Group #28371967 — subgroups

Subgroups — #28371967

Order 13952.

364 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 229 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	#20295157	C₂	1	4	yes	maximal, complemented
2	Q₈×#11724784	C₂	1	4	yes	maximal, complemented
3	#20295053	C₂	1	2	yes	maximal
4	#20295050	C₂	1	1	yes	maximal, characteristic
5	#20294982	C₂	1	2	yes	maximal
6	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182013	C₂²	1	2	yes	—
9	C₂²×#11724784	C₂²	1	1	yes	characteristic
10	C₂²×#11724784	C₂²	1	1	yes	characteristic
11	C₄⋊#11724784	C₂²	1	4	yes	—
12	C₄⋊#11724784	C₂²	1	8	yes	—
13	#16812610	C₂²	1	4	yes	—
14	C₄×#11724784	C₂²	1	2	yes	—
15	C₄×#11724784	C₂²	1	2	yes	—
16	C₄×#11724784	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₀₉	C₄	1	2	yes	—
18	#125×C₁₀₉	C₄	1	2	yes	—
19	C₂²⋊Q₈×C₁₀₉	C₄	1	4	yes	—
20	C₄×Q₈×C₁₀₉	C₂²	1	4	yes	—
21	#117×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
23	#13182013	—	2	4	no	—
24	C₂×#11724784	—	2	2	no	—
25	C₂×#11724784	Q₈	1	4	yes	complemented
26	C₂×#11724784	C₂³	1	2	yes	—
27	C₂×#11724784	C₂³	1	4	yes	—
28	C₂×#11724784	C₂³	1	1	yes	characteristic
29	C₂×#11724784	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
34	C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
35	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
36	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
37	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₀₉	C₂³	1	2	yes	—
40	C₄²×C₁₀₉	C₂³	1	4	yes	—
41	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₁₀₉	—	2	8	no	—
43	C₂×C₄×C₁₀₉	—	2	4	no	—
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₀₉	#41	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
51	#11724784	—	2	2	no	—
52	#11724784	—	2	4	no	—
53	#11724784	C₂×Q₈	1	2	yes	—
54	#11724784	C₂×Q₈	1	2	yes	—
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₀₉	—	2	2	no	—
59	C₄×C₁₀₉	—	2	8	no	—
60	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
61	C₄×C₁₀₉	#131	1	4	yes	—
62	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
64	C₂×C₁₀₉	—	2	2	no	—
65	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₀₉	C₂×#131	1	2	yes	—
67	#2960	—	109	1	no	maximal, Sylow
68	C₁₀₉	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	D₁₀₉	1	1	yes	characteristic
70	#446	—	109	4	no	—
71	C₈×Q₈	—	109	4	no	—
72	#424	—	109	2	no	—
73	#423	—	109	1	no	—
74	#405	—	109	2	no	—
75	C₂×C₄×C₈	—	109	1	no	—
76	C₂×M₁₆	—	109	2	no	—
77	C₂²×C₈	—	109	1	no	—
78	C₂²×C₈	—	109	1	no	—
79	C₄⋊Q₈	Dic₁₀₉	1	2	yes	—
80	#125	Dic₁₀₉	1	2	yes	—
81	C₂²⋊Q₈	Dic₁₀₉	1	4	yes	—
82	C₄×Q₈	C₂×D₁₀₉	1	4	yes	—
83	#117	C₂×D₁₀₉	1	2	yes	—
84	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
85	C₄⋊C₈	—	109	4	no	—
86	C₄⋊C₈	—	109	8	no	—
87	#97	—	109	4	no	—
88	C₄×C₈	—	109	2	no	—
89	C₄×C₈	—	109	2	no	—
90	C₄×C₈	—	109	4	no	—
91	C₂×Q₈	C₂×Dic₁₀₉	1	4	yes	—
92	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
94	M₁₆	—	218	4	no	—
95	C₂×C₈	—	109	4	no	—
96	C₂×C₈	—	109	1	no	—
97	C₂×C₈	—	109	1	no	—
98	C₂×C₈	—	109	2	no	—
99	C₂×C₈	—	109	4	no	—
100	C₂×C₈	—	218	2	no	—
101	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
102	C₄⋊C₄	C₂×Dic₁₀₉	1	8	yes	—
103	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
104	C₂²⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
105	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
106	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
107	C₄²	C₂²×D₁₀₉	1	2	yes	—
108	C₄²	C₂²×D₁₀₉	1	4	yes	—
109	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₀₉	1	8	yes	—
114	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
116	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
117	C₂×C₄	#13182030	1	2	yes	—
118	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
119	C₈	—	109	2	no	—
120	C₈	—	109	2	no	—
121	C₈	—	218	2	no	—
122	C₈	—	218	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₁₀₉	1	2	yes	—
125	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₁₀₉	1	2	yes	—
129	C₄	#16812706	1	4	yes	—
130	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
131	C₄	C₂×#13182030	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₁₀₉	1	1	yes	characteristic
134	C₂	C₂×#16812706	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295157

