Subgroups — #31011395

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570796	C₂	1	1	yes	maximal, characteristic, complemented
2	#22570789	C₂	1	1	yes	maximal, characteristic
3	#22562465	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562438	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562434	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562306	C₂	1	4	yes	maximal, complemented
7	#22562060	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2254	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863951	—	2	1	no	—
11	#16863947	—	2	1	no	—
12	C₂×#14523988	C₂²	1	4	yes	—
13	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
14	#16863776	C₂²	1	1	yes	characteristic
15	#16863773	C₂²	1	1	yes	characteristic
16	#16863748	—	2	1	no	—
17	#16863744	—	2	1	no	—
18	#16863638	—	2	1	no	—
19	#16863638	C₂²	1	2	yes	—
20	#16863633	—	2	1	no	—
21	#16863633	C₂²	1	2	yes	—
22	#16863490	—	2	1	no	—
23	#16863490	C₂²	1	2	yes	—
24	#16863478	—	2	1	no	—
25	#16863478	C₂²	1	2	yes	—
26	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
27	#16863199	C₂²	1	4	yes	—
28	#16863196	C₂²	1	4	yes	—
29	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₆₇	C₂²	1	2	yes	—
31	#500×C₆₇	C₂²	1	1	yes	characteristic
32	#500×C₆₇	C₂²	1	1	yes	characteristic
33	#367×C₆₇	C₂²	1	4	yes	—
34	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#11761629	—	4	1	no	—
36	C₂×Dic₂₆₈	—	2	1	no	—
37	C₂×Dic₂₆₈	C₂³	1	2	yes	—
38	C₂×D₅₃₆	—	2	1	no	—
39	C₂×D₅₃₆	C₂³	1	2	yes	—
40	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
41	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
42	#14524043	—	2	1	no	—
43	#14524040	—	2	1	no	—
44	#14524034	—	4	1	no	—
45	#14524032	—	4	1	no	—
46	#14524023	—	2	2	no	—
47	#14524019	—	4	1	no	—
48	#14524018	—	2	2	no	—
49	#14524018	—	4	1	no	—
50	Q₈×Dic₆₇	—	4	1	no	—
51	#14524008	—	2	2	no	—
52	#14524005	—	2	2	no	—
53	#14524005	—	4	1	no	—
54	#14524001	—	4	1	no	—
55	C₄²×D₆₇	D₄	1	1	yes	characteristic
56	#14523988	—	2	8	no	—
57	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
58	C₁₆×D₆₇	—	2	4	no	—
59	#14523966	—	2	2	no	—
60	#14523966	—	2	2	no	—
61	#14523963	—	2	2	no	—
62	#14523963	—	2	2	no	—
63	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
64	C₈×Dic₆₇	D₄	1	1	yes	characteristic
65	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₇	—	2	1	no	—
67	C₄⋊Q₈×C₆₇	—	2	1	no	—
68	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
69	#107×C₆₇	—	2	1	no	—
70	#107×C₆₇	—	2	1	no	—
71	#107×C₆₇	C₂³	1	2	yes	—
72	#107×C₆₇	C₂³	1	2	yes	—
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	#11761629	—	8	4	no	—
75	C₂×Dic₁₃₄	—	2	2	no	—
76	C₂×D₂₆₈	—	2	2	no	—
77	C₂×D₂₆₈	—	4	2	no	—
78	C₂×D₂₆₈	—	4	1	no	—
79	C₂×D₂₆₈	—	4	1	no	—
80	C₂×C₄×D₆₇	—	4	2	no	—
81	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
82	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
83	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
84	Dic₂₆₈	—	2	4	no	—
85	Dic₂₆₈	—	4	2	no	—
86	D₅₃₆	—	2	4	no	—
87	D₅₃₆	—	4	2	no	—
88	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₈×D₆₇	—	2	2	no	—
91	C₈×D₆₇	—	2	2	no	—
92	C₄⋊Dic₆₇	—	4	1	no	—
93	C₄⋊Dic₆₇	—	4	1	no	—
94	C₄⋊Dic₆₇	—	4	2	no	—
95	Dic₆₇⋊C₄	—	4	2	no	—
96	(C₂×D₆₇)⋊C₄	—	4	2	no	—
97	(C₂×D₆₇)⋊C₄	—	4	2	no	—
98	(C₂×D₆₇)⋊C₄	—	4	2	no	—
99	C₄×Dic₆₇	—	4	2	no	—
