Group #28349863 — subgroups

Subgroups — #28349863

Order 13696.

996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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	#20293256	C₂	1	12	yes	maximal, complemented
2	#20293253	C₂	1	3	yes	maximal
3	#20293179	C₂	1	12	yes	maximal, complemented
4	C₂×#16812065	C₂	1	3	yes	maximal
5	#543×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₀₇	C₂²	1	1	yes	characteristic
7	#16812108	C₂²	1	12	yes	complemented
8	#16812107	C₂²	1	12	yes	complemented
9	#16812099	C₂²	1	24	yes	complemented
10	#16812088	C₂²	1	12	yes	—
11	#16812086	C₂²	1	24	yes	—
12	Q₈×Dic₁₀₇	C₂²	1	4	yes	complemented
13	#16812074	C₂²	1	12	yes	complemented
14	#16812065	C₂²	1	12	yes	—
15	#16812065	C₂²	1	12	yes	—
16	C₂×Dic₁₀₇⋊C₄	C₂²	1	6	yes	—
17	C₂×C₂²⋊Dic₁₀₇	C₂²	1	3	yes	—
18	C₂×C₄×Dic₁₀₇	C₂²	1	6	yes	—
19	C₂²⋊Q₈×C₁₀₇	C₂²	1	12	yes	—
20	C₂×C₂²⋊C₄×C₁₀₇	C₂²	1	3	yes	—
21	C₂×Dic₂₁₄	C₂³	1	12	yes	—
22	C₂²×Dic₁₀₇	Q₈	1	8	yes	complemented
23	C₂²×Dic₁₀₇	C₂³	1	6	yes	—
24	C₂²×Dic₁₀₇	C₂³	1	3	yes	—
25	C₂²×Dic₁₀₇	C₂³	1	3	yes	—
26	C₄⋊Dic₁₀₇	C₂³	1	12	yes	—
27	Dic₁₀₇⋊C₄	C₂³	1	24	yes	—
28	Dic₁₀₇⋊C₄	C₂³	1	24	yes	—
29	C₂²⋊Dic₁₀₇	C₂³	1	12	yes	—
30	C₄×Dic₁₀₇	C₂³	1	12	yes	—
31	C₄×Dic₁₀₇	C₂³	1	12	yes	—
32	C₂⁴×C₁₀₇	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₀₇	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₀₇	C₂³	1	6	yes	—
35	C₄⋊C₄×C₁₀₇	C₂³	1	12	yes	—
36	C₂²⋊C₄×C₁₀₇	C₂³	1	12	yes	—
37	C₂³×C₁₀₇	—	4	2	no	—
38	C₂³×C₁₀₇	C₂×Q₈	1	4	yes	—
39	C₂³×C₁₀₇	C₂⁴	1	3	yes	—
40	Q₈×C₁₀₇	—	4	4	no	—
41	C₂×C₄×C₁₀₇	—	4	6	no	—
42	C₂×C₄×C₁₀₇	C₂⁴	1	12	yes	—
43	C₂×Dic₁₀₇	—	4	6	no	—
44	C₂×Dic₁₀₇	C₂×Q₈	1	12	yes	—
45	C₂×Dic₁₀₇	C₂×Q₈	1	12	yes	—
46	C₂×Dic₁₀₇	C₂⁴	1	12	yes	—
47	C₂×Dic₁₀₇	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₁₀₇	C₂⁴	1	3	yes	—
49	Dic₂₁₄	—	4	12	no	—
50	C₂²×C₁₀₇	—	2	6	no	—
51	C₂²×C₁₀₇	—	4	4	no	—
52	C₂²×C₁₀₇	C₂²×Q₈	1	6	yes	—
