Group #23765081 — subgroups

Subgroups — C₂×#19098362 #23765081

Order 10112.

240 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 77 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×#14579080	C₂	1	2	yes	maximal
2	#19098362	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#11929608	—	2	2	no	—
5	#14579080	—	2	2	no	—
6	#14579080	C₂²	1	4	yes	—
7	C₂²×Q₈×C₇₉	C₄	1	1	yes	characteristic
8	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₇₉	—	2	2	no	—
10	C₄⋊Q₈×C₇₉	C₄	1	2	yes	—
11	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
13	C₂×C₄²×C₇₉	C₄	1	1	yes	characteristic
14	#11929608	—	2	4	no	—
15	#11929608	—	4	2	no	—
16	C₂×C₇₉⋊C₈	—	4	2	no	—
17	C₂×Q₈×C₇₉	—	2	1	no	—
18	C₂×Q₈×C₇₉	—	4	1	no	—
19	C₂×Q₈×C₇₉	—	4	1	no	—
20	C₂×Q₈×C₇₉	—	4	1	no	—
21	C₂×Q₈×C₇₉	—	4	1	no	—
22	C₂×Q₈×C₇₉	C₂×C₄	1	2	yes	—
23	C₂×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₇₉	D₄	1	2	yes	—
25	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₇₉	—	2	1	no	—
27	C₂²×C₄×C₇₉	—	2	1	no	—
28	C₂²×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₉	D₄	1	2	yes	—
30	C₄⋊C₄×C₇₉	—	2	4	no	—
31	C₄⋊C₄×C₇₉	—	2	4	no	—
32	C₄²×C₇₉	—	2	1	no	—
33	C₄²×C₇₉	C₂×C₄	1	2	yes	—
34	C₂³×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₇₉	—	4	1	no	—
36	Q₈×C₇₉	—	4	2	no	—
37	Q₈×C₇₉	—	4	1	no	—
38	Q₈×C₇₉	—	4	2	no	—
39	Q₈×C₇₉	—	4	2	no	—
40	C₂×C₄×C₇₉	—	2	4	no	—
41	C₂×C₄×C₇₉	—	2	2	no	—
42	C₂×C₄×C₇₉	—	2	2	no	—
43	C₂×C₄×C₇₉	—	2	1	no	—
44	C₂×C₄×C₇₉	—	2	1	no	—
45	C₂×C₄×C₇₉	—	2	2	no	—
46	C₂×C₄×C₇₉	—	2	2	no	—
47	C₂×C₄×C₇₉	—	4	1	no	—
48	C₂×C₄×C₇₉	—	4	1	no	—
49	C₂×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
53	C₇₉⋊C₈	—	4	4	no	—
54	C₂²×C₇₉	—	2	2	no	—
55	C₂²×C₇₉	#99	1	1	yes	characteristic
56	C₂²×C₇₉	#99	1	1	yes	characteristic
57	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₇₉	—	2	4	no	—
59	C₄×C₇₉	—	2	2	no	—
60	C₄×C₇₉	—	2	1	no	—
61	C₄×C₇₉	—	2	1	no	—
62	C₄×C₇₉	—	2	2	no	—
63	C₄×C₇₉	—	4	2	no	—
64	C₂×C₇₉	—	2	1	no	—
65	C₂×C₇₉	—	2	1	no	—
66	C₂×C₇₉	#356	1	2	yes	complemented
67	C₂×C₇₉	C₂×#99	1	1	yes	characteristic
68	C₂×#356	—	79	1	no	maximal, Sylow
69	C₇₉	C₂×#356	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₄⋊Q₈	D₇₉	1	1	yes	characteristic
71	C₂×#101	—	79	2	no	—
72	#356	—	79	4	no	—
73	C₂²×Q₈	Dic₇₉	1	1	yes	characteristic
74	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
75	C₂×M₁₆	—	158	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	Dic₇₉	1	2	yes	—
78	C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	2	no	—
80	C₂×C₄²	Dic₇₉	1	1	yes	characteristic
81	#101	—	79	4	no	—
82	#101	—	158	2	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	4	1	no	—
85	C₂×Q₈	—	4	1	no	—
86	C₂×Q₈	—	4	1	no	—
87	C₂×Q₈	—	4	1	no	—
88	C₂×Q₈	C₂×Dic₇₉	1	2	yes	—
89	C₂×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
90	C₂×Q₈	C₂²⋊D₇₉	1	2	yes	—
91	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₂×Dic₇₉	1	1	yes	characteristic
95	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
96	M₁₆	—	316	2	no	—
97	M₁₆	—	158	4	no	—
98	C₂×C₈	—	316	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂×Dic₇₉	1	2	yes	—
103	C₂³	C₂²⋊Dic₇₉	1	1	yes	characteristic
104	Q₈	—	4	2	no	—
105	Q₈	—	4	1	no	—
106	Q₈	—	4	1	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	4	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
119	C₂×C₄	C₂²⋊Dic₇₉	1	2	yes	—
120	C₂×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
122	C₈	—	316	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#14579076	1	1	yes	characteristic
125	C₂²	#14579076	1	1	yes	characteristic
126	C₂²	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	2	no	—
131	C₄	—	2	1	no	—
132	C₄	—	4	2	no	—
133	C₂	—	2	1	no	—
134	C₂	—	2	1	no	—
135	C₂	#19098362	1	2	yes	direct product factor
136	C₂	C₂×#14579076	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579080

