Group #28377859 — subgroups

Subgroups — #28377859

Order 13952.

208 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 62 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	C₄²×#89137	C₂	1	1	yes	maximal, characteristic
3	#20295078	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#89137	—	2	1	no	—
5	C₂×C₄×#89137	—	2	1	no	—
6	C₂×C₄×#89137	C₄	1	2	yes	—
7	C₄⋊#11724786	C₄	1	2	yes	complemented
8	#16812756	—	2	1	no	—
9	C₄×#11724786	—	2	1	no	—
10	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊#11724784	C₄	1	1	yes	characteristic, complemented
12	#16812611	—	2	1	no	—
13	C₈×Dic₁₀₉	—	2	1	no	—
14	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×#89137	—	2	2	no	—
16	C₂×#11724786	—	2	2	no	—
17	C₄×#89137	—	2	2	no	—
18	C₄×#89137	—	2	2	no	—
19	C₄×#89137	—	2	4	no	—
20	C₄×#89137	—	2	2	no	—
21	C₄×#89137	C₈	1	4	yes	complemented
22	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
25	C₂×#11724784	—	2	1	no	—
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
28	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	—	2	1	no	—
30	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×#89137	—	2	4	no	—
33	C₂×#89137	—	2	4	no	—
34	C₂×#89137	—	2	4	no	—
35	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄×D₁₀₉	—	2	1	no	—
42	C₄×D₁₀₉	—	2	1	no	—
43	C₄×D₁₀₉	—	2	2	no	—
44	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
45	C₄×D₁₀₉	M₁₆	1	2	yes	—
46	#11724786	—	4	2	no	—
47	C₈×C₁₀₉	—	4	1	no	—
48	#11724784	—	4	1	no	—
49	C₂²×C₁₀₉	#95	1	1	yes	characteristic
50	C₂×D₁₀₉	—	2	1	no	—
51	C₂×D₁₀₉	—	2	1	no	—
52	C₂×D₁₀₉	C₂²⋊C₈	1	2	yes	—
53	#89137	—	2	8	no	—
54	C₄×C₁₀₉	—	2	1	no	—
55	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
56	C₄×C₁₀₉	#97	1	1	yes	characteristic
57	C₄×C₁₀₉	#104	1	1	yes	characteristic
58	C₄×C₁₀₉	#104	1	1	yes	characteristic
59	Dic₁₀₉	—	2	1	no	—
60	Dic₁₀₉	#104	1	2	yes	—
61	Dic₁₀₉	C₄⋊C₈	1	2	yes	—
62	C₂×C₁₀₉	#336	1	1	yes	characteristic
63	C₂×C₁₀₉	#339	1	1	yes	characteristic
64	C₂×C₁₀₉	#339	1	1	yes	characteristic
65	D₁₀₉	—	2	2	no	—
66	#1276	—	109	1	no	maximal, Sylow
67	C₁₀₉	#1276	1	1	yes	characteristic, complemented, Sylow
68	#431	—	109	1	no	—
69	#431	—	109	1	no	—
70	C₄³	—	109	1	no	—
71	C₂×C₄²	—	109	1	no	—
72	C₂×C₄²	—	109	2	no	—
73	C₂×C₄²	—	218	1	no	—
74	C₂×C₄²	—	218	1	no	—
75	C₄⋊C₈	—	109	2	no	—
76	C₄⋊C₈	—	109	1	no	—
77	C₄⋊C₈	#89137	1	1	yes	characteristic, complemented
78	C₂²⋊C₈	—	218	1	no	—
79	C₂²⋊C₈	—	218	1	no	—
80	C₄×C₈	—	218	1	no	—
81	C₄×C₈	—	218	1	no	—
82	C₂²×C₄	—	109	1	no	—
83	C₂²×C₄	—	109	1	no	—
84	C₂²×C₄	—	109	1	no	—
85	C₂²×C₄	—	218	2	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	218	2	no	—
88	C₂×C₈	—	218	1	no	—
89	C₄²	—	109	1	no	—
90	C₄²	—	109	1	no	—
91	C₄²	—	109	1	no	—
92	C₄²	—	109	4	no	—
93	C₄²	—	218	2	no	—
94	C₄²	—	218	4	no	—
95	C₄²	—	218	2	no	—
96	C₄²	—	218	2	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	2	no	—
102	C₂×C₄	—	109	2	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	218	2	no	—
105	C₂×C₄	—	218	4	no	—
106	C₂×C₄	—	218	4	no	—
107	C₂×C₄	—	218	4	no	—
108	C₂×C₄	—	218	1	no	—
109	C₂×C₄	—	218	1	no	—
110	C₂×C₄	C₄×#89137	1	1	yes	characteristic
111	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
112	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
113	C₈	—	4	1	no	—
114	C₈	—	436	2	no	—
115	C₈	—	436	1	no	—
116	C₂²	—	109	2	no	—
117	C₂²	—	218	1	no	—
118	C₂²	—	218	1	no	—
119	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	109	2	no	—
122	C₄	—	109	2	no	—
123	C₄	—	218	8	no	—
124	C₄	—	218	1	no	—
125	C₄	C₈×#89137	1	1	yes	characteristic
126	C₄	#16812753	1	1	yes	characteristic
127	C₄	#16812763	1	1	yes	characteristic
128	C₄	#16812764	1	1	yes	characteristic
129	C₂	—	218	2	no	—
130	C₂	(C₂×#89137)⋊C₈	1	1	yes	characteristic
131	C₂	#20295896	1	1	yes	characteristic
132	C₂	#20295898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296045

