Subgroups — #31888295

Order 18176.

462 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986129	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556051	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985729	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
5	#22586181	C₂	1	1	yes	maximal, characteristic
6	#22586174	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1770	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
9	#17986129	—	2	2	no	—
10	#17986129	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₇₁⋊C₈	—	2	1	no	—
13	C₂×#14556051	C₄	1	2	yes	—
14	C₂×#14556038	—	2	1	no	—
15	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	#17985761	C₄	1	2	yes	—
18	#17985729	C₄	1	4	yes	complemented
19	#17985721	—	2	2	no	—
20	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×#97×C₇₁	—	2	1	no	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
24	#336×C₇₁	—	2	1	no	—
25	C₂²×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇₁⋊C₈	—	2	1	no	—
27	C₂²×C₇₁⋊C₈	—	2	1	no	—
28	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₄₂	—	2	1	no	—
30	C₄×Dic₁₄₂	—	2	1	no	—
31	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×Dic₇₁⋊C₄	—	2	1	no	—
35	C₂×C₄×Dic₇₁	—	2	1	no	—
36	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
37	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
38	Dic₇₁⋊C₈	—	2	4	no	—
39	#14556051	—	2	4	no	—
40	#14556038	—	2	4	no	—
41	C₄×C₇₁⋊C₈	—	2	1	no	—
42	C₄×C₇₁⋊C₈	—	2	1	no	—
43	C₄×C₇₁⋊C₈	—	2	1	no	—
44	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₇₁	—	2	1	no	—
46	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₇₁	—	2	1	no	—
50	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₇₁	—	2	1	no	—
52	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
53	#97×C₇₁	—	4	2	no	—
54	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₇₁	D₄	1	2	yes	—
56	#95×C₇₁	—	2	1	no	—
57	C₂×Dic₁₄₂	—	2	2	no	—
58	C₂×Dic₁₄₂	—	2	2	no	—
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	C₄²	1	2	yes	—
61	C₂²×Dic₇₁	—	2	1	no	—
62	C₂²×Dic₇₁	—	2	1	no	—
63	C₂×C₇₁⋊C₈	—	2	2	no	—
64	C₂×C₇₁⋊C₈	—	2	2	no	—
65	C₂×C₇₁⋊C₈	—	2	2	no	—
66	C₂×C₇₁⋊C₈	—	2	2	no	—
67	C₂×C₇₁⋊C₈	—	4	2	no	—
68	C₂×C₇₁⋊C₈	—	4	1	no	—
69	C₂×C₇₁⋊C₈	—	4	1	no	—
70	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
71	C₄⋊Dic₇₁	—	2	1	no	—
72	C₄⋊Dic₇₁	C₄²	1	2	yes	—
73	Dic₇₁⋊C₄	—	2	2	no	—
74	Dic₇₁⋊C₄	—	2	2	no	—
75	C₄×Dic₇₁	—	2	2	no	—
76	C₄×Dic₇₁	—	2	2	no	—
77	C₂²×C₄×C₇₁	—	2	2	no	—
78	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
81	C₂×C₈×C₇₁	—	2	1	no	—
82	C₂×C₈×C₇₁	—	2	1	no	—
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	1	no	—
85	C₂×C₈×C₇₁	—	2	2	no	—
86	C₂×C₈×C₇₁	—	4	1	no	—
87	C₂×C₈×C₇₁	—	4	1	no	—
88	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
90	C₄⋊C₄×C₇₁	—	2	2	no	—
91	C₄⋊C₄×C₇₁	—	2	2	no	—
92	C₄²×C₇₁	—	4	1	no	—
93	C₄²×C₇₁	—	4	1	no	—
94	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇₁	#41	1	1	yes	characteristic
98	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇₁	—	2	2	no	—
