Subgroups — #31011350

Order 17152.

514 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570720	C₂	1	1	yes	maximal, characteristic
2	#500×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22562493	C₂	1	4	yes	maximal, complemented
4	#22562491	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562489	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562482	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561994	C₂	1	4	yes	maximal
8	C₄×#14523982	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863951	—	2	1	no	—
11	C₄⋊Q₈×D₆₇	—	2	1	no	—
12	C₂×#14523982	C₂²	1	2	yes	—
13	#16863801	C₂²	1	1	yes	characteristic
14	#16863801	C₂²	1	1	yes	characteristic
15	#16863800	—	2	1	no	—
16	#16863800	C₂²	1	2	yes	—
17	#16863799	—	2	1	no	—
18	#16863797	—	2	1	no	—
19	#16863797	C₂²	1	2	yes	—
20	#16863792	—	2	1	no	—
21	#16863433	—	2	1	no	—
22	#16863433	C₂²	1	2	yes	—
23	#107×D₆₇	—	2	1	no	—
24	#107×D₆₇	C₂²	1	2	yes	—
25	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
26	#16863224	C₂²	1	4	yes	—
27	#16863223	C₂²	1	4	yes	—
28	#16863221	C₂²	1	4	yes	—
29	#16863171	C₂²	1	2	yes	—
30	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
31	#500×C₆₇	C₂²	1	1	yes	characteristic
32	#500×C₆₇	C₂²	1	1	yes	characteristic
33	#366×C₆₇	C₂²	1	4	yes	—
34	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₆₇	—	4	1	no	—
36	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
37	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
38	#14524043	—	2	1	no	—
39	#14524043	—	2	1	no	—
40	#14524042	—	2	2	no	—
41	#14524042	—	4	1	no	—
42	#14524041	—	4	1	no	—
43	#14524034	—	2	2	no	—
44	#14524023	—	4	1	no	—
45	#14524019	—	4	1	no	—
46	Q₈×Dic₆₇	—	4	1	no	—
47	#14524008	—	4	1	no	—
48	#14524001	—	2	2	no	—
49	C₄⋊C₄×D₆₇	—	2	2	no	—
50	C₄⋊C₄×D₆₇	—	4	1	no	—
51	C₄²×D₆₇	D₄	1	1	yes	characteristic
52	#14523982	Q₈	1	8	yes	complemented
53	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
54	#14523977	—	2	1	no	—
55	#14523977	—	2	1	no	—
56	#14523977	C₂³	1	2	yes	—
57	#14523977	C₂³	1	2	yes	—
58	#14523976	—	2	2	no	—
59	#14523976	—	2	2	no	—
60	#14523974	—	2	2	no	—
61	#14523974	—	2	2	no	—
62	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
63	C₈×Dic₆₇	D₄	1	1	yes	characteristic
64	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₇	—	2	1	no	—
66	C₄⋊Q₈×C₆₇	—	2	1	no	—
67	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
68	#107×C₆₇	—	2	1	no	—
69	#107×C₆₇	—	2	1	no	—
70	#107×C₆₇	C₂³	1	2	yes	—
71	#107×C₆₇	C₂³	1	2	yes	—
72	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×Dic₁₃₄	—	4	1	no	—
74	C₂×Dic₁₃₄	—	4	1	no	—
75	C₂×Dic₁₃₄	—	4	2	no	—
76	Q₈×D₆₇	—	8	4	no	—
77	C₂×C₄×D₆₇	—	4	2	no	—
78	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
79	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
80	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
83	C₈×D₆₇	Q₁₆	1	4	yes	—
84	C₈×D₆₇	C₂×Q₈	1	4	yes	—
85	C₄⋊Dic₆₇	—	2	2	no	—
86	C₄⋊Dic₆₇	—	2	2	no	—
87	C₄⋊Dic₆₇	—	4	1	no	—
88	C₄⋊Dic₆₇	—	4	1	no	—
89	C₄⋊Dic₆₇	—	4	2	no	—
90	Dic₆₇⋊C₄	—	4	2	no	—
91	Dic₆₇⋊C₄	—	4	2	no	—
92	Dic₆₇⋊C₄	—	4	2	no	—
93	(C₂×D₆₇)⋊C₄	—	4	2	no	—
