Group #28377924 — subgroups

Subgroups — #28377924

Order 13952.

148 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 56 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	#20296045	C₂	1	1	yes	maximal, characteristic
2	#20296030	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182039	—	2	1	no	—
5	C₄⋊#11724786	—	2	1	no	—
6	C₄⋊#11724786	—	2	1	no	—
7	C₄⋊#11724786	C₄	1	2	yes	—
8	#16812756	C₄	1	2	yes	—
9	C₄×#11724786	—	2	1	no	—
10	C₂×C₈×D₁₀₉	—	2	1	no	—
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊#11724784	C₄	1	1	yes	characteristic
13	#16812628	—	2	1	no	—
14	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	#13182039	—	4	1	no	—
16	#13182039	—	4	1	no	—
17	C₂×#11724786	—	2	2	no	—
18	C₂×#11724786	—	2	2	no	—
19	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
21	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
22	C₂×#11724784	—	2	1	no	—
23	C₈×D₁₀₉	—	4	1	no	—
24	C₈×D₁₀₉	—	4	1	no	—
25	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	—	2	1	no	—
29	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
37	C₄×D₁₀₉	—	2	1	no	—
38	C₄×D₁₀₉	—	2	1	no	—
39	C₄×D₁₀₉	—	4	1	no	—
40	C₄×D₁₀₉	D₈	1	1	yes	characteristic
41	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
42	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
43	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
44	#11724786	—	4	2	no	—
45	#11724786	—	4	2	no	—
46	C₈×C₁₀₉	—	4	1	no	—
47	#11724784	—	4	1	no	—
48	C₂²×C₁₀₉	#95	1	1	yes	characteristic
49	C₂×D₁₀₉	—	2	1	no	—
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	#106	1	1	yes	characteristic
52	C₂×D₁₀₉	#107	1	1	yes	characteristic
53	C₄×C₁₀₉	—	2	1	no	—
54	C₄×C₁₀₉	#102	1	1	yes	characteristic
55	C₄×C₁₀₉	#103	1	1	yes	characteristic
56	C₄×C₁₀₉	#108	1	1	yes	characteristic
57	C₄×C₁₀₉	#108	1	1	yes	characteristic
58	Dic₁₀₉	—	2	1	no	—
59	Dic₁₀₉	—	2	1	no	—
60	Dic₁₀₉	#102	1	1	yes	characteristic
61	Dic₁₀₉	#103	1	1	yes	characteristic
62	C₂×C₁₀₉	#340	1	1	yes	characteristic
63	C₂×C₁₀₉	#341	1	1	yes	characteristic
64	C₂×C₁₀₉	#344	1	1	yes	characteristic
65	D₁₀₉	—	2	1	no	—
66	D₁₀₉	—	2	1	no	—
67	#1301	—	109	1	no	maximal, Sylow
68	C₁₀₉	#1301	1	1	yes	characteristic, complemented, Sylow
69	#431	—	109	1	no	—
70	#423	—	109	1	no	—
71	C₂×C₄⋊C₈	—	109	1	no	—
72	C₂×M₁₆	—	218	1	no	—
73	C₂²×C₈	—	218	1	no	—
74	C₂×C₄²	—	109	1	no	—
75	C₄⋊C₈	—	109	2	no	—
76	C₄⋊C₈	—	109	1	no	—
77	C₄⋊C₈	—	218	1	no	—
78	C₄⋊C₈	—	218	1	no	—
79	C₄⋊C₈	—	218	1	no	—
80	C₄⋊C₈	#89137	1	1	yes	characteristic
81	C₂²⋊C₈	—	109	2	no	—
82	C₄×C₈	—	218	1	no	—
83	C₂²×C₄	—	109	1	no	—
84	C₂²×C₄	—	109	1	no	—
85	C₂²×C₄	—	109	1	no	—
86	M₁₆	—	436	1	no	—
87	M₁₆	—	436	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	436	1	no	—
90	C₂×C₈	—	436	1	no	—
91	C₂×C₈	—	218	2	no	—
92	C₂×C₈	—	218	2	no	—
93	C₂×C₈	—	218	1	no	—
94	C₄²	—	109	1	no	—
95	C₄²	—	109	1	no	—
96	C₄²	—	109	1	no	—
97	C₄²	C₂×#89137	1	1	yes	characteristic
98	C₂³	—	109	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	109	1	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	109	1	no	—
106	C₂×C₄	—	436	1	no	—
107	C₂×C₄	—	218	1	no	—
108	C₂×C₄	—	218	1	no	—
109	C₂×C₄	C₄×#89137	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
111	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
112	C₈	—	4	1	no	—
113	C₈	—	436	2	no	—
114	C₈	—	436	2	no	—
115	C₈	—	436	1	no	—
116	C₂²	—	109	1	no	—
117	C₂²	—	109	1	no	—
118	C₂²	—	218	1	no	—
119	C₂²	—	218	1	no	—
120	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	109	1	no	—
123	C₄	—	109	1	no	—
124	C₄	—	218	1	no	—
125	C₄	—	218	1	no	—
126	C₄	#16812761	1	1	yes	characteristic
127	C₄	#16812762	1	1	yes	characteristic
128	C₄	#16812769	1	1	yes	characteristic
129	C₄	#16812770	1	1	yes	characteristic
130	C₂	—	218	1	no	—
131	C₂	—	218	1	no	—
132	C₂	#20295899	1	1	yes	characteristic
133	C₂	#20295902	1	1	yes	characteristic
134	C₂	#20295909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296045

