Group #27195945 — subgroups

Subgroups — #27195945

Order 13184.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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	C₂×#15692944	C₂	1	4	yes	maximal
3	C₂×C₄×C₁₀₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#384×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀₆	C₂²	1	2	yes	—
6	C₂²×C₁₀₃⋊C₈	C₂²	1	2	yes	—
7	#15693021	—	2	2	no	—
8	#15693021	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₀₃	—	2	2	no	—
10	#15692944	—	2	8	no	—
11	C₄×C₁₀₃⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₀₆	—	2	4	no	—
16	C₂×Dic₂₀₆	—	2	4	no	—
17	C₂×Dic₂₀₆	—	2	2	no	—
18	C₂×Dic₂₀₆	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₀₃	—	2	2	no	—
20	C₂×C₁₀₃⋊C₈	—	2	2	no	—
21	C₂×C₁₀₃⋊C₈	—	2	2	no	—
22	C₂×C₁₀₃⋊C₈	D₄	1	4	yes	—
23	C₄⋊Dic₁₀₃	—	2	8	no	—
24	C₂²×C₄×C₁₀₃	—	2	2	no	—
25	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₃	—	2	4	no	—
28	C₄⋊C₄×C₁₀₃	—	4	4	no	—
29	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₃	D₄	1	2	yes	—
31	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	—	4	2	no	—
33	C₂×C₄×C₁₀₃	—	4	4	no	—
34	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₀₃	SD₁₆	1	4	yes	—
36	C₂×C₄×C₁₀₃	Q₁₆	1	4	yes	—
37	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₃	#41	1	2	yes	—
41	C₂×Dic₁₀₃	—	2	4	no	—
42	C₂×Dic₁₀₃	—	4	2	no	—
43	C₂×Dic₁₀₃	—	4	2	no	—
44	Dic₂₀₆	—	2	8	no	—
45	Dic₂₀₆	—	4	4	no	—
46	C₁₀₃⋊C₈	—	2	4	no	—
47	C₂²×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₀₃	#124	1	1	yes	characteristic
49	C₂²×C₁₀₃	#127	1	1	yes	characteristic
50	C₂²×C₁₀₃	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₁₀₃	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₀₃	—	8	2	no	—
53	C₄×C₁₀₃	#103	1	8	yes	complemented
54	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
55	C₄×C₁₀₃	#121	1	2	yes	—
56	Dic₁₀₃	—	4	4	no	—
57	C₂×C₁₀₃	#390	1	1	yes	characteristic
58	C₂×C₁₀₃	C₂×#103	1	2	yes	—
59	C₂×C₁₀₃	#487	1	2	yes	—
60	C₂×C₁₀₃	#492	1	1	yes	characteristic
61	C₂×C₁₀₃	#494	1	1	yes	characteristic
62	#1956	—	103	1	no	maximal, Sylow
63	C₁₀₃	#1956	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	103	1	no	—
65	C₂×#103	—	103	4	no	—
66	C₂×C₄×C₈	—	103	1	no	—
67	#384	D₁₀₃	1	1	yes	characteristic
68	C₂²×Q₈	—	103	2	no	—
69	C₂²×C₈	—	103	2	no	—
70	C₄⋊Q₈	—	103	4	no	—
71	C₄⋊Q₈	—	206	2	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	206	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
75	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
76	#103	—	206	8	no	—
77	C₄×C₈	—	206	2	no	—
78	C₂×Q₈	—	103	4	no	—
79	C₂×Q₈	—	206	4	no	—
80	C₂×Q₈	—	206	4	no	—
81	C₂×Q₈	—	206	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	206	2	no	—
84	C₂²×C₄	C₂²⋊D₁₀₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
86	C₂×C₈	—	103	4	no	—
87	C₂×C₈	—	206	2	no	—
88	C₂×C₈	—	206	2	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	206	8	no	—
92	C₄²	C₄×D₁₀₃	1	2	yes	—
93	C₄²	D₄₁₂	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
95	Q₈	—	412	4	no	—
96	Q₈	—	206	8	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	412	2	no	—
100	C₂×C₄	—	412	2	no	—
101	C₂×C₄	—	206	4	no	—
102	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	4	yes	—
103	C₂×C₄	#13110093	1	4	yes	—
104	C₂×C₄	#13110095	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
106	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
107	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
108	C₂×C₄	#13110106	1	2	yes	—
109	C₈	—	206	4	no	—
110	C₂²	C₂×(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
111	C₂²	#15693010	1	1	yes	characteristic
112	C₂²	#15693017	1	1	yes	characteristic
113	C₂²	C₂×#13110093	1	2	yes	—
114	C₂²	C₂×#13110095	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	412	4	no	—
117	C₄	#15692944	1	8	yes	complemented
118	C₄	#15692983	1	2	yes	—
119	C₄	#15692996	1	2	yes	—
120	C₂	#20267692	1	1	yes	characteristic
121	C₂	C₂×#15692944	1	2	yes	—
122	C₂	#20268143	1	2	yes	—
123	C₂	#20268163	1	1	yes	characteristic
124	C₂	#20268170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15693021

