Subgroups — C₇₁×#2977 #22585539

Order 9088.

422 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#525×C₇₁	C₂	1	1	yes	maximal, characteristic
2	#435×C₇₁	C₂	1	4	yes	maximal, complemented
3	C₈×D₄×C₇₁	C₂	1	4	yes	maximal, complemented
4	#433×C₇₁	C₂	1	2	yes	maximal, complemented
5	#408×C₇₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₈×C₇₁	C₂	1	2	yes	maximal, complemented
7	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₇₁	C₂²	1	2	yes	complemented
10	C₂²×C₈×C₇₁	C₂²	1	4	yes	—
11	C₂²×C₈×C₇₁	C₂²	1	2	yes	complemented
12	#123×C₇₁	C₄	1	2	yes	—
13	C₂²⋊Q₈×C₇₁	C₄	1	2	yes	—
14	#121×C₇₁	C₄	1	2	yes	—
15	#120×C₇₁	C₄	1	2	yes	—
16	C₄×D₄×C₇₁	C₂²	1	4	yes	—
17	#117×C₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₇₁	C₂²	1	4	yes	complemented
19	C₂²⋊C₈×C₇₁	C₂²	1	4	yes	—
20	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
21	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₇₁	C₂²	1	4	yes	complemented
23	#97×C₇₁	C₂²	1	2	yes	complemented
24	C₄×C₈×C₇₁	C₂²	1	2	yes	complemented
25	C₂⁴×C₇₁	C₂×C₄	1	1	yes	characteristic
26	#41×C₇₁	—	4	2	no	—
27	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₇₁	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₇₁	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	—	4	2	no	—
32	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₇₁	C₂³	1	2	yes	—
37	M₁₆×C₇₁	—	4	2	no	—
38	C₂×C₈×C₇₁	—	4	4	no	—
39	C₂×C₈×C₇₁	—	4	2	no	—
40	C₂×C₈×C₇₁	C₂³	1	4	yes	—
41	C₂×C₈×C₇₁	C₂³	1	4	yes	—
42	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
44	C₂²⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
45	C₂²⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
46	C₂²⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
47	C₄²×C₇₁	C₂³	1	2	yes	—
48	C₂³×C₇₁	—	2	4	no	—
49	C₂³×C₇₁	—	4	1	no	—
50	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂³×C₇₁	C₂²×C₄	1	2	yes	—
52	C₂³×C₇₁	C₂²×C₄	1	2	yes	—
53	Q₈×C₇₁	—	4	1	no	—
54	D₄×C₇₁	—	2	8	no	—
55	D₄×C₇₁	—	4	1	no	—
56	D₄×C₇₁	—	4	2	no	—
57	C₂×C₄×C₇₁	—	2	4	no	—
58	C₂×C₄×C₇₁	—	2	2	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	—	4	1	no	—
62	C₂×C₄×C₇₁	—	4	2	no	—
63	C₂×C₄×C₇₁	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
68	C₈×C₇₁	—	2	4	no	—
69	C₈×C₇₁	—	2	4	no	—
70	C₂²×C₇₁	—	2	2	no	—
71	C₂²×C₇₁	—	2	2	no	—
72	C₂²×C₇₁	—	2	1	no	—
73	C₂²×C₇₁	—	2	2	no	—
74	C₂²×C₇₁	—	2	4	no	—
75	C₂²×C₇₁	—	2	2	no	—
76	C₂²×C₇₁	—	4	4	no	—
77	C₂²×C₇₁	#131	1	4	yes	—
78	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
79	C₄×C₇₁	—	2	2	no	—
80	C₄×C₇₁	—	2	1	no	—
81	C₄×C₇₁	—	2	2	no	—
82	C₄×C₇₁	—	2	4	no	—
83	C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
84	C₂×C₇₁	—	2	4	no	—
85	C₂×C₇₁	—	4	2	no	—
86	C₂×C₇₁	—	4	1	no	—
87	C₂×C₇₁	#518	1	1	yes	characteristic
88	C₂×C₇₁	C₂×#131	1	2	yes	—
89	#2977	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₇₁	#2977	1	1	yes	characteristic, direct product factor, Sylow
91	#525	C₂×C₇₁	1	1	yes	characteristic
92	#435	C₂×C₇₁	1	4	yes	complemented
93	C₈×D₄	C₂×C₇₁	1	4	yes	complemented
94	#433	C₂×C₇₁	1	2	yes	complemented
95	#408	C₂×C₇₁	1	2	yes	complemented
96	C₂×C₂²⋊C₈	C₂×C₇₁	1	2	yes	complemented
97	C₂×#41	C₂²×C₇₁	1	1	yes	characteristic
98	C₂³×C₄	C₂²×C₇₁	1	1	yes	characteristic
99	C₂×M₁₆	C₂²×C₇₁	1	2	yes	complemented
100	C₂²×C₈	C₂²×C₇₁	1	4	yes	—
