Group #28364724 — subgroups

Subgroups — #28364724

Order 13952.

262 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 73 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	#20295010	C₂	1	2	yes	maximal, complemented
2	#20294823	C₂	1	4	yes	maximal, complemented
3	#535×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182013	—	2	2	no	—
5	#16812618	C₂²	1	4	yes	complemented
6	#16812616	—	2	2	no	—
7	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₁₀₉	C₄	1	1	yes	characteristic
9	#127×C₁₀₉	C₄	1	2	yes	—
10	#124×C₁₀₉	C₂²	1	2	yes	—
11	#121×C₁₀₉	—	2	2	no	—
12	#120×C₁₀₉	—	2	2	no	—
13	#117×C₁₀₉	C₄	1	1	yes	characteristic
14	#13182013	—	2	4	no	—
15	#13182013	—	4	2	no	—
16	C₂×#11724784	—	4	2	no	—
17	C₂⁴×C₁₀₉	—	2	1	no	—
18	#41×C₁₀₉	—	4	2	no	—
19	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₁₀₉	—	2	4	no	—
21	C₂×D₄×C₁₀₉	—	2	1	no	—
22	C₂×D₄×C₁₀₉	—	4	2	no	—
23	C₂×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×D₄×C₁₀₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
28	C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
31	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂³×C₁₀₉	—	2	2	no	—
33	C₂³×C₁₀₉	—	4	2	no	—
34	C₂³×C₁₀₉	—	8	2	no	—
35	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂³×C₁₀₉	C₂²⋊C₄	1	2	yes	—
37	Q₈×C₁₀₉	—	4	1	no	—
38	D₄×C₁₀₉	—	4	1	no	—
39	D₄×C₁₀₉	—	4	2	no	—
40	D₄×C₁₀₉	—	4	2	no	—
41	D₄×C₁₀₉	—	4	2	no	—
42	D₄×C₁₀₉	—	8	4	no	—
43	C₂×C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄×C₁₀₉	—	2	2	no	—
45	C₂×C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄×C₁₀₉	—	2	2	no	—
47	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
50	#11724784	—	4	4	no	—
51	C₂²×C₁₀₉	—	2	1	no	—
52	C₂²×C₁₀₉	—	2	2	no	—
53	C₂²×C₁₀₉	—	4	2	no	—
54	C₂²×C₁₀₉	—	4	1	no	—
55	C₂²×C₁₀₉	—	4	4	no	—
56	C₂²×C₁₀₉	—	4	2	no	—
57	C₂²×C₁₀₉	—	8	4	no	—
58	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₁₀₉	—	2	2	no	—
60	C₄×C₁₀₉	—	2	1	no	—
61	C₄×C₁₀₉	—	4	2	no	—
62	C₄×C₁₀₉	#99	1	2	yes	—
63	C₂×C₁₀₉	—	2	1	no	—
64	C₂×C₁₀₉	—	4	2	no	—
65	C₂×C₁₀₉	—	4	1	no	—
66	C₂×C₁₀₉	—	8	2	no	—
67	C₂×C₁₀₉	C₂×#99	1	1	yes	characteristic
68	#2134	—	109	1	no	maximal, Sylow
69	C₁₀₉	#2134	1	1	yes	characteristic, complemented, Sylow
70	#535	D₁₀₉	1	1	yes	characteristic
71	#413	—	109	2	no	—
72	#355	—	109	4	no	—
73	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
74	C₂²×D₄	Dic₁₀₉	1	1	yes	characteristic
75	C₂×M₁₆	—	218	2	no	—
76	#127	Dic₁₀₉	1	2	yes	—
77	#124	C₂×D₁₀₉	1	2	yes	—
78	#121	—	2	2	no	—
79	#120	—	2	2	no	—
80	#117	Dic₁₀₉	1	1	yes	characteristic
81	#101	—	109	4	no	—
82	#100	—	218	2	no	—
83	C₂⁴	—	2	1	no	—
84	#41	—	4	2	no	—
85	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
86	C₂×D₄	—	2	4	no	—
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	C₂×Dic₁₀₉	1	2	yes	—
90	C₂×D₄	C₂×Dic₁₀₉	1	1	yes	characteristic
91	C₂×D₄	C₂²⋊D₁₀₉	1	2	yes	—
92	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
94	M₁₆	—	436	2	no	—
95	M₁₆	—	218	4	no	—
96	C₂×C₈	—	436	2	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₂²⋊C₄	—	2	1	no	—
99	C₂²⋊C₄	—	2	4	no	—
100	C₄²	C₂×Dic₁₀₉	1	2	yes	—
101	C₂³	—	2	2	no	—
102	C₂³	—	4	2	no	—
103	C₂³	—	8	2	no	—
104	C₂³	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
105	C₂³	C₂²⋊Dic₁₀₉	1	2	yes	—
106	Q₈	—	4	1	no	—
107	D₄	—	4	1	no	—
108	D₄	—	4	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
119	C₈	—	436	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	4	2	no	—
123	C₂²	—	4	1	no	—
124	C₂²	—	4	4	no	—
125	C₂²	—	4	2	no	—
126	C₂²	—	8	4	no	—
127	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	4	2	no	—
131	C₄	#16812614	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	4	1	no	—
134	C₂	—	4	2	no	—
135	C₂	—	8	2	no	—
136	C₂	C₂×#16812614	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295010

