Group #28376607 — subgroups

Subgroups — C₂×#20295756 #28376607

Order 13952.

1112 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 581 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	#20295756	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16812698	C₂	1	4	yes	maximal
3	C₂×#16812697	C₂	1	4	yes	maximal
4	C₂×C₄×Dic₂₁₈	C₂	1	2	yes	maximal
5	C₂×#16812664	C₂	1	4	yes	maximal
6	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₈	C₂²	1	4	yes	—
8	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
9	#16812698	C₂²	1	32	yes	—
10	#16812697	C₂²	1	32	yes	—
11	C₄×Dic₂₁₈	C₂²	1	16	yes	—
12	#16812664	C₂²	1	32	yes	—
13	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
14	C₂×#13182008	C₂²	1	8	yes	—
15	C₂×#13182008	C₂²	1	4	yes	—
16	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
18	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₀₉	C₂²	1	8	yes	—
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
21	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₁₈	—	4	8	no	—
23	C₂×Dic₂₁₈	Q₈	1	16	yes	complemented
24	C₂×Dic₂₁₈	C₂³	1	16	yes	—
25	C₂×Dic₂₁₈	C₂³	1	8	yes	—
26	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
27	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
28	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
29	#13182008	C₂³	1	32	yes	—
30	#13182008	C₂³	1	16	yes	—
31	C₄×Dic₁₀₉	C₂³	1	16	yes	—
32	C₄×Dic₁₀₉	C₂³	1	16	yes	—
33	C₂×Q₈×C₁₀₉	—	4	4	no	—
34	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
35	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
39	C₄²×C₁₀₉	C₂³	1	4	yes	—
40	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
41	Q₈×C₁₀₉	—	4	8	no	—
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	—	2	8	no	—
44	C₂×C₄×C₁₀₉	C₂×Q₈	1	8	yes	—
45	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
47	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
48	C₂×Dic₁₀₉	—	2	8	no	—
49	C₂×Dic₁₀₉	C₂×Q₈	1	16	yes	—
50	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
51	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
52	Dic₂₁₈	—	4	16	no	—
53	Dic₂₁₈	C₂×Q₈	1	32	yes	complemented
54	C₂²×C₁₀₉	C₂²×Q₈	1	4	yes	—
55	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
56	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
57	C₄×C₁₀₉	—	2	8	no	—
58	C₄×C₁₀₉	—	2	2	no	—
59	C₄×C₁₀₉	C₂²×Q₈	1	8	yes	—
60	Dic₁₀₉	—	2	8	no	—
61	Dic₁₀₉	C₂²×Q₈	1	16	yes	—
62	C₂×C₁₀₉	Q₈²	1	4	yes	complemented
63	C₂×C₁₀₉	C₂³×Q₈	1	2	yes	—
64	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×Q₈²	—	109	1	no	maximal, Sylow
66	C₁₀₉	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
67	Q₈²	—	109	16	no	—
68	C₂×C₄⋊Q₈	—	109	4	no	—
69	C₂×C₄⋊Q₈	—	109	4	no	—
70	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic
71	C₂×C₄×Q₈	—	109	4	no	—
72	C₂×C₄×Q₈	—	109	2	no	—
73	C₂²×Q₈	—	109	2	no	—
74	C₂²×Q₈	—	109	4	no	—
75	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
76	C₄⋊Q₈	—	109	32	no	—
77	C₄⋊Q₈	—	109	32	no	—
78	C₄⋊Q₈	C₂×D₁₀₉	1	8	yes	—
79	C₄×Q₈	—	109	32	no	—
80	C₄×Q₈	—	109	16	no	—
81	C₂×C₄⋊C₄	—	109	8	no	—
82	C₂×C₄⋊C₄	—	109	4	no	—
83	C₂×C₄⋊C₄	—	109	2	no	—
84	C₂×C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
85	C₂×C₄²	—	109	4	no	—
86	C₂×C₄²	—	109	4	no	—
87	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
88	C₂×Q₈	—	4	4	no	—
89	C₂×Q₈	—	109	16	no	—
90	C₂×Q₈	—	109	8	no	—
91	C₂×Q₈	—	109	16	no	—
92	C₂×Q₈	—	436	8	no	—
93	C₂×Q₈	C₂²×D₁₀₉	1	8	yes	—
94	C₂²×C₄	—	109	4	no	—
95	C₂²×C₄	—	109	4	no	—
96	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
97	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
99	C₄⋊C₄	—	109	32	no	—
100	C₄⋊C₄	—	109	16	no	—
101	C₄⋊C₄	—	109	8	no	—
102	C₄⋊C₄	C₂²×D₁₀₉	1	16	yes	—
103	C₄²	—	109	16	no	—
104	C₄²	—	109	16	no	—
105	C₄²	C₂²×D₁₀₉	1	4	yes	—
106	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
107	Q₈	—	4	8	no	—
108	Q₈	—	109	32	no	—
109	Q₈	—	436	16	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	109	16	no	—
113	C₂×C₄	—	109	8	no	—
114	C₂×C₄	—	109	8	no	—
115	C₂×C₄	—	218	8	no	—
116	C₂×C₄	Q₈×D₁₀₉	1	8	yes	—
117	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
118	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
119	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
120	C₂²	C₂×Q₈×D₁₀₉	1	4	yes	—
121	C₂²	#16812744	1	2	yes	—
122	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	8	no	—
125	C₄	—	109	16	no	—
126	C₄	—	218	8	no	—
127	C₄	C₂×Q₈×D₁₀₉	1	8	yes	—
128	C₂	#20295756	1	4	yes	direct product factor
129	C₂	C₂²×Q₈×D₁₀₉	1	2	yes	—
130	C₂	C₂×#16812744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295756

