Group #28346802 — subgroups

Subgroups — C₂×#20292997 #28346802

Order 13696.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812109	C₂	1	1	yes	maximal, characteristic
2	#20292997	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812032	C₂	1	4	yes	maximal
4	C₂×#16812020	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₄	C₂²	1	2	yes	—
7	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
8	#16812109	—	2	2	no	—
9	#16812109	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₀₇	—	2	2	no	—
11	#16812032	C₂²	1	16	yes	—
12	#16812020	C₂²	1	4	yes	—
13	#125×C₁₀₇	—	2	2	no	—
14	#125×C₁₀₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
16	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₁₄	—	2	8	no	—
19	C₂×Dic₂₁₄	—	2	2	no	—
20	C₂×Dic₂₁₄	C₂³	1	4	yes	—
21	C₂²×Dic₁₀₇	—	2	2	no	—
22	C₂×C₁₀₇⋊C₈	—	2	4	no	—
23	C₂×C₁₀₇⋊C₈	C₂³	1	4	yes	—
24	C₄⋊Dic₁₀₇	—	2	8	no	—
25	C₂²×C₄×C₁₀₇	—	2	2	no	—
26	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₇	—	2	8	no	—
29	C₄⋊C₄×C₁₀₇	—	2	2	no	—
30	C₄⋊C₄×C₁₀₇	C₂³	1	4	yes	—
31	C₄²×C₁₀₇	D₄	1	2	yes	—
32	C₄²×C₁₀₇	C₂³	1	2	yes	—
33	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₇	—	2	4	no	—
35	C₂×C₄×C₁₀₇	—	2	4	no	—
36	C₂×C₄×C₁₀₇	—	4	4	no	—
37	C₂×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	#41	1	4	yes	—
40	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
41	C₂×Dic₁₀₇	—	2	4	no	—
42	C₂×Dic₁₀₇	—	4	4	no	—
43	Dic₂₁₄	—	2	8	no	—
44	Dic₂₁₄	—	4	4	no	—
45	C₁₀₇⋊C₈	—	2	4	no	—
46	C₂²×C₁₀₇	#124	1	2	yes	—
47	C₂²×C₁₀₇	#137	1	4	yes	—
48	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₁₀₇	—	2	4	no	—
50	C₄×C₁₀₇	—	4	4	no	—
51	C₄×C₁₀₇	#124	1	2	yes	—
52	C₄×C₁₀₇	C₂×#41	1	2	yes	—
53	Dic₁₀₇	—	4	4	no	—
54	C₂×C₁₀₇	#491	1	4	yes	complemented
55	C₂×C₁₀₇	C₂×#124	1	1	yes	characteristic
56	C₂×C₁₀₇	C₂×#137	1	2	yes	—
57	C₂×#491	—	107	1	no	maximal, Sylow
58	C₁₀₇	C₂×#491	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄⋊Q₈	—	107	1	no	—
60	C₂×#125	D₁₀₇	1	1	yes	characteristic
61	#491	—	107	8	no	—
62	C₂×#103	—	107	4	no	—
63	C₂×#97	—	107	1	no	—
64	C₂²×Q₈	—	107	2	no	—
65	C₂²×C₈	—	107	2	no	—
66	C₄⋊Q₈	—	107	4	no	—
67	C₄⋊Q₈	—	214	2	no	—
68	#125	—	2	2	no	—
69	#125	C₂×D₁₀₇	1	4	yes	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	214	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
73	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
74	#103	—	107	16	no	—
75	#97	—	107	4	no	—
76	C₂×Q₈	—	107	4	no	—
77	C₂×Q₈	—	214	8	no	—
78	C₂×Q₈	—	214	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	214	2	no	—
81	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
83	C₂×C₈	—	107	4	no	—
84	C₂×C₈	—	214	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	214	8	no	—
88	C₄⋊C₄	C₂²×D₁₀₇	1	4	yes	—
89	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
90	C₄²	C₂²×D₁₀₇	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
92	Q₈	—	428	4	no	—
93	Q₈	—	214	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	428	4	no	—
98	C₂×C₄	—	214	4	no	—
99	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
101	C₂×C₄	#13133819	1	4	yes	—
102	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
103	C₈	—	214	4	no	—
104	C₂²	#16812098	1	2	yes	—
105	C₂²	#16812139	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	428	4	no	—
110	C₄	#16812098	1	2	yes	—
111	C₄	C₂×#13133819	1	2	yes	—
112	C₂	#20292997	1	4	yes	direct product factor
113	C₂	C₂×#16812098	1	1	yes	characteristic
114	C₂	C₂×#16812139	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812109

