Subgroups — #31013539

Order 17152.

484 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	#22573222	C₂	1	1	yes	maximal, characteristic
2	#22573101	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562441	C₂	1	2	yes	maximal, complemented
4	#22562433	C₂	1	1	yes	maximal, characteristic
5	#22562367	C₂	1	2	yes	maximal, complemented
6	#22562352	C₂	1	1	yes	maximal, characteristic, complemented
7	#22562272	C₂	1	2	yes	maximal, complemented
8	#22562250	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14523988	C₂	1	2	yes	maximal, complemented
11	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864131	—	2	1	no	—
13	#16864124	—	2	1	no	—
14	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
15	C₂×#14523988	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	#16863704	—	2	2	no	—
21	#16863682	—	2	1	no	—
22	#16863612	—	2	2	no	—
23	#16863597	—	2	1	no	—
24	C₄×Dic₂₆₈	C₂²	1	2	yes	—
25	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₆₈	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	1	yes	characteristic
31	#16863201	C₂²	1	1	yes	characteristic
32	#16863201	C₂²	1	2	yes	—
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×Dic₂₆₈	C₂³	1	1	yes	characteristic
43	C₂×#11761617	—	2	2	no	—
44	C₂×D₅₃₆	—	2	1	no	—
45	C₂×D₅₃₆	C₂³	1	2	yes	—
46	#14524043	—	2	1	no	—
47	#14524041	—	4	1	no	—
48	#14524040	—	2	1	no	—
49	#14524035	—	2	2	no	—
50	#14524033	—	4	2	no	—
51	#14524032	—	4	1	no	—
52	#14524031	—	2	2	no	—
53	#14524028	—	4	2	no	—
54	C₄×Dic₁₃₄	—	2	1	no	—
55	C₄×Dic₁₃₄	—	2	1	no	—
56	C₄×Dic₁₃₄	—	2	2	no	—
57	#14524008	—	4	1	no	—
58	C₄×D₂₆₈	—	2	2	no	—
59	#14524005	—	4	1	no	—
60	Dic₅₃₆	—	2	2	no	—
61	#14523988	—	2	4	no	—
62	#14523977	—	2	1	no	—
63	#14523977	C₂³	1	1	yes	characteristic
64	#14523977	C₂³	1	2	yes	—
65	#14523977	C₂³	1	1	yes	characteristic
66	#14523975	—	2	2	no	—
67	#14523968	—	2	1	no	—
68	#14523968	—	2	2	no	—
69	#14523968	—	2	1	no	—
70	#14523965	—	2	2	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	—	2	1	no	—
81	C₂×Dic₁₃₄	—	4	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	2	no	—
86	Dic₂₆₈	—	2	1	no	—
87	Dic₂₆₈	—	2	1	no	—
88	Dic₂₆₈	—	2	1	no	—
89	Dic₂₆₈	#41	1	2	yes	—
90	#11761617	—	8	2	no	—
91	D₅₃₆	—	2	2	no	—
92	D₅₃₆	—	2	2	no	—
93	D₅₃₆	—	8	1	no	—
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	Dic₆₇⋊C₄	—	4	2	no	—
99	Dic₆₇⋊C₄	—	4	1	no	—
100	Dic₆₇⋊C₄	—	4	1	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	2	no	—
106	C₄×Dic₆₇	—	4	1	no	—
107	C₄×Dic₆₇	—	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	2	no	—
121	Q₈×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₆₇	—	4	1	no	—
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	—	4	2	no	—
133	D₂₆₈	—	4	2	no	—
134	D₂₆₈	—	8	2	no	—
135	D₂₆₈	—	16	1	no	—
136	C₄×D₆₇	—	16	2	no	—
137	Dic₁₃₄	—	2	2	no	—
138	Dic₁₃₄	—	4	1	no	—
139	Dic₁₃₄	—	4	1	no	—
140	Dic₁₃₄	—	4	1	no	—