C₂

yes

maximal

#19098362

C₂

yes

maximal, direct product factor

C₂×C₄⋊Q₈×C₇₉

C₂

yes

maximal, characteristic

C₂×#11929608

—

#14579080

—

#14579080

C₂²

yes

—

C₂²×Q₈×C₇₉

C₄

yes

characteristic

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₄

yes

—

C₄⋊Q₈×C₇₉

C₂²

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₄

yes

characteristic

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂×C₄

yes

—

C₂×Q₈×C₇₉

C₂×C₄

yes

characteristic

C₂×Q₈×C₇₉

D₄

yes

—

C₂×Q₈×C₇₉

C₂³

yes

characteristic

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₄

yes

—

C₂³×C₇₉

C₂²⋊C₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#356

yes

complemented

C₂×C₇₉

C₂×#99

yes

—

C₇₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₇₉

yes

characteristic

C₂×#101

—

#356

—

C₂²×Q₈

Dic₇₉

yes

characteristic

C₂²×Q₈

C₂×D₇₉

yes

characteristic

C₂×M₁₆

—

158

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₇₉

yes

—

C₄⋊Q₈

C₂×D₇₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

Dic₇₉

yes

characteristic

#101

—

#101

—

158

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₇₉

yes

—

C₂×Q₈

C₂×Dic₇₉

yes

characteristic

C₂×Q₈

C₂²⋊D₇₉

yes

—

C₂×Q₈

C₂²×D₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₇₉

yes

—

M₁₆

—

316

—

M₁₆

—

158

—

C₂×C₈

—

316

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×Dic₇₉

yes

—

103

C₂³

C₂²⋊Dic₇₉

yes

characteristic

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²⋊Dic₇₉

yes

characteristic

119

C₂×C₄

C₂²⋊Dic₇₉

yes

—

120

C₂×C₄

C₂²×Dic₇₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₇₉

yes

—

122

C₈

—

316

—

123

C₂²

—

124

C₂²

#14579076

yes

characteristic

125

C₂²

#14579076

yes

characteristic

126

C₂²

C₂×C₂²⋊Dic₇₉

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₂

—

134

C₂

—

135

C₂

#19098362

yes

direct product factor

136

C₂

C₂×#14579076

yes

characteristic

	subgroup	G/H
center	C₂²	#14579076
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₇₉
Fitting	C₂×C₄⋊Q₈×C₇₉	C₂
socle	C₂²×C₇₉	#99
prime radical	C₂³×C₇₉	C₂²⋊C₄
cosocle	C₂×Q₈×C₇₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₇₉
hypocenter	C₇₉	C₂×#356
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#14579076

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₇₉

Fitting

C₂×C₄⋊Q₈×C₇₉

C₂

socle

C₂²×C₇₉

#99

prime radical

C₂³×C₇₉

C₂²⋊C₄

cosocle

C₂×Q₈×C₇₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₇₉

hypocenter

C₇₉

C₂×#356

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #19098362	C₂	direct (N × H)
66. C₂×C₇₉	#356	semidirect
69. C₇₉	C₂×#356	semidirect
135. C₂	#19098362	direct (N × H)

H ≅ G/N

type

2. #19098362

C₂

direct (N × H)

66. C₂×C₇₉

#356

semidirect

69. C₇₉

C₂×#356

semidirect

135. C₂

#19098362

direct (N × H)

Subgroups — C₂×#19098362 #23765081

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19098362 #23765081

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19098362 #23765081