Group #23773815 — subgroups

Subgroups — C₂×#19098942 #23773815

Order 10112.

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

normal subgroups only

This group has 552 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	C₂²×#11929614	C₂	1	4	yes	maximal
2	C₂×#14579161	C₂	1	1	yes	maximal, characteristic
3	#19098942	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₈	C₂²	1	2	yes	—
7	C₂×#11929614	—	2	16	no	—
8	C₂×#11929614	C₂²	1	16	yes	—
9	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
10	#14579161	—	2	2	no	—
11	#14579161	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₇₉	—	2	2	no	—
13	C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
14	C₂²×Q₈×C₇₉	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₇₉	—	2	2	no	—
16	C₄⋊Q₈×C₇₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
18	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₅₈	—	2	8	no	—
20	C₂×Dic₁₅₈	—	2	2	no	—
21	C₂×Dic₁₅₈	C₂³	1	4	yes	—
22	C₂²×Dic₇₉	—	2	2	no	—
23	#11929614	—	2	32	no	—
24	C₂×C₇₉⋊C₈	D₄	1	8	yes	—
25	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₇₉	—	2	8	no	—
27	C₂×Q₈×C₇₉	—	2	2	no	—
28	C₂×Q₈×C₇₉	—	2	8	no	—
29	C₂×Q₈×C₇₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₇₉	—	2	2	no	—
31	C₂²×C₄×C₇₉	D₄	1	2	yes	—
32	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₇₉	—	2	8	no	—
34	C₄²×C₇₉	D₄	1	2	yes	—
35	C₄²×C₇₉	C₂³	1	2	yes	—
36	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₇₉	—	2	8	no	—
38	Q₈×C₇₉	—	4	4	no	—
39	C₂×C₄×C₇₉	—	2	4	no	—
40	C₂×C₄×C₇₉	—	4	4	no	—
41	C₂×C₄×C₇₉	Q₁₆	1	8	yes	complemented
42	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₇₉	—	2	4	no	—
47	C₂×Dic₇₉	—	4	4	no	—
48	Dic₁₅₈	—	2	8	no	—
49	Dic₁₅₈	—	4	4	no	—
50	C₇₉⋊C₈	C₂×D₄	1	8	yes	—
51	C₂²×C₇₉	#127	1	2	yes	—
52	C₂²×C₇₉	C₂×Q₁₆	1	4	yes	—
53	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
54	C₄×C₇₉	—	4	4	no	—
55	C₄×C₇₉	#127	1	2	yes	—
56	C₄×C₇₉	C₂×Q₁₆	1	8	yes	complemented
57	C₄×C₇₉	C₂²×D₄	1	2	yes	—
58	Dic₇₉	—	4	4	no	—
59	C₂×C₇₉	#494	1	4	yes	complemented
60	C₂×C₇₉	C₂×#127	1	1	yes	characteristic
61	C₂×C₇₉	C₂²×Q₁₆	1	2	yes	—
62	C₂×#494	—	79	1	no	maximal, Sylow
63	C₇₉	C₂×#494	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	—	79	4	no	—
65	C₂×C₄⋊Q₈	—	79	1	no	—
66	C₂×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
67	#494	—	79	8	no	—
68	C₂×C₄×C₈	—	79	1	no	—
69	C₂²×Q₈	—	79	2	no	—
70	C₂²×Q₈	C₂×D₇₉	1	2	yes	—
71	C₂×Q₁₆	—	79	16	no	—
72	C₂×Q₁₆	—	158	16	no	—
73	C₂²×C₈	—	79	2	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	79	4	no	—
76	C₄⋊Q₈	—	158	2	no	—
77	C₄⋊Q₈	C₂×D₇₉	1	4	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄⋊C₄	—	158	2	no	—
80	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
81	C₄×C₈	—	79	4	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	8	no	—
84	C₂×Q₈	—	79	4	no	—
85	C₂×Q₈	—	158	8	no	—
86	C₂×Q₈	—	158	2	no	—
87	C₂×Q₈	C₂²×D₇₉	1	4	yes	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	158	2	no	—
90	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
91	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
92	Q₁₆	—	158	32	no	—
93	C₂×C₈	—	79	8	no	—
94	C₂×C₈	—	79	4	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	158	8	no	—
97	C₄²	C₂²⋊D₇₉	1	2	yes	—
98	C₄²	C₂²×D₇₉	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
100	Q₈	—	2	8	no	—
101	Q₈	—	4	4	no	—
102	Q₈	—	316	4	no	—
103	Q₈	—	158	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	316	4	no	—
107	C₂×C₄	—	158	4	no	—
108	C₂×C₄	#11929614	1	8	yes	complemented
109	C₂×C₄	D₄×D₇₉	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₇₉	1	4	yes	—
112	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
113	C₈	—	79	8	no	—
114	C₂²	#14579157	1	2	yes	—
115	C₂²	C₂×#11929614	1	4	yes	—
116	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
117	C₄	—	4	4	no	—
118	C₄	—	316	4	no	—
119	C₄	#14579157	1	2	yes	—
120	C₄	C₂×#11929614	1	8	yes	complemented
121	C₄	C₂×D₄×D₇₉	1	2	yes	—
122	C₂	#19098942	1	4	yes	direct product factor
123	C₂	C₂×#14579157	1	1	yes	characteristic
124	C₂	C₂²×#11929614	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11929614