53	C₂²×C₁₀₇	C₂⁵	1	1	yes	characteristic
54	C₄×C₁₀₇	—	2	12	no	—
55	Dic₁₀₇	—	2	12	no	—
56	Dic₁₀₇	C₂²×Q₈	1	2	yes	—
57	Dic₁₀₇	C₂²×Q₈	1	6	yes	—
58	C₂×C₁₀₇	—	2	6	no	—
59	C₂×C₁₀₇	C₂³×Q₈	1	1	yes	characteristic
60	C₂×C₁₀₇	#585	1	2	yes	—
61	#3458	—	107	1	no	maximal, Sylow
62	C₁₀₇	#3458	1	1	yes	characteristic, complemented, Sylow
63	#544	—	107	12	no	—
64	#543	—	107	3	no	—
65	#543	D₁₀₇	1	1	yes	characteristic
66	#533	—	107	12	no	—
67	C₂×#117	—	107	3	no	—
68	C₂³×C₄	—	107	1	no	—
69	C₄⋊Q₈	—	107	12	no	—
70	C₄⋊Q₈	—	107	12	no	—
71	#125	—	107	24	no	—
72	C₂²⋊Q₈	—	107	12	no	—
73	C₂²⋊Q₈	—	107	24	no	—
74	C₂²⋊Q₈	C₂×D₁₀₇	1	12	yes	—
75	C₄×Q₈	—	107	4	no	—
76	C₄×Q₈	—	107	12	no	—
77	#117	—	107	12	no	—
78	#117	—	107	12	no	—
79	C₂×C₄⋊C₄	—	107	6	no	—
80	C₂×C₂²⋊C₄	—	107	3	no	—
81	C₂×C₂²⋊C₄	C₂×D₁₀₇	1	3	yes	—
82	C₂×C₄²	—	107	6	no	—
83	C₂⁴	C₂²×D₁₀₇	1	1	yes	characteristic
84	C₂×Q₈	—	107	12	no	—
85	C₂×Q₈	C₂²×D₁₀₇	1	4	yes	—
86	C₂²×C₄	—	107	8	no	—
87	C₂²×C₄	—	107	3	no	—
88	C₂²×C₄	—	107	3	no	—
89	C₂²×C₄	—	107	6	no	—
90	C₂²×C₄	C₂²×D₁₀₇	1	6	yes	—
91	C₄⋊C₄	—	107	24	no	—
92	C₄⋊C₄	—	107	24	no	—
93	C₄⋊C₄	—	107	12	no	—
94	C₄⋊C₄	C₂²×D₁₀₇	1	12	yes	—
95	C₂²⋊C₄	—	107	12	no	—
96	C₂²⋊C₄	C₂²×D₁₀₇	1	12	yes	—
97	C₄²	—	107	12	no	—
98	C₄²	—	107	12	no	—
99	C₂³	—	4	2	no	—
100	C₂³	Q₈×D₁₀₇	1	4	yes	—
101	C₂³	C₂³×D₁₀₇	1	3	yes	—
102	Q₈	—	4	4	no	—
103	Q₈	—	428	12	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	107	12	no	—
106	C₂×C₄	—	107	12	no	—
107	C₂×C₄	—	107	1	no	—
108	C₂×C₄	—	107	3	no	—
109	C₂×C₄	—	107	12	no	—
110	C₂×C₄	—	428	6	no	—
111	C₂×C₄	C₂³×D₁₀₇	1	12	yes	—
112	C₂²	—	2	6	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×Q₈×D₁₀₇	1	6	yes	—
115	C₂²	C₂⁴×D₁₀₇	1	1	yes	characteristic
116	C₄	—	2	12	no	—
117	C₄	—	107	2	no	—
118	C₄	—	107	6	no	—
119	C₄	—	214	12	no	—
120	C₂	—	2	6	no	—
121	C₂	C₂²×Q₈×D₁₀₇	1	1	yes	characteristic
122	C₂	#20293462	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20293256