101	C₂²×C₈	C₂²×C₇₁	1	2	yes	complemented
102	#123	C₄×C₇₁	1	2	yes	—
103	C₂²⋊Q₈	C₄×C₇₁	1	2	yes	—
104	#121	C₄×C₇₁	1	2	yes	—
105	#120	C₄×C₇₁	1	2	yes	—
106	C₄×D₄	C₂²×C₇₁	1	4	yes	—
107	#117	C₂²×C₇₁	1	1	yes	characteristic
108	C₄⋊C₈	C₂²×C₇₁	1	4	yes	complemented
109	C₂²⋊C₈	C₂²×C₇₁	1	4	yes	—
110	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
111	C₂²⋊C₈	C₂²×C₇₁	1	2	yes	—
112	C₂²⋊C₈	C₂²×C₇₁	1	4	yes	complemented
113	#97	C₂²×C₇₁	1	2	yes	complemented
114	C₄×C₈	C₂²×C₇₁	1	2	yes	complemented
115	C₂⁴	C₂×C₄×C₇₁	1	1	yes	characteristic
116	#41	—	4	2	no	—
117	C₂×Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
118	C₂×D₄	C₂×C₄×C₇₁	1	4	yes	—
119	C₂×D₄	C₂×C₄×C₇₁	1	2	yes	—
120	C₂×D₄	C₂×C₄×C₇₁	1	1	yes	characteristic
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	C₂×C₄×C₇₁	1	2	yes	—
123	C₂²×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₇₁	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
126	C₂²×C₄	C₂³×C₇₁	1	2	yes	—
127	M₁₆	—	4	2	no	—
128	C₂×C₈	—	4	4	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	C₂³×C₇₁	1	4	yes	—
131	C₂×C₈	C₂³×C₇₁	1	4	yes	—
132	C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
133	C₄⋊C₄	C₂×C₄×C₇₁	1	4	yes	—
134	C₂²⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
135	C₂²⋊C₄	C₂×C₄×C₇₁	1	4	yes	—
136	C₂²⋊C₄	C₂×C₄×C₇₁	1	4	yes	—
137	C₄²	C₂³×C₇₁	1	2	yes	—
138	C₂³	—	2	4	no	—
139	C₂³	—	4	1	no	—
140	C₂³	C₂²×C₄×C₇₁	1	1	yes	characteristic
141	C₂³	C₂²×C₄×C₇₁	1	2	yes	—
142	C₂³	C₂²×C₄×C₇₁	1	2	yes	—
143	Q₈	—	4	1	no	—
144	D₄	—	2	8	no	—
145	D₄	—	4	1	no	—
146	D₄	—	4	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂²×C₄×C₇₁	1	4	yes	—
154	C₂×C₄	C₂²×C₄×C₇₁	1	2	yes	—
155	C₂×C₄	C₂²×C₄×C₇₁	1	2	yes	—
156	C₂×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
157	C₂×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	4	4	no	—
167	C₂²	#131×C₇₁	1	4	yes	—
168	C₂²	C₂³×C₄×C₇₁	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	2	no	—
172	C₄	—	2	4	no	—
173	C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
174	C₂	—	2	4	no	—
175	C₂	—	4	1	no	—
176	C₂	—	4	2	no	—
177	C₂	#518×C₇₁	1	1	yes	characteristic
178	C₂	C₂×#131×C₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₇₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₇₁
socle	C₂²×C₇₁	C₂³×C₄
prime radical	#525×C₇₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #435×C₇₁	C₂	semidirect
3. C₈×D₄×C₇₁	C₂	semidirect
4. #433×C₇₁	C₂	semidirect
5. #408×C₇₁	C₂	semidirect
6. C₂×C₂²⋊C₈×C₇₁	C₂	semidirect
9. C₂×M₁₆×C₇₁	C₂²	semidirect
11. C₂²×C₈×C₇₁	C₂²	semidirect
18. C₄⋊C₈×C₇₁	C₂²	semidirect
22. C₂²⋊C₈×C₇₁	C₂²	semidirect
23. #97×C₇₁	C₂²	semidirect
24. C₄×C₈×C₇₁	C₂²	semidirect
89. #2977	C₇₁	direct (N × H)
90. C₇₁	#2977	direct (N × H)
92. #435	C₂×C₇₁	semidirect
93. C₈×D₄	C₂×C₇₁	semidirect
94. #433	C₂×C₇₁	semidirect
95. #408	C₂×C₇₁	semidirect
96. C₂×C₂²⋊C₈	C₂×C₇₁	semidirect
99. C₂×M₁₆	C₂²×C₇₁	semidirect
101. C₂²×C₈	C₂²×C₇₁	semidirect
108. C₄⋊C₈	C₂²×C₇₁	semidirect
112. C₂²⋊C₈	C₂²×C₇₁	semidirect
113. #97	C₂²×C₇₁	semidirect
114. C₄×C₈	C₂²×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
#525×C₇₁	2
#435×C₇₁	2
C₈×D₄×C₇₁	2
#433×C₇₁	2
#408×C₇₁	2
C₂×C₂²⋊C₈×C₇₁	2
#2977	71

Subgroups — C71×#2977 #22585539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#2977 #22585539