C₂

yes

maximal, characteristic

C₄²×#89137

C₂

yes

maximal, characteristic

#20295078

C₂

yes

maximal, characteristic

C₂×C₄×#89137

—

C₂×C₄×#89137

—

C₂×C₄×#89137

C₄

yes

—

C₄⋊#11724786

C₄

yes

complemented

#16812756

—

C₄×#11724786

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic, complemented

#16812611

—

C₈×Dic₁₀₉

—

C₄⋊C₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

—

—

—

—

—

C₈

yes

complemented

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×#11724784

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

—

—

—

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

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₈

yes

—

C₄×D₁₀₉

M₁₆

yes

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₈

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×C₈

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

#104

yes

—

Dic₁₀₉

C₄⋊C₈

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1276

—

109

maximal, Sylow

C₁₀₉

#1276

yes

characteristic, complemented, Sylow

#431

—

109

—

#431

—

109

—

C₄³

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

218

—

C₂×C₄²

—

218

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

#89137

yes

characteristic, complemented

C₂²⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₄²

—

218

—

C₄²

—

218

—

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

—

218

—

105

C₂×C₄

—

218

—

106

C₂×C₄

—

218

—

107

C₂×C₄

—

218

—

108

C₂×C₄

—

218

—

109

C₂×C₄

—

218

—

110

C₂×C₄

C₄×#89137

yes

characteristic

111

C₂×C₄

C₂²⋊#89137

yes

characteristic

112

C₂×C₄

C₄⋊#89137

yes

characteristic

113

C₈

—

114

C₈

—

436

—

115

C₈

—

436

—

116

C₂²

—

109

—

117

C₂²

—

218

—

118

C₂²

—

218

—

119

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

120

C₄

—

121

C₄

—

109

—

122

C₄

—

109

—

123

C₄

—

218

—

124

C₄

—

218

—

125

C₄

C₈×#89137

yes

characteristic

126

C₄

#16812753

yes

characteristic

127

C₄

#16812763

yes

characteristic

128

C₄

#16812764

yes

characteristic

129

C₂

—

218

—

130

C₂

(C₂×#89137)⋊C₈

yes

characteristic

131

C₂

#20295896

yes

characteristic

132

C₂

#20295898

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₄×C₈
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂²×D₁₀₉	C₂²⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#1276
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊#89137

commutator

C₄×C₁₀₉

C₄×C₈

Frattini

C₄²

C₂×#89137

Fitting

C₄⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂²×D₁₀₉

C₂²⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₄⋊C₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
7. C₄⋊#11724786	C₄	semidirect
11. C₄⋊#11724784	C₄	semidirect
14. C₄⋊C₈×C₁₀₉	C₄	semidirect
21. C₄×#89137	C₈	semidirect
67. C₁₀₉	#1276	semidirect
77. C₄⋊C₈	#89137	semidirect

H ≅ G/N

type

7. C₄⋊#11724786

C₄

semidirect

11. C₄⋊#11724784

C₄

semidirect

14. C₄⋊C₈×C₁₀₉

C₄

semidirect

21. C₄×#89137

C₈

semidirect

67. C₁₀₉

#1276

semidirect

77. C₄⋊C₈

#89137

semidirect

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

subgroup	index
#20296045	2
C₄²×#89137	2
#20295078	2
#1276	109

subgroup

index

#20296045

C₄²×#89137

#20295078

#1276

109

Subgroups — #28377859

Conjugacy classes of subgroups

Complements

Maximal subgroups