C₂

yes

maximal, complemented

#20293253

C₂

yes

maximal

#20293179

C₂

yes

maximal, complemented

C₂×#16812065

C₂

yes

maximal

#543×C₁₀₇

C₂

yes

maximal, characteristic

C₂³×Dic₁₀₇

C₂²

yes

characteristic

#16812108

C₂²

yes

complemented

#16812107

C₂²

yes

complemented

#16812099

C₂²

yes

complemented

#16812088

C₂²

yes

—

#16812086

C₂²

yes

—

Q₈×Dic₁₀₇

C₂²

yes

complemented

#16812074

C₂²

yes

complemented

#16812065

C₂²

yes

—

#16812065

C₂²

yes

—

C₂×Dic₁₀₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₇

C₂²

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

—

C₂²⋊Q₈×C₁₀₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₇

C₂²

yes

—

C₂×Dic₂₁₄

C₂³

yes

—

C₂²×Dic₁₀₇

Q₈

yes

complemented

C₂²×Dic₁₀₇

C₂³

yes

—

C₂²×Dic₁₀₇

C₂³

yes

—

C₂²×Dic₁₀₇

C₂³

yes

—

C₄⋊Dic₁₀₇

C₂³

yes

—

Dic₁₀₇⋊C₄

C₂³

yes

—

Dic₁₀₇⋊C₄

C₂³

yes

—

C₂²⋊Dic₁₀₇

C₂³

yes

—

C₄×Dic₁₀₇

C₂³

yes

—

C₄×Dic₁₀₇

C₂³

yes

—

C₂⁴×C₁₀₇

C₂³

yes

characteristic

C₂×Q₈×C₁₀₇

C₂³

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

—

C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₂²⋊C₄×C₁₀₇

C₂³

yes

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₂×Q₈

yes

—

C₂³×C₁₀₇

C₂⁴

yes

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂⁴

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₂×Q₈

yes

—

C₂×Dic₁₀₇

C₂×Q₈

yes

—

C₂×Dic₁₀₇

C₂⁴

yes

—

C₂×Dic₁₀₇

C₂⁴

yes

characteristic

C₂×Dic₁₀₇

C₂⁴

yes

—

Dic₂₁₄

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂²×Q₈

yes

—

C₂²×C₁₀₇

C₂⁵

yes

characteristic

C₄×C₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

C₂²×Q₈

yes

—

Dic₁₀₇

C₂²×Q₈

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂³×Q₈

yes

characteristic

C₂×C₁₀₇

#585

yes

—

#3458

—

107

maximal, Sylow

C₁₀₇

#3458

yes

characteristic, complemented, Sylow

#544

—

107

—

#543

—

107

—

#543

D₁₀₇

yes

characteristic

#533

—

107

—

C₂×#117

—

107

—

C₂³×C₄

—

107

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

—

107

—

#125

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

—

107

—

C₂²⋊Q₈

C₂×D₁₀₇

yes

—

C₄×Q₈

—

107

—

C₄×Q₈

—

107

—

#117

—

107

—

#117

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₂²⋊C₄

—

107

—

C₂×C₂²⋊C₄

C₂×D₁₀₇

yes

—

C₂×C₄²

—

107

—

C₂⁴

C₂²×D₁₀₇

yes

characteristic

C₂×Q₈

—

107

—

C₂×Q₈

C₂²×D₁₀₇

yes

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂²×D₁₀₇

yes

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₄⋊C₄

C₂²×D₁₀₇

yes

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

C₂²×D₁₀₇

yes

—

C₄²

—

107

—

C₄²

—

107

—

C₂³

—

100

C₂³

Q₈×D₁₀₇

yes

—

101

C₂³

C₂³×D₁₀₇

yes

—

102

Q₈

—

103

Q₈

—

428

—

104

C₂×C₄

—

105

C₂×C₄

—

107

—

106

C₂×C₄

—

107

—

107

C₂×C₄

—

107

—

108

C₂×C₄

—

107

—

109

C₂×C₄

—

107

—

110

C₂×C₄

—

428

—

111

C₂×C₄

C₂³×D₁₀₇

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×Q₈×D₁₀₇

yes

—

115

C₂²

C₂⁴×D₁₀₇

yes

characteristic

116

C₄

—

117

C₄

—

107

—

118

C₄

—

107

—

119

C₄

—

214

—

120

C₂

—

121

C₂

C₂²×Q₈×D₁₀₇

yes

characteristic

122

C₂

#20293462

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₇
commutator, socle, cosocle	C₂²×C₁₀₇	C₂⁵
Fitting	#543×C₁₀₇	C₂
prime radical	C₂⁴×C₁₀₇	C₂³
hypercenter	#543	D₁₀₇
hypocenter	C₁₀₇	#3458
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ #543

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₀₇

commutator, socle, cosocle

C₂²×C₁₀₇

C₂⁵

Fitting

#543×C₁₀₇

C₂

prime radical

C₂⁴×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂² ◁ #543

N	H ≅ G/N	type
1. #20293256	C₂	semidirect
3. #20293179	C₂	semidirect
7. #16812108	C₂²	semidirect
8. #16812107	C₂²	semidirect
9. #16812099	C₂²	semidirect
12. Q₈×Dic₁₀₇	C₂²	semidirect
13. #16812074	C₂²	semidirect
22. C₂²×Dic₁₀₇	Q₈	semidirect
62. C₁₀₇	#3458	semidirect

H ≅ G/N

type

1. #20293256

C₂

semidirect

3. #20293179

C₂

semidirect

7. #16812108

C₂²

semidirect

8. #16812107

C₂²

semidirect

9. #16812099

C₂²

semidirect

12. Q₈×Dic₁₀₇

C₂²

semidirect

13. #16812074

C₂²

semidirect

22. C₂²×Dic₁₀₇

Q₈

semidirect

62. C₁₀₇

#3458

semidirect

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

subgroup	index
#20293256	2
#20293253	2
#20293179	2
C₂×#16812065	2
#543×C₁₀₇	2
#3458	107

subgroup

index

#543×C₁₀₇

107

Subgroups — #28349863

Conjugacy classes of subgroups

Complements

Maximal subgroups