Subgroups — #32155202

Order 18176.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601329	C₂	1	1	yes	maximal, characteristic
2	#22590579	C₂	1	2	yes	maximal
3	#22590569	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590484	C₂	1	1	yes	maximal, characteristic
5	#22590479	C₂	1	2	yes	maximal, complemented
6	#22590402	C₂	1	2	yes	maximal
7	#22590390	C₂	1	1	yes	maximal, characteristic, complemented
8	#22590249	C₂	1	1	yes	maximal, characteristic, complemented
9	#22590209	C₂	1	2	yes	maximal, complemented
10	C₄×#14556071	C₂	1	1	yes	maximal, characteristic
11	Q₈×Q₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986722	—	2	1	no	—
13	C₂×#14556074	C₂²	1	2	yes	—
14	C₂×#14556071	C₂²	1	1	yes	characteristic
15	#17986383	C₂²	1	2	yes	—
16	#17986379	C₂²	1	1	yes	characteristic
17	#17986222	—	2	2	no	—
18	#17986205	—	2	1	no	—
19	C₄×Dic₂₈₄	C₂²	1	2	yes	—
20	C₄×D₅₆₈	C₂²	1	1	yes	characteristic
21	#17985826	C₂²	1	2	yes	—
22	#17985826	C₂²	1	1	yes	characteristic
23	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
24	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#17985803	C₂²	1	1	yes	characteristic
26	#17985802	C₂²	1	2	yes	—
27	#17985802	C₂²	1	2	yes	—
28	#17985802	C₂²	1	2	yes	—
29	#17985799	C₂²	1	2	yes	—
30	#17985799	C₂²	1	1	yes	characteristic
31	#17985775	C₂²	1	2	yes	—
32	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
33	Q₈²×C₇₁	—	2	1	no	—
34	#500×C₇₁	C₂²	1	2	yes	—
35	#500×C₇₁	C₂²	1	1	yes	characteristic
36	#477×C₇₁	—	2	1	no	—
37	#477×C₇₁	—	2	2	no	—
38	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
40	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₈₄	—	2	1	no	—
42	C₂×Dic₂₈₄	C₂³	1	2	yes	—
43	C₂×#11765854	—	2	2	no	—
44	C₂×D₅₆₈	C₂³	1	1	yes	characteristic
45	#14556128	—	2	1	no	—
46	#14556123	—	4	2	no	—
47	#14556116	—	2	2	no	—
48	#14556104	—	4	1	no	—
49	C₄×Dic₁₄₂	—	2	2	no	—
50	C₄×D₂₈₄	—	2	1	no	—
51	#14556086	—	4	1	no	—
52	#14556074	—	2	4	no	—
53	#14556071	—	2	2	no	—
54	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14556062	C₂³	1	1	yes	characteristic
58	#14556053	—	2	2	no	—
59	#14556050	—	2	1	no	—
60	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₇₁	—	2	1	no	—
62	C₄⋊Q₈×C₇₁	—	2	2	no	—
63	C₄⋊Q₈×C₇₁	—	4	1	no	—
64	C₄⋊Q₈×C₇₁	—	4	2	no	—
65	C₄×Q₈×C₇₁	—	2	1	no	—
66	C₄×Q₈×C₇₁	—	2	2	no	—
67	C₄×Q₈×C₇₁	—	4	1	no	—
68	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
69	#107×C₇₁	—	2	1	no	—
70	#107×C₇₁	—	2	2	no	—
71	#107×C₇₁	C₂³	1	2	yes	—
72	#107×C₇₁	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
75	#103×C₇₁	—	2	1	no	—
76	#103×C₇₁	—	2	2	no	—
77	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₇₁	C₂³	1	2	yes	—
79	C₂×Dic₁₄₂	—	2	2	no	—
80	C₂×D₂₈₄	—	2	1	no	—
81	C₂×C₄×D₇₁	—	4	1	no	—
82	Dic₂₈₄	—	2	2	no	—
83	Dic₂₈₄	—	2	2	no	—
84	Dic₂₈₄	—	8	1	no	—
85	#11765854	—	8	2	no	—
86	D₅₆₈	—	2	1	no	—
87	D₅₆₈	—	2	1	no	—
88	C₄⋊Dic₇₁	—	2	1	no	—
89	C₄⋊Dic₇₁	—	4	1	no	—
90	C₄⋊Dic₇₁	—	4	1	no	—
91	Dic₇₁⋊C₄	—	4	2	no	—
92	(C₂×D₇₁)⋊C₄	—	4	2	no	—
93	(C₂×D₇₁)⋊C₄	—	4	1	no	—
94	(C₂×D₇₁)⋊C₄	—	4	2	no	—
95	C₄×Dic₇₁	—	4	2	no	—
96	C₇₁⋊C₁₆	—	2	1	no	—
97	C₇₁⋊C₁₆	—	2	2	no	—
98	C₇₁⋊C₁₆	—	2	1	no	—
99	C₂×Q₈×C₇₁	—	2	1	no	—