100	C₂×C₄×C₇₁	—	4	2	no	—
101	C₂×C₄×C₇₁	—	4	2	no	—
102	C₂×C₄×C₇₁	—	4	2	no	—
103	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₇₁	#131	1	2	yes	—
111	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
113	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
114	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
115	C₂×Dic₇₁	—	2	2	no	—
116	C₂×Dic₇₁	—	2	2	no	—
117	C₂×Dic₇₁	—	4	2	no	—
118	C₂×Dic₇₁	—	4	2	no	—
119	Dic₁₄₂	—	2	2	no	—
120	Dic₁₄₂	—	2	2	no	—
121	Dic₁₄₂	—	4	2	no	—
122	C₈×C₇₁	—	2	2	no	—
123	C₈×C₇₁	—	2	2	no	—
124	C₈×C₇₁	—	4	2	no	—
125	C₇₁⋊C₈	—	2	4	no	—
126	C₇₁⋊C₈	—	4	2	no	—
127	C₇₁⋊C₈	—	4	2	no	—
128	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₇₁	#436	1	2	yes	—
130	C₂²×C₇₁	#440	1	1	yes	characteristic
131	C₂²×C₇₁	#441	1	1	yes	characteristic
132	C₂²×C₇₁	#444	1	2	yes	—
133	C₄×C₇₁	—	2	1	no	—
134	C₄×C₇₁	—	2	1	no	—
135	C₄×C₇₁	—	8	2	no	—
136	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
137	C₄×C₇₁	#405	1	1	yes	characteristic
138	C₄×C₇₁	#417	1	1	yes	characteristic
139	C₄×C₇₁	#419	1	1	yes	characteristic
140	C₄×C₇₁	#435	1	2	yes	—
141	C₄×C₇₁	#440	1	1	yes	characteristic
142	C₄×C₇₁	#441	1	1	yes	characteristic
143	Dic₇₁	—	4	2	no	—
144	Dic₇₁	—	4	2	no	—
145	#8209	—	71	1	no	maximal, Sylow
146	C₂×C₇₁	#1592	1	2	yes	—
147	C₂×C₇₁	#1593	1	2	yes	—
148	C₂×C₇₁	#1753	1	1	yes	characteristic
149	C₂×C₇₁	#1763	1	1	yes	characteristic
150	C₂×C₇₁	#1765	1	1	yes	characteristic
151	C₂×#446	—	71	1	no	—
152	#1770	D₇₁	1	1	yes	characteristic
153	C₄×#103	—	71	1	no	—
154	C₂×Q₈⋊C₈	—	71	1	no	—
155	C₂×#322	—	71	1	no	—
156	#1279	—	71	1	no	—
157	#1277	—	71	1	no	—
158	C₇₁	#8209	1	1	yes	characteristic, complemented, Sylow
159	C₂×C₄×Q₈	—	71	1	no	—
160	#446	—	71	4	no	—
161	#446	—	142	2	no	—
162	C₂×C₄⋊C₈	—	71	1	no	—
163	C₂×C₄⋊C₈	—	142	1	no	—
164	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
165	C₂×#103	—	71	2	no	—
166	C₂×#97	—	2	1	no	—
167	C₂×#97	—	142	1	no	—
168	C₂×C₄×C₈	—	71	1	no	—
169	C₂×C₄×C₈	—	71	1	no	—
170	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
171	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
172	#340	—	71	2	no	—
173	#336	—	2	1	no	—
174	Q₈⋊C₈	—	71	4	no	—
175	#322	—	142	2	no	—
176	C₂²×Q₈	—	71	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	71	2	no	—
179	C₂²×C₈	—	142	1	no	—
180	C₂²×C₈	—	142	1	no	—
181	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
183	C₄×Q₈	—	71	2	no	—
184	C₄×Q₈	—	71	2	no	—
185	C₄×Q₈	—	142	1	no	—
186	C₄×Q₈	—	142	1	no	—
187	C₂×C₄⋊C₄	—	71	1	no	—
188	C₂×C₄⋊C₄	—	142	1	no	—
189	C₂×C₄⋊C₄	C₄×D₇₁	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	142	1	no	—
192	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	71	2	no	—
195	C₄⋊C₈	—	142	4	no	—
196	C₄⋊C₈	—	142	1	no	—
197	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
198	#103	—	142	4	no	—
199	#97	—	4	2	no	—
200	#97	—	142	4	no	—
201	C₄×C₈	—	71	2	no	—
202	C₄×C₈	—	142	1	no	—
203	C₄×C₈	—	142	1	no	—
204	C₄×C₈	—	142	1	no	—
205	C₄×C₈	C₄×D₇₁	1	2	yes	—
206	C₄×C₈	D₂₈₄	1	2	yes	—
207	#95	—	2	1	no	—
