Group #28336056 — subgroups

Subgroups — #28336056

Order 13696.

180 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20293030	C₂	1	1	yes	maximal, characteristic
2	#20293025	C₂	1	1	yes	maximal, characteristic
3	#20293006	C₂	1	1	yes	maximal, characteristic
4	#20292979	C₂	1	1	yes	maximal, characteristic
5	#20292977	C₂	1	1	yes	maximal, characteristic
6	C₈×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133808	—	2	2	no	—
9	#16812109	C₂²	1	1	yes	characteristic
10	#16812109	C₂²	1	1	yes	characteristic
11	#16812042	—	2	2	no	—
12	#16812041	—	2	2	no	—
13	#16812033	—	2	2	no	—
14	#16812032	—	2	2	no	—
15	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
19	#106×C₁₀₇	—	2	2	no	—
20	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₁₄	—	2	1	no	—
22	C₂×Dic₂₁₄	—	2	1	no	—
23	#13133808	—	4	4	no	—
24	C₂×C₁₀₇⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₀₇⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₁₀₇	—	2	1	no	—
27	C₄⋊Dic₁₀₇	—	2	1	no	—
28	C₄⋊Dic₁₀₇	—	2	1	no	—
29	C₄⋊Dic₁₀₇	—	2	1	no	—
30	C₂×Q₈×C₁₀₇	—	2	1	no	—
31	C₂×Q₈×C₁₀₇	—	2	1	no	—
32	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₀₇	—	2	1	no	—
34	C₄⋊C₄×C₁₀₇	—	2	1	no	—
35	C₄⋊C₄×C₁₀₇	—	2	1	no	—
36	C₄⋊C₄×C₁₀₇	—	2	1	no	—
37	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
38	Q₈×C₁₀₇	—	4	2	no	—
39	Q₈×C₁₀₇	—	4	2	no	—
40	C₂×C₄×C₁₀₇	—	4	1	no	—
41	C₂×C₄×C₁₀₇	—	4	1	no	—
42	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₁₀₇	—	4	1	no	—
46	C₂×Dic₁₀₇	—	4	1	no	—
47	Dic₂₁₄	—	4	2	no	—
48	Dic₂₁₄	—	4	2	no	—
49	C₈×C₁₀₇	Q₁₆	1	4	yes	complemented
50	C₁₀₇⋊C₈	—	2	2	no	—
51	C₁₀₇⋊C₈	SD₁₆	1	4	yes	—
52	C₂²×C₁₀₇	#127	1	1	yes	characteristic
53	C₄×C₁₀₇	—	8	1	no	—
54	C₄×C₁₀₇	—	8	1	no	—
55	C₄×C₁₀₇	C₂×SD₁₆	1	2	yes	—
56	C₄×C₁₀₇	C₂×Q₁₆	1	2	yes	—
57	C₄×C₁₀₇	#135	1	2	yes	—
58	Dic₁₀₇	—	8	1	no	—
59	Dic₁₀₇	—	8	1	no	—
60	C₂×C₁₀₇	#492	1	1	yes	characteristic
61	C₂×C₁₀₇	#494	1	1	yes	characteristic
62	C₂×C₁₀₇	#495	1	1	yes	characteristic
63	#1707	—	107	1	no	maximal, Sylow
64	C₁₀₇	#1707	1	1	yes	characteristic, complemented, Sylow
65	#500	—	107	1	no	—
66	#498	—	107	1	no	—
67	#498	D₁₀₇	1	1	yes	characteristic
68	#494	—	107	1	no	—
69	#487	—	107	1	no	—
70	#487	—	107	1	no	—
71	C₈²	—	107	1	no	—
72	C₂×Q₁₆	—	214	2	no	—
73	C₄⋊Q₈	—	107	1	no	—
74	C₄⋊Q₈	—	107	1	no	—
75	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
77	#107	—	214	2	no	—
78	#106	—	2	2	no	—
79	#106	—	214	2	no	—
80	#103	—	214	2	no	—
81	#103	—	214	2	no	—
82	C₄×C₈	—	107	1	no	—
83	C₄×C₈	—	107	1	no	—
84	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	214	1	no	—
88	C₂×Q₈	—	214	1	no	—
89	Q₁₆	—	428	4	no	—
90	C₂×C₈	—	107	2	no	—
91	C₂×C₈	—	107	2	no	—
92	C₂×C₈	C₂²⋊D₁₀₇	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	214	1	no	—
98	C₄⋊C₄	—	214	1	no	—
99	C₄⋊C₄	—	214	1	no	—
100	C₄⋊C₄	—	214	1	no	—
101	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	428	2	no	—
105	Q₈	—	428	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	428	1	no	—
109	C₂×C₄	—	428	1	no	—
110	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
113	C₈	—	107	4	no	—
114	C₈	—	214	2	no	—
115	C₈	#13133808	1	4	yes	complemented
116	C₂²	#16812105	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	856	1	no	—
120	C₄	—	856	1	no	—
121	C₄	SD₁₆×D₁₀₇	1	2	yes	—
122	C₄	C₂×#13133808	1	2	yes	—
123	C₄	#16812130	1	2	yes	—
124	C₂	#20292998	1	1	yes	characteristic
125	C₂	#20293006	1	1	yes	characteristic
126	C₂	#20293008	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293030

