Group #27196117 — subgroups

Subgroups — #27196117

Order 13184.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15693021	C₂	1	1	yes	maximal, characteristic
2	#20267657	C₂	1	1	yes	maximal, characteristic
3	#20267554	C₂	1	4	yes	maximal
4	C₂×#97×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀₆	—	2	1	no	—
6	#15693021	—	2	2	no	—
7	#15693021	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₀₃	—	2	1	no	—
9	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
11	C₁₀₃⋊#95	—	2	1	no	—
12	C₁₀₃⋊#95	—	2	1	no	—
13	C₂²×C₈×C₁₀₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	#97×C₁₀₃	—	2	2	no	—
16	C₂×Dic₂₀₆	—	2	4	no	—
17	C₂×Dic₂₀₆	—	4	4	no	—
18	C₂²×Dic₁₀₃	—	2	2	no	—
19	C₂²×Dic₁₀₃	—	2	2	no	—
20	C₄⋊Dic₁₀₃	—	2	4	no	—
21	C₄⋊Dic₁₀₃	—	2	2	no	—
22	C₄⋊Dic₁₀₃	—	2	4	no	—
23	C₄⋊Dic₁₀₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₀₃	—	2	2	no	—
28	C₂×C₈×C₁₀₃	—	2	2	no	—
29	C₂×C₈×C₁₀₃	Q₈	1	4	yes	complemented
30	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
31	C₄²×C₁₀₃	D₄	1	2	yes	—
32	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₃	—	2	2	no	—
34	C₂×C₄×C₁₀₃	—	2	2	no	—
35	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₀₃	#41	1	2	yes	—
41	C₂×Dic₁₀₃	—	2	4	no	—
42	C₂×Dic₁₀₃	—	4	4	no	—
43	C₂×Dic₁₀₃	—	4	2	no	—
44	C₂×Dic₁₀₃	—	4	2	no	—
45	C₂×Dic₁₀₃	—	4	2	no	—
46	Dic₂₀₆	—	4	8	no	—
47	C₈×C₁₀₃	—	2	4	no	—
48	C₂²×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₀₃	#124	1	1	yes	characteristic
50	C₂²×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₃	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₁₀₃	#137	1	2	yes	—
53	C₄×C₁₀₃	—	2	2	no	—
54	C₄×C₁₀₃	—	2	2	no	—
55	C₄×C₁₀₃	C₄×Q₈	1	2	yes	—
56	C₄×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
57	Dic₁₀₃	—	4	4	no	—
58	Dic₁₀₃	—	8	2	no	—
59	C₂×C₁₀₃	#391	1	1	yes	characteristic
60	C₂×C₁₀₃	#418	1	1	yes	characteristic
61	C₂×C₁₀₃	#419	1	1	yes	characteristic
62	C₂×C₁₀₃	#489	1	2	yes	—
63	C₂×C₁₀₃	#501	1	2	yes	—
64	#1993	—	103	1	no	maximal, Sylow
65	C₁₀₃	#1993	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄⋊Q₈	—	103	1	no	—
67	C₂×#97	D₁₀₃	1	1	yes	characteristic
68	#382	—	103	1	no	—
69	#340	—	103	4	no	—
70	C₂²×Q₈	—	206	1	no	—
71	C₂²×C₈	C₂×D₁₀₃	1	2	yes	—
72	C₄⋊Q₈	—	103	4	no	—
73	C₄⋊Q₈	—	206	2	no	—
74	C₂×C₄⋊C₄	—	103	2	no	—
75	C₂×C₄⋊C₄	—	103	2	no	—
76	C₂×C₄⋊C₄	—	206	1	no	—
77	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
78	#97	—	2	2	no	—
79	#95	—	206	1	no	—
80	#95	—	206	1	no	—
81	C₂×Q₈	—	412	4	no	—
82	C₂×Q₈	—	206	4	no	—
83	C₂²×C₄	—	206	2	no	—
84	C₂²×C₄	—	206	2	no	—
85	C₂²×C₄	D₄₁₂	1	1	yes	characteristic
86	C₂²×C₄	D₄₁₂	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₂₀₆	1	4	yes	complemented
91	C₄⋊C₄	—	103	4	no	—
92	C₄⋊C₄	—	206	4	no	—
93	C₄⋊C₄	—	206	4	no	—
94	C₄⋊C₄	—	206	2	no	—
95	C₄²	C₄×D₁₀₃	1	2	yes	—
96	C₄²	C₂²⋊D₁₀₃	1	2	yes	—
97	C₂³	C₂×D₄₁₂	1	1	yes	characteristic
98	Q₈	—	412	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	412	2	no	—
102	C₂×C₄	—	412	2	no	—
103	C₂×C₄	—	412	2	no	—
104	C₂×C₄	—	412	4	no	—
105	C₂×C₄	—	206	4	no	—
106	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
107	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
110	C₂×C₄	C₂×Dic₂₀₆	1	2	yes	—
111	C₂×C₄	#13110105	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₂²	C₂×(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
114	C₂²	#15693009	1	1	yes	characteristic
115	C₂²	#15693021	1	1	yes	characteristic
116	C₂²	#15693049	1	2	yes	—
117	C₂²	#15693054	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	824	2	no	—
121	C₄	—	412	4	no	—
122	C₄	C₄×Dic₂₀₆	1	2	yes	—
123	C₄	#15693000	1	2	yes	—
124	C₂	#20267697	1	1	yes	characteristic
125	C₂	#20267797	1	1	yes	characteristic
126	C₂	#20267801	1	1	yes	characteristic
127	C₂	#20268151	1	2	yes	—
128	C₂	#20268200	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15693021

