Group #23773712 — subgroups

Subgroups — C₂×#19098915 #23773712

Order 10112.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14579161	C₂	1	1	yes	maximal, characteristic
2	#19098915	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14579084	C₂	1	4	yes	maximal
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	C₂²×C₇₉⋊C₈	C₂²	1	2	yes	—
8	#14579161	—	2	2	no	—
9	#14579161	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₇₉	—	2	1	no	—
11	C₂×C₄⋊Dic₇₉	—	2	1	no	—
12	#14579084	C₂²	1	16	yes	—
13	C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
14	C₂²×Q₈×C₇₉	—	2	1	no	—
15	C₄⋊Q₈×C₇₉	—	2	2	no	—
16	C₄⋊Q₈×C₇₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
18	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₅₈	—	2	2	no	—
21	C₂×Dic₁₅₈	—	2	8	no	—
22	C₂×Dic₁₅₈	C₂³	1	4	yes	—
23	C₂²×Dic₇₉	—	2	2	no	—
24	C₂×C₇₉⋊C₈	—	2	4	no	—
25	C₂×C₇₉⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₇₉	—	2	4	no	—
27	C₄⋊Dic₇₉	—	2	4	no	—
28	C₂×Q₈×C₇₉	—	2	4	no	—
29	C₂×Q₈×C₇₉	—	4	4	no	—
30	C₂²×C₄×C₇₉	—	2	2	no	—
31	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₇₉	—	2	4	no	—
35	C₄⋊C₄×C₇₉	—	2	2	no	—
36	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
37	C₄²×C₇₉	D₄	1	2	yes	—
38	C₄²×C₇₉	C₂³	1	2	yes	—
39	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₇₉	—	4	8	no	—
41	C₂×C₄×C₇₉	—	2	4	no	—
42	C₂×C₄×C₇₉	—	4	4	no	—
43	C₂×C₄×C₇₉	SD₁₆	1	4	yes	—
44	C₂×C₄×C₇₉	Q₁₆	1	4	yes	—
45	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₇₉	#41	1	4	yes	—
49	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₇₉	—	2	4	no	—
51	C₂×Dic₇₉	—	4	4	no	—
52	Dic₁₅₈	—	2	8	no	—
53	Dic₁₅₈	—	4	4	no	—
54	C₇₉⋊C₈	—	2	4	no	—
55	C₂²×C₇₉	#124	1	2	yes	—
56	C₂²×C₇₉	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₇₉	C₂×Q₁₆	1	2	yes	—
58	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₇₉	—	4	4	no	—
60	C₄×C₇₉	#124	1	2	yes	—
61	C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
62	C₄×C₇₉	C₂×Q₁₆	1	4	yes	—
63	C₄×C₇₉	C₂×#41	1	2	yes	—
64	Dic₇₉	—	4	4	no	—
65	C₂×C₇₉	#487	1	4	yes	complemented
66	C₂×C₇₉	C₂×#124	1	1	yes	characteristic
67	C₂×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×#487	—	79	1	no	maximal, Sylow
70	C₇₉	C₂×#487	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	79	1	no	—
72	C₂×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
73	#487	—	79	8	no	—
74	C₂×#103	—	79	4	no	—
75	C₂×C₄×C₈	—	79	1	no	—
76	C₂²×Q₈	—	2	1	no	—
77	C₂²×Q₈	—	79	2	no	—
78	C₂²×C₈	—	79	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	79	4	no	—
81	C₄⋊Q₈	—	158	2	no	—
82	C₄⋊Q₈	C₂×D₇₉	1	4	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	158	1	no	—
85	C₂×C₄⋊C₄	—	158	1	no	—
86	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
87	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
88	#103	—	79	16	no	—
89	C₄×C₈	—	79	4	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×Q₈	—	4	4	no	—
92	C₂×Q₈	—	79	4	no	—
93	C₂×Q₈	—	158	8	no	—
94	C₂×Q₈	—	158	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	158	2	no	—
97	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
100	C₂×C₈	—	79	4	no	—
101	C₂×C₈	—	158	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	158	4	no	—
105	C₄⋊C₄	—	158	4	no	—
106	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
107	C₄²	C₂²⋊D₇₉	1	2	yes	—
108	C₄²	C₂²×D₇₉	1	2	yes	—
109	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
110	Q₈	—	4	8	no	—
111	Q₈	—	316	4	no	—
112	Q₈	—	158	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	316	4	no	—
116	C₂×C₄	—	158	4	no	—
117	C₂×C₄	#11929612	1	4	yes	—
118	C₂×C₄	#11929614	1	4	yes	—
119	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
122	C₂×C₄	#11929625	1	4	yes	—
123	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
124	C₈	—	158	4	no	—
125	C₂²	#14579150	1	2	yes	—
126	C₂²	C₂×#11929612	1	2	yes	—
127	C₂²	C₂×#11929614	1	2	yes	—
128	C₂²	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
129	C₄	—	4	4	no	—
130	C₄	—	316	4	no	—
131	C₄	#14579150	1	2	yes	—
132	C₄	C₂×#11929612	1	4	yes	—
133	C₄	C₂×#11929614	1	4	yes	—
134	C₄	C₂×#11929625	1	2	yes	—
135	C₂	#19098915	1	4	yes	direct product factor
136	C₂	C₂×#14579150	1	1	yes	characteristic
137	C₂	C₂²×#11929612	1	1	yes	characteristic
138	C₂	C₂²×#11929614	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579161

