Subgroups — C₇₁×#2805 #22585367

Order 9088.

368 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₇₁	C₂	1	2	yes	maximal
2	#400×C₇₁	C₂	1	2	yes	maximal
3	#395×C₇₁	C₂	1	2	yes	maximal
4	#391×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#389×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#389×C₇₁	C₂	1	1	yes	maximal, characteristic
7	#389×C₇₁	C₂	1	2	yes	maximal
8	#387×C₇₁	C₂	1	2	yes	maximal
9	#387×C₇₁	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₇₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
22	#95×C₇₁	C₂²	1	2	yes	—
23	#95×C₇₁	C₂²	1	2	yes	—
24	#95×C₇₁	C₂²	1	2	yes	—
25	#95×C₇₁	C₂²	1	2	yes	—
26	#95×C₇₁	C₂²	1	2	yes	—
27	#95×C₇₁	C₂²	1	2	yes	—
28	#95×C₇₁	C₂²	1	2	yes	—
29	#95×C₇₁	C₂²	1	2	yes	—
30	#95×C₇₁	C₂²	1	2	yes	—
31	C₂×Q₈×C₇₁	—	2	2	no	—
32	C₂×Q₈×C₇₁	—	4	1	no	—
33	C₂×Q₈×C₇₁	—	4	1	no	—
34	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
35	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₇₁	—	2	4	no	—
44	C₄⋊C₄×C₇₁	—	2	2	no	—
45	C₄⋊C₄×C₇₁	—	2	4	no	—
46	C₄⋊C₄×C₇₁	—	4	2	no	—
47	C₄⋊C₄×C₇₁	—	4	1	no	—
48	C₄⋊C₄×C₇₁	—	4	1	no	—
49	C₄⋊C₄×C₇₁	—	4	1	no	—
50	C₄⋊C₄×C₇₁	—	4	2	no	—
51	C₄⋊C₄×C₇₁	Q₈	1	4	yes	complemented
52	C₄²×C₇₁	—	4	1	no	—
53	C₄²×C₇₁	—	4	2	no	—
54	C₂³×C₇₁	C₂⁴	1	1	yes	characteristic
55	Q₈×C₇₁	—	4	4	no	—
56	C₂×C₄×C₇₁	—	2	2	no	—
57	C₂×C₄×C₇₁	—	2	1	no	—
58	C₂×C₄×C₇₁	—	2	1	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	1	no	—
61	C₂×C₄×C₇₁	—	2	1	no	—
62	C₂×C₄×C₇₁	—	2	1	no	—
63	C₂×C₄×C₇₁	—	2	1	no	—
64	C₂×C₄×C₇₁	—	2	2	no	—
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	—	2	2	no	—
73	C₂×C₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	2	2	no	—
78	C₂×C₄×C₇₁	—	2	2	no	—
79	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×Q₈	1	4	yes	—
81	C₂×C₄×C₇₁	#41	1	2	yes	—
82	C₂×C₄×C₇₁	#41	1	2	yes	—
83	C₂²×C₇₁	C₂²×Q₈	1	1	yes	characteristic
84	C₂²×C₇₁	C₂×#41	1	1	yes	characteristic
85	C₂²×C₇₁	C₂×#41	1	1	yes	characteristic
86	C₂²×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂²×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂²×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
89	C₂²×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
90	C₄×C₇₁	—	2	2	no	—
91	C₄×C₇₁	—	2	2	no	—
92	C₄×C₇₁	—	2	2	no	—
93	C₄×C₇₁	—	2	4	no	—
94	C₄×C₇₁	—	4	2	no	—
95	C₄×C₇₁	—	4	2	no	—
96	C₄×C₇₁	—	4	2	no	—
97	C₄×C₇₁	—	4	2	no	—
98	C₄×C₇₁	—	4	2	no	—
99	C₂×C₇₁	#538	1	1	yes	characteristic
100	C₂×C₇₁	#539	1	1	yes	characteristic
101	C₂×C₇₁	#544	1	1	yes	characteristic
102	C₂×C₇₁	#554	1	1	yes	characteristic
103	C₂×C₇₁	#556	1	1	yes	characteristic
104	C₂×C₇₁	#557	1	1	yes	characteristic
105	C₂×C₇₁	#563	1	1	yes	characteristic
