Group #79044 — subgroups

Subgroups — #79044

Order 384.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#5470	C₂	1	1	yes	maximal, characteristic
2	C₃×#423	C₂	1	1	yes	maximal, characteristic
3	#5122	C₂	1	2	yes	maximal
4	#5119	C₂	1	2	yes	maximal
5	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#4831	C₂	1	4	yes	maximal, complemented
7	#4830	C₂	1	4	yes	maximal, complemented
8	#2960	—	3	1	no	maximal, Sylow
9	C₂×C₃×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	#926	C₂²	1	2	yes	—
12	#924	C₄	1	4	yes	—
13	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	#870	C₄	1	2	yes	—
16	#869	C₄	1	2	yes	—
17	C₄×Dic₆	C₂²	1	4	yes	—
18	C₃×C₄⋊C₈	C₂²	1	4	yes	—
19	#815	C₂²	1	4	yes	—
20	Dic₃⋊C₈	C₂²	1	8	yes	—
21	C₈×Dic₃	C₂²	1	4	yes	—
22	C₄×C₃⋊C₈	C₂²	1	2	yes	—
23	C₄×C₃⋊C₈	C₂²	1	2	yes	—
24	#533	—	3	1	no	—
25	#446	—	3	4	no	—
26	C₈×Q₈	—	3	4	no	—
27	#424	—	3	2	no	—
28	#423	D₃	1	1	yes	characteristic
29	#405	—	3	2	no	—
30	C₂×C₄×C₈	—	3	1	no	—
31	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×Dic₆	C₂×C₄	1	4	yes	—
34	C₃×M₁₆	—	2	4	no	—
35	C₂×C₃×C₈	C₂³	1	4	yes	—
36	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²	C₂³	1	2	yes	—
39	#217	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
42	Dic₃⋊C₄	C₂×C₄	1	8	yes	—
43	C₄×Dic₃	C₂³	1	4	yes	—
44	C₂×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃⋊C₈	Q₈	1	4	yes	complemented
46	C₂×C₃⋊C₈	C₂³	1	2	yes	—
47	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×M₁₆	C₂×D₃	1	2	yes	—
50	C₂²×C₈	—	3	1	no	—
51	C₂²×C₈	—	3	1	no	—
52	C₄⋊Q₈	—	3	2	no	—
53	#125	—	3	2	no	—
54	C₂²⋊Q₈	—	3	4	no	—
55	C₄×Q₈	—	3	4	no	—
56	#117	—	3	2	no	—
57	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₄⋊C₈	—	3	8	no	—
59	C₄⋊C₈	C₂×D₃	1	4	yes	—
60	#97	—	3	4	no	—
61	C₄×C₈	—	3	4	no	—
62	C₄×C₈	—	3	2	no	—
63	C₄×C₈	—	3	2	no	—
64	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄	—	2	4	no	—
66	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
67	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
69	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
70	C₂×C₃×C₄	#41	1	2	yes	—
71	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₃	C₂²×C₄	1	8	yes	—
73	Dic₆	—	2	8	no	—
74	C₃×C₈	—	2	4	no	—
75	C₃⋊C₈	—	2	2	no	—
76	C₃⋊C₈	C₂×Q₈	1	2	yes	—
77	C₃⋊C₈	C₂×Q₈	1	2	yes	—
78	C₂×Q₈	—	3	4	no	—
79	C₂²×C₄	C₄×D₃	1	2	yes	—
80	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
81	M₁₆	—	2	4	no	—
82	C₂×C₈	—	3	4	no	—
83	C₂×C₈	—	3	1	no	—
84	C₂×C₈	—	3	1	no	—
85	C₂×C₈	—	3	2	no	—
86	C₂×C₈	—	6	2	no	—
87	C₂×C₈	C₂²×D₃	1	4	yes	—
88	C₄⋊C₄	—	3	4	no	—
89	C₄⋊C₄	—	3	8	no	—
90	C₄⋊C₄	—	3	4	no	—
91	C₂²⋊C₄	—	3	4	no	—
92	C₄²	—	3	4	no	—
93	C₄²	C₄×D₃	1	1	yes	characteristic
94	C₄²	C₄×D₃	1	1	yes	characteristic
95	C₄²	C₂²×D₃	1	2	yes	—
96	C₂²×C₃	—	2	2	no	—
97	C₂²×C₃	C₄×Q₈	1	2	yes	—
98	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
99	C₃×C₄	—	2	2	no	—
100	C₃×C₄	C₄×Q₈	1	2	yes	—
101	C₃×C₄	#131	1	4	yes	—
102	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
103	C₃×C₄	C₂×#41	1	1	yes	characteristic
104	Dic₃	—	2	8	no	—
105	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
106	Q₈	—	6	8	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	3	8	no	—
109	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
112	C₂×C₄	#237	1	2	yes	—
113	C₂×C₄	D₃×Q₈	1	2	yes	—
114	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₈	—	3	2	no	—
117	C₈	—	3	2	no	—
118	C₈	—	6	2	no	—
119	C₂×C₃	—	2	2	no	—
120	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₂×C₃	C₂×#131	1	2	yes	—
122	C₂²	—	2	2	no	—
123	C₂²	#887	1	2	yes	—
124	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	6	8	no	—
127	C₄	#887	1	2	yes	—
128	C₄	#907	1	4	yes	—
129	C₄	C₂×#237	1	1	yes	characteristic
130	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
131	C₃	#2960	1	1	yes	characteristic, complemented, Sylow
132	C₂	—	2	2	no	—
133	C₂	C₂×#887	1	1	yes	characteristic
134	C₂	C₂×#907	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#5470

