Group #28379577 — subgroups

Subgroups — #28379577

Order 13952.

424 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 152 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	(C₂×#89137)⋊Q₈	C₂	1	4	yes	maximal
2	#20295975	C₂	1	8	yes	maximal
3	#20295971	C₂	1	2	yes	maximal
4	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#89137	C₂²	1	4	yes	—
6	C₂×C₄×#89137	C₂²	1	4	yes	—
7	(C₂×#89137)⋊C₄	C₂²	1	8	yes	—
8	(C₂×#89137)⋊C₄	C₂²	1	4	yes	—
9	(C₂×#89137)⋊C₄	C₂²	1	8	yes	—
10	C₂×Q₈×D₁₀₉	C₂²	1	2	yes	—
11	#16812699	C₄	1	1	yes	characteristic, complemented
12	#16812698	C₄	1	4	yes	complemented
13	#16812697	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₁₀₉	C₂²	1	4	yes	—
15	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
17	C₂²×#89137	C₂³	1	8	yes	—
18	C₄⋊#89137	—	4	4	no	—
19	C₄×#89137	—	4	4	no	—
20	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₁₈	C₂×C₄	1	4	yes	—
22	Q₈×D₁₀₉	—	4	2	no	—
23	Q₈×D₁₀₉	—	4	2	no	—
24	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
25	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
27	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
28	#13182008	C₂×C₄	1	8	yes	—
29	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
33	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₁₀₉	C₂⁴	1	1	yes	characteristic
35	C₂×#89137	—	2	8	no	—
36	C₂×#89137	—	2	16	no	—
37	Q₈×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
41	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
42	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
44	C₄×D₁₀₉	—	2	2	no	—
45	C₄×D₁₀₉	—	2	4	no	—
46	C₄×D₁₀₉	—	2	4	no	—
47	C₄×D₁₀₉	—	2	4	no	—
48	Dic₂₁₈	—	4	4	no	—
49	Dic₂₁₈	—	4	2	no	—
50	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₁₀₉	H(5,𝔽₂)	1	2	yes	—
52	C₂×D₁₀₉	2₋¹⁺⁴	1	4	yes	—
53	#89137	—	4	8	no	—
54	C₄×C₁₀₉	—	2	1	no	—
55	C₄×C₁₀₉	—	2	4	no	—
56	C₄×C₁₀₉	—	2	2	no	—
57	Dic₁₀₉	—	2	1	no	—
58	Dic₁₀₉	—	2	4	no	—
59	Dic₁₀₉	—	2	2	no	—
60	C₂×C₁₀₉	#518	1	1	yes	characteristic
61	C₂×C₁₀₉	#519	1	2	yes	—
62	D₁₀₉	#563	1	4	yes	—
63	#2381	—	109	1	no	maximal, Sylow
64	C₁₀₉	#2381	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄⋊Q₈	—	109	1	no	—
66	#391	—	109	4	no	—
67	#387	—	109	8	no	—
68	#383	—	109	2	no	—
69	C₂²×Q₈	—	109	2	no	—
70	C₄⋊Q₈	—	109	2	no	—
71	C₄⋊Q₈	—	109	4	no	—
72	C₄⋊Q₈	—	109	1	no	—
73	C₄⋊Q₈	#89137	1	1	yes	characteristic, complemented
74	C₂×C₄⋊C₄	—	109	4	no	—
75	C₂×C₄⋊C₄	—	109	4	no	—
76	C₂×C₄²	—	109	1	no	—
77	C₂×C₄²	—	109	4	no	—
78	#95	—	109	8	no	—
79	#95	—	109	4	no	—
80	#95	—	109	8	no	—
81	C₂×Q₈	—	109	4	no	—
82	C₂×Q₈	—	109	2	no	—
83	C₂×Q₈	—	436	2	no	—
84	C₂×Q₈	—	436	2	no	—
85	C₂×Q₈	C₂×#89137	1	2	yes	—
86	C₂²×C₄	—	109	2	no	—
87	C₂²×C₄	—	109	1	no	—
88	C₂²×C₄	—	109	4	no	—
89	C₂²×C₄	—	109	8	no	—
90	C₄⋊C₄	—	109	8	no	—
91	C₄⋊C₄	—	109	4	no	—
92	C₄⋊C₄	—	436	4	no	—
93	C₄⋊C₄	C₂×#89137	1	4	yes	—
94	C₄²	—	109	1	no	—
95	C₄²	—	109	2	no	—
96	C₄²	—	436	4	no	—
97	C₄²	C₂×#89137	1	1	yes	characteristic
98	C₂³	—	109	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	436	2	no	—
101	Q₈	—	436	4	no	—
102	C₂×C₄	—	109	2	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	4	no	—
105	C₂×C₄	—	218	4	no	—
106	C₂×C₄	—	218	4	no	—
107	C₂×C₄	—	218	2	no	—
108	C₂×C₄	—	218	4	no	—
109	C₂×C₄	—	218	8	no	—
110	C₂×C₄	—	218	16	no	—
111	C₂×C₄	C₂²×#89137	1	2	yes	—
112	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
113	C₂×C₄	C₂²×#89137	1	4	yes	—
114	C₂²	—	109	2	no	—
115	C₂²	—	109	4	no	—
116	C₂²	C₂³×#89137	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	2	4	no	—
119	C₄	—	2	2	no	—
120	C₄	—	436	8	no	—
121	C₄	—	218	1	no	—
122	C₄	—	218	4	no	—
123	C₄	—	218	2	no	—
124	C₂	—	109	4	no	—
125	C₂	#20296093	1	1	yes	characteristic
126	C₂	#20296094	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

