Subgroups — #31890270

Order 18176.

496 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556128	C₂	1	1	yes	maximal, characteristic
2	#22588457	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985744	C₂	1	2	yes	maximal
4	#22586224	C₂	1	2	yes	maximal
5	C₇₁×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₄₂	—	2	1	no	—
8	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
9	#17985905	—	2	1	no	—
10	#17985897	—	2	1	no	—
11	#17985897	C₄	1	2	yes	—
12	#17985869	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₇₁	—	2	1	no	—
14	C₄×C₄⋊Dic₇₁	C₄	1	2	yes	—
15	#17985754	—	2	1	no	—
16	#17985754	—	2	1	no	—
17	#17985744	—	2	4	no	—
18	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
19	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
20	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
21	C₄³×C₇₁	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₄₂	—	2	1	no	—
23	C₂²×C₇₁⋊C₈	—	2	2	no	—
24	#14556128	—	2	1	no	—
25	#14556128	—	2	1	no	—
26	#14556128	D₄	1	2	yes	—
27	#14556128	Q₈	1	2	yes	—
28	C₄×Dic₁₄₂	—	4	4	no	—
29	C₂×C₄⋊Dic₇₁	—	2	1	no	—
30	C₂×C₄⋊Dic₇₁	—	2	1	no	—
31	C₂×C₄⋊Dic₇₁	—	2	1	no	—
32	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
34	C₂×Dic₇₁⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₇₁	—	2	2	no	—
36	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
38	C₇₁⋊#95	—	2	2	no	—
39	C₇₁⋊#95	—	2	2	no	—
40	C₂²×C₈×C₇₁	—	2	2	no	—
41	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₇₁	—	2	2	no	—
49	C₄⋊C₈×C₇₁	—	2	2	no	—
50	C₄⋊C₈×C₇₁	—	4	2	no	—
51	C₄⋊C₈×C₇₁	—	4	2	no	—
52	C₂×Dic₁₄₂	—	2	2	no	—
53	C₂×Dic₁₄₂	—	2	2	no	—
54	C₂×Dic₁₄₂	—	4	2	no	—
55	C₂×Dic₁₄₂	—	4	2	no	—
56	C₂²×Dic₇₁	—	2	2	no	—
57	C₂²×Dic₇₁	—	2	2	no	—
58	C₂×C₇₁⋊C₈	—	4	2	no	—
59	C₂×C₇₁⋊C₈	—	4	2	no	—
60	C₂×C₇₁⋊C₈	—	4	2	no	—
61	C₄⋊Dic₇₁	—	2	2	no	—
62	C₄⋊Dic₇₁	—	2	2	no	—
63	C₄⋊Dic₇₁	—	2	2	no	—
64	C₄⋊Dic₇₁	—	4	1	no	—
65	C₄⋊Dic₇₁	—	4	2	no	—
66	C₄⋊Dic₇₁	—	4	1	no	—
67	C₄⋊Dic₇₁	—	4	2	no	—
68	C₄⋊Dic₇₁	C₄⋊C₄	1	4	yes	—
69	Dic₇₁⋊C₄	—	4	4	no	—
70	C₄×Dic₇₁	—	4	4	no	—
71	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	—	4	2	no	—
79	C₂×C₈×C₇₁	—	4	2	no	—
80	C₂×C₈×C₇₁	—	4	2	no	—
81	C₄²×C₇₁	—	2	2	no	—
82	C₄²×C₇₁	—	2	2	no	—
83	C₄²×C₇₁	—	2	1	no	—
84	C₄²×C₇₁	—	2	2	no	—
85	C₄²×C₇₁	—	2	1	no	—
86	C₄²×C₇₁	—	4	1	no	—
87	C₄²×C₇₁	SD₁₆	1	2	yes	—
88	C₄²×C₇₁	Q₁₆	1	2	yes	—
89	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₇₁	#41	1	1	yes	characteristic
92	C₄²×C₇₁	#41	1	1	yes	characteristic
93	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₁	—	2	1	no	—
95	C₂×C₄×C₇₁	—	2	1	no	—
96	C₂×C₄×C₇₁	—	2	1	no	—
97	C₂×C₄×C₇₁	—	2	1	no	—
98	C₂×C₄×C₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	—	2	1	no	—
102	C₂×C₄×C₇₁	—	2	1	no	—
103	C₂×C₄×C₇₁	—	2	1	no	—
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	#101	1	2	yes	—
106	C₂×C₄×C₇₁	#103	1	2	yes	—
107	C₂×C₄×C₇₁	#103	1	2	yes	—