100	C₂×Q₈×C₇₁	—	4	2	no	—
101	C₂×Q₈×C₇₁	—	4	1	no	—
102	C₂×Q₈×C₇₁	D₈	1	1	yes	characteristic
103	Q₁₆×C₇₁	—	2	1	no	—
104	Q₁₆×C₇₁	#41	1	2	yes	—
105	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₇₁	—	2	1	no	—
109	C₄⋊C₄×C₇₁	—	2	2	no	—
110	C₄⋊C₄×C₇₁	—	4	2	no	—
111	C₄⋊C₄×C₇₁	—	4	1	no	—
112	C₄⋊C₄×C₇₁	—	4	2	no	—
113	C₄⋊C₄×C₇₁	—	4	1	no	—
114	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
115	C₄⋊C₄×C₇₁	D₈	1	2	yes	—
116	C₄²×C₇₁	—	4	2	no	—
117	C₄²×C₇₁	—	4	1	no	—
118	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇₁	C₂×D₄	1	2	yes	—
120	C₂²×D₇₁	—	4	1	no	—
121	Q₈×C₇₁	—	2	2	no	—
122	Q₈×C₇₁	—	2	2	no	—
123	Q₈×C₇₁	—	4	1	no	—
124	Q₈×C₇₁	—	16	2	no	—
125	Q₈×C₇₁	—	16	1	no	—
126	C₂×C₄×C₇₁	—	4	1	no	—
127	C₂×C₄×C₇₁	—	4	2	no	—
128	C₂×C₄×C₇₁	—	4	1	no	—
129	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₇₁	—	4	1	no	—
135	C₂×Dic₇₁	—	4	2	no	—
136	D₂₈₄	—	4	1	no	—
137	D₂₈₄	—	4	1	no	—
138	C₄×D₇₁	—	8	2	no	—
139	Dic₁₄₂	—	4	2	no	—
140	Dic₁₄₂	—	4	2	no	—
141	C₈×C₇₁	—	2	2	no	—
142	C₈×C₇₁	—	2	1	no	—
143	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
144	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₇₁	—	8	1	no	—
147	C₂×D₇₁	—	8	1	no	—
148	C₂×D₇₁	—	8	1	no	—
149	C₄×C₇₁	—	2	2	no	—
150	C₄×C₇₁	—	2	2	no	—
151	C₄×C₇₁	—	2	1	no	—
152	C₄×C₇₁	—	4	2	no	—
153	C₄×C₇₁	—	8	2	no	—
154	C₄×C₇₁	—	8	1	no	—
155	C₄×C₇₁	#510	1	2	yes	—
156	C₄×C₇₁	#548	1	1	yes	characteristic
157	C₄×C₇₁	#577	1	1	yes	characteristic
158	Dic₇₁	—	8	2	no	—
159	Dic₇₁	—	8	1	no	—
160	#33936	—	71	1	no	maximal, Sylow
161	C₂×C₇₁	#3358	1	1	yes	characteristic
162	C₂×C₇₁	C₂×#510	1	1	yes	characteristic
163	C₂×C₇₁	#3416	1	1	yes	characteristic
164	D₇₁	—	16	1	no	—
165	#3393	—	71	1	no	—
166	Q₈×Q₁₆	D₇₁	1	1	yes	characteristic, complemented
167	#2243	—	71	2	no	—
168	#2241	—	71	1	no	—
169	#2226	—	71	1	no	—
170	#2225	—	71	2	no	—
171	#2211	—	71	2	no	—
172	#2209	—	71	1	no	—
173	#2183	—	71	1	no	—
174	#2176	—	71	2	no	—
175	C₄×SD₃₂	—	71	1	no	—
176	C₇₁	#33936	1	1	yes	characteristic, complemented, Sylow
177	Q₈²	—	2	1	no	—
178	#556	—	142	1	no	—
179	C₂×Q₃₂	—	71	2	no	—
180	C₂×SD₃₂	—	71	1	no	—
181	#500	C₂×D₇₁	1	2	yes	—
182	#500	C₂×D₇₁	1	1	yes	characteristic
183	#495	—	71	2	no	—
184	#494	—	71	1	no	—
185	#477	—	2	1	no	—
186	#477	—	2	2	no	—
187	#464	—	142	2	no	—
188	#461	—	142	1	no	—
189	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
190	C₄×Q₁₆	—	71	2	no	—
191	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
192	C₄×Q₁₆	C₂×D₇₁	1	2	yes	—
193	C₄×D₈	—	71	1	no	—
194	#367	—	71	1	no	—
195	#367	—	71	2	no	—
196	C₄⋊C₁₆	—	71	1	no	—
197	C₄⋊C₁₆	—	71	2	no	—
198	#358	—	71	1	no	—
199	#358	—	71	2	no	—
200	#358	—	71	2	no	—
201	#358	—	71	2	no	—
202	#357	—	71	1	no	—
203	#357	—	71	2	no	—
204	#346	—	71	2	no	—
205	C₄×C₁₆	—	71	1	no	—
206	C₂×Q₁₆	—	71	2	no	—
207	C₂×Q₁₆	—	142	1	no	—
208	C₂×Q₁₆	C₂²×D₇₁	1	1	yes	characteristic
209	C₂×SD₁₆	—	142	2	no	—
210	C₂×D₈	—	71	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	4	1	no	—
214	C₄⋊Q₈	—	4	2	no	—
215	C₄⋊Q₈	—	142	1	no	—
216	#126	—	284	2	no	—
217	#124	—	142	2	no	—
218	C₂²⋊Q₈	—	284	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	4	1	no	—
222	C₄×Q₈	—	142	2	no	—