141	Dic₁₃₄	—	8	2	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	—	2	2	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	—	8	2	no	—
148	C₂×D₆₇	—	8	2	no	—
149	C₂×D₆₇	—	8	2	no	—
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	#508	1	2	yes	—
154	C₄×C₆₇	#548	1	1	yes	characteristic
155	C₄×C₆₇	#577	1	1	yes	characteristic
156	Dic₆₇	—	4	2	no	—
157	Dic₆₇	—	8	1	no	—
158	Dic₆₇	—	8	2	no	—
159	Dic₆₇	—	8	1	no	—
160	Dic₆₇	—	8	1	no	—
161	#33932	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	#3358	1	1	yes	characteristic
163	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
164	C₂×C₆₇	#3416	1	1	yes	characteristic
165	D₆₇	—	16	2	no	—
166	#3396	—	67	1	no	—
167	#3383	—	67	1	no	—
168	#2242	—	67	2	no	—
169	#2240	—	67	1	no	—
170	#2228	—	67	2	no	—
171	#2225	—	67	1	no	—
172	#2211	—	67	2	no	—
173	#2207	—	67	1	no	—
174	Q₈×C₁₆	D₆₇	1	1	yes	characteristic, complemented
175	C₄×Q₃₂	—	67	1	no	—
176	C₄×SD₃₂	—	67	2	no	—
177	C₆₇	#33932	1	1	yes	characteristic, complemented, Sylow
178	#559	—	134	1	no	—
179	#557	—	134	1	no	—
180	C₂×Q₃₂	—	67	1	no	—
181	C₂×SD₃₂	—	67	2	no	—
182	#500	—	67	1	no	—
183	#499	—	67	2	no	—
184	#495	—	67	2	no	—
185	#493	—	67	1	no	—
186	#479	—	134	2	no	—
187	#475	—	134	1	no	—
188	#463	—	134	2	no	—
189	#460	—	134	1	no	—
190	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
191	C₄×Q₁₆	—	67	2	no	—
192	C₄×Q₁₆	—	67	1	no	—
193	C₄×Q₁₆	—	67	1	no	—
194	C₄×D₈	—	67	2	no	—
195	#367	—	67	2	no	—
196	#366	—	67	1	no	—
197	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
198	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
199	#358	—	67	1	no	—
200	#358	—	67	1	no	—
201	#358	—	67	2	no	—
202	#357	—	67	2	no	—
203	#357	—	67	2	no	—
204	#357	—	67	2	no	—
205	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
206	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
207	C₂×Q₁₆	—	67	1	no	—
208	C₂×Q₁₆	—	67	1	no	—
209	C₂×SD₁₆	—	134	2	no	—
210	C₂×D₈	—	67	2	no	—
211	C₂×D₈	—	134	1	no	—
212	C₄⋊Q₈	—	268	1	no	—
213	C₄⋊Q₈	—	134	1	no	—
214	#127	—	134	1	no	—
215	#125	—	268	2	no	—
216	#125	—	134	2	no	—
217	#124	—	268	1	no	—
218	#124	—	134	2	no	—
219	#123	—	268	2	no	—
220	C₄×Q₈	—	268	1	no	—
221	C₄×Q₈	—	134	1	no	—
222	C₄×Q₈	—	134	2	no	—
223	C₄×Q₈	—	134	1	no	—
224	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
225	C₄×D₄	—	268	1	no	—
226	C₄×D₄	—	134	2	no	—
227	Q₃₂	—	134	2	no	—
228	SD₃₂	—	134	4	no	—
229	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
232	#107	—	67	1	no	—
233	#107	—	67	2	no	—
234	#107	—	67	1	no	—
235	#107	—	134	1	no	—
236	#106	—	134	2	no	—
237	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
238	C₄⋊C₈	D₂₆₈	1	2	yes	—
239	#103	—	134	1	no	—
240	#103	—	134	2	no	—
241	#103	—	134	1	no	—
242	#102	—	134	2	no	—
243	C₄×C₈	C₂²×D₆₇	1	2	yes	—
244	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
245	C₂×Q₈	—	268	1	no	—
