Group #28379966 — subgroups

Subgroups — #28379966

Order 13952.

394 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 168 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	Q₈×#11724786	C₂	1	2	yes	maximal, complemented
2	#20296060	C₂	1	6	yes	maximal, complemented
3	#20296045	C₂	1	3	yes	maximal, complemented
4	#20296030	C₂	1	3	yes	maximal, complemented
5	C₄×#13182030	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₂²	1	6	yes	—
7	C₄⋊#11724786	C₂²	1	3	yes	—
8	C₄⋊#11724786	C₂²	1	6	yes	—
9	C₄⋊#11724786	C₂²	1	1	yes	characteristic
10	#16812756	C₂²	1	6	yes	—
11	C₄×#11724786	C₂²	1	6	yes	—
12	C₂×#13182030	C₄	1	1	yes	characteristic
13	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×D₄₃₆	C₄	1	3	yes	—
15	#16812661	C₂²	1	3	yes	—
16	#16812657	C₄	1	3	yes	—
17	C₄²×D₁₀₉	C₂²	1	3	yes	—
18	C₄×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
19	#13182039	—	4	6	no	—
20	C₂×#11724786	C₂³	1	2	yes	—
21	C₂×#11724786	C₂³	1	6	yes	—
22	#13182030	—	2	4	no	—
23	C₂×D₄₃₆	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
27	C₄⋊Dic₁₀₉	C₂×C₄	1	3	yes	—
28	#13182006	C₂×C₄	1	6	yes	—
29	C₄×Dic₁₀₉	C₂×C₄	1	3	yes	—
30	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
31	C₄×Dic₁₀₉	C₂³	1	3	yes	—
32	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₉	C₂×C₄	1	3	yes	—
34	C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
35	C₂²×D₁₀₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₁₀₉	M₁₆	1	4	yes	complemented
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
42	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
43	D₄₃₆	—	2	6	no	—
44	C₄×D₁₀₉	—	2	6	no	—
45	C₄×D₁₀₉	—	2	6	no	—
46	C₄×D₁₀₉	—	2	3	no	—
47	C₄×D₁₀₉	—	2	3	no	—
48	#11724786	—	2	6	no	—
49	#11724786	—	2	2	no	—
50	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₁₀₉	—	2	3	no	—
52	C₂×D₁₀₉	—	2	3	no	—
53	C₂×D₁₀₉	—	2	3	no	—
54	C₄×C₁₀₉	—	2	3	no	—
55	C₄×C₁₀₉	C₂×M₁₆	1	6	yes	—
56	C₄×C₁₀₉	#131	1	2	yes	—
57	Dic₁₀₉	—	2	1	no	—
58	Dic₁₀₉	—	2	3	no	—
59	Dic₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
60	Dic₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂×C₁₀₉	#520	1	1	yes	characteristic
62	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
63	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
64	D₁₀₉	—	4	3	no	—
65	#2991	—	109	1	no	maximal, Sylow
66	C₁₀₉	#2991	1	1	yes	characteristic, complemented, Sylow
67	C₄×#41	—	109	1	no	—
68	C₈×Q₈	—	109	2	no	—
69	#436	—	109	6	no	—
70	#431	—	109	3	no	—
71	#423	—	109	3	no	—
72	C₂×#41	—	109	1	no	—
73	C₂×M₁₆	—	109	6	no	—
74	C₄×Q₈	—	109	1	no	—
75	C₄×Q₈	#89137	1	1	yes	characteristic
76	C₄×D₄	—	109	3	no	—
77	C₄×D₄	—	109	3	no	—
78	#117	—	109	3	no	—
79	C₂×C₄²	—	109	3	no	—
80	C₄⋊C₈	—	109	3	no	—
81	C₄⋊C₈	—	109	6	no	—
82	C₄⋊C₈	—	109	1	no	—
83	C₂²⋊C₈	—	109	6	no	—
84	C₄×C₈	—	109	6	no	—
85	#41	—	218	4	no	—
86	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
87	C₂×D₄	—	109	3	no	—
88	C₂²×C₄	—	109	3	no	—
89	C₂²×C₄	—	109	3	no	—
90	C₂²×C₄	—	109	3	no	—
91	M₁₆	—	436	6	no	—
92	C₂×C₈	—	109	2	no	—
93	C₂×C₈	—	109	6	no	—
94	C₄⋊C₄	—	109	3	no	—
95	C₄⋊C₄	C₂×#89137	1	3	yes	—
96	C₂²⋊C₄	—	109	6	no	—
97	C₄²	—	109	3	no	—
98	C₄²	—	109	3	no	—
99	C₄²	—	109	1	no	—
100	C₄²	C₂×#89137	1	3	yes	—
101	C₂³	—	109	3	no	—
102	Q₈	#13182039	1	4	yes	complemented
103	D₄	—	218	6	no	—
104	C₂×C₄	—	109	1	no	—
105	C₂×C₄	—	109	1	no	—
106	C₂×C₄	—	109	3	no	—
107	C₂×C₄	—	218	3	no	—
108	C₂×C₄	—	218	3	no	—
109	C₂×C₄	—	218	6	no	—
110	C₂×C₄	—	218	6	no	—
111	C₂×C₄	C₂²×#89137	1	3	yes	—
112	C₂×C₄	C₂²×#89137	1	3	yes	—
113	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
114	C₈	—	218	6	no	—
115	C₈	—	218	2	no	—
116	C₂²	—	218	3	no	—
117	C₂²	—	218	3	no	—
118	C₂²	—	218	3	no	—
119	C₂²	C₂³×#89137	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	109	1	no	—
122	C₄	—	109	1	no	—
123	C₄	—	218	1	no	—
124	C₄	—	218	3	no	—
125	C₄	C₂×#13182039	1	6	yes	—
126	C₄	#16812787	1	2	yes	—
127	C₂	—	436	3	no	—
128	C₂	#20296095	1	1	yes	characteristic
129	C₂	C₂²×#13182039	1	1	yes	characteristic
130	C₂	C₂×#16812787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11724786