C₂

yes

maximal, direct product factor

C₂×#16812698

C₂

yes

maximal

C₂×#16812697

C₂

yes

maximal

C₂×C₄×Dic₂₁₈

C₂

yes

maximal

C₂×#16812664

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₂²

yes

—

C₂²×Dic₂₁₈

C₂²

yes

—

#16812698

C₂²

yes

—

#16812697

C₂²

yes

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812664

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×#13182008

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

Q₈

yes

complemented

C₂×Dic₂₁₈

C₂³

yes

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

C₂³

yes

—

#13182008

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×Q₈

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

—

Dic₂₁₈

—

Dic₂₁₈

C₂×Q₈

yes

complemented

C₂²×C₁₀₉

C₂²×Q₈

yes

—

C₂²×C₁₀₉

H(5,𝔽₂)

yes

—

C₂²×C₁₀₉

C₂⁵

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

C₂²×Q₈

yes

—

C₂×C₁₀₉

Q₈²

yes

complemented

C₂×C₁₀₉

C₂³×Q₈

yes

—

C₂×C₁₀₉

C₂×H(5,𝔽₂)

yes

characteristic

C₂×Q₈²

—

109

maximal, Sylow

C₁₀₉

C₂×Q₈²

yes

characteristic, complemented, Sylow

Q₈²

—

109

—

C₂×C₄⋊Q₈

—

109

—

C₂×C₄⋊Q₈

—

109

—

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic

C₂×C₄×Q₈

—

109

—

C₂×C₄×Q₈

—

109

—

C₂²×Q₈

—

109

—

C₂²×Q₈

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

436

—

C₂×Q₈

C₂²×D₁₀₉

yes

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

109

—

100

C₄⋊C₄

—

109

—

101

C₄⋊C₄

—

109

—

102

C₄⋊C₄

C₂²×D₁₀₉

yes

—

103

C₄²

—

109

—

104

C₄²

—

109

—

105

C₄²

C₂²×D₁₀₉

yes

—

106

C₂³

C₂³×D₁₀₉

yes

characteristic

107

Q₈

—

108

Q₈

—

109

—

109

Q₈

—

436

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

109

—

113

C₂×C₄

—

109

—

114

C₂×C₄

—

109

—

115

C₂×C₄

—

218

—

116

C₂×C₄

Q₈×D₁₀₉

yes

—

117

C₂×C₄

C₂³×D₁₀₉

yes

—

118

C₂×C₄

C₂³×D₁₀₉

yes

—

119

C₂×C₄

C₂³×D₁₀₉

yes

—

120

C₂²

C₂×Q₈×D₁₀₉

yes

—

121

C₂²

#16812744

yes

—

122

C₂²

C₂⁴×D₁₀₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

109

—

126

C₄

—

218

—

127

C₄

C₂×Q₈×D₁₀₉

yes

—

128

C₂

#20295756

yes

direct product factor

129

C₂

C₂²×Q₈×D₁₀₉

yes

—

130

C₂

C₂×#16812744

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×Q₈²
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₁₀₉

commutator, cosocle

C₂²×C₁₀₉

C₂⁵

Frattini

C₂²

C₂⁴×D₁₀₉

Fitting

C₂×C₄⋊Q₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₁₀₉

hypocenter

C₁₀₉

C₂×Q₈²

LCS

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #20295756	C₂	direct (N × H)
23. C₂×Dic₂₁₈	Q₈	semidirect
53. Dic₂₁₈	C₂×Q₈	semidirect
62. C₂×C₁₀₉	Q₈²	semidirect
66. C₁₀₉	C₂×Q₈²	semidirect
128. C₂	#20295756	direct (N × H)

H ≅ G/N

type

1. #20295756

C₂

direct (N × H)

23. C₂×Dic₂₁₈

Q₈

semidirect

53. Dic₂₁₈

C₂×Q₈

semidirect

62. C₂×C₁₀₉

Q₈²

semidirect

66. C₁₀₉

C₂×Q₈²

semidirect

128. C₂

#20295756

direct (N × H)

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

subgroup	index
#20295756	2
C₂×#16812698	2
C₂×#16812697	2
C₂×C₄×Dic₂₁₈	2
C₂×#16812664	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂×Q₈²	109

subgroup

index

C₂×C₄×Dic₂₁₈

C₂×C₄⋊Q₈×C₁₀₉

C₂×Q₈²

109

Subgroups — C₂×#20295756 #28376607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295756 #28376607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295756 #28376607