246	C₂×Q₈	—	134	1	no	—
247	C₂×Q₈	—	134	1	no	—
248	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
249	C₂×D₄	—	268	1	no	—
250	C₂×D₄	—	268	1	no	—
251	C₂×D₄	—	134	2	no	—
252	C₂²×C₄	—	268	2	no	—
253	Q₁₆	—	67	2	no	—
254	Q₁₆	—	134	1	no	—
255	Q₁₆	—	134	1	no	—
256	Q₁₆	—	134	1	no	—
257	SD₁₆	—	536	2	no	—
258	D₈	—	536	1	no	—
259	D₈	—	134	2	no	—
260	D₈	—	134	2	no	—
261	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
262	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
264	C₄⋊C₄	—	268	2	no	—
265	C₄⋊C₄	—	268	1	no	—
266	C₄⋊C₄	—	268	1	no	—
267	C₄⋊C₄	—	268	2	no	—
268	C₄⋊C₄	—	268	1	no	—
269	C₄⋊C₄	—	268	2	no	—
270	C₄⋊C₄	—	134	1	no	—
271	C₄⋊C₄	—	134	1	no	—
272	C₄⋊C₄	—	134	2	no	—
273	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
274	C₄⋊C₄	D₅₃₆	1	2	yes	—
275	C₂²⋊C₄	—	268	2	no	—
276	C₂²⋊C₄	—	268	2	no	—
277	C₂²⋊C₄	—	268	2	no	—
278	C₄²	—	268	1	no	—
279	C₄²	—	268	2	no	—
280	C₄²	—	268	1	no	—
281	C₄²	C₂×D₂₆₈	1	2	yes	—
282	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₂³	—	268	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	268	1	no	—
289	Q₈	—	268	1	no	—
290	Q₈	—	268	1	no	—
291	Q₈	—	536	2	no	—
292	Q₈	—	134	2	no	—
293	D₄	—	268	2	no	—
294	D₄	—	268	2	no	—
295	D₄	—	536	2	no	—
296	D₄	—	1072	1	no	—
297	C₂×C₄	—	268	1	no	—
298	C₂×C₄	—	268	1	no	—
299	C₂×C₄	—	268	1	no	—
300	C₂×C₄	—	268	1	no	—
301	C₂×C₄	—	268	2	no	—
302	C₂×C₄	—	1072	2	no	—
303	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
304	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
305	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
306	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	#41×D₆₇	1	1	yes	characteristic
311	C₈	#14524091	1	1	yes	characteristic
312	C₂²	—	536	2	no	—
313	C₂²	—	536	2	no	—
314	C₂²	—	536	2	no	—
315	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	268	2	no	—
320	C₄	—	536	2	no	—
321	C₄	—	536	1	no	—
322	C₄	—	536	1	no	—
323	C₄	—	536	1	no	—
324	C₄	#16863828	1	2	yes	—
325	C₄	#16864058	1	1	yes	characteristic
326	C₄	#16864208	1	1	yes	characteristic
327	C₂	—	1072	2	no	—
328	C₂	#22572845	1	1	yes	characteristic
329	C₂	C₂×#16863828	1	1	yes	characteristic
330	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	#16863773	C₂²
hypercenter	Q₈×C₁₆	D₆₇
hypocenter	C₆₇	#33932
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
2. #22573101	C₂	semidirect
3. #22562441	C₂	semidirect
5. #22562367	C₂	semidirect
6. #22562352	C₂	semidirect
7. #22562272	C₂	semidirect
9. C₄×Dic₅₃₆	C₂	semidirect
10. C₄×#14523988	C₂	semidirect
11. Q₈×C₁₆×C₆₇	C₂	semidirect
174. Q₈×C₁₆	D₆₇	semidirect
177. C₆₇	#33932	semidirect

Maximal subgroups

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

subgroup	index
#22573222	2
#22573101	2
#22562441	2
#22562433	2
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C₄×Dic₅₃₆	2
C₄×#14523988	2
Q₈×C₁₆×C₆₇	2
#33932	67