Group #23771758 — subgroups

Subgroups — #23771758

Order 10112.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14579158	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579150	C₂	1	4	yes	maximal, complemented
3	#19098464	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₁₆	C₂²	1	4	yes	—
7	C₂²×D₃₁₆	C₂²	1	2	yes	—
8	#14579158	—	2	4	no	—
9	#14579150	—	2	16	no	—
10	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	16	yes	—
11	C₂×C₄×Dic₇₉	C₂²	1	4	yes	—
12	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
13	C₂²×Q₈×C₇₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₇₉	—	2	4	no	—
15	C₂×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
16	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₂³×D₇₉	C₂³	1	4	yes	—
18	C₂×D₃₁₆	—	2	8	no	—
19	C₂×D₃₁₆	—	2	8	no	—
20	C₂×D₃₁₆	—	2	4	no	—
21	C₂×D₃₁₆	—	2	16	no	—
22	C₂²×Dic₇₉	C₂³	1	4	yes	—
23	(C₂×D₇₉)⋊C₄	—	2	32	no	—
24	C₄×Dic₇₉	—	2	8	no	—
25	C₄×Dic₇₉	D₄	1	8	yes	complemented
26	C₂×Q₈×C₇₉	—	2	4	no	—
27	C₂×Q₈×C₇₉	—	2	8	no	—
28	C₂²×C₄×C₇₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
31	C₄⋊C₄×C₇₉	—	2	8	no	—
32	C₄²×C₇₉	D₄	1	4	yes	—
33	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
34	C₂²×D₇₉	—	2	8	no	—
35	C₂²×D₇₉	—	2	4	no	—
36	C₂²×D₇₉	—	2	16	no	—
37	Q₈×C₇₉	—	4	8	no	—
38	C₂×C₄×C₇₉	—	2	8	no	—
39	C₂×C₄×C₇₉	—	2	4	no	—
40	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₇₉	#41	1	8	yes	—
44	C₂×Dic₇₉	—	2	8	no	—
45	C₂×Dic₇₉	C₂×D₄	1	8	yes	—
46	D₃₁₆	—	4	16	no	—
47	D₃₁₆	—	4	8	no	—
48	C₂²×C₇₉	C₂²×D₄	1	2	yes	—
49	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₇₉	C₂×#41	1	4	yes	—
51	C₂×D₇₉	—	4	8	no	—
52	C₂×D₇₉	—	4	4	no	—
53	C₂×D₇₉	—	4	16	no	—
54	C₄×C₇₉	—	4	4	no	—
55	C₄×C₇₉	#124	1	8	yes	complemented
56	C₄×C₇₉	#127	1	4	yes	—
57	Dic₇₉	—	2	8	no	—
58	C₂×C₇₉	C₂×#124	1	2	yes	—
59	C₂×C₇₉	C₂×#127	1	1	yes	characteristic
60	C₂×C₇₉	#532	1	4	yes	—
61	D₇₉	—	8	4	no	—
62	#2858	—	79	1	no	maximal, Sylow
63	C₇₉	#2858	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	D₇₉	1	1	yes	characteristic, complemented
65	C₂×#127	—	79	1	no	—
66	C₂×#124	—	79	4	no	—
67	#390	—	79	8	no	—
68	C₄³	—	79	1	no	—
69	C₂²×Q₈	C₂×D₇₉	1	2	yes	—
70	C₂²×D₄	—	79	2	no	—
71	C₂²×D₄	—	79	4	no	—
72	C₄⋊Q₈	—	2	4	no	—
73	#127	—	158	4	no	—
74	#124	—	158	16	no	—
75	C₂×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
76	C₂×C₂²⋊C₄	—	79	16	no	—
77	C₂×C₄²	—	79	2	no	—
78	C₂×C₄²	—	79	4	no	—
79	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
80	C₂⁴	—	79	4	no	—
81	C₂×Q₈	—	2	8	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×D₄	—	158	8	no	—
84	C₂×D₄	—	158	8	no	—
85	C₂×D₄	—	158	4	no	—
86	C₂×D₄	—	158	16	no	—
87	C₂²×C₄	—	79	4	no	—
88	C₂²×C₄	C₂²×D₇₉	1	4	yes	—
89	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
91	C₄⋊C₄	—	2	8	no	—
92	C₂²⋊C₄	—	158	32	no	—
93	C₄²	—	79	8	no	—
94	C₄²	—	158	8	no	—
95	C₄²	C₂²⋊D₇₉	1	4	yes	—
96	C₂³	—	158	16	no	—
97	C₂³	—	158	8	no	—
98	C₂³	—	158	4	no	—
99	C₂³	C₂³×D₇₉	1	1	yes	characteristic
100	Q₈	—	4	8	no	—
101	D₄	—	316	16	no	—
102	D₄	—	316	8	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	79	8	no	—
106	C₂×C₄	—	158	8	no	—
107	C₂×C₄	D₄×D₇₉	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₇₉	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
110	C₂×C₄	#11929625	1	8	yes	—
111	C₂²	—	316	8	no	—
112	C₂²	—	316	4	no	—
113	C₂²	—	316	16	no	—
114	C₂²	C₂×D₄×D₇₉	1	2	yes	—
115	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
116	C₂²	C₂×#11929625	1	4	yes	—
117	C₄	—	4	4	no	—
118	C₄	—	158	8	no	—
119	C₄	#14579150	1	8	yes	complemented
120	C₄	#14579157	1	4	yes	—
121	C₂	—	632	4	no	—
122	C₂	C₂×#14579150	1	2	yes	—
123	C₂	C₂×#14579157	1	1	yes	characteristic
124	C₂	#19099114	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579158

