Subgroups — C₂×#22590634 #32155561

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 118 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590634	C₂	1	8	yes	maximal, direct product factor
2	C₂×#17985833	C₂	1	3	yes	maximal, complemented
3	C₂²×C₇₁⋊C₃₂	C₂	1	3	yes	maximal, complemented
4	C₂×#504×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17985833	C₂²	1	12	yes	complemented
6	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₇₁⋊C₃₂	C₂²	1	3	yes	—
8	C₂×C₇₁⋊C₃₂	C₂²	1	12	yes	complemented
9	C₂×#131×C₇₁	C₄	1	1	yes	characteristic
10	#504×C₇₁	C₄	1	4	yes	—
11	#504×C₇₁	C₂²	1	4	yes	—
12	C₂×M₃₂×C₇₁	C₄	1	3	yes	—
13	C₂²×C₁₆×C₇₁	C₂²	1	3	yes	—
14	C₇₁⋊C₃₂	C₂³	1	2	yes	—
15	C₇₁⋊C₃₂	C₂³	1	6	yes	—
16	C₂×#41×C₇₁	C₈	1	1	yes	characteristic
17	#131×C₇₁	C₈	1	4	yes	—
18	#131×C₇₁	C₂×C₄	1	4	yes	—
19	C₂×M₁₆×C₇₁	C₈	1	3	yes	—
20	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
21	M₃₂×C₇₁	C₂×C₄	1	12	yes	—
22	C₂×C₁₆×C₇₁	C₂×C₄	1	3	yes	—
23	C₂×C₁₆×C₇₁	C₂×C₄	1	6	yes	—
24	C₂×C₁₆×C₇₁	C₂³	1	6	yes	—
25	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
26	#41×C₇₁	C₁₆	1	4	yes	—
27	#41×C₇₁	C₂×C₈	1	4	yes	—
28	C₂×Q₈×C₇₁	C₁₆	1	1	yes	characteristic
29	C₂×D₄×C₇₁	C₁₆	1	3	yes	—
30	C₂²×C₄×C₇₁	C₂×C₈	1	3	yes	—
31	M₁₆×C₇₁	C₂×C₈	1	12	yes	—
32	C₂×C₈×C₇₁	C₂×C₈	1	3	yes	—
33	C₂×C₈×C₇₁	C₂×C₈	1	6	yes	—
34	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₂²×C₄	1	6	yes	—
36	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₁₆×C₇₁	C₂²×C₄	1	6	yes	—
38	C₁₆×C₇₁	C₂⁴	1	1	yes	characteristic
39	C₂³×C₇₁	C₂×C₁₆	1	3	yes	—
40	Q₈×C₇₁	C₂×C₁₆	1	4	yes	—
41	D₄×C₇₁	C₂×C₁₆	1	12	yes	—
42	C₂×C₄×C₇₁	C₂×C₁₆	1	3	yes	—
43	C₂×C₄×C₇₁	C₂×C₁₆	1	6	yes	—
44	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂²×C₈	1	6	yes	—
46	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₇₁	C₂²×C₈	1	6	yes	—
48	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
49	C₂²×C₇₁	—	2	6	no	—
50	C₂²×C₇₁	C₂²×C₁₆	1	6	yes	—
51	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
52	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
53	C₄×C₇₁	C₂²×C₁₆	1	6	yes	—
54	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
55	C₂×#2258	—	71	1	no	maximal, Sylow
56	C₂×C₇₁	—	2	6	no	—
57	C₂×C₇₁	#2258	1	2	yes	complemented
58	C₂×C₇₁	C₂³×C₁₆	1	1	yes	characteristic
59	C₂×#504	D₇₁	1	1	yes	characteristic
60	#2258	—	71	8	no	—
61	C₂×M₆₄	—	71	3	no	—
62	C₂²×C₃₂	—	71	3	no	—
63	C₇₁	C₂×#2258	1	1	yes	characteristic, complemented, Sylow
64	C₂×#131	Dic₇₁	1	1	yes	characteristic
65	#504	Dic₇₁	1	4	yes	—
66	#504	C₂×D₇₁	1	4	yes	—
67	C₂×M₃₂	Dic₇₁	1	3	yes	—
