Subgroups — #31013503

Order 17152.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573222	C₂	1	1	yes	maximal, characteristic, complemented
2	#22573077	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562441	C₂	1	2	yes	maximal, complemented
4	#22562433	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562367	C₂	1	2	yes	maximal, complemented
6	#22562362	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562265	C₂	1	2	yes	maximal, complemented
8	#22562245	C₂	1	1	yes	maximal, characteristic
9	C₄×#14523988	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₀₇₂	C₂	1	2	yes	maximal, complemented
11	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864131	—	2	1	no	—
13	#16864106	—	2	1	no	—
14	C₂×#14523988	C₂²	1	1	yes	characteristic
15	C₂×D₁₀₇₂	C₂²	1	2	yes	—
16	#16863801	C₂²	1	1	yes	characteristic
17	#16863798	C₂²	1	2	yes	—
18	#16863780	C₂²	1	2	yes	—
19	#16863773	C₂²	1	1	yes	characteristic
20	#16863697	—	2	2	no	—
21	#16863687	—	2	1	no	—
22	#16863612	—	2	2	no	—
23	#16863597	—	2	1	no	—
24	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
25	C₄×D₅₃₆	C₂²	1	2	yes	—
26	C₄×D₅₃₆	C₂²	1	2	yes	—
27	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
28	#16863224	C₂²	1	1	yes	characteristic
29	#16863222	C₂²	1	2	yes	—
30	#16863201	C₂²	1	1	yes	characteristic
31	#16863201	C₂²	1	2	yes	—
32	#16863198	C₂²	1	1	yes	characteristic
33	#16863198	C₂²	1	2	yes	—
34	#16863198	C₂²	1	2	yes	—
35	#16863198	C₂²	1	2	yes	—
36	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
42	C₂×#11761617	—	2	2	no	—
43	C₂×D₅₃₆	—	2	1	no	—
44	C₂×D₅₃₆	C₂³	1	2	yes	—
45	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
46	#14524043	—	2	1	no	—
47	#14524040	—	2	1	no	—
48	#14524035	—	2	2	no	—
49	#14524032	—	4	1	no	—
50	#14524031	—	2	2	no	—
51	#14524028	—	4	2	no	—
52	#14524021	—	4	2	no	—
53	#14524019	—	4	1	no	—
54	C₄×Dic₁₃₄	—	2	1	no	—
55	C₄×D₂₆₈	—	2	1	no	—
56	C₄×D₂₆₈	—	2	2	no	—
57	C₄×D₂₆₈	—	2	2	no	—
58	#14524005	—	4	1	no	—
59	C₄⋊C₄×D₆₇	—	4	1	no	—
60	#14523988	—	2	2	no	—
61	D₁₀₇₂	—	2	4	no	—
62	#14523977	—	2	1	no	—
63	#14523977	C₂³	1	1	yes	characteristic
64	#14523977	C₂³	1	1	yes	characteristic
65	#14523977	C₂³	1	2	yes	—
66	#14523975	—	2	2	no	—
67	#14523968	—	2	1	no	—
68	#14523965	—	2	2	no	—
69	#14523965	—	2	2	no	—
70	#14523965	—	2	1	no	—
71	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂³	1	2	yes	—
79	C₂×Dic₁₃₄	—	2	1	no	—
80	C₂×Dic₁₃₄	—	4	1	no	—
81	C₂×D₂₆₈	—	2	1	no	—
82	C₂×D₂₆₈	—	2	2	no	—
83	C₂×D₂₆₈	—	4	1	no	—
84	C₂×D₂₆₈	—	4	1	no	—
85	C₂×C₄×D₆₇	—	4	1	no	—
86	C₂×C₄×D₆₇	—	4	2	no	—
87	C₂×C₄×D₆₇	—	4	2	no	—
88	Dic₂₆₈	—	2	1	no	—
89	Dic₂₆₈	—	2	1	no	—
90	#11761617	—	8	2	no	—
91	D₅₃₆	—	2	2	no	—
92	D₅₃₆	—	2	1	no	—
93	D₅₃₆	—	2	2	no	—
94	D₅₃₆	—	8	1	no	—
95	D₅₃₆	#41	1	2	yes	—
96	C₄⋊Dic₆₇	—	2	1	no	—
97	C₄⋊Dic₆₇	—	2	1	no	—
98	C₄⋊Dic₆₇	—	2	2	no	—
99	C₄⋊Dic₆₇	—	4	1	no	—
100	Dic₆₇⋊C₄	—	4	2	no	—
101	Dic₆₇⋊C₄	—	4	1	no	—
102	Dic₆₇⋊C₄	—	4	2	no	—
103	(C₂×D₆₇)⋊C₄	—	4	2	no	—
104	(C₂×D₆₇)⋊C₄	—	4	2	no	—
105	(C₂×D₆₇)⋊C₄	—	4	1	no	—
106	(C₂×D₆₇)⋊C₄	—	4	2	no	—
107	(C₂×D₆₇)⋊C₄	—	4	2	no	—
108	C₄×Dic₆₇	—	4	1	no	—
109	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
115	C₄²×C₆₇	C₂×D₄	1	2	yes	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₇	—	2	2	no	—
