Group #25008344 — subgroups

Subgroups — #25008344

Order 11008.

222 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 87 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×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₁₇₂	C₂	1	2	yes	maximal
3	#19132669	C₂	1	1	yes	maximal, characteristic
4	#19132560	C₂	1	2	yes	maximal
5	C₄₃×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₄₄	C₄	1	1	yes	characteristic
7	C₈×Dic₈₆	—	2	2	no	—
8	C₄×Dic₁₇₂	C₂²	1	2	yes	—
9	#14582725	C₄	1	1	yes	characteristic
10	#14582704	C₄	1	2	yes	—
11	#14582652	C₂²	1	2	yes	—
12	#14582631	—	2	2	no	—
13	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
15	C₈²×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
17	C₄×Dic₈₆	—	2	2	no	—
18	Dic₃₄₄	C₈	1	4	yes	complemented
19	#13063259	C₂×C₄	1	2	yes	—
20	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
21	Dic₄₃⋊C₈	—	4	2	no	—
22	#13063250	—	2	2	no	—
23	C₈×Dic₄₃	—	4	2	no	—
24	C₂×C₁₆×C₄₃	—	2	1	no	—
25	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₈₆	—	2	2	no	—
31	C₂×C₈×C₄₃	—	2	1	no	—
32	C₂×C₈×C₄₃	D₈	1	2	yes	—
33	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
35	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₄₃	—	2	2	no	—
37	Dic₄₃⋊C₄	—	4	2	no	—
38	C₄×Dic₄₃	—	4	2	no	—
39	C₂×C₄₃⋊C₈	—	4	2	no	—
40	Dic₁₇₂	—	2	2	no	—
41	Dic₁₇₂	C₂×C₈	1	4	yes	—
42	C₁₆×C₄₃	—	2	2	no	—
43	C₁₆×C₄₃	—	2	1	no	—
44	C₁₆×C₄₃	C₂×C₈	1	2	yes	—
45	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
48	C₂×Dic₄₃	—	4	2	no	—
49	C₂×Dic₄₃	—	4	2	no	—
50	Dic₈₆	—	2	4	no	—
51	Dic₈₆	—	4	2	no	—
52	C₈×C₄₃	—	2	1	no	—
53	C₈×C₄₃	—	2	2	no	—
54	C₈×C₄₃	—	4	1	no	—
55	C₈×C₄₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₄₃	#131	1	1	yes	characteristic
57	C₄₃⋊C₈	—	8	2	no	—
58	#12044	—	43	1	no	maximal, Sylow
59	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
60	C₄×C₄₃	—	2	1	no	—
61	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₄₃	#443	1	1	yes	characteristic
63	C₄×C₄₃	#510	1	2	yes	—
64	Dic₄₃	—	4	4	no	—
65	Dic₄₃	—	8	2	no	—
66	C₄×Q₃₂	—	43	1	no	—
67	C₈×Q₁₆	—	43	2	no	—
68	#1372	—	43	1	no	—
69	#1332	—	43	2	no	—
70	#1311	D₄₃	1	1	yes	characteristic
71	C₂×C₄₃	C₈×D₈	1	1	yes	characteristic
72	C₂×C₄₃	#2176	1	1	yes	characteristic
73	C₂×C₄₃	#2179	1	1	yes	characteristic
74	C₂×Q₃₂	—	43	1	no	—
75	C₈×Q₈	—	86	2	no	—
76	C₄×Q₁₆	—	43	2	no	—
77	#366	—	43	1	no	—
78	#358	—	43	2	no	—
79	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
80	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
81	#335	—	43	2	no	—
82	Q₈⋊C₈	—	86	2	no	—
83	C₈²	C₂×D₄₃	1	1	yes	characteristic
84	C₄₃	#12044	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	43	2	no	—
86	C₄×Q₈	—	86	2	no	—
87	Q₃₂	—	43	4	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
90	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
91	#107	—	43	2	no	—
92	C₄⋊C₈	—	86	2	no	—
93	C₄⋊C₈	—	172	2	no	—
94	#103	—	86	2	no	—
95	C₄×C₈	—	172	2	no	—
96	C₄×C₈	D₁₇₂	1	1	yes	characteristic
97	C₄×C₈	D₁₇₂	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
99	C₂×Q₈	—	86	2	no	—
100	Q₁₆	—	43	4	no	—
101	Q₁₆	—	86	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	172	2	no	—
104	C₂×C₈	D₃₄₄	1	2	yes	—
105	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
106	C₂×C₈	#10621898	1	1	yes	characteristic
107	C₄⋊C₄	—	86	2	no	—
108	C₄⋊C₄	—	172	2	no	—
109	C₄²	—	172	2	no	—
110	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	C₈×D₄₃	1	2	yes	—
114	Q₈	—	86	4	no	—
115	Q₈	—	172	2	no	—
116	C₂×C₄	—	172	2	no	—
117	C₂×C₄	—	172	2	no	—
118	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
119	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
120	C₂×C₄	#13063347	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	344	2	no	—
125	C₈	C₂×C₈×D₄₃	1	1	yes	characteristic
126	C₈	#13063331	1	1	yes	characteristic
127	C₂²	C₄×D₃₄₄	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	344	2	no	—
130	C₄	—	172	4	no	—
131	C₄	C₈×D₁₇₂	1	1	yes	characteristic
132	C₄	#14583017	1	1	yes	characteristic
133	C₄	#14583342	1	2	yes	—
134	C₂	C₈×D₃₄₄	1	1	yes	characteristic
135	C₂	#19136467	1	1	yes	characteristic
136	C₂	#19136482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₃₄₄

