Group #23776652 — subgroups

Subgroups — C₂²×#14579152 #23776652

Order 10112.

1192 conjugacy classes of subgroups; only the 645 normal subgroups are shown.

normal subgroups only

This group has 1192 conjugacy classes of subgroups — too many to draw. Only its 645 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14579152	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊Dic₇₉	C₂	1	2	yes	maximal
3	C₂²×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₇₉⋊C₄	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₇₉	C₂²	1	1	yes	characteristic
8	C₂³×Dic₇₉	C₂²	1	1	yes	characteristic
9	C₂³×Dic₇₉	C₂²	1	2	yes	—
10	#14579152	C₂²	1	64	yes	direct product factor
11	C₂×C₄⋊Dic₇₉	C₂²	1	24	yes	—
12	C₂×C₄⋊Dic₇₉	C₂²	1	12	yes	—
13	C₂×Dic₇₉⋊C₄	C₂²	1	24	yes	—
14	C₂×C₄×Dic₇₉	C₂²	1	12	yes	—
15	C₂³×C₄×C₇₉	C₂²	1	2	yes	—
16	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₇₉	C₂²	1	12	yes	—
18	C₂²×Dic₇₉	—	2	4	no	—
19	C₂²×Dic₇₉	—	2	4	no	—
20	C₂²×Dic₇₉	—	2	8	no	—
21	C₂²×Dic₇₉	C₂³	1	6	yes	—
22	C₂²×Dic₇₉	C₂³	1	6	yes	—
23	C₂²×Dic₇₉	C₂³	1	12	yes	—
24	C₄⋊Dic₇₉	C₂³	1	32	yes	—
25	C₄⋊Dic₇₉	C₂³	1	16	yes	—
26	Dic₇₉⋊C₄	C₂³	1	32	yes	—
27	C₄×Dic₇₉	C₂³	1	16	yes	—
28	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	—	2	8	no	—
30	C₂²×C₄×C₇₉	Q₈	1	8	yes	—
31	C₂²×C₄×C₇₉	C₂³	1	12	yes	—
32	C₂²×C₄×C₇₉	C₂³	1	6	yes	—
33	C₄⋊C₄×C₇₉	C₂³	1	16	yes	—
34	C₂³×C₇₉	C₂×Q₈	1	4	yes	—
35	C₂³×C₇₉	#41	1	4	yes	—
36	C₂³×C₇₉	#41	1	4	yes	—
37	C₂³×C₇₉	C₂⁴	1	3	yes	—
38	C₂×C₄×C₇₉	—	2	24	no	—
39	C₂×C₄×C₇₉	C₂×Q₈	1	24	yes	—
40	C₂×C₄×C₇₉	C₂⁴	1	8	yes	—
41	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
42	C₂×Dic₇₉	—	2	12	no	—
43	C₂×Dic₇₉	—	2	12	no	—
44	C₂×Dic₇₉	—	2	24	no	—
45	C₂×Dic₇₉	C₂⁴	1	4	yes	—
46	C₂×Dic₇₉	C₂⁴	1	4	yes	—
47	C₂×Dic₇₉	C₂⁴	1	8	yes	—
48	C₂²×C₇₉	#125	1	16	yes	complemented
49	C₂²×C₇₉	C₂²×Q₈	1	6	yes	—
50	C₂²×C₇₉	C₂×#41	1	6	yes	—
51	C₂²×C₇₉	C₂×#41	1	6	yes	—
52	C₂²×C₇₉	C₂⁵	1	1	yes	characteristic
53	C₄×C₇₉	—	2	8	no	—
54	C₄×C₇₉	C₂²×Q₈	1	8	yes	—
55	Dic₇₉	—	2	4	no	—
56	Dic₇₉	—	2	4	no	—
57	Dic₇₉	—	2	8	no	—