118	C₁₆×C₆₇	—	2	1	no	—
119	C₁₆×C₆₇	—	2	1	no	—
120	C₂²×D₆₇	—	4	1	no	—
121	C₂²×D₆₇	—	4	2	no	—
122	Q₈×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₆₇	—	4	1	no	—
129	C₂×Dic₆₇	—	4	1	no	—
130	C₂×Dic₆₇	—	4	1	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	D₂₆₈	—	2	2	no	—
133	D₂₆₈	—	4	2	no	—
134	D₂₆₈	—	4	1	no	—
135	D₂₆₈	—	4	2	no	—
136	D₂₆₈	—	8	2	no	—
137	D₂₆₈	—	16	1	no	—
138	C₄×D₆₇	—	8	2	no	—
139	C₄×D₆₇	—	16	2	no	—
140	C₄×D₆₇	—	16	2	no	—
141	Dic₁₃₄	—	4	1	no	—
142	Dic₁₃₄	—	4	1	no	—
143	Dic₁₃₄	—	16	1	no	—
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	2	2	no	—
146	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₆₇	—	4	2	no	—
150	C₂×D₆₇	—	8	1	no	—
151	C₂×D₆₇	—	8	1	no	—
152	C₂×D₆₇	—	8	2	no	—
153	C₂×D₆₇	—	8	2	no	—
154	C₂×D₆₇	—	8	2	no	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	#508	1	2	yes	—
159	C₄×C₆₇	#548	1	1	yes	characteristic
160	C₄×C₆₇	#577	1	1	yes	characteristic
161	Dic₆₇	—	8	1	no	—
162	Dic₆₇	—	8	1	no	—
163	Dic₆₇	—	8	2	no	—
164	Dic₆₇	—	8	1	no	—
165	#33928	—	67	1	no	maximal, Sylow
166	C₂×C₆₇	#3358	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
168	C₂×C₆₇	#3415	1	1	yes	characteristic
169	D₆₇	—	8	2	no	—
170	D₆₇	—	16	2	no	—
171	#3396	—	67	1	no	—
172	#3380	—	67	1	no	—
173	#2242	—	67	2	no	—
174	#2240	—	67	1	no	—
175	#2228	—	67	2	no	—
176	#2227	—	67	1	no	—
177	#2210	—	67	2	no	—
178	#2206	—	67	1	no	—
179	Q₈×C₁₆	D₆₇	1	1	yes	characteristic, complemented
180	C₄×SD₃₂	—	67	1	no	—
181	C₄×D₁₆	—	67	2	no	—
182	C₆₇	#33928	1	1	yes	characteristic, complemented, Sylow
183	#559	—	134	1	no	—
184	#554	—	134	1	no	—
185	C₂×SD₃₂	—	67	1	no	—
186	C₂×D₁₆	—	67	2	no	—
187	#500	—	67	1	no	—
188	#499	—	67	2	no	—
189	#495	—	67	2	no	—
190	#493	—	67	1	no	—
191	#478	—	134	2	no	—
192	#476	—	134	1	no	—
193	#463	—	134	2	no	—
194	#460	—	134	1	no	—
195	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
196	C₄×Q₁₆	—	67	1	no	—
197	C₄×D₈	—	67	2	no	—
198	C₄×D₈	—	67	1	no	—
199	C₄×D₈	—	67	2	no	—
200	#367	—	67	1	no	—
201	#366	—	67	2	no	—
202	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
204	#358	—	67	1	no	—
205	#358	—	67	2	no	—
206	#357	—	67	1	no	—
207	#357	—	67	2	no	—
208	#357	—	67	2	no	—
209	#357	—	67	2	no	—
210	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
211	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
212	C₂×Q₁₆	—	67	1	no	—
213	C₂×SD₁₆	—	134	2	no	—
214	C₂×D₈	—	67	1	no	—
215	C₂×D₈	—	67	2	no	—
216	C₂×D₈	—	134	1	no	—
217	C₄⋊Q₈	—	134	1	no	—
218	#127	—	134	1	no	—
219	#125	—	134	2	no	—
220	#124	—	268	1	no	—
221	#124	—	134	2	no	—
222	#123	—	268	2	no	—
223	C₂²⋊Q₈	—	268	1	no	—
224	C₂²⋊Q₈	—	268	2	no	—
225	C₄×Q₈	—	134	1	no	—
226	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
227	C₄×D₄	—	268	1	no	—
228	C₄×D₄	—	134	1	no	—
229	C₄×D₄	—	134	2	no	—
230	C₄×D₄	—	134	2	no	—
231	C₂×C₄⋊C₄	—	268	1	no	—
232	SD₃₂	—	134	2	no	—
233	D₁₆	—	134	4	no	—
234	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
237	#107	—	67	2	no	—
238	#107	—	67	1	no	—
239	#107	—	67	1	no	—
240	#107	—	134	1	no	—
241	#106	—	134	2	no	—
242	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
243	C₄⋊C₈	D₂₆₈	1	2	yes	—
244	#103	—	134	1	no	—
245	#102	—	134	2	no	—
246	#102	—	134	2	no	—
247	#102	—	134	1	no	—
