Subgroups — #32151141

Order 18176.

532 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11765865	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₄₂	C₂	1	2	yes	maximal, complemented
3	#22590173	C₂	1	2	yes	maximal, complemented
4	#22590169	C₂	1	4	yes	maximal, complemented
5	#22590138	C₂	1	2	yes	maximal, complemented
6	#22590096	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590094	C₂	1	2	yes	maximal, complemented
8	C₇₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#11765865	C₄	1	1	yes	characteristic
10	C₂×#14556068	C₂²	1	2	yes	—
11	C₂×#14556067	C₂²	1	4	yes	—
12	C₈×Dic₁₄₂	C₄	1	1	yes	characteristic
13	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
14	C₈×D₂₈₄	C₄	1	1	yes	characteristic
15	C₈×D₂₈₄	C₂²	1	1	yes	characteristic
16	C₈×C₂²⋊D₇₁	C₄	1	2	yes	—
17	C₈×C₂²⋊D₇₁	C₂²	1	2	yes	—
18	#17986052	C₄	1	1	yes	characteristic
19	#17986050	C₄	1	2	yes	—
20	C₄×C₈×D₇₁	C₂²	1	2	yes	—
21	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
22	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
23	Dic₇₁⋊C₁₆	C₂²	1	4	yes	—
24	(C₂×D₇₁)⋊C₁₆	C₂²	1	4	yes	—
25	C₁₆×Dic₇₁	C₂²	1	4	yes	—
26	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
27	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
29	C₂²⋊C₁₆×C₇₁	C₂²	1	2	yes	—
30	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
31	C₂×#11765865	C₈	1	1	yes	characteristic
32	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₇₁	C₂³	1	2	yes	—
34	C₄×Dic₁₄₂	C₈	1	1	yes	characteristic
35	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
36	C₄×D₂₈₄	C₈	1	1	yes	characteristic
37	C₄×D₂₈₄	C₂×C₄	1	1	yes	characteristic
38	C₄×C₂²⋊D₇₁	C₈	1	2	yes	—
39	C₄×C₂²⋊D₇₁	C₂×C₄	1	2	yes	—
40	#14556087	C₈	1	1	yes	characteristic
41	#14556085	C₈	1	2	yes	—
42	C₄²×D₇₁	C₂×C₄	1	2	yes	—
43	#14556068	—	4	2	no	—
44	#14556067	—	4	4	no	—
45	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
46	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	Dic₇₁⋊C₈	C₂×C₄	1	2	yes	—
49	Dic₇₁⋊C₈	C₂×C₄	1	2	yes	—
50	C₂²⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
51	(C₂×D₇₁)⋊C₈	C₂×C₄	1	2	yes	—
52	(C₂×D₇₁)⋊C₈	C₂×C₄	1	2	yes	—
53	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
54	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
55	C₈×Dic₇₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
60	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
61	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
65	#11765865	—	2	4	no	—
66	C₂×Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
67	C₂×D₂₈₄	C₂×C₈	1	1	yes	characteristic
68	C₂×C₂²⋊D₇₁	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₇₁	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₇₁	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₇₁	C₂²×C₄	1	2	yes	—
72	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
74	C₈×D₇₁	—	2	2	no	—
75	C₈×D₇₁	—	2	2	no	—
76	C₈×D₇₁	—	4	2	no	—
77	C₄⋊Dic₇₁	C₂×C₈	1	1	yes	characteristic
78	C₄⋊Dic₇₁	C₂×C₈	1	1	yes	characteristic
79	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
80	Dic₇₁⋊C₄	C₂×C₈	1	2	yes	—
81	C₂²⋊Dic₇₁	C₂×C₈	1	2	yes	—
82	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
83	(C₂×D₇₁)⋊C₄	C₂×C₈	1	2	yes	—
84	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
85	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