C₂

yes

maximal, characteristic

C₂×#15692944

C₂

yes

maximal

C₂×C₄×C₁₀₃⋊C₈

C₂

yes

maximal, characteristic

#384×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

C₂²

yes

—

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

#15693021

—

#15693021

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₃

—

#15692944

—

C₄×C₁₀₃⋊C₈

—

C₂×C₄⋊C₄×C₁₀₃

—

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

C₂×C₄

yes

—

C₂²×Dic₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₄⋊Dic₁₀₃

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

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

SD₁₆

yes

—

C₂×C₄×C₁₀₃

Q₁₆

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

C₂×SD₁₆

yes

—

C₂²×C₁₀₃

C₂×Q₁₆

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#103

yes

complemented

C₄×C₁₀₃

C₄×D₄

yes

—

C₄×C₁₀₃

#121

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

#390

yes

characteristic

C₂×C₁₀₃

C₂×#103

yes

—

C₂×C₁₀₃

#487

yes

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#1956

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

103

—

C₂×#103

—

103

—

C₂×C₄×C₈

—

103

—

#384

D₁₀₃

yes

characteristic

C₂²×Q₈

—

103

—

C₂²×C₈

—

103

—

C₄⋊Q₈

—

103

—

C₄⋊Q₈

—

206

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

206

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#103

—

206

—

C₄×C₈

—

206

—

C₂×Q₈

—

103

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂²×C₄

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²⋊D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

103

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

206

—

C₄²

C₄×D₁₀₃

yes

—

C₄²

D₄₁₂

yes

—

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

Q₈

—

412

—

Q₈

—

206

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

412

—

100

C₂×C₄

—

412

—

101

C₂×C₄

—

206

—

102

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

103

C₂×C₄

#13110093

yes

—

104

C₂×C₄

#13110095

yes

—

105

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

106

C₂×C₄

D₄×D₁₀₃

yes

—

107

C₂×C₄

C₂×D₄₁₂

yes

characteristic

108

C₂×C₄

#13110106

yes

—

109

C₈

—

206

—

110

C₂²

C₂×(C₂×D₁₀₃)⋊C₄

yes

characteristic

111

C₂²

#15693010

yes

characteristic

112

C₂²

#15693017

yes

characteristic

113

C₂²

C₂×#13110093

yes

—

114

C₂²

C₂×#13110095

yes

—

115

C₄

—

116

C₄

—

412

—

117

C₄

#15692944

yes

complemented

118

C₄

#15692983

yes

—

119

C₄

#15692996

yes

—

120

C₂

#20267692

yes

characteristic

121

C₂

C₂×#15692944

yes

—

122

C₂

#20268143

yes

—

123

C₂

#20268163

yes

characteristic

124

C₂

#20268170

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₃
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#384×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	#384	D₁₀₃
hypocenter	C₁₀₃	#1956
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₃

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#384×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
8. #15693021	C₄	semidirect
53. C₄×C₁₀₃	#103	semidirect
63. C₁₀₃	#1956	semidirect
117. C₄	#15692944	semidirect

H ≅ G/N

type

8. #15693021

C₄

semidirect

53. C₄×C₁₀₃

#103

semidirect

63. C₁₀₃

#1956

semidirect

117. C₄

#15692944

semidirect

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

subgroup	index
C₂×#15693021	2
C₂×#15692944	2
C₂×C₄×C₁₀₃⋊C₈	2
#384×C₁₀₃	2
#1956	103

subgroup

index

C₂×#15693021

C₂×#15692944

C₂×C₄×C₁₀₃⋊C₈

#384×C₁₀₃

#1956

103

Subgroups — #27195945

Conjugacy classes of subgroups

Complements

Maximal subgroups