Subgroups — C₇₁×#10796 #31831668

Order 18176.

838 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#376×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#336×C₇₁	C₂	1	2	yes	maximal
3	C₂×#336×C₇₁	C₂	1	2	yes	maximal
4	C₂×#336×C₇₁	C₂	1	2	yes	maximal
5	C₂³×C₈×C₇₁	C₂²	1	4	yes	—
6	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
9	#376×C₇₁	—	2	4	no	—
10	C₂×#95×C₇₁	C₄	1	4	yes	—
11	#336×C₇₁	—	2	4	no	—
12	#336×C₇₁	—	2	4	no	—
13	#336×C₇₁	C₄	1	8	yes	complemented
14	C₂³×C₄×C₇₁	C₂×C₄	1	2	yes	—
15	C₂³×C₄×C₇₁	C₂×C₄	1	2	yes	—
16	C₂³×C₄×C₇₁	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₇₁	—	2	4	no	—
19	C₂²×C₈×C₇₁	—	2	4	no	—
20	C₂²×C₈×C₇₁	—	2	4	no	—
21	C₂²×C₈×C₇₁	—	2	4	no	—
22	C₂²×C₈×C₇₁	—	2	4	no	—
23	C₂²×C₈×C₇₁	—	2	4	no	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₇₁	—	2	2	no	—
26	C₂×C₄²×C₇₁	—	2	2	no	—
27	C₂×C₄²×C₇₁	—	2	2	no	—
28	C₂×C₄²×C₇₁	—	2	2	no	—
29	C₂×C₄²×C₇₁	—	2	2	no	—
30	C₂×C₄²×C₇₁	—	2	2	no	—
31	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₇₁	D₄	1	4	yes	—
33	C₂×C₄²×C₇₁	D₄	1	2	yes	—
34	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
35	#95×C₇₁	—	2	8	no	—
36	C₂⁴×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	—	2	2	no	—
38	C₂²×C₄×C₇₁	—	2	2	no	—
39	C₂²×C₄×C₇₁	—	2	2	no	—
40	C₂²×C₄×C₇₁	—	2	2	no	—
41	C₂²×C₄×C₇₁	—	2	2	no	—
42	C₂²×C₄×C₇₁	—	2	2	no	—
43	C₂²×C₄×C₇₁	—	2	8	no	—
44	C₂²×C₄×C₇₁	—	2	8	no	—
45	C₂²×C₄×C₇₁	—	2	8	no	—
46	C₂²×C₄×C₇₁	C₄²	1	4	yes	—
47	C₂²×C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
48	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	#41	1	2	yes	—
54	C₂²×C₄×C₇₁	#41	1	2	yes	—
55	C₂²×C₄×C₇₁	#41	1	2	yes	—
56	C₂²×C₄×C₇₁	#41	1	2	yes	—
57	C₂×C₈×C₇₁	—	2	8	no	—
58	C₂×C₈×C₇₁	—	2	8	no	—
59	C₂×C₈×C₇₁	—	2	8	no	—
60	C₂×C₈×C₇₁	—	4	4	no	—
61	C₂×C₈×C₇₁	—	4	4	no	—
62	C₂×C₈×C₇₁	—	4	4	no	—
63	C₂×C₈×C₇₁	—	4	4	no	—
64	C₄²×C₇₁	—	2	8	no	—
65	C₄²×C₇₁	—	2	4	no	—
66	C₄²×C₇₁	—	2	4	no	—
67	C₄²×C₇₁	—	2	8	no	—
68	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
69	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₇₁	#117	1	1	yes	characteristic
72	C₂³×C₇₁	#117	1	1	yes	characteristic
73	C₂³×C₇₁	#117	1	1	yes	characteristic
74	C₂³×C₇₁	#117	1	1	yes	characteristic
75	C₂³×C₇₁	#131	1	8	yes	—
76	C₂×C₄×C₇₁	—	2	4	no	—
77	C₂×C₄×C₇₁	—	2	4	no	—
78	C₂×C₄×C₇₁	—	2	4	no	—
79	C₂×C₄×C₇₁	—	2	8	no	—
80	C₂×C₄×C₇₁	—	2	8	no	—
81	C₂×C₄×C₇₁	—	2	8	no	—
82	C₂×C₄×C₇₁	—	4	4	no	—
83	C₂×C₄×C₇₁	—	4	4	no	—
84	C₂×C₄×C₇₁	—	4	4	no	—
85	C₂×C₄×C₇₁	—	4	4	no	—