68	C₂²×C₁₆	C₂×D₇₁	1	3	yes	—
69	M₆₄	—	71	12	no	—
70	C₂×C₃₂	—	71	12	no	—
71	C₂×C₃₂	—	71	1	no	—
72	C₂×C₃₂	—	71	3	no	—
73	C₂×#41	C₇₁⋊C₈	1	1	yes	characteristic
74	#131	C₇₁⋊C₈	1	4	yes	—
75	#131	C₂×Dic₇₁	1	4	yes	—
76	C₂×M₁₆	C₇₁⋊C₈	1	3	yes	—
77	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
78	M₃₂	C₂×Dic₇₁	1	12	yes	—
79	C₂×C₁₆	C₂×Dic₇₁	1	3	yes	—
80	C₂×C₁₆	C₂×Dic₇₁	1	6	yes	—
81	C₂×C₁₆	C₂²×D₇₁	1	6	yes	—
82	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
83	C₃₂	—	71	2	no	—
84	C₃₂	—	71	6	no	—
85	#41	C₇₁⋊C₁₆	1	4	yes	—
86	#41	C₂×C₇₁⋊C₈	1	4	yes	—
87	C₂×Q₈	C₇₁⋊C₁₆	1	1	yes	characteristic
88	C₂×D₄	C₇₁⋊C₁₆	1	3	yes	—
89	C₂²×C₄	C₂×C₇₁⋊C₈	1	3	yes	—
90	M₁₆	C₂×C₇₁⋊C₈	1	12	yes	—
91	C₂×C₈	C₂×C₇₁⋊C₈	1	3	yes	—
92	C₂×C₈	C₂×C₇₁⋊C₈	1	6	yes	—
93	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
94	C₂×C₈	C₂²×Dic₇₁	1	6	yes	—
95	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
96	C₁₆	C₂²×Dic₇₁	1	6	yes	—
97	C₁₆	C₂³×D₇₁	1	1	yes	characteristic
98	C₂³	C₂×C₇₁⋊C₁₆	1	3	yes	—
99	Q₈	C₂×C₇₁⋊C₁₆	1	4	yes	—
100	D₄	C₂×C₇₁⋊C₁₆	1	12	yes	—
101	C₂×C₄	C₂×C₇₁⋊C₁₆	1	3	yes	—
102	C₂×C₄	C₂×C₇₁⋊C₁₆	1	6	yes	—
103	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₇₁⋊C₈	1	6	yes	—
105	C₈	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
106	C₈	C₂²×C₇₁⋊C₈	1	6	yes	—
107	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
108	C₂²	—	2	6	no	—
109	C₂²	C₂²×C₇₁⋊C₁₆	1	6	yes	—
110	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
111	C₄	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
112	C₄	C₂²×C₇₁⋊C₁₆	1	6	yes	—
113	C₄	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
114	C₂	—	2	6	no	—
115	C₂	#22590634	1	2	yes	direct product factor
116	C₂	C₂³×C₇₁⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇₁
commutator	C₂×C₇₁	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₇₁
Fitting	C₂×#504×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₁₆
prime radical	C₂×#41×C₇₁	C₈
cosocle	C₁₆×C₇₁	C₂⁴
hypercenter	C₂×#504	D₇₁
hypocenter	C₇₁	C₂×#2258
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. #22590634	C₂	direct (N × H)
2. C₂×#17985833	C₂	semidirect
3. C₂²×C₇₁⋊C₃₂	C₂	semidirect
5. #17985833	C₂²	semidirect
8. C₂×C₇₁⋊C₃₂	C₂²	semidirect
57. C₂×C₇₁	#2258	semidirect
63. C₇₁	C₂×#2258	semidirect
115. C₂	#22590634	direct (N × H)

Maximal subgroups

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

subgroup	index
#22590634	2
C₂×#17985833	2
C₂²×C₇₁⋊C₃₂	2
C₂×#504×C₇₁	2
C₂×#2258	71

Subgroups — C2×#22590634 #32155561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590634 #32155561