223	C₄×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
224	C₄×D₄	—	142	1	no	—
225	#117	—	284	1	no	—
226	Q₃₂	—	142	4	no	—
227	SD₃₂	—	142	2	no	—
228	C₂×C₁₆	—	71	1	no	—
229	C₂×C₁₆	—	71	1	no	—
230	C₂×C₁₆	—	71	2	no	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#107	—	71	1	no	—
234	#107	C₂²×D₇₁	1	1	yes	characteristic
235	#107	C₂²×D₇₁	1	2	yes	—
236	C₄⋊C₈	C₂²⋊D₇₁	1	1	yes	characteristic
237	C₄⋊C₈	C₂²⋊D₇₁	1	2	yes	—
238	#103	—	2	1	no	—
239	#103	—	2	2	no	—
240	#103	—	142	2	no	—
241	#102	—	142	1	no	—
242	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₇₁	1	2	yes	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	142	2	no	—
248	C₂×Q₈	#11765850	1	1	yes	characteristic
249	C₂×D₄	—	142	1	no	—
250	C₂²×C₄	—	284	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	568	1	no	—
253	Q₁₆	—	142	2	no	—
254	Q₁₆	—	142	2	no	—
255	Q₁₆	#11765866	1	2	yes	—
256	SD₁₆	—	568	2	no	—
257	D₈	—	142	1	no	—
258	D₈	—	142	1	no	—
259	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
261	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	284	2	no	—
269	C₄⋊C₄	—	284	1	no	—
270	C₄⋊C₄	—	284	1	no	—
271	C₄⋊C₄	—	142	1	no	—
272	C₄⋊C₄	#11765850	1	1	yes	characteristic
273	C₄⋊C₄	#11765850	1	2	yes	—
274	C₂²⋊C₄	—	284	2	no	—
275	C₂²⋊C₄	—	284	1	no	—
276	C₂²⋊C₄	—	284	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	284	2	no	—
280	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
282	C₁₆	—	142	1	no	—
283	C₁₆	—	142	2	no	—
284	C₁₆	—	142	1	no	—
285	C₂³	—	284	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	16	2	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	284	2	no	—
292	Q₈	—	284	2	no	—
293	D₄	—	284	1	no	—
294	D₄	—	284	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	284	1	no	—
299	C₂×C₄	—	284	2	no	—
300	C₂×C₄	—	568	2	no	—
301	C₂×C₄	C₂×#11765850	1	2	yes	—
302	C₂×C₄	C₂×#11765850	1	2	yes	—
303	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
304	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×#11765866	1	1	yes	characteristic
309	C₈	#14556176	1	1	yes	characteristic
310	C₂²	—	568	1	no	—
311	C₂²	—	568	1	no	—
312	C₂²	—	568	1	no	—
313	C₂²	C₂²×#11765850	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	—	568	2	no	—
321	C₄	—	568	1	no	—
322	C₄	#17986440	1	2	yes	—
323	C₄	#17986660	1	1	yes	characteristic
324	C₄	#17986807	1	1	yes	characteristic
325	C₂	—	1136	1	no	—
326	C₂	#22600975	1	1	yes	characteristic
327	C₂	C₂×#17986440	1	1	yes	characteristic
328	C₂	#22601487	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	Q₈×Q₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂×D₅₆₈	C₂³
hypercenter	Q₈×Q₁₆	D₇₁
hypocenter	C₇₁	#33936
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #22590569	C₂	semidirect
5. #22590479	C₂	semidirect
7. #22590390	C₂	semidirect
8. #22590249	C₂	semidirect
9. #22590209	C₂	semidirect
11. Q₈×Q₁₆×C₇₁	C₂	semidirect
166. Q₈×Q₁₆	D₇₁	semidirect
176. C₇₁	#33936	semidirect

Maximal subgroups

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

subgroup	index
#22601329	2
#22590579	2
#22590569	2
#22590484	2
#22590479	2
#22590402	2
#22590390	2
#22590249	2
#22590209	2
C₄×#14556071	2
Q₈×Q₁₆×C₇₁	2
#33936	71