Group #26128024 — subgroups

Subgroups — #26128024

Order 12032.

222 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 99 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	Q₁₆×C₄₇⋊C₈	C₂	1	2	yes	maximal
2	#20241462	C₂	1	1	yes	maximal, characteristic
3	#20241352	C₂	1	2	yes	maximal
4	#20241304	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₄₇⋊C₈	—	2	2	no	—
7	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#15688876	C₂²	1	2	yes	—
9	Q₈⋊(C₄₇⋊C₈)	—	2	2	no	—
10	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₄₇	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#366	C₄	1	1	yes	characteristic
14	C₄₇×#358	C₄	1	2	yes	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄₇⋊C₁₆	—	2	1	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
18	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
19	C₄×C₄₇⋊C₈	—	4	2	no	—
20	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
21	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
23	C₄×Q₈×C₄₇	—	2	2	no	—
24	Q₃₂×C₄₇	C₈	1	4	yes	complemented
25	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
27	#107×C₄₇	C₂×C₄	1	2	yes	—
28	#103×C₄₇	—	2	2	no	—
29	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₄₇	—	2	2	no	—
31	Q₁₆×C₄₇	—	2	2	no	—
32	Q₁₆×C₄₇	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₄⋊C₄×C₄₇	—	4	2	no	—
37	C₄²×C₄₇	—	4	2	no	—
38	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	—	4	2	no	—
41	C₂×C₄₇⋊C₈	D₈	1	2	yes	—
42	C₁₆×C₄₇	—	2	1	no	—
43	C₁₆×C₄₇	C₂×C₈	1	2	yes	—
44	C₄₇⋊C₁₆	—	2	2	no	—
45	Q₈×C₄₇	—	2	4	no	—
46	Q₈×C₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
52	C₈×C₄₇	—	2	1	no	—
53	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₄₇	#131	1	1	yes	characteristic
55	C₄₇⋊C₈	—	2	2	no	—
56	C₄₇⋊C₈	—	4	1	no	—
57	C₄₇⋊C₈	—	8	2	no	—
58	#12044	—	47	1	no	maximal, Sylow
59	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	4	4	no	—
62	C₄×C₄₇	—	8	2	no	—
63	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
64	C₄×C₄₇	#443	1	1	yes	characteristic
65	C₄×C₄₇	#510	1	2	yes	—
66	C₄×Q₃₂	D₄₇	1	1	yes	characteristic
67	C₈×Q₁₆	—	47	2	no	—
68	#1372	—	47	1	no	—
69	#1332	—	47	2	no	—
70	#1311	—	47	1	no	—
71	C₂×C₄₇	C₈×D₈	1	1	yes	characteristic
72	C₂×C₄₇	#2176	1	1	yes	characteristic
73	C₂×C₄₇	#2179	1	1	yes	characteristic
74	C₂×Q₃₂	Dic₄₇	1	1	yes	characteristic
75	C₈×Q₈	—	94	2	no	—
76	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
77	#366	Dic₄₇	1	1	yes	characteristic
78	#358	Dic₄₇	1	2	yes	—
79	C₄×C₁₆	—	47	1	no	—
80	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
81	#335	—	47	2	no	—
82	Q₈⋊C₈	—	94	2	no	—
83	C₈²	—	47	1	no	—
84	C₄₇	#12044	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
86	C₄×Q₈	—	2	2	no	—
87	Q₃₂	C₄₇⋊C₈	1	4	yes	complemented
88	C₂×C₁₆	—	94	1	no	—
89	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
90	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
91	#107	C₂×Dic₄₇	1	2	yes	—
92	C₄⋊C₈	—	94	2	no	—
93	C₄⋊C₈	—	188	2	no	—
94	#103	—	2	2	no	—
95	C₄×C₈	—	47	1	no	—
96	C₄×C₈	—	47	1	no	—
97	C₄×C₈	—	188	2	no	—
98	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	C₂×C₄₇⋊C₈	1	4	yes	—
102	C₂×C₈	—	47	2	no	—
103	C₂×C₈	—	94	1	no	—
104	C₂×C₈	—	188	2	no	—
105	C₂×C₈	#10625352	1	1	yes	characteristic
106	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	—	4	2	no	—
110	C₄²	D₄×D₄₇	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	—	94	2	no	—
113	C₁₆	C₂×C₄₇⋊C₈	1	2	yes	—
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
119	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
120	C₂×C₄	#13094043	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	94	2	no	—
123	C₈	—	376	2	no	—
124	C₈	—	188	1	no	—
125	C₈	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
126	C₈	#13094031	1	1	yes	characteristic
127	C₂²	D₈×Dic₄₇	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	4	4	no	—
130	C₄	—	8	2	no	—
131	C₄	D₄×C₄₇⋊C₈	1	1	yes	characteristic
132	C₄	#15689240	1	1	yes	characteristic
133	C₄	#15689565	1	2	yes	—
134	C₂	D₈×C₄₇⋊C₈	1	1	yes	characteristic
135	C₂	#20245259	1	1	yes	characteristic
136	C₂	#20245274	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×C₄₇⋊C₈