C₂

yes

maximal, characteristic

C₃×#423

C₂

yes

maximal, characteristic

#5122

C₂

yes

maximal

#5119

C₂

yes

maximal

C₂×C₄×C₃⋊C₈

C₂

yes

maximal, characteristic

#4831

C₂

yes

maximal, complemented

#4830

C₂

yes

maximal, complemented

#2960

—

maximal, Sylow

C₂×C₃×M₁₆

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

#926

C₂²

yes

—

#924

C₄

yes

—

C₂²×C₃⋊C₈

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

characteristic

#870

C₄

yes

—

#869

C₄

yes

—

C₄×Dic₆

C₂²

yes

—

C₃×C₄⋊C₈

C₂²

yes

—

#815

C₂²

yes

—

Dic₃⋊C₈

C₂²

yes

—

C₈×Dic₃

C₂²

yes

—

C₄×C₃⋊C₈

C₂²

yes

—

C₄×C₃⋊C₈

C₂²

yes

—

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

D₃

yes

characteristic

#405

—

C₂×C₄×C₈

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×Dic₆

C₂×C₄

yes

—

C₃×M₁₆

—

C₂×C₃×C₈

C₂³

yes

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂³

yes

—

#217

C₂×C₄

yes

—

C₄⋊Dic₃

C₂×C₄

yes

—

C₄⋊Dic₃

C₂×C₄

yes

—

Dic₃⋊C₄

C₂×C₄

yes

—

C₄×Dic₃

C₂³

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

Q₈

yes

complemented

C₂×C₃⋊C₈

C₂³

yes

—

C₂×C₃⋊C₈

C₂³

yes

characteristic

C₂×C₃⋊C₈

C₂³

yes

characteristic

C₂×M₁₆

C₂×D₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

C₂⁴

yes

characteristic

C₂×Dic₃

C₂²×C₄

yes

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

C₂×Q₈

yes

—

C₃⋊C₈

C₂×Q₈

yes

—

C₂×Q₈

—

C₂²×C₄

C₄×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₃

yes

characteristic

C₄²

C₄×D₃

yes

characteristic

C₄²

C₂²×D₃

yes

—

C₂²×C₃

—

C₂²×C₃

C₄×Q₈

yes

—

C₂²×C₃

C₂³×C₄

yes

characteristic

C₃×C₄

—

100

C₃×C₄

C₄×Q₈

yes

—

101

C₃×C₄

#131

yes

—

102

C₃×C₄

C₂²×Q₈

yes

characteristic

103

C₃×C₄

C₂×#41

yes

characteristic

104

Dic₃

—

105

C₂³

C₂×C₄×D₃

yes

characteristic

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×C₄×D₃

yes

—

110

C₂×C₄

C₂×C₄×D₃

yes

characteristic

111

C₂×C₄

C₂×C₄×D₃

yes

—

112

C₂×C₄

#237

yes

—

113

C₂×C₄

D₃×Q₈

yes

—

114

C₂×C₄

C₂³×D₃

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₂×C₃

—

120

C₂×C₃

C₂×C₄×Q₈

yes

characteristic

121

C₂×C₃

C₂×#131

yes

—

122

C₂²

—

123

C₂²

#887

yes

—

124

C₂²

C₂²×C₄×D₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

#887

yes

—

128

C₄

#907

yes

—

129

C₄

C₂×#237

yes

characteristic

130

C₄

C₂×D₃×Q₈

yes

characteristic

131

C₃

#2960

yes

characteristic, complemented, Sylow

132

C₂

—

133

C₂

C₂×#887

yes

characteristic

134

C₂

C₂×#907

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₃×#423	C₂
prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#423	D₃
hypocenter	C₃	#2960
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃

commutator, socle

C₂²×C₃

C₂³×C₄

Fitting

C₃×#423

C₂

prime radical

C₂³×C₃

C₂²×C₄

cosocle

C₂×C₃×C₄

C₂⁴

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
6. #4831	C₂	semidirect
7. #4830	C₂	semidirect
45. C₂×C₃⋊C₈	Q₈	semidirect
131. C₃	#2960	semidirect

H ≅ G/N

type

6. #4831

C₂

semidirect

7. #4830

C₂

semidirect

45. C₂×C₃⋊C₈

Q₈

semidirect

131. C₃

#2960

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#5470	2
C₃×#423	2
#5122	2
#5119	2
C₂×C₄×C₃⋊C₈	2
#4831	2
#4830	2
#2960	3

subgroup

index

C₂×C₄×C₃⋊C₈

Subgroups — #79044

Conjugacy classes of subgroups

Complements

Maximal subgroups