100	C₄×Dic₆₇	D₈	1	1	yes	characteristic
101	C₄×Dic₆₇	D₈	1	1	yes	characteristic
102	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
103	C₆₇⋊C₁₆	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	4	1	no	—
105	C₂×Q₈×C₆₇	—	4	1	no	—
106	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	4	1	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₆₇	#41	1	4	yes	complemented
114	C₂²×D₆₇	—	4	2	no	—
115	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₆₇	—	8	2	no	—
117	Q₈×C₆₇	—	8	2	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₆₇	—	4	2	no	—
124	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
127	D₂₆₈	—	4	2	no	—
128	D₂₆₈	—	4	2	no	—
129	D₂₆₈	—	8	4	no	—
130	D₂₆₈	—	8	2	no	—
131	D₂₆₈	—	8	2	no	—
132	C₄×D₆₇	—	2	1	no	—
133	C₄×D₆₇	—	2	1	no	—
134	C₄×D₆₇	—	2	2	no	—
135	C₄×D₆₇	—	8	4	no	—
136	C₄×D₆₇	SD₃₂	1	4	yes	complemented
137	Dic₁₃₄	—	4	2	no	—
138	Dic₁₃₄	—	4	2	no	—
139	C₈×C₆₇	#135	1	2	yes	—
140	C₈×C₆₇	C₂×#41	1	2	yes	—
141	C₆₇⋊C₈	—	2	1	no	—
142	C₆₇⋊C₈	—	2	1	no	—
143	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₆₇	—	2	1	no	—
145	C₂×D₆₇	—	2	1	no	—
146	C₂×D₆₇	—	8	2	no	—
147	C₂×D₆₇	—	8	2	no	—
148	C₂×D₆₇	—	8	2	no	—
149	C₂×D₆₇	C₂×SD₃₂	1	2	yes	—
150	C₄×C₆₇	—	8	2	no	—
151	C₄×C₆₇	—	8	2	no	—
152	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
153	C₄×C₆₇	#508	1	2	yes	—
154	C₄×C₆₇	#532	1	1	yes	characteristic
155	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
156	Dic₆₇	—	2	1	no	—
157	Dic₆₇	—	2	1	no	—
158	Dic₆₇	—	8	2	no	—
159	Dic₆₇	C₂×SD₃₂	1	2	yes	—
160	#33736	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	#3169	1	1	yes	characteristic
162	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
163	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
164	D₆₇	—	2	2	no	—
165	D₆₇	—	16	2	no	—
166	#3170	—	67	1	no	—
167	#3169	—	67	1	no	—
168	#2254	D₆₇	1	1	yes	characteristic, complemented
169	#2248	—	67	1	no	—
170	#2241	—	67	1	no	—
171	#2240	—	67	1	no	—
172	#2217	—	67	4	no	—
173	C₄×SD₃₂	—	67	4	no	—
174	C₂×C₄×C₁₆	—	67	1	no	—
175	C₆₇	#33736	1	1	yes	characteristic, complemented, Sylow
176	#533	—	134	1	no	—
177	#532	—	134	1	no	—
178	C₂×SD₃₂	—	67	4	no	—
179	C₂²×C₁₆	—	67	2	no	—
180	#500	C₂×D₆₇	1	1	yes	characteristic
181	#500	C₂×D₆₇	1	1	yes	characteristic
182	#494	—	67	1	no	—
183	#493	—	67	1	no	—
184	#487	—	134	1	no	—
185	#486	—	134	1	no	—
186	#467	—	67	2	no	—
187	#467	—	134	1	no	—
188	#466	—	67	2	no	—
189	#466	—	134	1	no	—
190	C₄×Q₁₆	—	67	2	no	—
191	C₄×Q₁₆	—	134	1	no	—
192	C₄×D₈	—	67	2	no	—
193	C₄×D₈	—	134	1	no	—
194	C₂×C₄×C₈	—	67	1	no	—
195	#367	C₂×D₆₇	1	4	yes	—
196	#358	—	67	4	no	—
197	#357	—	67	4	no	—
198	C₄×C₁₆	—	67	2	no	—
199	C₄×C₁₆	—	67	1	no	—
200	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
201	C₂×#41	—	268	1	no	—
202	C₂×Q₁₆	—	67	2	no	—
203	C₂×Q₁₆	—	134	1	no	—
204	C₂×D₈	—	67	2	no	—
205	C₂×D₈	—	134	1	no	—
206	C₂²×C₈	—	67	1	no	—
207	C₂²×C₈	—	67	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	134	1	no	—
211	#127	—	134	1	no	—
212	#125	—	268	1	no	—
213	#124	—	268	1	no	—
214	C₂²⋊Q₈	—	268	1	no	—
215	C₂²⋊Q₈	—	134	2	no	—
216	#121	—	268	1	no	—
217	#121	—	134	2	no	—
218	C₄×Q₈	—	268	1	no	—
219	C₄×Q₈	—	134	2	no	—
220	C₄×D₄	—	268	1	no	—
221	C₄×D₄	—	134	2	no	—
222	#117	—	268	1	no	—
223	C₂×C₄²	—	67	1	no	—
224	SD₃₂	—	134	8	no	—
225	C₂×C₁₆	—	67	2	no	—
226	C₂×C₁₆	—	134	4	no	—