C₂

yes

maximal, complemented

#20296060

C₂

yes

maximal, complemented

#20296045

C₂

yes

maximal, complemented

#20296030

C₂

yes

maximal, complemented

C₄×#13182030

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

characteristic

#16812756

C₂²

yes

—

C₄×#11724786

C₂²

yes

—

C₂×#13182030

C₄

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×D₄₃₆

C₄

yes

—

#16812661

C₂²

yes

—

#16812657

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

C₂×#11724786

C₂³

yes

—

C₂×#11724786

C₂³

yes

—

#13182030

—

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182006

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

C₂²×C₄

yes

—

Q₈×C₁₀₉

M₁₆

yes

complemented

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

characteristic

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

C₄×C₁₀₉

#131

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

H(5,𝔽₂)

yes

characteristic

Dic₁₀₉

2₋¹⁺⁴

yes

characteristic

C₂×C₁₀₉

#520

yes

characteristic

C₂×C₁₀₉

C₂²×M₁₆

yes

characteristic

C₂×C₁₀₉

C₂×#131

yes

characteristic

D₁₀₉

—

#2991

—

109

maximal, Sylow

C₁₀₉

#2991

yes

characteristic, complemented, Sylow

C₄×#41

—

109

—

C₈×Q₈

—

109

—

#436

—

109

—

#431

—

109

—

#423

—

109

—

C₂×#41

—

109

—

C₂×M₁₆

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

#89137

yes

characteristic

C₄×D₄

—

109

—

C₄×D₄

—

109

—

#117

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₄×C₈

—

109

—

#41

—

218

—

C₂×Q₈

C₂×#89137

yes

characteristic

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

436

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×#89137

yes

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

100

C₄²

C₂×#89137

yes

—

101

C₂³

—

109

—

102

Q₈

#13182039

yes

complemented

103

D₄

—

218

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

218

—

108

C₂×C₄

—

218

—

109

C₂×C₄

—

218

—

110

C₂×C₄

—

218

—

111

C₂×C₄

C₂²×#89137

yes

—

112

C₂×C₄

C₂²×#89137

yes

—

113

C₂×C₄

C₂²×#89137

yes

characteristic

114

C₈

—

218

—

115

C₈

—

218

—

116

C₂²

—

218

—

117

C₂²

—

218

—

118

C₂²

—

218

—

119

C₂²

C₂³×#89137

yes

characteristic

120

C₄

—

121

C₄

—

109

—

122

C₄

—

109

—

123

C₄

—

218

—

124

C₄

—

218

—

125

C₄

C₂×#13182039

yes

—

126

C₄

#16812787

yes

—

127

C₂

—

436

—

128

C₂

#20296095

yes

characteristic

129

C₂

C₂²×#13182039

yes

characteristic

130

C₂

C₂×#16812787

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₄×Q₈×C₁₀₉	C₄
prime radical	C₂×#13182030	C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	C₄×Q₈	#89137
hypocenter	C₁₀₉	#2991
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#89137

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

C₄×Q₈×C₁₀₉

C₄

prime radical

C₂×#13182030

C₄

cosocle

C₂×Dic₁₀₉

C₂⁴

C₄×Q₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. Q₈×#11724786	C₂	semidirect
2. #20296060	C₂	semidirect
3. #20296045	C₂	semidirect
4. #20296030	C₂	semidirect
36. Q₈×C₁₀₉	M₁₆	semidirect
66. C₁₀₉	#2991	semidirect
102. Q₈	#13182039	semidirect

H ≅ G/N

type

1. Q₈×#11724786

C₂

semidirect

2. #20296060

C₂

semidirect

3. #20296045

C₂

semidirect

4. #20296030

C₂

semidirect

36. Q₈×C₁₀₉

M₁₆

semidirect

66. C₁₀₉

#2991

semidirect

102. Q₈

#13182039

semidirect

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

subgroup	index
Q₈×#11724786	2
#20296060	2
#20296045	2
#20296030	2
C₄×#13182030	2
#2991	109

subgroup

index

109

Subgroups — #28379966

Conjugacy classes of subgroups

Complements

Maximal subgroups