86	C₄×Dic₇₁	C₂²×C₄	1	2	yes	—
87	C₇₁⋊C₁₆	—	2	4	no	—
88	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₁	—	2	1	no	—
92	C₂×C₈×C₇₁	—	2	1	no	—
93	C₂×C₈×C₇₁	—	4	1	no	—
94	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
98	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
100	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
102	C₁₆×C₇₁	—	2	2	no	—
103	C₁₆×C₇₁	—	2	2	no	—
104	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
105	C₂²×D₇₁	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	2	no	—
109	C₂×C₄×C₇₁	—	2	2	no	—
110	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
116	C₂²⋊D₇₁	—	2	4	no	—
117	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
118	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
119	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
120	D₂₈₄	—	2	2	no	—
121	C₄×D₇₁	—	2	2	no	—
122	C₄×D₇₁	—	2	2	no	—
123	C₄×D₇₁	—	2	4	no	—
124	C₄×D₇₁	—	2	4	no	—
125	Dic₁₄₂	M₃₂	1	4	yes	complemented
126	C₈×C₇₁	—	2	1	no	—
127	C₈×C₇₁	—	2	1	no	—
128	C₈×C₇₁	—	2	1	no	—
129	C₈×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
130	C₈×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
131	C₇₁⋊C₈	—	2	2	no	—
132	C₇₁⋊C₈	—	2	2	no	—
133	C₂²×C₇₁	—	2	1	no	—
134	C₂²×C₇₁	—	2	1	no	—
135	C₂²×C₇₁	—	2	1	no	—
136	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
137	C₂×D₇₁	—	2	2	no	—
138	C₂×D₇₁	—	2	2	no	—
139	C₂×D₇₁	—	2	2	no	—
140	C₄×C₇₁	—	2	1	no	—
141	C₄×C₇₁	—	2	1	no	—
142	C₄×C₇₁	—	2	1	no	—
143	C₄×C₇₁	C₂×M₃₂	1	2	yes	—
144	C₄×C₇₁	#504	1	2	yes	—
145	C₄×C₇₁	#520	1	1	yes	characteristic
146	C₄×C₇₁	#568	1	1	yes	characteristic
147	Dic₇₁	—	2	2	no	—
148	Dic₇₁	—	2	2	no	—
149	Dic₇₁	C₂×M₃₂	1	4	yes	—
150	#33505	—	71	1	no	maximal, Sylow
151	C₂×C₇₁	—	4	1	no	—
152	C₂×C₇₁	#2988	1	1	yes	characteristic
153	C₂×C₇₁	C₂²×M₃₂	1	1	yes	characteristic
154	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
155	D₇₁	—	4	2	no	—
156	C₈×#41	—	71	1	no	—
157	Q₈×C₁₆	—	71	2	no	—
158	#2169	—	71	4	no	—
159	#2169	—	71	2	no	—
160	#2162	—	71	2	no	—
161	#2162	D₇₁	1	1	yes	characteristic, complemented
162	#2150	—	71	1	no	—
163	#2150	—	71	2	no	—
164	C₇₁	#33505	1	1	yes	characteristic, complemented, Sylow
165	C₄×#41	—	71	1	no	—
166	C₂×M₃₂	—	71	2	no	—
167	C₂×M₃₂	—	71	4	no	—
168	C₈×Q₈	—	71	1	no	—
169	C₈×Q₈	—	71	1	no	—
170	C₈×D₄	—	71	2	no	—
171	C₈×D₄	—	71	2	no	—
172	C₈×D₄	—	71	1	no	—
173	C₈×D₄	—	71	1	no	—
174	#431	—	71	1	no	—
175	#431	—	71	2	no	—
176	C₂×C₄×C₈	—	71	2	no	—
177	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
178	C₄⋊C₁₆	—	71	4	no	—
179	C₄⋊C₁₆	—	71	2	no	—
180	C₄⋊C₁₆	—	71	2	no	—
181	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
182	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
183	C₂²⋊C₁₆	—	71	4	no	—
184	C₂²⋊C₁₆	C₂×D₇₁	1	2	yes	—
185	C₄×C₁₆	—	71	4	no	—
186	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
187	C₂×#41	—	71	1	no	—
188	C₂²×C₈	—	71	2	no	—
189	C₂²×C₈	—	71	2	no	—
190	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
192	C₄×Q₈	—	71	1	no	—
193	C₄×Q₈	—	71	1	no	—
194	C₄×D₄	—	71	2	no	—
195	C₄×D₄	—	71	2	no	—
196	C₄×D₄	—	71	1	no	—
197	C₄×D₄	—	71	1	no	—
198	#117	—	71	1	no	—
199	#117	—	71	2	no	—
200	C₂×C₄²	—	71	2	no	—
201	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
202	M₃₂	—	284	4	no	—
203	M₃₂	—	284	2	no	—
