Group #28390558 — subgroups

Subgroups — #28390558

Order 14464.

312 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄⋊#11744751	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#11744751	C₂	1	2	yes	maximal
3	#20298353	C₂	1	2	yes	maximal, complemented
4	C₂×#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11744751	C₂²	1	4	yes	—
6	C₄⋊#11744751	—	2	4	no	—
7	C₂²⋊#11744751	—	2	4	no	—
8	C₂³×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
9	#117×C₁₁₃	—	2	4	no	—
10	C₂×C₄⋊C₄×C₁₁₃	C₄	1	2	yes	—
11	C₂×C₂²⋊C₄×C₁₁₃	C₄	1	2	yes	—
12	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×#11744751	—	2	4	no	—
15	C₂×#11744751	—	2	4	no	—
16	C₂×#11744751	—	2	4	no	—
17	C₂⁴×C₁₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₁₃	—	2	2	no	—
19	C₂²×C₄×C₁₁₃	—	2	2	no	—
20	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
24	C₂²×C₄×C₁₁₃	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₁₃	—	2	4	no	—
27	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
28	C₄²×C₁₁₃	—	2	2	no	—
29	C₄²×C₁₁₃	D₄	1	4	yes	—
30	C₂³×C₁₁₃	—	2	1	no	—
31	C₂³×C₁₁₃	—	2	1	no	—
32	C₂³×C₁₁₃	—	2	4	no	—
33	C₂³×C₁₁₃	C₄⋊C₄	1	2	yes	—
34	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₁₃	—	2	1	no	—
36	C₂×C₄×C₁₁₃	—	2	1	no	—
37	C₂×C₄×C₁₁₃	—	2	4	no	—
38	C₂×C₄×C₁₁₃	—	2	4	no	—
39	C₂×C₄×C₁₁₃	—	2	4	no	—
40	C₂×C₄×C₁₁₃	—	2	2	no	—
41	C₂×C₄×C₁₁₃	—	2	4	no	—
42	C₂×C₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₁₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₁₃	#41	1	2	yes	—
48	#11744751	—	4	4	no	—
49	C₂²×C₁₁₃	—	2	2	no	—
50	C₂²×C₁₁₃	—	2	4	no	—
51	C₂²×C₁₁₃	—	4	2	no	—
52	C₂²×C₁₁₃	—	4	2	no	—
53	C₂²×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₁₃	#117	1	1	yes	characteristic
56	C₂²×C₁₁₃	#131	1	4	yes	—
57	C₄×C₁₁₃	—	2	2	no	—
58	C₄×C₁₁₃	—	2	4	no	—
59	C₄×C₁₁₃	—	4	2	no	—
60	C₄×C₁₁₃	#121	1	2	yes	—
61	C₄×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₁₁₃	—	4	2	no	—
63	C₂×C₁₁₃	#380	1	1	yes	characteristic
64	C₂×C₁₁₃	#408	1	2	yes	—
65	C₂×C₁₁₃	#424	1	2	yes	—
66	C₂×C₁₁₃	#433	1	2	yes	—
67	#1798	—	113	1	no	maximal, Sylow
68	C₁₁₃	#1798	1	1	yes	characteristic, complemented, Sylow
69	C₂×#117	D₁₁₃	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	113	2	no	—
71	C₂×C₂²⋊C₈	—	113	2	no	—
72	#336	—	113	2	no	—
73	C₂³×C₄	C₂×D₁₁₃	1	1	yes	characteristic
74	C₂²×C₈	—	113	4	no	—
75	#117	—	2	4	no	—
76	C₂×C₄⋊C₄	Dic₁₁₃	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₁₁₃	1	2	yes	—
78	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
80	C₄⋊C₈	—	226	4	no	—
81	C₂²⋊C₈	—	226	4	no	—
82	C₂⁴	C₂×Dic₁₁₃	1	1	yes	characteristic
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	Dic₂₂₆	1	2	yes	—
86	C₂²×C₄	D₄₅₂	1	2	yes	—
87	C₂²×C₄	C₂×Dic₁₁₃	1	2	yes	—
88	C₂²×C₄	C₂×Dic₁₁₃	1	2	yes	—
89	C₂²×C₄	C₂×Dic₁₁₃	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
91	C₂×C₈	—	226	4	no	—
92	C₂×C₈	—	226	4	no	—
93	C₂×C₈	—	226	4	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²⋊D₁₁₃	1	4	yes	—
98	C₂³	—	2	1	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	2	4	no	—
101	C₂³	C₄⋊Dic₁₁₃	1	2	yes	—
102	C₂³	C₂²×Dic₁₁₃	1	1	yes	characteristic
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	C₂²⋊Dic₁₁₃	1	4	yes	—
111	C₂×C₄	C₄⋊Dic₁₁₃	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₁₃	1	2	yes	—
113	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
115	C₂×C₄	#14268647	1	2	yes	—
116	C₈	—	452	4	no	—
117	C₂²	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	2	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×C₂²⋊Dic₁₁₃	1	1	yes	characteristic
122	C₂²	C₂×C₄⋊Dic₁₁₃	1	1	yes	characteristic
123	C₂²	#16813581	1	1	yes	characteristic
124	C₂²	#16813629	1	4	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	2	2	no	—
127	C₄	—	4	2	no	—
128	C₄	#16813596	1	2	yes	—
129	C₄	#16813601	1	2	yes	—
130	C₂	—	4	2	no	—
131	C₂	#20298458	1	1	yes	characteristic
132	C₂	#20298565	1	2	yes	—
133	C₂	#20298625	1	2	yes	—
134	C₂	#20298659	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11744751

