Group #28336590 — subgroups

Subgroups — #28336590

Order 13696.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₈×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
2	#20292492	C₂	1	2	yes	maximal
3	#20292390	C₂	1	1	yes	maximal, characteristic
4	#20292389	C₂	1	2	yes	maximal
5	#340×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₀₇	C₄	1	4	yes	complemented
10	C₁₀₇⋊#95	—	2	2	no	—
11	C₁₀₇⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
14	C₂²×Dic₁₀₇	—	2	2	no	—
15	C₂²×Dic₁₀₇	D₄	1	1	yes	characteristic
16	C₂²×Dic₁₀₇	D₄	1	1	yes	characteristic
17	C₂×C₁₀₇⋊C₈	—	2	2	no	—
18	C₂×C₁₀₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₁₀₇	—	2	2	no	—
20	C₄⋊Dic₁₀₇	—	2	2	no	—
21	C₄×Dic₁₀₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₀₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₀₇	—	2	2	no	—
24	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₀₇	—	2	2	no	—
26	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₁₀₇	—	2	2	no	—
28	C₄⋊C₄×C₁₀₇	—	2	2	no	—
29	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₇	—	4	2	no	—
31	C₂×C₄×C₁₀₇	—	4	2	no	—
32	C₂×C₄×C₁₀₇	—	4	2	no	—
33	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	#41	1	2	yes	—
36	C₂×C₄×C₁₀₇	#41	1	2	yes	—
37	C₂×Dic₁₀₇	—	2	1	no	—
38	C₂×Dic₁₀₇	—	2	2	no	—
39	C₂×Dic₁₀₇	—	2	1	no	—
40	C₂×Dic₁₀₇	—	4	2	no	—
41	C₂×Dic₁₀₇	—	4	2	no	—
42	C₂×Dic₁₀₇	—	4	2	no	—
43	C₂×Dic₁₀₇	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₁₀₇	C₄⋊C₄	1	2	yes	—
45	C₈×C₁₀₇	—	2	2	no	—
46	C₁₀₇⋊C₈	—	2	2	no	—
47	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₀₇	#124	1	2	yes	—
49	C₂²×C₁₀₇	#135	1	1	yes	characteristic
50	C₂²×C₁₀₇	#135	1	2	yes	—
51	C₂²×C₁₀₇	#135	1	1	yes	characteristic
52	C₄×C₁₀₇	—	8	2	no	—
53	C₄×C₁₀₇	#117	1	1	yes	characteristic
54	C₄×C₁₀₇	#117	1	1	yes	characteristic
55	C₄×C₁₀₇	#125	1	2	yes	—
56	Dic₁₀₇	—	2	2	no	—
57	Dic₁₀₇	—	2	2	no	—
58	Dic₁₀₇	—	8	2	no	—
59	C₂×C₁₀₇	#382	1	1	yes	characteristic
60	C₂×C₁₀₇	#427	1	1	yes	characteristic
61	C₂×C₁₀₇	#427	1	1	yes	characteristic
62	C₂×C₁₀₇	#488	1	2	yes	—
63	C₂×C₁₀₇	#488	1	2	yes	—
64	#1835	—	107	1	no	maximal, Sylow
65	C₁₀₇	#1835	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄×C₈	—	107	1	no	—
67	#382	—	107	2	no	—
68	#340	—	107	1	no	—
69	#340	—	107	2	no	—
70	#340	D₁₀₇	1	1	yes	characteristic
71	C₂²×C₈	—	107	1	no	—
72	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	107	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
75	C₂×C₄²	—	107	1	no	—
76	C₄×C₈	—	107	4	no	—
77	#95	—	214	2	no	—
78	#95	—	214	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	107	1	no	—
81	C₂²×C₄	—	107	1	no	—
82	C₂²×C₄	—	214	2	no	—
83	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	107	2	no	—
86	C₂×C₈	—	214	2	no	—
87	C₂×C₈	C₄×D₁₀₇	1	2	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	214	2	no	—
91	C₄⋊C₄	—	214	2	no	—
92	C₄²	—	107	2	no	—
93	C₄²	—	107	2	no	—
94	C₂³	D₄×D₁₀₇	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	107	2	no	—
99	C₂×C₄	—	107	2	no	—
100	C₂×C₄	—	428	2	no	—
101	C₂×C₄	—	428	2	no	—
102	C₂×C₄	—	428	2	no	—
103	C₂×C₄	—	214	1	no	—
104	C₂×C₄	—	214	2	no	—
105	C₂×C₄	—	214	1	no	—
106	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₁₀₇	1	1	yes	characteristic
108	C₂×C₄	#13133817	1	2	yes	—
109	C₂×C₄	#13133818	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	214	2	no	—
112	C₂²	C₄⋊C₄×D₁₀₇	1	1	yes	characteristic
113	C₂²	#16812095	1	2	yes	—
114	C₂²	#16812128	1	1	yes	characteristic
115	C₂²	#16812130	1	2	yes	—
116	C₂²	#16812132	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	856	2	no	—
119	C₄	—	214	2	no	—
120	C₄	—	214	2	no	—
121	C₄	#16812065	1	1	yes	characteristic
122	C₄	#16812066	1	1	yes	characteristic
123	C₄	#16812099	1	2	yes	—
124	C₂	#20292491	1	1	yes	characteristic
125	C₂	#20292662	1	1	yes	characteristic
126	C₂	#20292664	1	1	yes	characteristic
127	C₂	#20292980	1	2	yes	—
128	C₂	#20292983	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₀₇