86	C₂×C₄×C₇₁	—	4	4	no	—
87	C₂×C₄×C₇₁	—	4	4	no	—
88	C₂×C₄×C₇₁	#117	1	4	yes	—
89	C₂×C₄×C₇₁	C₄×D₄	1	4	yes	—
90	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
92	C₂×C₄×C₇₁	#121	1	2	yes	—
93	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₇₁	#123	1	4	yes	—
95	C₂×C₄×C₇₁	#124	1	2	yes	—
96	C₂×C₄×C₇₁	#125	1	2	yes	—
97	C₂×C₄×C₇₁	#126	1	4	yes	—
98	C₈×C₇₁	—	4	8	no	—
99	C₂²×C₇₁	#376	1	1	yes	characteristic
100	C₂²×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₇₁	#380	1	1	yes	characteristic
103	C₂²×C₇₁	#381	1	1	yes	characteristic
104	C₂²×C₇₁	#382	1	1	yes	characteristic
105	C₂²×C₇₁	#383	1	1	yes	characteristic
106	C₂²×C₇₁	#405	1	4	yes	—
107	C₂²×C₇₁	#408	1	4	yes	—
108	C₂²×C₇₁	#424	1	4	yes	—
109	C₂²×C₇₁	#433	1	4	yes	—
110	C₂²×C₇₁	#433	1	4	yes	—
111	C₂²×C₇₁	#433	1	4	yes	—
112	C₂²×C₇₁	#433	1	4	yes	—
113	C₄×C₇₁	—	4	4	no	—
114	C₄×C₇₁	—	4	4	no	—
115	C₄×C₇₁	—	4	4	no	—
116	C₄×C₇₁	#387	1	4	yes	—
117	C₄×C₇₁	#388	1	4	yes	—
118	#10796	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₇₁	#1434	1	1	yes	characteristic
120	C₂×C₇₁	#1746	1	2	yes	—
121	C₂×C₇₁	#1750	1	2	yes	—
122	C₂×C₇₁	#1768	1	2	yes	—
123	C₂×C₇₁	#1798	1	2	yes	—
124	C₂×C₇₁	#1818	1	2	yes	—
125	C₂×C₇₁	#1832	1	2	yes	—
126	C₂×C₇₁	#1840	1	2	yes	—
127	C₂×#376	C₂×C₇₁	1	1	yes	characteristic
128	C₂×#336	C₂×C₇₁	1	2	yes	—
129	C₂×#336	C₂×C₇₁	1	2	yes	—
130	C₂×#336	C₂×C₇₁	1	2	yes	—
131	C₇₁	#10796	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₈	C₂²×C₇₁	1	4	yes	—
133	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
134	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
135	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
136	#376	—	2	4	no	—
137	C₂×#95	C₄×C₇₁	1	4	yes	—
138	#336	—	2	4	no	—
139	#336	—	2	4	no	—
140	#336	C₄×C₇₁	1	8	yes	complemented
141	C₂³×C₄	C₂×C₄×C₇₁	1	2	yes	—
142	C₂³×C₄	C₂×C₄×C₇₁	1	2	yes	—
143	C₂³×C₄	C₂×C₄×C₇₁	1	2	yes	—
144	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	—	2	4	no	—
151	C₂²×C₈	C₂×C₄×C₇₁	1	8	yes	—
152	C₂×C₄²	—	2	2	no	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₂×C₄×C₇₁	1	4	yes	—
159	C₂×C₄²	D₄×C₇₁	1	4	yes	—
160	C₂×C₄²	D₄×C₇₁	1	2	yes	—
161	C₂×C₄²	Q₈×C₇₁	1	2	yes	—
162	#95	—	2	8	no	—
163	C₂⁴	C₂²×C₄×C₇₁	1	1	yes	characteristic
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	2	8	no	—
172	C₂²×C₄	—	2	8	no	—
173	C₂²×C₄	C₄²×C₇₁	1	4	yes	—
174	C₂²×C₄	C₂²⋊C₄×C₇₁	1	4	yes	—
175	C₂²×C₄	C₄⋊C₄×C₇₁	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