C₂

yes

maximal, characteristic

#20267657

C₂

yes

maximal, characteristic

#20267554

C₂

yes

maximal

C₂×#97×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

—

#15693021

—

#15693021

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₃

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₁₀₃⋊#95

—

C₁₀₃⋊#95

—

C₂²×C₈×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

#97×C₁₀₃

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂²×Dic₁₀₃

—

C₂²×Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

Q₈

yes

complemented

C₄²×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

D₄

yes

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₂²×C₁₀₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₃

#124

yes

characteristic

C₂²×C₁₀₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₃

#137

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄×Q₈

yes

—

C₄×C₁₀₃

C₂²⋊Q₈

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

C₂×C₁₀₃

#501

yes

—

#1993

—

103

maximal, Sylow

C₁₀₃

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

103

—

C₂×#97

D₁₀₃

yes

characteristic

#382

—

103

—

#340

—

103

—

C₂²×Q₈

—

206

—

C₂²×C₈

C₂×D₁₀₃

yes

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

206

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

206

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#97

—

#95

—

206

—

#95

—

206

—

C₂×Q₈

—

412

—

C₂×Q₈

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

D₄₁₂

yes

characteristic

C₂²×C₄

D₄₁₂

yes

characteristic

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₀₆

yes

complemented

C₄⋊C₄

—

103

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄²

C₄×D₁₀₃

yes

—

C₄²

C₂²⋊D₁₀₃

yes

—

C₂³

C₂×D₄₁₂

yes

characteristic

Q₈

—

412

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

412

—

102

C₂×C₄

—

412

—

103

C₂×C₄

—

412

—

104

C₂×C₄

—

412

—

105

C₂×C₄

—

206

—

106

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

107

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

108

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

110

C₂×C₄

C₂×Dic₂₀₆

yes

—

111

C₂×C₄

#13110105

yes

—

112

C₈

—

113

C₂²

C₂×(C₂×D₁₀₃)⋊C₄

yes

characteristic

114

C₂²

#15693009

yes

characteristic

115

C₂²

#15693021

yes

characteristic

116

C₂²

#15693049

yes

—

117

C₂²

#15693054

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

824

—

121

C₄

—

412

—

122

C₄

C₄×Dic₂₀₆

yes

—

123

C₄

#15693000

yes

—

124

C₂

#20267697

yes

characteristic

125

C₂

#20267797

yes

characteristic

126

C₂

#20267801

yes

characteristic

127

C₂

#20268151

yes

—

128

C₂

#20268200

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₁₂
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	C₂×#97×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	C₂×#97	D₁₀₃
hypocenter	C₁₀₃	#1993
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₄₁₂

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

C₂×#97×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

hypercenter

C₂×#97

D₁₀₃

hypocenter

C₁₀₃

#1993

LCS

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #15693021	C₄	semidirect
29. C₂×C₈×C₁₀₃	Q₈	semidirect
65. C₁₀₃	#1993	semidirect
90. C₂×C₈	Dic₂₀₆	semidirect

H ≅ G/N

type

7. #15693021

C₄

semidirect

29. C₂×C₈×C₁₀₃

Q₈

semidirect

65. C₁₀₃

#1993

semidirect

90. C₂×C₈

Dic₂₀₆

semidirect

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

subgroup	index
C₂×#15693021	2
#20267657	2
#20267554	2
C₂×#97×C₁₀₃	2
#1993	103

subgroup

index

C₂×#97×C₁₀₃

103

Subgroups — #27196117

Conjugacy classes of subgroups

Complements

Maximal subgroups