C₂

yes

maximal

C₂×#14579161

C₂

yes

maximal, characteristic

#19098942

C₂

yes

maximal, direct product factor

C₂×C₄×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

C₂×#11929614

—

C₂×#11929614

C₂²

yes

—

C₂²×C₇₉⋊C₈

C₂²

yes

—

#14579161

—

#14579161

C₂²

yes

—

C₂×C₄⋊Dic₇₉

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂²×Q₈×C₇₉

C₂²

yes

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

#11929614

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

Q₁₆

yes

complemented

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₇₉⋊C₈

C₂×D₄

yes

—

C₂²×C₇₉

#127

yes

—

C₂²×C₇₉

C₂×Q₁₆

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#127

yes

—

C₄×C₇₉

C₂×Q₁₆

yes

complemented

C₄×C₇₉

C₂²×D₄

yes

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

C₂²×Q₁₆

yes

—

C₂×#494

—

maximal, Sylow

C₇₉

C₂×#494

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₇₉

yes

characteristic

#494

—

C₂×C₄×C₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₉

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

158

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂×Q₈

C₂²×D₇₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²⋊D₇₉

yes

—

C₂²×C₄

C₂²×D₇₉

yes

characteristic

Q₁₆

—

158

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

158

—

C₄²

C₂²⋊D₇₉

yes

—

C₄²

C₂²×D₇₉

yes

—

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

316

—

103

Q₈

—

158

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

316

—

107

C₂×C₄

—

158

—

108

C₂×C₄

#11929614

yes

complemented

109

C₂×C₄

D₄×D₇₉

yes

—

110

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

112

C₂×C₄

C₂³×D₇₉

yes

characteristic

113

C₈

—

114

C₂²

#14579157

yes

—

115

C₂²

C₂×#11929614

yes

—

116

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

117

C₄

—

118

C₄

—

316

—

119

C₄

#14579157

yes

—

120

C₄

C₂×#11929614

yes

complemented

121

C₄

C₂×D₄×D₇₉

yes

—

122

C₂

#19098942

yes

direct product factor

123

C₂

C₂×#14579157

yes

characteristic

124

C₂

C₂²×#11929614

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₂×C₄⋊Q₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	C₂×#494
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₂×C₄⋊Q₈×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₇₉

hypocenter

C₇₉

C₂×#494

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #19098942	C₂	direct (N × H)
41. C₂×C₄×C₇₉	Q₁₆	semidirect
56. C₄×C₇₉	C₂×Q₁₆	semidirect
59. C₂×C₇₉	#494	semidirect
63. C₇₉	C₂×#494	semidirect
108. C₂×C₄	#11929614	semidirect
120. C₄	C₂×#11929614	semidirect
122. C₂	#19098942	direct (N × H)

H ≅ G/N

type

3. #19098942

C₂

direct (N × H)

41. C₂×C₄×C₇₉

Q₁₆

semidirect

56. C₄×C₇₉

C₂×Q₁₆

semidirect

59. C₂×C₇₉

#494

semidirect

63. C₇₉

C₂×#494

semidirect

108. C₂×C₄

#11929614

semidirect

120. C₄

C₂×#11929614

semidirect

122. C₂

#19098942

direct (N × H)

Subgroups — C₂×#19098942 #23773815

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098942 #23773815

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098942 #23773815