C₂

yes

maximal, characteristic

#20292997

C₂

yes

maximal, direct product factor

C₂×#16812032

C₂

yes

maximal

C₂×#16812020

C₂

yes

maximal, characteristic

C₂×#125×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

#16812109

—

#16812109

C₂²

yes

—

C₂×C₄⋊Dic₁₀₇

—

#16812032

C₂²

yes

—

#16812020

C₂²

yes

—

#125×C₁₀₇

—

#125×C₁₀₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

C₂³

yes

—

C₂²×Dic₁₀₇

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

C₂³

yes

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

C₂⁴

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

#124

yes

—

C₂²×C₁₀₇

#137

yes

—

C₂²×C₁₀₇

C₂²×D₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#124

yes

—

C₄×C₁₀₇

C₂×#41

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

C₂×#491

—

107

maximal, Sylow

C₁₀₇

C₂×#491

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

107

—

C₂×#125

D₁₀₇

yes

characteristic

#491

—

107

—

C₂×#103

—

107

—

C₂×#97

—

107

—

C₂²×Q₈

—

107

—

C₂²×C₈

—

107

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

—

214

—

#125

—

#125

C₂×D₁₀₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

214

—

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#103

—

107

—

#97

—

107

—

C₂×Q₈

—

107

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂²×C₄

—

C₂²×C₄

—

214

—

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

214

—

C₄⋊C₄

C₂²×D₁₀₇

yes

—

C₄²

C₂²⋊D₁₀₇

yes

—

C₄²

C₂²×D₁₀₇

yes

—

C₂³

C₂×C₂²⋊D₁₀₇

yes

characteristic

Q₈

—

428

—

Q₈

—

214

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

428

—

C₂×C₄

—

214

—

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

101

C₂×C₄

#13133819

yes

—

102

C₂×C₄

C₂³×D₁₀₇

yes

characteristic

103

C₈

—

214

—

104

C₂²

#16812098

yes

—

105

C₂²

#16812139

yes

—

106

C₂²

C₂²×C₂²⋊D₁₀₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

428

—

110

C₄

#16812098

yes

—

111

C₄

C₂×#13133819

yes

—

112

C₂

#20292997

yes

direct product factor

113

C₂

C₂×#16812098

yes

characteristic

114

C₂

C₂×#16812139

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₇
commutator, cosocle	C₂×C₄×C₁₀₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting	C₂×#125×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂×D₄
hypercenter	C₂×#125	D₁₀₇
hypocenter	C₁₀₇	C₂×#491
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₇

commutator, cosocle

C₂×C₄×C₁₀₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₇

Fitting

C₂×#125×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂×D₄

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #20292997	C₂	direct (N × H)
54. C₂×C₁₀₇	#491	semidirect
58. C₁₀₇	C₂×#491	semidirect
112. C₂	#20292997	direct (N × H)

H ≅ G/N

type

2. #20292997

C₂

direct (N × H)

54. C₂×C₁₀₇

#491

semidirect

58. C₁₀₇

C₂×#491

semidirect

112. C₂

#20292997

direct (N × H)

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

subgroup	index
C₂×#16812109	2
#20292997	2
C₂×#16812032	2
C₂×#16812020	2
C₂×#125×C₁₀₇	2
C₂×#491	107

subgroup

index

C₂×#125×C₁₀₇

107

Subgroups — C₂×#20292997 #28346802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20292997 #28346802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20292997 #28346802