204	C₂×C₁₆	—	71	4	no	—
205	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
206	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
207	C₄⋊C₈	—	71	2	no	—
208	C₄⋊C₈	—	71	2	no	—
209	C₄⋊C₈	—	71	1	no	—
210	C₄⋊C₈	—	71	1	no	—
211	C₂²⋊C₈	—	71	2	no	—
212	C₂²⋊C₈	—	71	2	no	—
213	C₂²⋊C₈	—	71	2	no	—
214	C₄×C₈	—	71	2	no	—
215	C₄×C₈	—	71	2	no	—
216	C₄×C₈	—	71	2	no	—
217	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
218	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
221	#41	—	142	4	no	—
222	C₂×Q₈	—	71	1	no	—
223	C₂×D₄	—	71	2	no	—
224	C₂×D₄	—	71	1	no	—
225	C₂²×C₄	—	71	2	no	—
226	C₂²×C₄	—	71	2	no	—
227	C₂²×C₄	—	71	2	no	—
228	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
229	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	284	2	no	—
235	C₂×C₈	—	71	2	no	—
236	C₂×C₈	—	71	2	no	—
237	C₂×C₈	—	142	2	no	—
238	C₂×C₈	—	142	2	no	—
239	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
240	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
242	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
243	C₄⋊C₄	—	71	2	no	—
244	C₄⋊C₄	—	71	2	no	—
245	C₄⋊C₄	—	71	1	no	—
246	C₄⋊C₄	—	71	1	no	—
247	C₂²⋊C₄	—	71	2	no	—
248	C₂²⋊C₄	—	71	2	no	—
249	C₂²⋊C₄	—	71	2	no	—
250	C₄²	—	71	2	no	—
251	C₄²	—	71	2	no	—
252	C₄²	—	71	2	no	—
253	C₄²	C₈×D₇₁	1	1	yes	characteristic
254	C₄²	C₈×D₇₁	1	1	yes	characteristic
255	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
256	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
257	C₁₆	—	2	2	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	142	4	no	—
260	C₂³	—	71	2	no	—
261	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
262	Q₈	—	71	4	no	—
263	D₄	—	142	4	no	—
264	D₄	—	142	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	71	2	no	—
270	C₂×C₄	—	71	2	no	—
271	C₂×C₄	—	71	2	no	—
272	C₂×C₄	—	142	4	no	—
273	C₂×C₄	—	142	4	no	—
274	C₂×C₄	—	142	2	no	—
275	C₂×C₄	—	142	2	no	—
276	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
278	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	142	2	no	—
286	C₈	—	142	2	no	—
287	C₈	#14556174	1	1	yes	characteristic
288	C₈	#14556175	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	142	2	no	—
293	C₂²	—	142	2	no	—
294	C₂²	—	142	2	no	—
295	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	71	4	no	—
300	C₄	—	142	2	no	—
301	C₄	—	142	2	no	—
302	C₄	M₃₂×D₇₁	1	2	yes	—
303	C₄	#17986415	1	2	yes	—
304	C₄	#17986479	1	1	yes	characteristic
305	C₄	#17986765	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	284	2	no	—
308	C₂	#22597448	1	1	yes	characteristic
309	C₂	C₂×M₃₂×D₇₁	1	1	yes	characteristic
310	C₂	C₂×#17986415	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₇₁×#2162	C₂
prime radical	C₂×#11765865	C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2162	D₇₁
hypocenter	C₇₁	#33505
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. C₁₆×Dic₁₄₂	C₂	semidirect
3. #22590173	C₂	semidirect
4. #22590169	C₂	semidirect
5. #22590138	C₂	semidirect
6. #22590096	C₂	semidirect
7. #22590094	C₂	semidirect
8. C₇₁×#2162	C₂	semidirect
125. Dic₁₄₂	M₃₂	semidirect
161. #2162	D₇₁	semidirect
164. C₇₁	#33505	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11765865	2
C₁₆×Dic₁₄₂	2
#22590173	2
#22590169	2
#22590138	2
#22590096	2
#22590094	2
C₇₁×#2162	2
#33505	71