Subgroups — C₇₁×#10279 #31831151

Order 18176.

510 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#477×C₇₁	C₂	1	2	yes	maximal
3	C₇₁×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×#333×C₇₁	C₂	1	2	yes	maximal
5	C₇₁×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
7	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
8	#500×C₇₁	—	2	4	no	—
9	#477×C₇₁	—	2	4	no	—
10	#477×C₇₁	—	2	4	no	—
11	C₂×#107×C₇₁	—	2	2	no	—
12	C₂×#107×C₇₁	—	2	2	no	—
13	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×#103×C₇₁	—	2	2	no	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#336×C₇₁	—	2	2	no	—
18	#333×C₇₁	—	2	4	no	—
19	C₂²×Q₈×C₇₁	—	2	2	no	—
20	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	—	2	2	no	—
22	C₂²×C₈×C₇₁	—	2	1	no	—
23	C₂²×C₈×C₇₁	D₄	1	2	yes	—
24	C₄⋊Q₈×C₇₁	—	2	2	no	—
25	C₄⋊Q₈×C₇₁	—	2	2	no	—
26	C₄⋊Q₈×C₇₁	—	2	2	no	—
27	C₄⋊Q₈×C₇₁	D₄	1	4	yes	—
28	C₄×Q₈×C₇₁	—	2	2	no	—
29	C₄×Q₈×C₇₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	—	2	1	no	—
35	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
36	#107×C₇₁	—	2	8	no	—
37	#107×C₇₁	—	2	8	no	—
38	C₄⋊C₈×C₇₁	—	2	2	no	—
39	C₄⋊C₈×C₇₁	—	2	2	no	—
40	C₄⋊C₈×C₇₁	D₄	1	4	yes	—
41	#103×C₇₁	—	2	8	no	—
42	C₄×C₈×C₇₁	—	2	2	no	—
43	C₂×Q₈×C₇₁	—	2	1	no	—
44	C₂×Q₈×C₇₁	—	2	2	no	—
45	C₂×Q₈×C₇₁	—	2	4	no	—
46	C₂×Q₈×C₇₁	—	2	2	no	—
47	C₂×Q₈×C₇₁	—	4	2	no	—
48	C₂×Q₈×C₇₁	—	8	4	no	—
49	C₂×Q₈×C₇₁	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₇₁	—	2	1	no	—
51	C₂²×C₄×C₇₁	—	2	1	no	—
52	C₂²×C₄×C₇₁	—	4	2	no	—
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	—	2	4	no	—
59	C₂×C₈×C₇₁	—	4	2	no	—
60	C₂×C₈×C₇₁	—	4	2	no	—
61	C₂×C₈×C₇₁	—	4	1	no	—
62	C₂×C₈×C₇₁	—	4	2	no	—
63	C₂×C₈×C₇₁	Q₁₆	1	4	yes	—
64	C₄⋊C₄×C₇₁	—	2	4	no	—
65	C₄⋊C₄×C₇₁	—	2	4	no	—
66	C₄⋊C₄×C₇₁	—	2	2	no	—
67	C₄⋊C₄×C₇₁	—	4	2	no	—
68	C₄⋊C₄×C₇₁	—	4	2	no	—
69	C₄⋊C₄×C₇₁	—	4	2	no	—
70	C₄²×C₇₁	—	4	1	no	—
71	C₄²×C₇₁	—	4	1	no	—
72	C₄²×C₇₁	C₂×D₄	1	2	yes	—
73	C₄²×C₇₁	C₂×D₄	1	2	yes	—
74	C₂³×C₇₁	#121	1	1	yes	characteristic
75	Q₈×C₇₁	—	2	4	no	—
76	Q₈×C₇₁	—	4	2	no	—
77	Q₈×C₇₁	—	8	4	no	—
78	Q₈×C₇₁	—	8	4	no	—
79	C₂×C₄×C₇₁	—	2	2	no	—
80	C₂×C₄×C₇₁	—	2	2	no	—
81	C₂×C₄×C₇₁	—	4	1	no	—
82	C₂×C₄×C₇₁	—	4	1	no	—
83	C₂×C₄×C₇₁	—	4	1	no	—
84	C₂×C₄×C₇₁	—	4	4	no	—
85	C₂×C₄×C₇₁	—	4	2	no	—
86	C₂×C₄×C₇₁	—	8	4	no	—
87	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
88	C₂×C₄×C₇₁	#120	1	2	yes	—
89	C₂×C₄×C₇₁	#121	1	2	yes	—
90	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
94	C₂×C₄×C₇₁	#137	1	2	yes	—
95	C₂×C₄×C₇₁	#137	1	2	yes	—
96	C₈×C₇₁	—	2	4	no	—
97	C₈×C₇₁	—	4	4	no	—
98	C₈×C₇₁	—	8	1	no	—
99	C₂²×C₇₁	#392	1	1	yes	characteristic
100	C₂²×C₇₁	#462	1	2	yes	—
101	C₂²×C₇₁	#467	1	1	yes	characteristic
102	C₂²×C₇₁	#470	1	1	yes	characteristic
103	C₂²×C₇₁	#472	1	2	yes	—
104	C₄×C₇₁	—	2	2	no	—
105	C₄×C₇₁	—	4	2	no	—
106	C₄×C₇₁	—	8	1	no	—
107	C₄×C₇₁	—	8	4	no	—
108	C₄×C₇₁	#451	1	2	yes	—
109	C₄×C₇₁	#451	1	2	yes	—
110	C₄×C₇₁	#452	1	2	yes	—
111	C₄×C₇₁	#494	1	1	yes	characteristic
112	C₄×C₇₁	#497	1	1	yes	characteristic
113	#10279	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂×C₇₁	#1676	1	2	yes	—