208	C₂×Q₈	—	71	2	no	—
209	C₂×Q₈	—	142	2	no	—
210	C₂×Q₈	—	142	2	no	—
211	C₂×Q₈	—	142	1	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	142	1	no	—
214	C₂²×C₄	—	142	1	no	—
215	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
217	C₂²×C₄	#11765865	1	1	yes	characteristic
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	284	2	no	—
226	C₂×C₈	—	284	1	no	—
227	C₂×C₈	—	284	1	no	—
228	C₂×C₈	—	71	4	no	—
229	C₂×C₈	—	142	2	no	—
230	C₂×C₈	—	142	2	no	—
231	C₂×C₈	—	142	2	no	—
232	C₂×C₈	—	142	2	no	—
233	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
234	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	71	2	no	—
238	C₄⋊C₄	—	142	2	no	—
239	C₄⋊C₄	—	142	2	no	—
240	C₄⋊C₄	—	142	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	142	2	no	—
244	C₄²	—	142	2	no	—
245	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
247	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
248	C₄²	#11765865	1	1	yes	characteristic
249	C₂³	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
250	Q₈	—	284	2	no	—
251	Q₈	—	142	2	no	—
252	Q₈	—	142	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	284	2	no	—
258	C₂×C₄	—	284	2	no	—
259	C₂×C₄	—	142	2	no	—
260	C₂×C₄	—	142	2	no	—
261	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
262	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	#14556085	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
265	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
266	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
267	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
268	C₂×C₄	#14556136	1	2	yes	—
269	C₂×C₄	#14556154	1	1	yes	characteristic
270	C₂×C₄	#14556158	1	1	yes	characteristic
271	C₂×C₄	#14556158	1	1	yes	characteristic
272	C₂×C₄	#14556159	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	284	2	no	—
277	C₈	—	284	2	no	—
278	C₈	—	142	4	no	—
279	C₂²	C₄×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
280	C₂²	#17986075	1	2	yes	—
281	C₂²	#17986099	1	1	yes	characteristic
282	C₂²	#17986104	1	1	yes	characteristic
283	C₂²	#17986119	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	8	2	no	—
287	C₄	—	284	2	no	—
288	C₄	—	284	2	no	—
289	C₄	#17985951	1	1	yes	characteristic
290	C₄	#17985956	1	1	yes	characteristic
291	C₄	#17985998	1	1	yes	characteristic
292	C₄	#17986007	1	1	yes	characteristic
293	C₄	#17986072	1	2	yes	—
294	C₄	#17986099	1	1	yes	characteristic
295	C₄	#17986104	1	1	yes	characteristic
296	C₂	#22587242	1	2	yes	—
297	C₂	#22587247	1	2	yes	—
298	C₂	#22588125	1	1	yes	characteristic
299	C₂	#22588173	1	1	yes	characteristic
300	C₂	#22588186	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1770	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1770	D₇₁
hypocenter	C₇₁	#8209
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
10. #17986129	C₄	semidirect
18. #17985729	C₄	semidirect
158. C₇₁	#8209	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986129	2
C₄×#14556051	2
C₂×#17985729	2
C₂×#17985721	2
#22586181	2
#22586174	2
C₇₁×#1770	2
#8209	71