Subgroups — C₂×C₈×D₄×C₇₁ #22585488

Order 9088.

616 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₈×C₇₁	C₂²
commutator	C₂	C₂³×C₈×C₇₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₇₁
socle	C₂³×C₇₁	C₂²×C₄
prime radical	C₂²×D₄×C₇₁	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×C₇₁	C₂	semidirect
3. C₈×D₄×C₇₁	C₂	direct (N × H)
4. C₂×C₄⋊C₈×C₇₁	C₂	semidirect
5. C₂×C₂²⋊C₈×C₇₁	C₂	semidirect
6. C₂×C₄×C₈×C₇₁	C₂	semidirect
13. C₂²×C₈×C₇₁	C₂²	semidirect
19. C₄⋊C₈×C₇₁	C₂²	semidirect
20. C₂²⋊C₈×C₇₁	C₂²	semidirect
21. C₄×C₈×C₇₁	C₂²	semidirect
23. C₂×D₄×C₇₁	C₈	direct (N × H)
36. C₂×C₈×C₇₁	D₄	direct (N × H)
46. C₂³×C₇₁	C₂×C₈	semidirect
49. D₄×C₇₁	C₂×C₈	direct (N × H)
55. C₂×C₄×C₇₁	C₂×C₈	semidirect
65. C₈×C₇₁	C₂×D₄	direct (N × H)
71. C₂²×C₇₁	C₂²×C₈	semidirect
77. C₄×C₇₁	C₂²×C₈	semidirect
81. C₂×C₇₁	C₈×D₄	direct (N × H)
85. C₂×C₈×D₄	C₇₁	direct (N × H)
86. C₇₁	C₂×C₈×D₄	direct (N × H)
87. C₂³×C₈	C₂×C₇₁	semidirect
89. C₈×D₄	C₂×C₇₁	direct (N × H)
90. C₂×C₄⋊C₈	C₂×C₇₁	semidirect
91. C₂×C₂²⋊C₈	C₂×C₇₁	semidirect
92. C₂×C₄×C₈	C₂×C₇₁	semidirect
99. C₂²×C₈	C₂²×C₇₁	semidirect
105. C₄⋊C₈	C₂²×C₇₁	semidirect
106. C₂²⋊C₈	C₂²×C₇₁	semidirect
107. C₄×C₈	C₂²×C₇₁	semidirect
109. C₂×D₄	C₈×C₇₁	direct (N × H)
122. C₂×C₈	D₄×C₇₁	direct (N × H)
132. C₂³	C₂×C₈×C₇₁	semidirect
135. D₄	C₂×C₈×C₇₁	direct (N × H)
141. C₂×C₄	C₂×C₈×C₇₁	semidirect
151. C₈	C₂×D₄×C₇₁	direct (N × H)
157. C₂²	C₂²×C₈×C₇₁	semidirect
163. C₄	C₂²×C₈×C₇₁	semidirect
167. C₂	C₈×D₄×C₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₇₁	2
C₂×C₄×D₄×C₇₁	2
C₈×D₄×C₇₁	2
C₂×C₄⋊C₈×C₇₁	2
C₂×C₂²⋊C₈×C₇₁	2
C₂×C₄×C₈×C₇₁	2
C₂×C₈×D₄	71

Subgroups — C2×C8×D4×C71 #22585488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×D₄×C₇₁ #22585488