C₂

yes

maximal, characteristic

#20296030

C₂

yes

maximal, characteristic

C₄⋊C₈×D₁₀₉

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#16812756

C₄

yes

—

C₄×#11724786

—

C₂×C₈×D₁₀₉

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic

#16812628

—

C₄⋊C₈×C₁₀₉

C₄

yes

—

—

—

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×#11724784

—

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

#11724786

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

D₁₀₉

—

#1301

—

109

maximal, Sylow

C₁₀₉

#1301

yes

characteristic, complemented, Sylow

#431

—

109

—

#423

—

109

—

C₂×C₄⋊C₈

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₄⋊C₈

#89137

yes

characteristic

C₂²⋊C₈

—

109

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

436

—

M₁₆

—

436

—

C₂×C₈

—

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

436

—

107

C₂×C₄

—

218

—

108

C₂×C₄

—

218

—

109

C₂×C₄

C₄×#89137

yes

characteristic

110

C₂×C₄

C₂²⋊#89137

yes

characteristic

111

C₂×C₄

C₄⋊#89137

yes

characteristic

112

C₈

—

113

C₈

—

436

—

114

C₈

—

436

—

115

C₈

—

436

—

116

C₂²

—

109

—

117

C₂²

—

109

—

118

C₂²

—

218

—

119

C₂²

—

218

—

120

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

121

C₄

—

122

C₄

—

109

—

123

C₄

—

109

—

124

C₄

—

218

—

125

C₄

—

218

—

126

C₄

#16812761

yes

characteristic

127

C₄

#16812762

yes

characteristic

128

C₄

#16812769

yes

characteristic

129

C₄

#16812770

yes

characteristic

130

C₂

—

218

—

131

C₂

—

218

—

132

C₂

#20295899

yes

characteristic

133

C₂

#20295902

yes

characteristic

134

C₂

#20295909

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#89137)⋊C₄
commutator	C₂×C₄×C₁₀₉	C₄²
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂²×D₁₀₉	C₄⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#1301
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

(C₂×#89137)⋊C₄

commutator

C₂×C₄×C₁₀₉

C₄²

Frattini

C₄²

C₂×#89137

Fitting

C₄⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂²×D₁₀₉

C₄⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₄⋊C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
68. C₁₀₉	#1301	semidirect

H ≅ G/N

type

68. C₁₀₉

#1301

semidirect

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

subgroup	index
#20296045	2
#20296030	2
C₄⋊C₈×D₁₀₉	2
#1301	109

subgroup

index

#20296045

#20296030

C₄⋊C₈×D₁₀₉

#1301

109

Subgroups — #28377924

Conjugacy classes of subgroups

Complements

Maximal subgroups