C₂

yes

maximal, complemented

Q₈×#11724784

C₂

yes

maximal, complemented

#20295053

C₂

yes

maximal

#20295050

C₂

yes

maximal, characteristic

#20294982

C₂

yes

maximal

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

#533×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

#16812610

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₄

yes

—

#125×C₁₀₉

C₄

yes

—

C₂²⋊Q₈×C₁₀₉

C₄

yes

—

C₄×Q₈×C₁₀₉

C₂²

yes

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

—

Q₈

yes

complemented

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

—

C₂×Q₈

yes

—

#11724784

C₂×Q₈

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

#131

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#131

yes

—

#2960

—

109

maximal, Sylow

C₁₀₉

#2960

yes

characteristic, complemented, Sylow

#533

D₁₀₉

yes

characteristic

#446

—

109

—

C₈×Q₈

—

109

—

#424

—

109

—

#423

—

109

—

#405

—

109

—

C₂×C₄×C₈

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

Dic₁₀₉

yes

—

#125

Dic₁₀₉

yes

—

C₂²⋊Q₈

Dic₁₀₉

yes

—

C₄×Q₈

C₂×D₁₀₉

yes

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

100

C₂×C₈

—

218

—

101

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

102

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

103

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

104

C₂²⋊C₄

C₂×Dic₁₀₉

yes

—

105

C₄²

C₂×Dic₁₀₉

yes

characteristic

106

C₄²

C₂×Dic₁₀₉

yes

characteristic

107

C₄²

C₂²×D₁₀₉

yes

—

108

C₄²

C₂²×D₁₀₉

yes

—

109

C₂³

C₂²×Dic₁₀₉

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×Dic₁₀₉

yes

—

113

C₂×C₄

C₂²×Dic₁₀₉

yes

—

114

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

115

C₂×C₄

C₂²×Dic₁₀₉

yes

—

116

C₂×C₄

Q₈×D₁₀₉

yes

—

117

C₂×C₄

#13182030

yes

—

118

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

119

C₈

—

109

—

120

C₈

—

109

—

121

C₈

—

218

—

122

C₈

—

218

—

123

C₂²

—

124

C₂²

Q₈×Dic₁₀₉

yes

—

125

C₂²

C₂³×Dic₁₀₉

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₁₀₉

yes

—

129

C₄

#16812706

yes

—

130

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

131

C₄

C₂×#13182030

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₁₀₉

yes

characteristic

134

C₂

C₂×#16812706

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#533×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#533	D₁₀₉
hypocenter	C₁₀₉	#2960
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

#533×C₁₀₉

C₂

prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂×C₄×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #20295157	C₂	semidirect
2. Q₈×#11724784	C₂	semidirect
25. C₂×#11724784	Q₈	semidirect
68. C₁₀₉	#2960	semidirect

H ≅ G/N

type

1. #20295157

C₂

semidirect

2. Q₈×#11724784

C₂

semidirect

25. C₂×#11724784

Q₈

semidirect

68. C₁₀₉

#2960

semidirect

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

subgroup	index
#20295157	2
Q₈×#11724784	2
#20295053	2
#20295050	2
#20294982	2
C₂×C₄×#11724784	2
#533×C₁₀₉	2
#2960	109

subgroup

index

#533×C₁₀₉

109

Subgroups — #28371967

Conjugacy classes of subgroups

Complements

Maximal subgroups