58	C₂×C₇₉	C₂×#125	1	12	yes	complemented
59	C₂×C₇₉	C₂³×Q₈	1	1	yes	characteristic
60	C₂×C₇₉	C₂²×#41	1	1	yes	characteristic
61	C₂×C₇₉	C₂²×#41	1	1	yes	characteristic
62	C₂²×#125	—	79	1	no	maximal, Sylow
63	C₇₉	C₂²×#125	1	1	yes	characteristic, complemented, Sylow
64	C₂×#125	—	79	24	no	—
65	C₂²×C₄⋊C₄	—	79	2	no	—
66	C₂²×C₄⋊C₄	—	79	1	no	—
67	C₂²×C₄⋊C₄	—	79	2	no	—
68	C₂²×C₄⋊C₄	D₇₉	1	1	yes	characteristic
69	C₂²×C₄²	—	79	1	no	—
70	C₂³×C₄	—	79	2	no	—
71	C₂³×C₄	—	79	1	no	—
72	C₂³×C₄	—	79	1	no	—
73	C₂³×C₄	C₂×D₇₉	1	2	yes	—
74	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
75	#125	—	79	64	no	—
76	C₂×C₄⋊C₄	—	79	24	no	—
77	C₂×C₄⋊C₄	—	79	24	no	—
78	C₂×C₄⋊C₄	—	79	12	no	—
79	C₂×C₄⋊C₄	C₂×D₇₉	1	12	yes	—
80	C₂×C₄²	—	79	12	no	—
81	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
82	C₂²×C₄	—	2	8	no	—
83	C₂²×C₄	—	79	6	no	—
84	C₂²×C₄	—	79	6	no	—
85	C₂²×C₄	—	79	12	no	—
86	C₂²×C₄	—	158	4	no	—
87	C₂²×C₄	—	158	8	no	—
88	C₂²×C₄	—	158	4	no	—
89	C₂²×C₄	Dic₁₅₈	1	8	yes	—
90	C₂²×C₄	C₂²×D₇₉	1	12	yes	—
91	C₂²×C₄	C₂²×D₇₉	1	6	yes	—
92	C₄⋊C₄	—	79	32	no	—
93	C₄⋊C₄	—	79	32	no	—
94	C₄⋊C₄	—	79	16	no	—
95	C₄⋊C₄	C₂²×D₇₉	1	16	yes	—
96	C₄²	—	79	16	no	—
97	C₂³	C₂×Dic₁₅₈	1	4	yes	—
98	C₂³	#11929623	1	4	yes	—
99	C₂³	#11929625	1	4	yes	—
100	C₂³	C₂³×D₇₉	1	3	yes	—
101	C₂×C₄	—	2	24	no	—
102	C₂×C₄	—	79	4	no	—
103	C₂×C₄	—	79	4	no	—
104	C₂×C₄	—	79	8	no	—
105	C₂×C₄	—	158	12	no	—
106	C₂×C₄	—	158	12	no	—
107	C₂×C₄	—	158	24	no	—
108	C₂×C₄	C₂×Dic₁₅₈	1	24	yes	—
109	C₂×C₄	C₂³×D₇₉	1	8	yes	—
110	C₂×C₄	C₂³×D₇₉	1	4	yes	—
111	C₂²	#14579152	1	16	yes	direct product factor
112	C₂²	C₂²×Dic₁₅₈	1	6	yes	—
113	C₂²	C₂×#11929623	1	6	yes	—
114	C₂²	C₂×#11929625	1	6	yes	—
115	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
116	C₄	—	2	8	no	—
117	C₄	—	158	4	no	—
118	C₄	—	158	4	no	—
119	C₄	—	158	8	no	—
120	C₄	C₂²×Dic₁₅₈	1	8	yes	—
121	C₂	C₂×#14579152	1	12	yes	direct product factor
122	C₂	C₂³×Dic₁₅₈	1	1	yes	characteristic
123	C₂	C₂²×#11929623	1	1	yes	characteristic
124	C₂	C₂²×#11929625	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579152