108	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
110	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
111	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₄×C₇₁	#135	1	2	yes	—
117	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
120	C₂×Dic₇₁	—	2	4	no	—
121	C₂×Dic₇₁	—	4	2	no	—
122	C₂×Dic₇₁	—	4	2	no	—
123	C₂×Dic₇₁	—	4	2	no	—
124	C₂×Dic₇₁	—	4	2	no	—
125	C₂×Dic₇₁	—	4	2	no	—
126	Dic₁₄₂	—	4	4	no	—
127	Dic₁₄₂	—	4	4	no	—
128	C₈×C₇₁	—	8	2	no	—
129	C₇₁⋊C₈	—	8	2	no	—
130	C₂²×C₇₁	#380	1	1	yes	characteristic
131	C₂²×C₇₁	C₂×#101	1	1	yes	characteristic
132	C₂²×C₇₁	#413	1	1	yes	characteristic
133	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
134	C₂²×C₇₁	#416	1	1	yes	characteristic
135	C₂²×C₇₁	#417	1	1	yes	characteristic
136	C₂²×C₇₁	#418	1	1	yes	characteristic
137	C₄×C₇₁	—	2	2	no	—
138	C₄×C₇₁	—	2	1	no	—
139	C₄×C₇₁	—	2	1	no	—
140	C₄×C₇₁	—	2	2	no	—
141	C₄×C₇₁	—	2	2	no	—
142	C₄×C₇₁	—	4	1	no	—
143	C₄×C₇₁	#460	1	1	yes	characteristic
144	C₄×C₇₁	#462	1	1	yes	characteristic
145	C₄×C₇₁	#463	1	2	yes	—
146	C₄×C₇₁	#475	1	1	yes	characteristic
147	C₄×C₇₁	#477	1	1	yes	characteristic
148	C₄×C₇₁	#478	1	2	yes	—
149	Dic₇₁	—	4	4	no	—
150	Dic₇₁	—	8	2	no	—
151	#8591	—	71	1	no	maximal, Sylow
152	C₂×C₇₁	#1542	1	1	yes	characteristic
153	C₂×C₇₁	#1549	1	1	yes	characteristic
154	C₂×C₇₁	#1552	1	1	yes	characteristic
155	C₂×C₇₁	#1557	1	1	yes	characteristic
156	C₂×C₇₁	#1801	1	1	yes	characteristic
157	C₂×C₇₁	#1803	1	1	yes	characteristic
158	C₂×C₇₁	#1804	1	1	yes	characteristic
159	C₄×C₄⋊Q₈	—	71	1	no	—
160	#1842	D₇₁	1	1	yes	characteristic
161	#1831	—	71	1	no	—
162	C₂×#332	—	71	2	no	—
163	#1296	—	71	2	no	—
164	C₇₁	#8591	1	1	yes	characteristic, complemented, Sylow
165	C₂×C₄⋊Q₈	—	71	1	no	—
166	C₂×C₄×Q₈	—	142	1	no	—
167	C₂×C₄⋊C₈	—	2	1	no	—
168	C₂×C₄⋊C₈	—	2	1	no	—
169	C₂×C₄⋊C₈	—	71	2	no	—
170	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
171	#391	—	142	1	no	—
172	#389	—	71	2	no	—
173	#389	—	142	1	no	—
174	#384	—	71	1	no	—
175	C₄×C₄⋊C₄	—	71	2	no	—
176	C₄×C₄⋊C₄	—	142	1	no	—
177	C₄³	C₂×D₇₁	1	1	yes	characteristic
178	#336	—	142	1	no	—
179	#336	—	142	1	no	—
180	#332	—	142	4	no	—
181	C₂²×Q₈	—	142	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	142	2	no	—
184	C₄⋊Q₈	—	71	2	no	—
185	C₄⋊Q₈	—	71	2	no	—
186	C₄⋊Q₈	—	142	1	no	—
187	C₄⋊Q₈	—	142	1	no	—
188	C₄×Q₈	—	284	4	no	—
189	C₂×C₄⋊C₄	—	71	2	no	—
190	C₂×C₄⋊C₄	—	71	2	no	—
191	C₂×C₄⋊C₄	—	142	2	no	—
192	C₂×C₄⋊C₄	—	142	1	no	—
193	C₂×C₄⋊C₄	—	142	1	no	—
194	C₂×C₄⋊C₄	—	142	1	no	—
195	C₂×C₄²	—	142	2	no	—
196	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
198	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
199	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	4	2	no	—
206	C₄⋊C₈	—	4	2	no	—
207	C₄⋊C₈	—	142	2	no	—
208	C₄⋊C₈	—	142	2	no	—
209	#95	—	142	2	no	—
210	#95	—	142	2	no	—
211	C₂×Q₈	—	284	2	no	—