C₂

yes

maximal, complemented

#20294823

C₂

yes

maximal, complemented

#535×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

#16812618

C₂²

yes

complemented

#16812616

—

C₂×#41×C₁₀₉

C₂²

yes

characteristic

C₂²×D₄×C₁₀₉

C₄

yes

characteristic

#127×C₁₀₉

C₄

yes

—

#124×C₁₀₉

C₂²

yes

—

#121×C₁₀₉

—

#120×C₁₀₉

—

#117×C₁₀₉

C₄

yes

—

—

—

C₂⁴×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂³

yes

characteristic

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂×D₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂³×C₁₀₉

C₂²⋊C₄

yes

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#99

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#99

yes

characteristic

#2134

—

109

maximal, Sylow

C₁₀₉

#2134

yes

characteristic, complemented, Sylow

#535

D₁₀₉

yes

characteristic

#413

—

109

—

#355

—

109

—

C₂×#41

C₂×D₁₀₉

yes

characteristic

C₂²×D₄

Dic₁₀₉

yes

characteristic

C₂×M₁₆

—

218

—

#127

Dic₁₀₉

yes

—

#124

C₂×D₁₀₉

yes

—

#121

—

#120

—

#117

Dic₁₀₉

yes

characteristic

#101

—

109

—

#100

—

218

—

C₂⁴

—

#41

—

C₂×Q₈

C₂²×D₁₀₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₀₉

yes

—

C₂×D₄

C₂×Dic₁₀₉

yes

characteristic

C₂×D₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

M₁₆

—

436

—

M₁₆

—

218

—

C₂×C₈

—

436

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

C₂×Dic₁₀₉

yes

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

C₂²⋊Dic₁₀₉

yes

characteristic

105

C₂³

C₂²⋊Dic₁₀₉

yes

—

106

Q₈

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂²⋊Dic₁₀₉

yes

characteristic

117

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

119

C₈

—

436

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#16812614

yes

—

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

C₂×#16812614

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#16812614
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₁₀₉
Fitting, prime radical	#535×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×#99
cosocle	C₂×Q₈×C₁₀₉	C₂³
hypercenter	#535	D₁₀₉
hypocenter	C₁₀₉	#2134
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #535

subgroup

G/H

center

C₂

C₂×#16812614

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₁₀₉

Fitting, prime radical

#535×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂×#99

cosocle

C₂×Q₈×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #535

N	H ≅ G/N	type
1. #20295010	C₂	semidirect
2. #20294823	C₂	semidirect
5. #16812618	C₂²	semidirect
69. C₁₀₉	#2134	semidirect

H ≅ G/N

type

1. #20295010

C₂

semidirect

2. #20294823

C₂

semidirect

5. #16812618

C₂²

semidirect

69. C₁₀₉

#2134

semidirect

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

subgroup	index
#20295010	2
#20294823	2
#535×C₁₀₉	2
#2134	109

subgroup

index

#20295010

#20294823

#535×C₁₀₉

#2134

109

Subgroups — #28364724

Conjugacy classes of subgroups

Complements

Maximal subgroups