Subgroups — #31013492

Order 17152.

486 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573198	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	#22562437	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562374	C₂	1	2	yes	maximal, complemented
6	#22562357	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562265	C₂	1	2	yes	maximal
8	#22562255	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14523988	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₀₇₂	C₂	1	1	yes	maximal, characteristic
11	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864119	—	2	1	no	—
13	#16864106	—	2	1	no	—
14	C₂×#14523988	C₂²	1	2	yes	—
15	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic
16	#16863801	C₂²	1	1	yes	characteristic
17	#16863798	C₂²	1	2	yes	—
18	#16863780	C₂²	1	2	yes	—
19	#16863776	C₂²	1	1	yes	characteristic
20	#16863697	—	2	2	no	—
21	#16863687	—	2	1	no	—
22	#16863619	—	2	2	no	—
23	#16863602	—	2	1	no	—
24	C₄×Dic₂₆₈	C₂²	1	2	yes	—
25	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
26	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
27	C₄×D₅₃₆	C₂²	1	2	yes	—
28	#16863224	C₂²	1	2	yes	—
29	#16863222	C₂²	1	1	yes	characteristic
30	#16863201	C₂²	1	2	yes	—
31	#16863201	C₂²	1	2	yes	—
32	#16863201	C₂²	1	2	yes	—
33	#16863198	C₂²	1	1	yes	characteristic
34	#16863198	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
40	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
41	C₂×Dic₂₆₈	—	2	1	no	—
42	C₂×Dic₂₆₈	C₂³	1	2	yes	—
43	C₂×#11761617	—	2	2	no	—
44	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
45	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
46	#14524043	—	2	1	no	—
47	#14524043	—	2	1	no	—
48	#14524038	—	4	2	no	—
49	#14524035	—	2	2	no	—
50	#14524031	—	2	2	no	—
51	#14524021	—	4	2	no	—
52	#14524019	—	4	1	no	—
53	#14524019	—	4	1	no	—
54	C₄×Dic₁₃₄	—	2	2	no	—
55	C₄×D₂₆₈	—	2	1	no	—
56	C₄×D₂₆₈	—	2	2	no	—
57	C₄×D₂₆₈	—	2	1	no	—
58	#14524001	—	4	1	no	—
59	C₄⋊C₄×D₆₇	—	4	1	no	—
60	#14523988	—	2	4	no	—
61	D₁₀₇₂	—	2	2	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	2	no	—
68	#14523965	—	2	1	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	1	yes	characteristic
73	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
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	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₃₄	—	2	2	no	—
80	C₂×Dic₁₃₄	—	4	1	no	—
81	C₂×D₂₆₈	—	2	1	no	—
82	C₂×D₂₆₈	—	2	1	no	—
83	C₂×C₄×D₆₇	—	4	1	no	—
84	C₂×C₄×D₆₇	—	4	1	no	—
85	C₂×C₄×D₆₇	—	4	2	no	—
86	Dic₂₆₈	—	2	2	no	—
87	Dic₂₆₈	—	2	2	no	—
88	Dic₂₆₈	—	8	1	no	—
89	#11761617	—	8	2	no	—
90	D₅₃₆	—	2	1	no	—
91	D₅₃₆	—	2	1	no	—
92	D₅₃₆	—	2	1	no	—
93	D₅₃₆	#41	1	2	yes	—
94	C₄⋊Dic₆₇	—	2	1	no	—
95	C₄⋊Dic₆₇	—	2	1	no	—
96	C₄⋊Dic₆₇	—	2	2	no	—
97	C₄⋊Dic₆₇	—	4	1	no	—
98	C₄⋊Dic₆₇	—	4	1	no	—
99	C₄⋊Dic₆₇	—	4	1	no	—
100	Dic₆₇⋊C₄	—	4	2	no	—
101	Dic₆₇⋊C₄	—	4	2	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	1	no	—
107	(C₂×D₆₇)⋊C₄	—	4	2	no	—
108	C₄×Dic₆₇	—	4	2	no	—
109	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
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	1	yes	characteristic
116	C₄²×C₆₇	C₂×D₄	1	2	yes	—
117	C₁₆×C₆₇	—	2	1	no	—
118	C₁₆×C₆₇	—	2	2	no	—
119	C₁₆×C₆₇	—	2	1	no	—
120	C₂²×D₆₇	—	4	1	no	—