(C₂×#89137)⋊Q₈

C₂

yes

maximal

#20295975

C₂

yes

maximal

#20295971

C₂

yes

maximal

C₄⋊Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊#89137

C₂²

yes

—

C₂×C₄×#89137

C₂²

yes

—

(C₂×#89137)⋊C₄

C₂²

yes

—

(C₂×#89137)⋊C₄

C₂²

yes

—

(C₂×#89137)⋊C₄

C₂²

yes

—

C₂×Q₈×D₁₀₉

C₂²

yes

—

#16812699

C₄

yes

characteristic, complemented

#16812698

C₄

yes

complemented

#16812697

C₄

yes

complemented

C₄⋊C₄×D₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

C₂³

yes

—

C₄⋊#89137

—

C₄×#89137

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

Q₈×D₁₀₉

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂³

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×C₄×D₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂⁴

yes

characteristic

C₂×#89137

—

C₂×#89137

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×D₁₀₉

H(5,𝔽₂)

yes

—

C₂×D₁₀₉

2₋¹⁺⁴

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#518

yes

characteristic

C₂×C₁₀₉

#519

yes

—

D₁₀₉

#563

yes

—

#2381

—

109

maximal, Sylow

C₁₀₉

#2381

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

#391

—

109

—

#387

—

109

—

#383

—

109

—

C₂²×Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

#89137

yes

characteristic, complemented

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

#95

—

109

—

#95

—

109

—

#95

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

436

—

C₂×Q₈

—

436

—

C₂×Q₈

C₂×#89137

yes

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

436

—

C₄⋊C₄

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

436

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

Q₈

—

100

Q₈

—

436

—

101

Q₈

—

436

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

218

—

106

C₂×C₄

—

218

—

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

characteristic

113

C₂×C₄

C₂²×#89137

yes

—

114

C₂²

—

109

—

115

C₂²

—

109

—

116

C₂²

C₂³×#89137

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

436

—

121

C₄

—

218

—

122

C₄

—

218

—

123

C₄

—

218

—

124

C₂

—

109

—

125

C₂

#20296093

yes

characteristic

126

C₂

#20296094

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₄⋊Q₈×C₁₀₉	C₄
prime radical, cosocle	C₂²×D₁₀₉	C₂⁴
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#2381
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, cosocle

C₂²×D₁₀₉

C₂⁴

C₄⋊Q₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #16812699	C₄	semidirect
12. #16812698	C₄	semidirect
13. #16812697	C₄	semidirect
16. C₄⋊Q₈×C₁₀₉	C₄	semidirect
64. C₁₀₉	#2381	semidirect
73. C₄⋊Q₈	#89137	semidirect

H ≅ G/N

type

11. #16812699

C₄

semidirect

12. #16812698

C₄

semidirect

13. #16812697

C₄

semidirect

16. C₄⋊Q₈×C₁₀₉

C₄

semidirect

64. C₁₀₉

#2381

semidirect

73. C₄⋊Q₈

#89137

semidirect

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

subgroup	index
(C₂×#89137)⋊Q₈	2
#20295975	2
#20295971	2
C₄⋊Q₈×D₁₀₉	2
#2381	109

subgroup

index

C₄⋊Q₈×D₁₀₉

109

Subgroups — #28379577

Conjugacy classes of subgroups

Complements

Maximal subgroups