212	C₂×Q₈	—	284	2	no	—
213	C₂×Q₈	—	142	2	no	—
214	C₂×Q₈	—	142	2	no	—
215	C₂²×C₄	—	142	2	no	—
216	C₂²×C₄	—	142	2	no	—
217	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	284	2	no	—
228	C₂×C₈	—	284	2	no	—
229	C₂×C₈	—	284	2	no	—
230	C₄⋊C₄	—	284	2	no	—
231	C₄⋊C₄	—	284	4	no	—
232	C₄⋊C₄	—	284	1	no	—
233	C₄⋊C₄	—	284	2	no	—
234	C₄⋊C₄	—	284	1	no	—
235	C₄⋊C₄	—	71	4	no	—
236	C₄⋊C₄	—	142	2	no	—
237	C₄⋊C₄	—	142	2	no	—
238	C₄⋊C₄	—	142	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	284	4	no	—
246	C₄²	#11765853	1	2	yes	—
247	C₄²	#11765855	1	2	yes	—
248	C₄²	D₄×D₇₁	1	1	yes	characteristic
249	C₄²	Q₈×D₇₁	1	1	yes	characteristic
250	C₄²	#11765864	1	1	yes	characteristic
251	C₄²	#11765866	1	1	yes	characteristic
252	C₂³	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
253	Q₈	—	284	4	no	—
254	Q₈	—	284	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	284	2	no	—
267	C₂×C₄	—	284	2	no	—
268	C₂×C₄	—	284	2	no	—
269	C₂×C₄	—	284	2	no	—
270	C₂×C₄	—	284	2	no	—
271	C₂×C₄	—	142	4	no	—
272	C₂×C₄	#14556046	1	2	yes	—
273	C₂×C₄	#14556051	1	2	yes	—
274	C₂×C₄	#14556051	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
276	C₂×C₄	#14556086	1	1	yes	characteristic
277	C₂×C₄	#14556099	1	1	yes	characteristic
278	C₂×C₄	#14556103	1	1	yes	characteristic
279	C₂×C₄	#14556104	1	1	yes	characteristic
280	C₂×C₄	#14556108	1	1	yes	characteristic
281	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
282	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
283	C₂×C₄	#14556150	1	2	yes	—
284	C₂×C₄	#14556154	1	1	yes	characteristic
285	C₂×C₄	#14556156	1	2	yes	—
286	C₂×C₄	#14556159	1	1	yes	characteristic
287	C₈	—	8	2	no	—
288	C₈	—	568	2	no	—
289	C₂²	#17985856	1	1	yes	characteristic
290	C₂²	C₂×#14556046	1	1	yes	characteristic
291	C₂²	#17985981	1	1	yes	characteristic
292	C₂²	C₂×#14556051	1	1	yes	characteristic
293	C₂²	#17985996	1	1	yes	characteristic
294	C₂²	#17985998	1	1	yes	characteristic
295	C₂²	#17986005	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	284	4	no	—
303	C₄	—	568	2	no	—
304	C₄	#17986196	1	1	yes	characteristic
305	C₄	#17986206	1	1	yes	characteristic
306	C₄	#17986215	1	2	yes	—
307	C₄	#17986281	1	1	yes	characteristic
308	C₄	#17986291	1	1	yes	characteristic
309	C₄	#17986300	1	2	yes	—
310	C₂	#22586998	1	1	yes	characteristic
311	C₂	#22587026	1	1	yes	characteristic
312	C₂	#22587038	1	1	yes	characteristic
313	C₂	#22587060	1	1	yes	characteristic
314	C₂	#22588333	1	1	yes	characteristic
315	C₂	#22588342	1	1	yes	characteristic
316	C₂	#22588351	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1842	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1842	D₇₁
hypocenter	C₇₁	#8591
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
164. C₇₁	#8591	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556128	2
#22588457	2
C₂×#17985744	2
#22586224	2
C₇₁×#1842	2
#8591	71