C₂

yes

maximal, characteristic

#20292492

C₂

yes

maximal

#20292390

C₂

yes

maximal, characteristic

#20292389

C₂

yes

maximal

#340×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

C₂×C₄×Dic₁₀₇

C₂²

yes

characteristic

C₈×Dic₁₀₇

C₄

yes

complemented

C₁₀₇⋊#95

—

C₁₀₇⋊#95

—

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂²×Dic₁₀₇

—

C₂²×Dic₁₀₇

D₄

yes

characteristic

C₂²×Dic₁₀₇

D₄

yes

characteristic

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄×Dic₁₀₇

C₂×C₄

yes

—

C₄×Dic₁₀₇

Q₈

yes

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

C₄⋊C₄

yes

—

C₂×Dic₁₀₇

C₄⋊C₄

yes

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₇

#124

yes

—

C₂²×C₁₀₇

#135

yes

characteristic

C₂²×C₁₀₇

#135

yes

—

C₂²×C₁₀₇

#135

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

C₂×C₁₀₇

#488

yes

—

#1835

—

107

maximal, Sylow

C₁₀₇

#1835

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

107

—

#382

—

107

—

#340

—

107

—

#340

—

107

—

#340

D₁₀₇

yes

characteristic

C₂²×C₈

—

107

—

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

—

C₂×C₄²

—

107

—

C₄×C₈

—

107

—

#95

—

214

—

#95

—

214

—

C₂²×C₄

—

C₂²×C₄

—

107

—

C₂²×C₄

—

107

—

C₂²×C₄

—

214

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₂×C₈

C₄×D₁₀₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

—

107

—

C₄²

—

107

—

C₂³

D₄×D₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

107

—

C₂×C₄

—

107

—

100

C₂×C₄

—

428

—

101

C₂×C₄

—

428

—

102

C₂×C₄

—

428

—

103

C₂×C₄

—

214

—

104

C₂×C₄

—

214

—

105

C₂×C₄

—

214

—

106

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

107

C₂×C₄

Q₈×D₁₀₇

yes

characteristic

108

C₂×C₄

#13133817

yes

—

109

C₂×C₄

#13133818

yes

—

110

C₈

—

111

C₈

—

214

—

112

C₂²

C₄⋊C₄×D₁₀₇

yes

characteristic

113

C₂²

#16812095

yes

—

114

C₂²

#16812128

yes

characteristic

115

C₂²

#16812130

yes

—

116

C₂²

#16812132

yes

characteristic

117

C₄

—

118

C₄

—

856

—

119

C₄

—

214

—

120

C₄

—

214

—

121

C₄

#16812065

yes

characteristic

122

C₄

#16812066

yes

characteristic

123

C₄

#16812099

yes

—

124

C₂

#20292491

yes

characteristic

125

C₂

#20292662

yes

characteristic

126

C₂

#20292664

yes

characteristic

127

C₂

#20292980

yes

—

128

C₂

#20292983

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	#340×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	#340	D₁₀₇
hypocenter	C₁₀₇	#1835
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₀₇

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

#340×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂×D₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
9. C₈×Dic₁₀₇	C₄	semidirect
65. C₁₀₇	#1835	semidirect

H ≅ G/N

type

9. C₈×Dic₁₀₇

C₄

semidirect

65. C₁₀₇

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₁₀₇	2
#20292492	2
#20292390	2
#20292389	2
#340×C₁₀₇	2
#1835	107

subgroup

index

C₂×C₈×Dic₁₀₇

#340×C₁₀₇

107

Subgroups — #28336590

Conjugacy classes of subgroups

Complements

Maximal subgroups