C₂

yes

maximal, direct product factor

C₂²×C₄⋊Dic₇₉

C₂

yes

maximal

C₂²×C₄⋊Dic₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₇₉⋊C₄

C₂

yes

maximal

C₂²×C₄×Dic₇₉

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₄×C₇₉

C₂

yes

maximal, characteristic

C₂³×Dic₇₉

C₂²

yes

characteristic

C₂³×Dic₇₉

C₂²

yes

characteristic

C₂³×Dic₇₉

C₂²

yes

—

#14579152

C₂²

yes

direct product factor

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×C₄⋊Dic₇₉

C₂²

yes

—

C₂×Dic₇₉⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₂³×C₄×C₇₉

C₂²

yes

—

C₂³×C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₂²×Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

C₂³

yes

—

C₄×Dic₇₉

C₂³

yes

—

C₂⁴×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

Q₈

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×Q₈

yes

—

C₂³×C₇₉

#41

yes

—

C₂³×C₇₉

#41

yes

—

C₂³×C₇₉

C₂⁴

yes

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×Q₈

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂⁴

yes

—

C₂×Dic₇₉

C₂⁴

yes

—

C₂×Dic₇₉

C₂⁴

yes

—

C₂²×C₇₉

#125

yes

complemented

C₂²×C₇₉

C₂²×Q₈

yes

—

C₂²×C₇₉

C₂×#41

yes

—

C₂²×C₇₉

C₂×#41

yes

—

C₂²×C₇₉

C₂⁵

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₂²×Q₈

yes

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

C₂×#125

yes

complemented

C₂×C₇₉

C₂³×Q₈

yes

characteristic

C₂×C₇₉

C₂²×#41

yes

characteristic

C₂×C₇₉

C₂²×#41

yes

characteristic

C₂²×#125

—

maximal, Sylow

C₇₉

C₂²×#125

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

D₇₉

yes

characteristic

C₂²×C₄²

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₇₉

yes

—

C₂³×C₄

C₂×D₇₉

yes

characteristic

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

—

C₂⁴

C₂²×D₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

—

158

—

C₂²×C₄

—

158

—

C₂²×C₄

Dic₁₅₈

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₇₉

yes

—

C₄²

—

C₂³

C₂×Dic₁₅₈

yes

—

C₂³

#11929623

yes

—

C₂³

#11929625

yes

—

100

C₂³

C₂³×D₇₉

yes

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

158

—

106

C₂×C₄

—

158

—

107

C₂×C₄

—

158

—

108

C₂×C₄

C₂×Dic₁₅₈

yes

—

109

C₂×C₄

C₂³×D₇₉

yes

—

110

C₂×C₄

C₂³×D₇₉

yes

—

111

C₂²

#14579152

yes

direct product factor

112

C₂²

C₂²×Dic₁₅₈

yes

—

113

C₂²

C₂×#11929623

yes

—

114

C₂²

C₂×#11929625

yes

—

115

C₂²

C₂⁴×D₇₉

yes

characteristic

116

C₄

—

117

C₄

—

158

—

118

C₄

—

158

—

119

C₄

—

158

—

120

C₄

C₂²×Dic₁₅₈

yes

—

121

C₂

C₂×#14579152

yes

direct product factor

122

C₂

C₂³×Dic₁₅₈

yes

characteristic

123

C₂

C₂²×#11929623

yes

characteristic

124

C₂

C₂²×#11929625

yes

characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁵
Frattini	C₂²	C₂⁴×D₇₉
Fitting	C₂²×C₄⋊C₄×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₇₉
hypocenter	C₇₉	C₂²×#125
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂⁴

C₂²×D₇₉

commutator, cosocle

C₂²×C₇₉

C₂⁵

Frattini

C₂²

C₂⁴×D₇₉

Fitting

C₂²×C₄⋊C₄×C₇₉

C₂

socle, prime radical

C₂⁴×C₇₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₇₉

hypocenter

C₇₉

C₂²×#125

LCS

G ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
1. C₂×#14579152	C₂	direct (N × H)
10. #14579152	C₂²	direct (N × H)
48. C₂²×C₇₉	#125	semidirect
58. C₂×C₇₉	C₂×#125	semidirect
63. C₇₉	C₂²×#125	semidirect
111. C₂²	#14579152	direct (N × H)
121. C₂	C₂×#14579152	direct (N × H)

H ≅ G/N

type

1. C₂×#14579152

C₂

direct (N × H)

10. #14579152

C₂²

direct (N × H)

48. C₂²×C₇₉

#125

semidirect

58. C₂×C₇₉

C₂×#125

semidirect

63. C₇₉

C₂²×#125

semidirect

111. C₂²

#14579152

direct (N × H)

121. C₂

C₂×#14579152

direct (N × H)

Subgroups — C₂²×#14579152 #23776652

Conjugacy classes of subgroups

Complements

Subgroups — C22×#14579152 #23776652

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14579152 #23776652