227	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	C₂²×D₆₇	1	2	yes	—
231	#107	C₂²×D₆₇	1	2	yes	—
232	#103	—	134	2	no	—
233	#103	—	134	2	no	—
234	#102	—	134	2	no	—
235	#102	—	134	2	no	—
236	C₄×C₈	—	67	1	no	—
237	C₄×C₈	—	67	1	no	—
238	C₄×C₈	—	67	1	no	—
239	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
240	#41	—	536	4	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	134	2	no	—
244	C₂×D₄	—	268	2	no	—
245	C₂×D₄	—	268	1	no	—
246	C₂×D₄	—	268	1	no	—
247	C₂×D₄	—	134	2	no	—
248	C₂²×C₄	—	268	2	no	—
249	C₂²×C₄	—	67	1	no	—
250	C₂²×C₄	—	67	1	no	—
251	C₂²×C₄	—	67	1	no	—
252	Q₁₆	—	268	2	no	—
253	Q₁₆	—	134	4	no	—
254	D₈	—	268	2	no	—
255	D₈	—	134	4	no	—
256	C₂×C₈	—	67	1	no	—
257	C₂×C₈	—	67	1	no	—
258	C₂×C₈	—	134	2	no	—
259	C₂×C₈	—	134	2	no	—
260	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	268	2	no	—
267	C₄⋊C₄	—	268	2	no	—
268	C₄⋊C₄	—	268	1	no	—
269	C₄⋊C₄	—	268	1	no	—
270	C₂²⋊C₄	—	268	2	no	—
271	C₂²⋊C₄	—	268	2	no	—
272	C₂²⋊C₄	—	268	2	no	—
273	C₄²	—	268	2	no	—
274	C₄²	—	67	1	no	—
275	C₄²	—	67	1	no	—
276	C₄²	—	67	1	no	—
277	C₄²	D₄×D₆₇	1	1	yes	characteristic
278	C₁₆	—	134	2	no	—
279	C₁₆	#11761629	1	4	yes	complemented
280	C₂³	—	268	2	no	—
281	C₂³	—	67	1	no	—
282	Q₈	—	8	2	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	268	2	no	—
285	Q₈	—	268	2	no	—
286	D₄	—	268	2	no	—
287	D₄	—	268	2	no	—
288	D₄	—	536	4	no	—
289	D₄	—	536	2	no	—
290	D₄	—	536	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	268	2	no	—
294	C₂×C₄	—	536	4	no	—
295	C₂×C₄	—	67	1	no	—
296	C₂×C₄	—	67	4	no	—
297	C₂×C₄	—	67	1	no	—
298	C₂×C₄	—	67	1	no	—
299	C₂×C₄	—	134	2	no	—
300	C₂×C₄	—	134	1	no	—
301	C₂×C₄	—	134	1	no	—
302	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
303	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
305	C₈	—	134	1	no	—
306	C₈	—	134	1	no	—
307	C₈	#14524066	1	2	yes	—
308	C₈	C₂×#11761629	1	2	yes	—
309	C₂²	—	536	2	no	—
310	C₂²	—	536	2	no	—
311	C₂²	—	536	2	no	—
312	C₂²	—	67	2	no	—
313	C₂²	—	134	1	no	—
314	C₂²	—	134	1	no	—
315	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
316	C₄	—	8	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	536	2	no	—
319	C₄	—	67	2	no	—
320	C₄	—	134	1	no	—
321	C₄	—	134	1	no	—
322	C₄	SD₃₂×D₆₇	1	2	yes	—
323	C₄	#16863827	1	2	yes	—
324	C₄	#16863947	1	1	yes	characteristic
325	C₄	C₂×#14524066	1	1	yes	characteristic
326	C₂	—	1072	2	no	—
327	C₂	—	134	2	no	—
328	C₂	#22570789	1	1	yes	characteristic
329	C₂	C₂×SD₃₂×D₆₇	1	1	yes	characteristic
330	C₂	C₂×#16863827	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×#2254	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22570789	C₂
hypercenter	#2254	D₆₇
hypocenter	C₆₇	#33736
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
1. #22570796	C₂	semidirect
3. #22562465	C₂	semidirect
4. #22562438	C₂	semidirect
5. #22562434	C₂	semidirect
6. #22562306	C₂	semidirect
7. #22562060	C₂	semidirect
8. C₄×C₁₆×D₆₇	C₂	semidirect
9. C₆₇×#2254	C₂	semidirect
113. C₁₆×C₆₇	#41	semidirect
136. C₄×D₆₇	SD₃₂	semidirect
168. #2254	D₆₇	semidirect
175. C₆₇	#33736	semidirect
279. C₁₆	#11761629	semidirect

Maximal subgroups

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

subgroup	index
#22570796	2
#22570789	2
#22562465	2
#22562438	2
#22562434	2
#22562306	2
#22562060	2
C₄×C₁₆×D₆₇	2
C₆₇×#2254	2
#33736	67