115	C₂×C₇₁	#1696	1	2	yes	—
116	C₂×C₇₁	#2001	1	1	yes	characteristic
117	C₂×C₇₁	#2016	1	1	yes	characteristic
118	C₂×C₇₁	#2018	1	1	yes	characteristic
119	C₂×#500	C₂×C₇₁	1	1	yes	characteristic
120	C₂×#477	C₂×C₇₁	1	2	yes	—
121	#1984	C₂×C₇₁	1	1	yes	characteristic
122	C₂×#333	C₂×C₇₁	1	2	yes	—
123	#1277	C₂×C₇₁	1	1	yes	characteristic
124	C₇₁	#10279	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
126	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
127	#500	—	2	4	no	—
128	#477	—	2	4	no	—
129	#477	—	2	4	no	—
130	C₂×#107	—	2	2	no	—
131	C₂×#107	—	2	2	no	—
132	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
134	C₂×#103	—	2	2	no	—
135	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
136	#336	—	2	2	no	—
137	#333	—	2	4	no	—
138	C₂²×Q₈	—	2	2	no	—
139	C₂²×Q₈	D₄×C₇₁	1	1	yes	characteristic
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	D₄×C₇₁	1	2	yes	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	D₄×C₇₁	1	4	yes	—
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
155	#107	—	2	8	no	—
156	#107	—	2	8	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	D₄×C₇₁	1	4	yes	—
160	#103	—	2	8	no	—
161	C₄×C₈	—	2	2	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	8	4	no	—
168	C₂×Q₈	Q₁₆×C₇₁	1	2	yes	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
174	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	Q₁₆×C₇₁	1	4	yes	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄²	—	4	1	no	—
190	C₄²	—	4	1	no	—
191	C₄²	C₂×D₄×C₇₁	1	2	yes	—
192	C₄²	C₂×D₄×C₇₁	1	2	yes	—
193	C₂³	#121×C₇₁	1	1	yes	characteristic
194	Q₈	—	2	4	no	—
195	Q₈	—	4	2	no	—
196	Q₈	—	8	4	no	—
197	Q₈	—	8	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	8	4	no	—
206	C₂×C₄	#120×C₇₁	1	1	yes	characteristic
207	C₂×C₄	#120×C₇₁	1	2	yes	—
208	C₂×C₄	#121×C₇₁	1	2	yes	—
209	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
210	C₂×C₄	C₂×Q₁₆×C₇₁	1	1	yes	characteristic
211	C₂×C₄	C₂×Q₁₆×C₇₁	1	2	yes	—
212	C₂×C₄	#135×C₇₁	1	1	yes	characteristic
213	C₂×C₄	#137×C₇₁	1	2	yes	—
214	C₂×C₄	#137×C₇₁	1	2	yes	—
215	C₈	—	2	4	no	—
216	C₈	—	4	4	no	—
217	C₈	—	8	1	no	—
218	C₂²	#392×C₇₁	1	1	yes	characteristic
219	C₂²	#462×C₇₁	1	2	yes	—
220	C₂²	#467×C₇₁	1	1	yes	characteristic
221	C₂²	#470×C₇₁	1	1	yes	characteristic
222	C₂²	#472×C₇₁	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	8	1	no	—
226	C₄	—	8	4	no	—
227	C₄	#451×C₇₁	1	2	yes	—
228	C₄	#451×C₇₁	1	2	yes	—
229	C₄	#452×C₇₁	1	2	yes	—
230	C₄	#494×C₇₁	1	1	yes	characteristic
231	C₄	#497×C₇₁	1	1	yes	characteristic
232	C₂	C₇₁×#1676	1	2	yes	—
233	C₂	C₇₁×#1696	1	2	yes	—
234	C₂	C₇₁×#2001	1	1	yes	characteristic
235	C₂	C₇₁×#2016	1	1	yes	characteristic
236	C₂	C₇₁×#2018	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
113. #10279	C₇₁	direct (N × H)
124. C₇₁	#10279	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#500×C₇₁	2
C₂×#477×C₇₁	2
C₇₁×#1984	2
C₂×#333×C₇₁	2
C₇₁×#1277	2
#10279	71

Subgroups — C71×#10279 #31831151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#10279 #31831151