248	C₄×C₈	C₂²×D₆₇	1	2	yes	—
249	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
250	C₂×Q₈	—	268	1	no	—
251	C₂×Q₈	—	134	1	no	—
252	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
253	C₂×D₄	—	268	1	no	—
254	C₂×D₄	—	268	1	no	—
255	C₂×D₄	—	134	2	no	—
256	C₂×D₄	—	134	1	no	—
257	C₂²×C₄	—	268	1	no	—
258	C₂²×C₄	—	268	2	no	—
259	C₂²×C₄	—	268	2	no	—
260	Q₁₆	—	134	1	no	—
261	Q₁₆	—	134	1	no	—
262	SD₁₆	—	536	2	no	—
263	D₈	—	536	1	no	—
264	D₈	—	67	2	no	—
265	D₈	—	134	2	no	—
266	D₈	—	134	2	no	—
267	D₈	—	134	1	no	—
268	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
269	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	268	1	no	—
273	C₄⋊C₄	—	268	1	no	—
274	C₄⋊C₄	—	268	2	no	—
275	C₄⋊C₄	—	134	1	no	—
276	C₄⋊C₄	—	134	1	no	—
277	C₄⋊C₄	—	134	2	no	—
278	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
279	C₄⋊C₄	D₅₃₆	1	2	yes	—
280	C₂²⋊C₄	—	268	2	no	—
281	C₂²⋊C₄	—	268	1	no	—
282	C₂²⋊C₄	—	268	2	no	—
283	C₂²⋊C₄	—	268	2	no	—
284	C₂²⋊C₄	—	268	2	no	—
285	C₄²	—	268	1	no	—
286	C₄²	C₂×D₂₆₈	1	2	yes	—
287	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
288	C₁₆	—	2	2	no	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	1	no	—
291	C₂³	—	268	1	no	—
292	C₂³	—	268	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	268	1	no	—
295	Q₈	—	268	1	no	—
296	Q₈	—	1072	1	no	—
297	D₄	—	268	2	no	—
298	D₄	—	268	1	no	—
299	D₄	—	268	2	no	—
300	D₄	—	536	2	no	—
301	D₄	—	1072	1	no	—
302	D₄	—	134	2	no	—
303	C₂×C₄	—	268	1	no	—
304	C₂×C₄	—	268	1	no	—
305	C₂×C₄	—	268	1	no	—
306	C₂×C₄	—	268	2	no	—
307	C₂×C₄	—	536	2	no	—
308	C₂×C₄	—	1072	2	no	—
309	C₂×C₄	—	1072	2	no	—
310	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
311	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
312	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
313	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	#41×D₆₇	1	1	yes	characteristic
318	C₈	#14524091	1	1	yes	characteristic
319	C₂²	—	268	2	no	—
320	C₂²	—	536	1	no	—
321	C₂²	—	536	1	no	—
322	C₂²	—	536	2	no	—
323	C₂²	—	536	2	no	—
324	C₂²	—	536	2	no	—
325	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	536	1	no	—
330	C₄	—	536	1	no	—
331	C₄	—	536	2	no	—
332	C₄	—	536	1	no	—
333	C₄	#16863828	1	2	yes	—
334	C₄	#16864058	1	1	yes	characteristic
335	C₄	#16864208	1	1	yes	characteristic
336	C₂	—	536	2	no	—
337	C₂	—	1072	2	no	—
338	C₂	#22572845	1	1	yes	characteristic
339	C₂	C₂×#16863828	1	1	yes	characteristic
340	C₂	#22573350	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	Q₈×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22562245	C₂
hypercenter	Q₈×C₁₆	D₆₇
hypocenter	C₆₇	#33928
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #22573222	C₂	semidirect
2. #22573077	C₂	semidirect
3. #22562441	C₂	semidirect
4. #22562433	C₂	semidirect
5. #22562367	C₂	semidirect
6. #22562362	C₂	semidirect
7. #22562265	C₂	semidirect
9. C₄×#14523988	C₂	semidirect
10. C₄×D₁₀₇₂	C₂	semidirect
11. Q₈×C₁₆×C₆₇	C₂	semidirect
179. Q₈×C₁₆	D₆₇	semidirect
182. C₆₇	#33928	semidirect

Maximal subgroups

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

subgroup	index
#22573222	2
#22573077	2
#22562441	2
#22562433	2
#22562367	2
#22562362	2
#22562265	2
#22562245	2
C₄×#14523988	2
C₄×D₁₀₇₂	2
Q₈×C₁₆×C₆₇	2
#33928	67