94	C₄×Dic₆₇	—	4	2	no	—
95	C₄×Dic₆₇	D₈	1	1	yes	characteristic
96	C₄×Dic₆₇	D₈	1	1	yes	characteristic
97	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
98	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
99	C₂×Q₈×C₆₇	—	4	1	no	—
100	C₂×Q₈×C₆₇	—	4	1	no	—
101	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	4	1	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₁₆×C₆₇	C₂×Q₈	1	4	yes	complemented
109	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
110	Q₈×C₆₇	—	8	2	no	—
111	Q₈×C₆₇	—	8	2	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	2	no	—
114	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₆₇	—	4	2	no	—
118	C₂×Dic₆₇	—	4	2	no	—
119	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
120	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
121	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
122	C₄×D₆₇	—	2	2	no	—
123	C₄×D₆₇	—	2	2	no	—
124	C₄×D₆₇	—	8	4	no	—
125	C₄×D₆₇	C₄⋊Q₈	1	2	yes	—
126	C₄×D₆₇	C₂×Q₁₆	1	2	yes	—
127	Dic₁₃₄	—	8	4	no	—
128	Dic₁₃₄	—	8	2	no	—
129	Dic₁₃₄	—	8	2	no	—
130	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
131	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
132	C₆₇⋊C₈	C₄⋊Q₈	1	2	yes	—
133	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	—
134	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
135	C₂×D₆₇	—	2	1	no	—
136	C₂×D₆₇	—	2	1	no	—
137	C₂×D₆₇	#500	1	2	yes	—
138	C₄×C₆₇	—	8	2	no	—
139	C₄×C₆₇	—	8	2	no	—
140	C₄×C₆₇	#509	1	2	yes	—
141	C₄×C₆₇	#510	1	2	yes	—
142	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
144	Dic₆₇	—	2	1	no	—
145	Dic₆₇	—	2	1	no	—
146	Dic₆₇	—	8	2	no	—
147	Dic₆₇	—	8	2	no	—
148	Dic₆₇	#500	1	2	yes	—
149	#33731	—	67	1	no	maximal, Sylow
150	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
153	D₆₇	—	2	2	no	—
154	#3160	—	67	1	no	—
155	C₂×#500	—	67	1	no	—
156	#2255	—	67	4	no	—
157	#2254	—	67	1	no	—
158	#2254	—	67	1	no	—
159	#2252	—	67	1	no	—
160	#2252	D₆₇	1	1	yes	characteristic, complemented
161	#2159	—	67	4	no	—
162	C₄×M₃₂	—	67	1	no	—
163	C₆₇	#33731	1	1	yes	characteristic, complemented, Sylow
164	#533	—	134	1	no	—
165	C₂×C₄⋊Q₈	—	134	1	no	—
166	C₂×M₃₂	—	67	2	no	—
167	#500	—	67	2	no	—
168	#500	—	67	1	no	—
169	#500	—	67	1	no	—
170	#500	—	134	1	no	—
171	#500	—	134	1	no	—
172	#500	C₂×D₆₇	1	1	yes	characteristic
173	#500	C₂×D₆₇	1	1	yes	characteristic
174	#499	—	67	2	no	—
175	#499	—	134	1	no	—
176	#498	—	134	1	no	—
177	#427	—	67	2	no	—
178	#427	—	134	1	no	—
179	C₂×#107	—	67	2	no	—
180	C₂×#107	—	134	1	no	—
181	C₂×C₄×C₈	—	67	1	no	—
182	#367	—	67	4	no	—
183	#367	—	67	4	no	—
184	#366	—	67	4	no	—
185	#366	C₂×D₆₇	1	4	yes	—
186	#346	—	67	2	no	—
187	C₄×C₁₆	—	67	1	no	—
188	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
189	C₂²×Q₈	—	268	1	no	—
190	C₂²×C₈	—	67	1	no	—
191	C₂²×C₈	—	67	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	268	1	no	—
195	C₄⋊Q₈	—	268	1	no	—
196	C₄⋊Q₈	—	134	2	no	—
197	C₄⋊Q₈	—	134	1	no	—
198	C₄⋊Q₈	—	134	1	no	—
199	#125	—	134	2	no	—
200	C₂²⋊Q₈	—	268	1	no	—
201	C₂²⋊Q₈	—	268	1	no	—
202	C₄×Q₈	—	268	1	no	—
203	C₄×Q₈	—	268	1	no	—
204	#117	—	134	2	no	—