121	C₂²×D₆₇	—	4	1	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	2	no	—
130	C₂×Dic₆₇	—	4	1	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	D₂₆₈	—	2	2	no	—
133	D₂₆₈	—	4	1	no	—
134	D₂₆₈	—	4	1	no	—
135	D₂₆₈	—	4	1	no	—
136	C₄×D₆₇	—	8	2	no	—
137	C₄×D₆₇	—	8	2	no	—
138	C₄×D₆₇	—	16	2	no	—
139	Dic₁₃₄	—	4	2	no	—
140	Dic₁₃₄	—	4	2	no	—
141	Dic₁₃₄	—	16	1	no	—
142	C₈×C₆₇	—	2	2	no	—
143	C₈×C₆₇	—	2	1	no	—
144	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₆₇	—	4	2	no	—
148	C₂×D₆₇	—	8	1	no	—
149	C₂×D₆₇	—	8	1	no	—
150	C₂×D₆₇	—	8	1	no	—
151	C₂×D₆₇	—	8	1	no	—
152	C₂×D₆₇	—	8	1	no	—
153	C₄×C₆₇	—	2	2	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	#508	1	2	yes	—
157	C₄×C₆₇	#548	1	1	yes	characteristic
158	C₄×C₆₇	#577	1	1	yes	characteristic
159	Dic₆₇	—	8	2	no	—
160	Dic₆₇	—	8	1	no	—
161	Dic₆₇	—	8	1	no	—
162	Dic₆₇	—	8	2	no	—
163	#33927	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	#3358	1	1	yes	characteristic
165	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
166	C₂×C₆₇	#3416	1	1	yes	characteristic
167	D₆₇	—	8	2	no	—
168	D₆₇	—	16	1	no	—
169	#3393	—	67	1	no	—
170	#3380	—	67	1	no	—
171	#2242	—	67	2	no	—
172	#2241	—	67	1	no	—
173	#2229	—	67	2	no	—
174	#2226	—	67	1	no	—
175	#2210	—	67	2	no	—
176	#2208	—	67	1	no	—
177	Q₈×C₁₆	D₆₇	1	1	yes	characteristic, complemented
178	C₄×SD₃₂	—	67	2	no	—
179	C₄×D₁₆	—	67	1	no	—
180	C₆₇	#33927	1	1	yes	characteristic, complemented, Sylow
181	#556	—	134	1	no	—
182	#554	—	134	1	no	—
183	C₂×SD₃₂	—	67	2	no	—
184	C₂×D₁₆	—	67	1	no	—
185	#500	—	67	1	no	—
186	#499	—	67	2	no	—
187	#495	—	67	2	no	—
188	#494	—	67	1	no	—
189	#478	—	134	2	no	—
190	#476	—	134	1	no	—
191	#464	—	134	2	no	—
192	#461	—	134	1	no	—
193	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
194	C₄×Q₁₆	—	67	2	no	—
195	C₄×D₈	—	67	1	no	—
196	C₄×D₈	—	67	2	no	—
197	C₄×D₈	—	67	1	no	—
198	#367	—	67	2	no	—
199	#366	—	67	1	no	—
200	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
201	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
202	#358	—	67	2	no	—
203	#358	—	67	2	no	—
204	#358	—	67	2	no	—
205	#357	—	67	1	no	—
206	#357	—	67	1	no	—
207	#357	—	67	2	no	—
208	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
209	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
210	C₂×Q₁₆	—	67	2	no	—
211	C₂×Q₁₆	—	134	1	no	—
212	C₂×SD₁₆	—	134	2	no	—
213	C₂×D₈	—	67	1	no	—
214	C₂×D₈	—	67	1	no	—
215	C₄⋊Q₈	—	134	1	no	—
216	C₄⋊Q₈	—	134	1	no	—
217	#126	—	268	2	no	—
218	#125	—	134	2	no	—
219	#124	—	134	2	no	—
220	C₂²⋊Q₈	—	268	1	no	—
221	C₂²⋊Q₈	—	268	1	no	—
222	C₂²⋊Q₈	—	268	2	no	—
223	C₄×Q₈	—	134	2	no	—
224	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
225	C₄×D₄	—	134	1	no	—
226	C₄×D₄	—	134	2	no	—
227	C₄×D₄	—	134	1	no	—
228	#117	—	268	1	no	—
229	C₂×C₄⋊C₄	—	268	1	no	—
230	SD₃₂	—	134	4	no	—
231	D₁₆	—	134	2	no	—
232	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
233	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
234	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
235	#107	—	67	1	no	—
236	#107	—	67	2	no	—
237	#107	—	67	1	no	—
238	#107	—	134	1	no	—
239	#106	—	134	2	no	—
240	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
241	C₄⋊C₈	D₂₆₈	1	2	yes	—
242	#103	—	134	2	no	—