C₂

yes

maximal, complemented

C₂×C₂²⋊#11744751

C₂

yes

maximal

#20298353

C₂

yes

maximal, complemented

C₂×#117×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×#11744751

C₂²

yes

—

C₄⋊#11744751

—

C₂²⋊#11744751

—

C₂³×C₄×C₁₁₃

C₂²

yes

characteristic

#117×C₁₁₃

—

C₂×C₄⋊C₄×C₁₁₃

C₄

yes

—

C₂×C₂²⋊C₄×C₁₁₃

C₄

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₁₃

C₂²

yes

—

—

—

C₂⁴×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

Q₈

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₁₃

—

C₂²⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

—

C₄²×C₁₁₃

D₄

yes

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

C₄⋊C₄

yes

—

C₂³×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁₃

#41

yes

—

#11744751

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

C₂²×C₁₁₃

yes

C₂²×C₁₁₃

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#121

yes

—

C₄×C₁₁₃

C₂²⋊Q₈

yes

—

C₂×C₁₁₃

—

C₂×C₁₁₃

#380

yes

characteristic

C₂×C₁₁₃

#408

yes

—

C₂×C₁₁₃

#424

yes

—

C₂×C₁₁₃

#433

yes

—

#1798

—

113

maximal, Sylow

C₁₁₃

#1798

yes

characteristic, complemented, Sylow

C₂×#117

D₁₁₃

yes

characteristic

C₂×C₄⋊C₈

—

113

—

C₂×C₂²⋊C₈

—

113

—

#336

—

113

—

C₂³×C₄

C₂×D₁₁₃

yes

characteristic

C₂²×C₈

—

113

—

#117

—

C₂×C₄⋊C₄

Dic₁₁₃

yes

—

C₂×C₂²⋊C₄

Dic₁₁₃

yes

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

C₄⋊C₈

—

226

—

C₂²⋊C₈

—

226

—

C₂⁴

C₂×Dic₁₁₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₂₂₆

yes

—

C₂²×C₄

D₄₅₂

yes

—

C₂²×C₄

C₂×Dic₁₁₃

yes

—

C₂²×C₄

C₂×Dic₁₁₃

yes

—

C₂²×C₄

C₂×Dic₁₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊D₁₁₃

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₄⋊Dic₁₁₃

yes

—

102

C₂³

C₂²×Dic₁₁₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²⋊Dic₁₁₃

yes

—

111

C₂×C₄

C₄⋊Dic₁₁₃

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₁₃

yes

—

113

C₂×C₄

C₂×D₄₅₂

yes

characteristic

114

C₂×C₄

C₂×Dic₂₂₆

yes

characteristic

115

C₂×C₄

#14268647

yes

—

116

C₈

—

452

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₂²⋊Dic₁₁₃

yes

characteristic

122

C₂²

C₂×C₄⋊Dic₁₁₃

yes

characteristic

123

C₂²

#16813581

yes

characteristic

124

C₂²

#16813629

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#16813596

yes

—

129

C₄

#16813601

yes

—

130

C₂

—

131

C₂

#20298458

yes

characteristic

132

C₂

#20298565

yes

—

133

C₂

#20298625

yes

—

134

C₂

#20298659

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₁₃
commutator, socle	C₂³×C₁₁₃	C₂²×C₄
Fitting	C₂×#117×C₁₁₃	C₂
prime radical	C₂⁴×C₁₁₃	C₂×C₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	C₂×#117	D₁₁₃
hypocenter	C₁₁₃	#1798
LCS	G ▷ C₂³×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₁₃

commutator, socle

C₂³×C₁₁₃

C₂²×C₄

Fitting

C₂×#117×C₁₁₃

C₂

prime radical

C₂⁴×C₁₁₃

C₂×C₄

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂³×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. C₂×C₄⋊#11744751	C₂	semidirect
3. #20298353	C₂	semidirect
68. C₁₁₃	#1798	semidirect

H ≅ G/N

type

1. C₂×C₄⋊#11744751

C₂

semidirect

3. #20298353

C₂

semidirect

68. C₁₁₃

#1798

semidirect

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

subgroup	index
C₂×C₄⋊#11744751	2
C₂×C₂²⋊#11744751	2
#20298353	2
C₂×#117×C₁₁₃	2
#1798	113

subgroup

index

C₂×C₄⋊#11744751

C₂×C₂²⋊#11744751

#20298353

C₂×#117×C₁₁₃

#1798

113

Subgroups — #28390558

Conjugacy classes of subgroups

Complements

Maximal subgroups