C₂

yes

maximal

#20241462

C₂

yes

maximal, characteristic

#20241352

C₂

yes

maximal

#20241304

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₄₇

C₂

yes

maximal, characteristic

Q₈×C₄₇⋊C₈

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15688876

C₂²

yes

—

Q₈⋊(C₄₇⋊C₈)

—

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₄₇

C₄

yes

characteristic

C₄×Q₁₆×C₄₇

C₂²

yes

—

C₄₇×#366

C₄

yes

characteristic

C₄₇×#358

C₄

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₄₇

C₂×C₄

yes

—

C₄×Q₈×C₄₇

—

Q₃₂×C₄₇

C₈

yes

complemented

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

#107×C₄₇

C₂×C₄

yes

—

#103×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×C₈

yes

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

#131

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#12044

—

maximal, Sylow

C₂²×C₄₇

C₄×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₈×D₄

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×Q₃₂

D₄₇

yes

characteristic

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

—

C₂×C₄₇

C₈×D₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

Dic₄₇

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

C₂×D₄₇

yes

—

#366

Dic₄₇

yes

characteristic

#358

Dic₄₇

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

—

C₄₇

#12044

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×Dic₄₇

yes

—

C₄×Q₈

—

Q₃₂

C₄₇⋊C₈

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

#107

C₂×Dic₄₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

188

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

188

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

C₂×C₄₇⋊C₈

yes

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

188

—

105

C₂×C₈

#10625352

yes

characteristic

106

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

D₄×D₄₇

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₂×C₄₇⋊C₈

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

D₄×Dic₄₇

yes

characteristic

119

C₂×C₄

D₈×D₄₇

yes

characteristic

120

C₂×C₄

#13094043

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

376

—

124

C₈

—

188

—

125

C₈

C₂²×C₄₇⋊C₈

yes

characteristic

126

C₈

#13094031

yes

characteristic

127

C₂²

D₈×Dic₄₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

D₄×C₄₇⋊C₈

yes

characteristic

132

C₄

#15689240

yes

characteristic

133

C₄

#15689565

yes

—

134

C₂

D₈×C₄₇⋊C₈

yes

characteristic

135

C₂

#20245259

yes

characteristic

136

C₂

#20245274

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄×Q₃₂×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₄×D₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₄×Q₃₂	D₄₇
hypocenter	C₄₇	#12044
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂×C₄

D₈×D₄₇

commutator

C₈×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄×Q₃₂×C₄₇

C₂

socle, prime radical

C₂²×C₄₇

C₄×D₈

cosocle

C₄×C₈×C₄₇

C₂³

hypercenter

C₄×Q₃₂

D₄₇

hypocenter

C₄₇

#12044

LCS

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

N	H ≅ G/N	type
24. Q₃₂×C₄₇	C₈	semidirect
84. C₄₇	#12044	semidirect
87. Q₃₂	C₄₇⋊C₈	semidirect

H ≅ G/N

type

24. Q₃₂×C₄₇

C₈

semidirect

84. C₄₇

#12044

semidirect

87. Q₃₂

C₄₇⋊C₈

semidirect

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

subgroup	index
Q₁₆×C₄₇⋊C₈	2
#20241462	2
#20241352	2
#20241304	2
C₄×Q₃₂×C₄₇	2
#12044	47

subgroup

index

Q₁₆×C₄₇⋊C₈

C₄×Q₃₂×C₄₇

Subgroups — #26128024

Conjugacy classes of subgroups

Complements

Maximal subgroups