C₂

yes

maximal, characteristic

#20293025

C₂

yes

maximal, characteristic

#20293006

C₂

yes

maximal, characteristic

#20292979

C₂

yes

maximal, characteristic

#20292977

C₂

yes

maximal, characteristic

C₈×C₁₀₇⋊C₈

C₂

yes

maximal, characteristic

#498×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133808

—

#16812109

C₂²

yes

characteristic

#16812109

C₂²

yes

—

—

—

—

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

#106×C₁₀₇

—

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

#13133808

—

C₂×C₁₀₇⋊C₈

D₄

yes

—

C₂×C₁₀₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

C₈×C₁₀₇

Q₁₆

yes

complemented

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

SD₁₆

yes

—

C₂²×C₁₀₇

#127

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₄×C₁₀₇

C₂×Q₁₆

yes

—

C₄×C₁₀₇

#135

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

107

maximal, Sylow

C₁₀₇

#1707

yes

characteristic, complemented, Sylow

#500

—

107

—

#498

—

107

—

#498

D₁₀₇

yes

characteristic

#494

—

107

—

#487

—

107

—

#487

—

107

—

C₈²

—

107

—

C₂×Q₁₆

—

214

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

—

107

—

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

#107

—

214

—

#106

—

#106

—

214

—

#103

—

214

—

#103

—

214

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

Q₁₆

—

428

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

C₂²⋊D₁₀₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

100

C₄⋊C₄

—

214

—

101

C₄²

C₂²×D₁₀₇

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

428

—

105

Q₈

—

428

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

428

—

109

C₂×C₄

—

428

—

110

C₂×C₄

D₄×D₁₀₇

yes

characteristic

111

C₂×C₄

D₄×D₁₀₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

113

C₈

—

107

—

114

C₈

—

214

—

115

C₈

#13133808

yes

complemented

116

C₂²

#16812105

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

856

—

120

C₄

—

856

—

121

C₄

SD₁₆×D₁₀₇

yes

—

122

C₄

C₂×#13133808

yes

—

123

C₄

#16812130

yes

—

124

C₂

#20292998

yes

characteristic

125

C₂

#20293006

yes

characteristic

126

C₂

#20293008

yes

characteristic

	subgroup	G/H
center	C₂²	#16812105
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#498×C₁₀₇	C₂
socle, prime radical	C₂²×C₁₀₇	#127
hypercenter	#498	D₁₀₇
hypocenter	C₁₀₇	#1707
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#16812105

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#498×C₁₀₇

C₂

C₂²×C₁₀₇

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
49. C₈×C₁₀₇	Q₁₆	semidirect
64. C₁₀₇	#1707	semidirect
115. C₈	#13133808	semidirect

H ≅ G/N

type

49. C₈×C₁₀₇

Q₁₆

semidirect

64. C₁₀₇

#1707

semidirect

115. C₈

#13133808

semidirect

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

subgroup	index
#20293030	2
#20293025	2
#20293006	2
#20292979	2
#20292977	2
C₈×C₁₀₇⋊C₈	2
#498×C₁₀₇	2
#1707	107

subgroup

index

C₈×C₁₀₇⋊C₈

#498×C₁₀₇

#1707

107

Subgroups — #28336056

Conjugacy classes of subgroups

Complements

Maximal subgroups