179	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
180	C₂²×C₄	#41×C₇₁	1	2	yes	—
181	C₂²×C₄	#41×C₇₁	1	2	yes	—
182	C₂²×C₄	#41×C₇₁	1	2	yes	—
183	C₂²×C₄	#41×C₇₁	1	2	yes	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	4	4	no	—
190	C₂×C₈	—	4	4	no	—
191	C₄²	—	2	8	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	8	no	—
195	C₂³	C₂×C₄²×C₇₁	1	1	yes	characteristic
196	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
197	C₂³	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
198	C₂³	#117×C₇₁	1	1	yes	characteristic
199	C₂³	#117×C₇₁	1	1	yes	characteristic
200	C₂³	#117×C₇₁	1	1	yes	characteristic
201	C₂³	#117×C₇₁	1	1	yes	characteristic
202	C₂³	#131×C₇₁	1	8	yes	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	#117×C₇₁	1	4	yes	—
216	C₂×C₄	C₄×D₄×C₇₁	1	4	yes	—
217	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
218	C₂×C₄	C₄×Q₈×C₇₁	1	2	yes	—
219	C₂×C₄	#121×C₇₁	1	2	yes	—
220	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
221	C₂×C₄	#123×C₇₁	1	4	yes	—
222	C₂×C₄	#124×C₇₁	1	2	yes	—
223	C₂×C₄	#125×C₇₁	1	2	yes	—
224	C₂×C₄	#126×C₇₁	1	4	yes	—
225	C₈	—	4	8	no	—
226	C₂²	#376×C₇₁	1	1	yes	characteristic
227	C₂²	C₄×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
228	C₂²	C₄×C₄⋊C₄×C₇₁	1	1	yes	characteristic
229	C₂²	#380×C₇₁	1	1	yes	characteristic
230	C₂²	#381×C₇₁	1	1	yes	characteristic
231	C₂²	#382×C₇₁	1	1	yes	characteristic
232	C₂²	#383×C₇₁	1	1	yes	characteristic
233	C₂²	#405×C₇₁	1	4	yes	—
234	C₂²	#408×C₇₁	1	4	yes	—
235	C₂²	#424×C₇₁	1	4	yes	—
236	C₂²	#433×C₇₁	1	4	yes	—
237	C₂²	#433×C₇₁	1	4	yes	—
238	C₂²	#433×C₇₁	1	4	yes	—
239	C₂²	#433×C₇₁	1	4	yes	—
240	C₄	—	4	4	no	—
241	C₄	—	4	4	no	—
242	C₄	—	4	4	no	—
243	C₄	#387×C₇₁	1	4	yes	—
244	C₄	#388×C₇₁	1	4	yes	—
245	C₂	C₇₁×#1434	1	1	yes	characteristic
246	C₂	C₇₁×#1746	1	2	yes	—
247	C₂	C₇₁×#1750	1	2	yes	—
248	C₂	C₇₁×#1768	1	2	yes	—
249	C₂	C₇₁×#1798	1	2	yes	—
250	C₂	C₇₁×#1818	1	2	yes	—
251	C₂	C₇₁×#1832	1	2	yes	—
252	C₂	C₇₁×#1840	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇₁	C₂³
commutator	C₂³	C₂×C₄²×C₇₁
Frattini, cosocle	C₂³×C₄	C₂³×C₇₁
socle, prime radical	C₂⁴×C₇₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. #336×C₇₁	C₄	semidirect
118. #10796	C₇₁	direct (N × H)
131. C₇₁	#10796	direct (N × H)
140. #336	C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#376×C₇₁	2
C₂×#336×C₇₁	2
#10796	71

Subgroups — C71×#10796 #31831668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#10796 #31831668