106	#2805	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₇₁	#2805	1	1	yes	characteristic, direct product factor, Sylow
108	#400	C₂×C₇₁	1	2	yes	—
109	#400	C₂×C₇₁	1	2	yes	—
110	#395	C₂×C₇₁	1	2	yes	—
111	#391	C₂×C₇₁	1	1	yes	characteristic
112	#389	C₂×C₇₁	1	1	yes	characteristic
113	#389	C₂×C₇₁	1	1	yes	characteristic
114	#389	C₂×C₇₁	1	2	yes	—
115	#387	C₂×C₇₁	1	2	yes	—
116	#387	C₂×C₇₁	1	2	yes	—
117	C₂²×Q₈	C₂²×C₇₁	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
126	C₂×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
127	C₂×C₄²	C₂²×C₇₁	1	2	yes	—
128	C₂×C₄²	C₂²×C₇₁	1	1	yes	characteristic
129	#95	C₂²×C₇₁	1	2	yes	—
130	#95	C₂²×C₇₁	1	2	yes	—
131	#95	C₂²×C₇₁	1	2	yes	—
132	#95	C₂²×C₇₁	1	2	yes	—
133	#95	C₂²×C₇₁	1	2	yes	—
134	#95	C₂²×C₇₁	1	2	yes	—
135	#95	C₂²×C₇₁	1	2	yes	—
136	#95	C₂²×C₇₁	1	2	yes	—
137	#95	C₂²×C₇₁	1	2	yes	—
138	C₂×Q₈	—	2	2	no	—
139	C₂×Q₈	—	4	1	no	—
140	C₂×Q₈	—	4	1	no	—
141	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
142	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
143	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
144	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
145	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
146	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
148	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄⋊C₄	—	4	1	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	Q₈×C₇₁	1	4	yes	complemented
159	C₄²	—	4	1	no	—
160	C₄²	—	4	2	no	—
161	C₂³	C₂⁴×C₇₁	1	1	yes	characteristic
162	Q₈	—	4	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₂×Q₈×C₇₁	1	2	yes	—
187	C₂×C₄	C₂×Q₈×C₇₁	1	4	yes	—
188	C₂×C₄	#41×C₇₁	1	2	yes	—
189	C₂×C₄	#41×C₇₁	1	2	yes	—
190	C₂²	C₂²×Q₈×C₇₁	1	1	yes	characteristic
191	C₂²	C₂×#41×C₇₁	1	1	yes	characteristic
192	C₂²	C₂×#41×C₇₁	1	1	yes	characteristic
193	C₂²	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
194	C₂²	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
195	C₂²	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
196	C₂²	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₂	#538×C₇₁	1	1	yes	characteristic
207	C₂	#539×C₇₁	1	1	yes	characteristic
208	C₂	#544×C₇₁	1	1	yes	characteristic
209	C₂	#554×C₇₁	1	1	yes	characteristic
210	C₂	#556×C₇₁	1	1	yes	characteristic
211	C₂	#557×C₇₁	1	1	yes	characteristic
212	C₂	#563×C₇₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
51. C₄⋊C₄×C₇₁	Q₈	semidirect
106. #2805	C₇₁	direct (N × H)
107. C₇₁	#2805	direct (N × H)
158. C₄⋊C₄	Q₈×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
#400×C₇₁	2
#395×C₇₁	2
#391×C₇₁	2
#389×C₇₁	2
#387×C₇₁	2
#2805	71

Subgroups — C71×#2805 #22585367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#2805 #22585367