C₂

yes

maximal, characteristic

C₂×#14579150

C₂

yes

maximal, complemented

#19098464

C₂

yes

maximal, complemented

C₄²×Dic₇₉

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₁₆

C₂²

yes

—

C₂²×D₃₁₆

C₂²

yes

—

#14579158

—

#14579150

—

C₂×(C₂×D₇₉)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₂×C₄×Dic₇₉

C₂²

yes

—

C₂²×Q₈×C₇₉

C₂²

yes

—

C₄⋊Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂³×D₇₉

C₂³

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂²×Dic₇₉

C₂³

yes

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₄

yes

complemented

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₂³×C₇₉

C₂⁴

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

D₃₁₆

—

D₃₁₆

—

C₂²×C₇₉

C₂²×D₄

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂²×C₇₉

C₂×#41

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#124

yes

complemented

C₄×C₇₉

#127

yes

—

Dic₇₉

—

C₂×C₇₉

C₂×#124

yes

—

C₂×C₇₉

C₂×#127

yes

characteristic

C₂×C₇₉

#532

yes

—

D₇₉

—

#2858

—

maximal, Sylow

C₇₉

#2858

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₇₉

yes

characteristic, complemented

C₂×#127

—

C₂×#124

—

#390

—

C₄³

—

C₂²×Q₈

C₂×D₇₉

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₄⋊Q₈

—

#127

—

158

—

#124

—

158

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

C₂²×C₄

C₂²×D₇₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

158

—

C₄²

—

C₄²

—

158

—

C₄²

C₂²⋊D₇₉

yes

—

C₂³

—

158

—

C₂³

—

158

—

C₂³

—

158

—

C₂³

C₂³×D₇₉

yes

characteristic

100

Q₈

—

101

D₄

—

316

—

102

D₄

—

316

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

158

—

107

C₂×C₄

D₄×D₇₉

yes

—

108

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

109

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

110

C₂×C₄

#11929625

yes

—

111

C₂²

—

316

—

112

C₂²

—

316

—

113

C₂²

—

316

—

114

C₂²

C₂×D₄×D₇₉

yes

—

115

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

116

C₂²

C₂×#11929625

yes

—

117

C₄

—

118

C₄

—

158

—

119

C₄

#14579150

yes

complemented

120

C₄

#14579157

yes

—

121

C₂

—

632

—

122

C₂

C₂×#14579150

yes

—

123

C₂

C₂×#14579157

yes

characteristic

124

C₂

#19099114

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₇₉	C₂
prime radical	C₂×#14579158	C₂
hypercenter	C₂×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	#2858
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₉

commutator, socle, cosocle

C₂³×C₇₉

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₇₉

C₂

prime radical

C₂×#14579158

C₂

hypercenter

C₂×C₄⋊Q₈

D₇₉

C₇₉

G ▷ C₂³×C₇₉ ▷ C₇₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#14579150	C₂	semidirect
3. #19098464	C₂	semidirect
4. C₄²×Dic₇₉	C₂	semidirect
5. C₂×C₄⋊Q₈×C₇₉	C₂	semidirect
25. C₄×Dic₇₉	D₄	semidirect
55. C₄×C₇₉	#124	semidirect
63. C₇₉	#2858	semidirect
64. C₂×C₄⋊Q₈	D₇₉	semidirect
119. C₄	#14579150	semidirect

H ≅ G/N

type

2. C₂×#14579150

C₂

semidirect

3. #19098464

C₂

semidirect

4. C₄²×Dic₇₉

C₂

semidirect

5. C₂×C₄⋊Q₈×C₇₉

C₂

semidirect

25. C₄×Dic₇₉

D₄

semidirect

55. C₄×C₇₉

#124

semidirect

63. C₇₉

#2858

semidirect

64. C₂×C₄⋊Q₈

D₇₉

semidirect

119. C₄

#14579150

semidirect

Subgroups — #23771758

Conjugacy classes of subgroups

Complements