205	C₂×C₄⋊C₄	—	268	1	no	—
206	C₂×C₄⋊C₄	—	134	2	no	—
207	C₂×C₄²	—	67	1	no	—
208	M₃₂	—	67	8	no	—
209	C₂×C₁₆	—	67	2	no	—
210	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	67	2	no	—
214	#107	—	67	2	no	—
215	#107	—	134	2	no	—
216	#107	—	134	2	no	—
217	#107	—	134	1	no	—
218	#107	—	134	1	no	—
219	#107	C₂²×D₆₇	1	2	yes	—
220	#107	C₂²×D₆₇	1	2	yes	—
221	#106	—	134	2	no	—
222	#106	—	134	2	no	—
223	C₄×C₈	—	67	1	no	—
224	C₄×C₈	—	67	1	no	—
225	C₄×C₈	—	67	1	no	—
226	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	268	2	no	—
230	C₂×Q₈	—	268	1	no	—
231	C₂×Q₈	—	268	1	no	—
232	C₂×Q₈	—	536	4	no	—
233	C₂²×C₄	—	268	2	no	—
234	C₂²×C₄	—	67	1	no	—
235	C₂²×C₄	—	67	1	no	—
236	C₂²×C₄	—	67	1	no	—
237	C₂×C₈	—	67	1	no	—
238	C₂×C₈	—	67	1	no	—
239	C₂×C₈	—	67	4	no	—
240	C₂×C₈	—	67	4	no	—
241	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
242	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	268	2	no	—
248	C₄⋊C₄	—	268	2	no	—
249	C₄⋊C₄	—	268	2	no	—
250	C₄⋊C₄	—	268	2	no	—
251	C₄⋊C₄	—	268	1	no	—
252	C₄⋊C₄	—	268	1	no	—
253	C₄⋊C₄	—	134	2	no	—
254	C₄⋊C₄	—	134	2	no	—
255	C₂²⋊C₄	—	268	2	no	—
256	C₄²	—	268	2	no	—
257	C₄²	—	67	1	no	—
258	C₄²	—	67	1	no	—
259	C₄²	—	67	1	no	—
260	C₄²	D₄×D₆₇	1	1	yes	characteristic
261	C₁₆	—	67	4	no	—
262	C₁₆	Q₈×D₆₇	1	4	yes	complemented
263	C₂³	—	67	1	no	—
264	Q₈	—	8	2	no	—
265	Q₈	—	8	2	no	—
266	Q₈	—	536	4	no	—
267	Q₈	—	536	2	no	—
268	Q₈	—	536	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	268	2	no	—
272	C₂×C₄	—	268	2	no	—
273	C₂×C₄	—	536	4	no	—
274	C₂×C₄	—	67	1	no	—
275	C₂×C₄	—	67	2	no	—
276	C₂×C₄	—	67	2	no	—
277	C₂×C₄	—	67	1	no	—
278	C₂×C₄	—	67	1	no	—
279	C₂×C₄	—	134	2	no	—
280	C₂×C₄	—	134	2	no	—
281	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
282	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
284	C₈	—	67	2	no	—
285	C₈	—	67	2	no	—
286	C₈	Q₁₆×D₆₇	1	2	yes	—
287	C₈	C₂×Q₈×D₆₇	1	2	yes	—
288	C₂²	—	67	2	no	—
289	C₂²	—	134	1	no	—
290	C₂²	—	134	1	no	—
291	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	536	2	no	—
295	C₄	—	536	2	no	—
296	C₄	—	67	2	no	—
297	C₄	—	134	1	no	—
298	C₄	—	134	1	no	—
299	C₄	#16863830	1	2	yes	—
300	C₄	#16863838	1	2	yes	—
301	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
302	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
303	C₂	—	134	2	no	—
304	C₂	#500×D₆₇	1	1	yes	characteristic
305	C₂	C₂×#16863830	1	1	yes	characteristic
306	C₂	C₂×#16863838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2252	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₆₇	C₂×D₈
hypercenter	#2252	D₆₇
hypocenter	C₆₇	#33731
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
3. #22562493	C₂	semidirect
4. #22562491	C₂	semidirect
5. #22562489	C₂	semidirect
6. #22562482	C₂	semidirect
9. C₆₇×#2252	C₂	semidirect
52. #14523982	Q₈	semidirect
98. C₆₇⋊C₁₆	C₂×Q₈	semidirect
108. C₁₆×C₆₇	C₂×Q₈	semidirect
160. #2252	D₆₇	semidirect
163. C₆₇	#33731	semidirect
262. C₁₆	Q₈×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22570720	2
#500×D₆₇	2
#22562493	2
#22562491	2
#22562489	2
#22562482	2
#22561994	2
C₄×#14523982	2
C₆₇×#2252	2
#33731	67