Group #26128282 — subgroups

Subgroups — #26128282

Order 12032.

206 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₄×Dic₃₇₆	C₂	1	1	yes	maximal, characteristic
2	#20242286	C₂	1	2	yes	maximal
3	#20241462	C₂	1	1	yes	maximal, characteristic
4	#20241353	C₂	1	2	yes	maximal
5	C₄₇×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₇₆	C₄	1	1	yes	characteristic
7	#15689253	—	2	2	no	—
8	C₄×Dic₁₈₈	C₂²	1	2	yes	—
9	#15688949	C₄	1	1	yes	characteristic
10	#15688928	C₄	1	2	yes	—
11	#15688876	C₂²	1	2	yes	—
12	#15688855	—	2	2	no	—
13	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₈²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₉₄	—	2	2	no	—
18	Dic₃₇₆	—	2	2	no	—
19	#13093958	C₂×C₄	1	2	yes	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	Dic₄₇⋊C₈	—	4	2	no	—
22	#13093949	—	2	2	no	—
23	#13093934	—	4	2	no	—
24	C₂×C₁₆×C₄₇	—	2	1	no	—
25	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₉₄	—	2	2	no	—
31	C₂×C₈×C₄₇	—	2	1	no	—
32	C₂×C₈×C₄₇	D₈	1	2	yes	—
33	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
35	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₄₇	—	2	2	no	—
37	Dic₄₇⋊C₄	—	4	2	no	—
38	C₄×Dic₄₇	—	4	2	no	—
39	C₂×C₄₇⋊C₈	—	4	2	no	—
40	Dic₁₈₈	—	2	2	no	—
41	Dic₁₈₈	—	2	2	no	—
42	C₁₆×C₄₇	—	2	2	no	—
43	C₁₆×C₄₇	—	2	1	no	—
44	C₁₆×C₄₇	M₁₆	1	2	yes	—
45	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
48	C₂×Dic₄₇	—	4	2	no	—
49	C₂×Dic₄₇	—	4	2	no	—
50	Dic₉₄	—	4	2	no	—
51	Dic₉₄	—	4	2	no	—
52	C₈×C₄₇	—	2	1	no	—
53	C₈×C₄₇	—	2	2	no	—
54	C₈×C₄₇	—	4	1	no	—
55	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₄₇	#131	1	1	yes	characteristic
57	C₄₇⋊C₈	—	8	2	no	—
58	#12085	—	47	1	no	maximal, Sylow
59	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	#436	1	1	yes	characteristic
62	C₄×C₄₇	#443	1	1	yes	characteristic
63	C₄×C₄₇	#510	1	2	yes	—
64	Dic₄₇	—	8	2	no	—
65	Dic₄₇	—	8	2	no	—
66	C₄×Q₃₂	—	47	1	no	—
67	#1591	—	47	2	no	—
68	#1372	—	47	1	no	—
69	#1332	—	47	2	no	—
70	#1311	D₄₇	1	1	yes	characteristic
71	C₂×C₄₇	#1589	1	1	yes	characteristic
72	C₂×C₄₇	#2176	1	1	yes	characteristic
73	C₂×C₄₇	#2179	1	1	yes	characteristic
74	C₂×Q₃₂	—	47	1	no	—
75	#446	—	94	2	no	—
76	C₄×Q₁₆	—	47	2	no	—
77	#366	—	47	1	no	—
78	#358	—	47	2	no	—
79	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
80	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
81	#335	—	47	2	no	—
82	Q₈⋊C₈	—	94	2	no	—
83	C₈²	C₂×D₄₇	1	1	yes	characteristic
84	C₄₇	#12085	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	47	2	no	—
86	C₄×Q₈	—	94	2	no	—
87	Q₃₂	—	94	2	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
90	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
91	#107	—	47	2	no	—
92	C₄⋊C₈	—	94	2	no	—
93	C₄⋊C₈	—	188	2	no	—
94	#103	—	94	2	no	—
95	#97	—	188	2	no	—
96	C₄×C₈	D₁₈₈	1	1	yes	characteristic
97	C₄×C₈	D₁₈₈	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×Q₈	—	94	2	no	—
100	Q₁₆	—	94	2	no	—
101	Q₁₆	—	94	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	188	2	no	—
104	C₂×C₈	D₃₇₆	1	2	yes	—
105	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
106	C₂×C₈	#10625350	1	1	yes	characteristic
107	C₄⋊C₄	—	94	2	no	—
108	C₄⋊C₄	—	188	2	no	—
109	C₄²	—	188	2	no	—
110	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	#10625324	1	2	yes	—
114	Q₈	—	188	2	no	—
115	Q₈	—	188	2	no	—
116	C₂×C₄	—	188	2	no	—
117	C₂×C₄	—	188	2	no	—
118	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
119	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
120	C₂×C₄	#13094046	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	2	no	—
123	C₈	—	4	1	no	—
124	C₈	—	376	2	no	—
125	C₈	C₂×#10625324	1	1	yes	characteristic
126	C₈	#13094030	1	1	yes	characteristic
127	C₂²	C₄×D₃₇₆	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	376	2	no	—
130	C₄	—	376	2	no	—
131	C₄	#15689202	1	1	yes	characteristic
132	C₄	#15689241	1	1	yes	characteristic
133	C₄	#15689566	1	2	yes	—
134	C₂	#20242274	1	1	yes	characteristic
135	C₂	#20245260	1	1	yes	characteristic
136	C₂	#20245275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₃₇₆