C₂

yes

maximal, characteristic

#19098915

C₂

yes

maximal, direct product factor

C₂×#14579084

C₂

yes

maximal

C₂×C₄×C₇₉⋊C₈

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

C₂²×C₇₉⋊C₈

C₂²

yes

—

#14579161

—

#14579161

C₂²

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579084

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

—

C₂²×Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

SD₁₆

yes

—

C₂×C₄×C₇₉

Q₁₆

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

#41

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

#124

yes

—

C₂²×C₇₉

C₂×SD₁₆

yes

—

C₂²×C₇₉

C₂×Q₁₆

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

#124

yes

—

C₄×C₇₉

C₂×SD₁₆

yes

—

C₄×C₇₉

C₂×Q₁₆

yes

—

C₄×C₇₉

C₂×#41

yes

—

Dic₇₉

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

C₂²×SD₁₆

yes

characteristic

C₂×C₇₉

C₂²×Q₁₆

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₇₉

yes

characteristic

#487

—

C₂×#103

—

C₂×C₄×C₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

158

—

C₄⋊Q₈

C₂×D₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

—

158

—

C₂×C₄⋊C₄

C₂×D₇₉

yes

—

C₂×C₄²

C₂×D₇₉

yes

characteristic

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

158

—

C₂×Q₈

—

158

—

C₂²×C₄

—

C₂²×C₄

—

158

—

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

characteristic

C₂²×C₄

C₂²×D₇₉

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

158

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

158

—

105

C₄⋊C₄

—

158

—

106

C₄⋊C₄

C₂²×D₇₉

yes

—

107

C₄²

C₂²⋊D₇₉

yes

—

108

C₄²

C₂²×D₇₉

yes

—

109

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

110

Q₈

—

111

Q₈

—

316

—

112

Q₈

—

158

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

316

—

116

C₂×C₄

—

158

—

117

C₂×C₄

#11929612

yes

—

118

C₂×C₄

#11929614

yes

—

119

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

121

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

122

C₂×C₄

#11929625

yes

—

123

C₂×C₄

C₂³×D₇₉

yes

characteristic

124

C₈

—

158

—

125

C₂²

#14579150

yes

—

126

C₂²

C₂×#11929612

yes

—

127

C₂²

C₂×#11929614

yes

—

128

C₂²

C₂²×C₂²⋊D₇₉

yes

characteristic

129

C₄

—

130

C₄

—

316

—

131

C₄

#14579150

yes

—

132

C₄

C₂×#11929612

yes

—

133

C₄

C₂×#11929614

yes

—

134

C₄

C₂×#11929625

yes

—

135

C₂

#19098915

yes

direct product factor

136

C₂

C₂×#14579150

yes

characteristic

137

C₂

C₂²×#11929612

yes

characteristic

138

C₂

C₂²×#11929614

yes

characteristic

	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×#487
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×#487

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #19098915	C₂	direct (N × H)
65. C₂×C₇₉	#487	semidirect
70. C₇₉	C₂×#487	semidirect
135. C₂	#19098915	direct (N × H)

H ≅ G/N

type

2. #19098915

C₂

direct (N × H)

65. C₂×C₇₉

#487

semidirect

70. C₇₉

C₂×#487

semidirect

135. C₂

#19098915

direct (N × H)

Subgroups — C₂×#19098915 #23773712

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098915 #23773712

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098915 #23773712