243	#102	—	134	1	no	—
244	#102	—	134	2	no	—
245	#102	—	134	1	no	—
246	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₆₇	1	2	yes	—
248	C₂×Q₈	—	268	1	no	—
249	C₂×Q₈	—	134	2	no	—
250	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
251	C₂×D₄	—	134	1	no	—
252	C₂×D₄	—	134	1	no	—
253	C₂²×C₄	—	268	1	no	—
254	C₂²×C₄	—	268	1	no	—
255	C₂²×C₄	—	268	2	no	—
256	Q₁₆	—	536	1	no	—
257	Q₁₆	—	134	2	no	—
258	Q₁₆	—	134	2	no	—
259	SD₁₆	—	536	2	no	—
260	D₈	—	67	2	no	—
261	D₈	—	134	1	no	—
262	D₈	—	134	1	no	—
263	D₈	—	134	1	no	—
264	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
265	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
266	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
267	C₄⋊C₄	—	268	2	no	—
268	C₄⋊C₄	—	268	1	no	—
269	C₄⋊C₄	—	268	1	no	—
270	C₄⋊C₄	—	268	2	no	—
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	268	1	no	—
273	C₄⋊C₄	—	134	1	no	—
274	C₄⋊C₄	—	134	1	no	—
275	C₄⋊C₄	—	134	2	no	—
276	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
277	C₄⋊C₄	D₅₃₆	1	2	yes	—
278	C₂²⋊C₄	—	268	2	no	—
279	C₂²⋊C₄	—	268	1	no	—
280	C₂²⋊C₄	—	268	2	no	—
281	C₂²⋊C₄	—	268	1	no	—
282	C₂²⋊C₄	—	268	2	no	—
283	C₄²	—	268	2	no	—
284	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
285	C₄²	C₂×D₂₆₈	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	2	1	no	—
289	C₂³	—	268	1	no	—
290	C₂³	—	268	1	no	—
291	Q₈	—	2	2	no	—
292	Q₈	—	268	2	no	—
293	Q₈	—	268	2	no	—
294	Q₈	—	1072	1	no	—
295	D₄	—	268	1	no	—
296	D₄	—	268	1	no	—
297	D₄	—	268	1	no	—
298	D₄	—	134	2	no	—
299	C₂×C₄	—	268	1	no	—
300	C₂×C₄	—	268	2	no	—
301	C₂×C₄	—	268	1	no	—
302	C₂×C₄	—	268	2	no	—
303	C₂×C₄	—	536	2	no	—
304	C₂×C₄	—	536	2	no	—
305	C₂×C₄	—	1072	2	no	—
306	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
307	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
308	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
309	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	#41×D₆₇	1	1	yes	characteristic
314	C₈	#14524091	1	1	yes	characteristic
315	C₂²	—	268	2	no	—
316	C₂²	—	536	1	no	—
317	C₂²	—	536	1	no	—
318	C₂²	—	536	1	no	—
319	C₂²	—	536	1	no	—
320	C₂²	—	536	1	no	—
321	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	536	1	no	—
326	C₄	—	536	1	no	—
327	C₄	—	536	2	no	—
328	C₄	—	536	2	no	—
329	C₄	#16863828	1	2	yes	—
330	C₄	#16864058	1	1	yes	characteristic
331	C₄	#16864208	1	1	yes	characteristic
332	C₂	—	536	2	no	—
333	C₂	—	1072	1	no	—
334	C₂	#22572845	1	1	yes	characteristic
335	C₂	C₂×#16863828	1	1	yes	characteristic
336	C₂	#22573357	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	C₂×D₁₀₇₂	C₂²
hypercenter	Q₈×C₁₆	D₆₇
hypocenter	C₆₇	#33927
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. #22573198	C₂	semidirect
2. #22573077	C₂	semidirect
3. #22562441	C₂	semidirect
4. #22562437	C₂	semidirect
5. #22562374	C₂	semidirect
6. #22562357	C₂	semidirect
8. #22562255	C₂	semidirect
9. C₄×#14523988	C₂	semidirect
11. Q₈×C₁₆×C₆₇	C₂	semidirect
177. Q₈×C₁₆	D₆₇	semidirect
180. C₆₇	#33927	semidirect

Maximal subgroups

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

subgroup	index
#22573198	2
#22573077	2
#22562441	2
#22562437	2
#22562374	2
#22562357	2
#22562265	2
#22562255	2
C₄×#14523988	2
C₄×D₁₀₇₂	2
Q₈×C₁₆×C₆₇	2
#33927	67