C₂

yes

maximal, characteristic

#20242286

C₂

yes

maximal

#20241462

C₂

yes

maximal, characteristic

#20241353

C₂

yes

maximal

C₄₇×#1311

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

C₄

yes

characteristic

#15689253

—

C₄×Dic₁₈₈

C₂²

yes

—

#15688949

C₄

yes

characteristic

#15688928

C₄

yes

—

#15688876

C₂²

yes

—

#15688855

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

C₂×C₄

yes

—

C₄×Dic₉₄

—

Dic₃₇₆

—

#13093958

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093949

—

#13093934

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×M₁₆

yes

characteristic

C₈×C₄₇

#131

yes

characteristic

C₄₇⋊C₈

—

#12085

—

maximal, Sylow

C₂²×C₄₇

C₄×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

Dic₄₇

—

C₄×Q₃₂

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

—

#446

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12085

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₇

yes

characteristic

C₂×C₁₆

C₄×D₄₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

188

—

#103

—

#97

—

188

—

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

188

—

104

C₂×C₈

D₃₇₆

yes

—

105

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

106

C₂×C₈

#10625350

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

188

—

109

C₄²

—

188

—

110

C₄²

C₂×D₁₈₈

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

#10625324

yes

—

114

Q₈

—

188

—

115

Q₈

—

188

—

116

C₂×C₄

—

188

—

117

C₂×C₄

—

188

—

118

C₂×C₄

C₄×D₁₈₈

yes

characteristic

119

C₂×C₄

C₂×D₃₇₆

yes

characteristic

120

C₂×C₄

#13094046

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

376

—

125

C₈

C₂×#10625324

yes

characteristic

126

C₈

#13094030

yes

characteristic

127

C₂²

C₄×D₃₇₆

yes

characteristic

128

C₄

—

129

C₄

—

376

—

130

C₄

—

376

—

131

C₄

#15689202

yes

characteristic

132

C₄

#15689241

yes

characteristic

133

C₄

#15689566

yes

—

134

C₂

#20242274

yes

characteristic

135

C₂

#20245260

yes

characteristic

136

C₂

#20245275

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₇₆
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1311	C₂
socle, prime radical	C₂²×C₄₇	C₄×D₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1311	D₄₇
hypocenter	C₄₇	#12085
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

C₂×D₃₇₆

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1311

C₂

socle, prime radical

C₂²×C₄₇

C₄×D₈

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

N	H ≅ G/N	type
84. C₄₇	#12085	semidirect

H ≅ G/N

type

84. C₄₇

#12085

semidirect

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

subgroup	index
C₄×Dic₃₇₆	2
#20242286	2
#20241462	2
#20241353	2
C₄₇×#1311	2
#12085	47

subgroup

index

C₄×Dic₃₇₆

C₄₇×#1311

Subgroups — #26128282

Conjugacy classes of subgroups

Complements

Maximal subgroups