	no	—
549	C₂×C₈	—	134	1	no	—
550	C₂×C₈	—	134	1	no	—
551	C₂×C₈	—	134	1	no	—
552	C₂×C₈	—	134	1	no	—
553	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
554	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
555	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
556	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
557	C₄⋊C₄	—	268	2	no	—
558	C₄⋊C₄	—	67	1	no	—
559	C₄⋊C₄	—	67	1	no	—
560	C₄⋊C₄	—	67	1	no	—
561	C₄⋊C₄	—	67	1	no	—
562	C₄⋊C₄	—	134	1	no	—
563	C₄⋊C₄	—	134	1	no	—
564	C₄⋊C₄	—	134	1	no	—
565	C₄⋊C₄	—	134	1	no	—
566	C₂²⋊C₄	—	134	1	no	—
567	C₂²⋊C₄	—	134	1	no	—
568	C₂²⋊C₄	—	134	1	no	—
569	C₂²⋊C₄	—	134	1	no	—
570	C₂²⋊C₄	—	134	1	no	—
571	C₂²⋊C₄	—	134	1	no	—
572	C₄²	—	2	1	no	—
573	C₄²	—	2	1	no	—
574	C₄²	—	67	1	no	—
575	C₄²	—	67	1	no	—
576	C₄²	—	67	1	no	—
577	C₄²	—	67	1	no	—
578	C₄²	—	134	1	no	—
579	C₄²	—	134	1	no	—
580	C₄²	—	134	1	no	—
581	C₄²	—	134	1	no	—
582	C₄²	—	134	1	no	—
583	C₄²	—	134	1	no	—
584	C₄²	—	134	1	no	—
585	C₄²	—	134	1	no	—
586	C₂³	—	2	1	no	—
587	C₂³	—	2	1	no	—
588	C₂³	#41×D₆₇	1	1	yes	characteristic
589	Q₈	—	2	1	no	—
590	Q₈	—	2	1	no	—
591	Q₈	—	2	1	no	—
592	Q₈	#14523999	1	2	yes	—
593	D₄	—	2	2	no	—
594	D₄	—	2	2	no	—
595	D₄	—	2	1	no	—
596	D₄	—	2	1	no	—
597	D₄	—	2	1	no	—
598	D₄	—	4	1	no	—
599	D₄	—	4	1	no	—
600	D₄	#14523999	1	2	yes	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	268	2	no	—
611	C₂×C₄	—	268	2	no	—
612	C₂×C₄	—	268	1	no	—
613	C₂×C₄	—	268	1	no	—
614	C₂×C₄	—	268	1	no	—
615	C₂×C₄	—	268	1	no	—
616	C₂×C₄	—	536	1	no	—
617	C₂×C₄	—	536	1	no	—
618	C₂×C₄	—	67	1	no	—
619	C₂×C₄	—	67	1	no	—
620	C₂×C₄	—	67	1	no	—
621	C₂×C₄	—	67	1	no	—
622	C₂×C₄	—	134	1	no	—
623	C₂×C₄	—	134	1	no	—
624	C₂×C₄	—	134	1	no	—
625	C₂×C₄	—	134	1	no	—
626	C₂×C₄	—	134	2	no	—
627	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
628	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
629	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
630	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
631	C₂×C₄	#14524088	1	1	yes	characteristic
632	C₂×C₄	#14524090	1	1	yes	characteristic
633	C₈	—	2	1	no	—
634	C₈	—	2	1	no	—
635	C₈	—	2	1	no	—
636	C₈	—	2	1	no	—
637	C₈	—	4	1	no	—
638	C₈	—	4	1	no	—
639	C₈	—	268	1	no	—
640	C₈	—	268	1	no	—
641	C₈	—	134	1	no	—
642	C₈	—	134	1	no	—
643	C₈	—	134	1	no	—
644	C₈	—	134	1	no	—
645	C₂²	—	2	1	no	—
646	C₂²	—	2	1	no	—
647	C₂²	—	2	2	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	1	no	—
653	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
654	C₂²	#16864083	1	1	yes	characteristic
655	C₂²	#16864089	1	1	yes	characteristic
656	C₄	—	2	2	no	—
657	C₄	—	4	1	no	—
658	C₄	—	8	1	no	—
659	C₄	—	268	1	no	—
660	C₄	—	268	1	no	—
661	C₄	—	268	1	no	—
662	C₄	—	268	1	no	—
663	C₄	—	536	1	no	—
664	C₄	—	67	2	no	—
665	C₄	—	134	1	no	—
666	C₄	—	134	1	no	—
667	C₄	—	134	1	no	—
668	C₄	C₂×#14523999	1	1	yes	characteristic
669	C₄	#16863875	1	1	yes	characteristic
670	C₄	#16864033	1	1	yes	characteristic
671	C₄	#16864052	1	1	yes	characteristic
672	C₂	—	2	1	no	—
673	C₂	—	2	1	no	—
674	C₂	—	4	2	no	—
675	C₂	—	8	1	no	—
676	C₂	#22563136	1	1	yes	characteristic
677	C₂	#444×D₆₇	1	1	yes	characteristic
678	C₂	#22569121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1876	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1876	D₆₇
hypocenter	C₆₇	#22782
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1876

Complements

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

N	H ≅ G/N	type
2. #22568871	C₂	semidirect
3. #22568775	C₂	semidirect
4. #22560923	C₂	semidirect
5. #22560917	C₂	semidirect
6. #22560809	C₂	semidirect
7. #22560807	C₂	semidirect
9. #22560345	C₂	semidirect
10. #22560227	C₂	semidirect
11. #22560227	C₂	semidirect
12. #22560075	C₂	semidirect
13. #22560054	C₂	semidirect
14. #22560051	C₂	semidirect
19. Q₁₆×Dic₆₇	C₄	semidirect
20. Q₁₆×Dic₆₇	C₄	semidirect
21. SD₁₆×Dic₆₇	C₄	semidirect
22. SD₁₆×Dic₆₇	C₄	semidirect
23. SD₁₆×Dic₆₇	C₄	semidirect
24. SD₁₆×Dic₆₇	C₄	semidirect
25. D₈×Dic₆₇	C₄	semidirect
26. D₈×Dic₆₇	C₄	semidirect
355. C₆₇	#22782	semidirect

Maximal subgroups

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

subgroup	index
#135×Dic₆₇	2
#22568871	2
#22568775	2
#22560923	2
#22560917	2
#22560809	2
#22560807	2
#22560538	2
#22560345	2
#22560227	2
#22560075	2
#22560054	2
#22560051	2
C₆₇×#1876	2
#22782	67