C₂

yes

maximal, characteristic

C₈×Dic₁₇₂

C₂

yes

maximal

#19132669

C₂

yes

maximal, characteristic

#19132560

C₂

yes

maximal

C₄₃×#1311

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

C₄

yes

characteristic

C₈×Dic₈₆

—

C₄×Dic₁₇₂

C₂²

yes

—

#14582725

C₄

yes

characteristic

#14582704

C₄

yes

—

#14582652

C₂²

yes

—

#14582631

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

C₂×C₄

yes

—

C₄×Dic₈₆

—

Dic₃₄₄

C₈

yes

complemented

#13063259

C₂×C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

#13063250

—

C₈×Dic₄₃

—

C₂×C₁₆×C₄₃

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

C₂×C₈

yes

—

C₁₆×C₄₃

—

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₂²×C₈

yes

characteristic

C₈×C₄₃

#131

yes

characteristic

C₄₃⋊C₈

—

#12044

—

maximal, Sylow

C₂²×C₄₃

C₄×D₈

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

C₈×D₄

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

Dic₄₃

—

C₄×Q₃₂

—

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

D₄₃

yes

characteristic

C₂×C₄₃

C₈×D₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×Q₃₂

—

C₈×Q₈

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄×C₁₆

C₂×D₄₃

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12044

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₃

yes

characteristic

C₂×C₁₆

C₄×D₄₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

172

—

#103

—

C₄×C₈

—

172

—

C₄×C₈

D₁₇₂

yes

characteristic

C₄×C₈

D₁₇₂

yes

characteristic

C₄×C₈

C₂²×D₄₃

yes

characteristic

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

172

—

104

C₂×C₈

D₃₄₄

yes

—

105

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

106

C₂×C₈

#10621898

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

172

—

109

C₄²

—

172

—

110

C₄²

C₂×D₁₇₂

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₈×D₄₃

yes

—

114

Q₈

—

115

Q₈

—

172

—

116

C₂×C₄

—

172

—

117

C₂×C₄

—

172

—

118

C₂×C₄

C₄×D₁₇₂

yes

characteristic

119

C₂×C₄

C₂×D₃₄₄

yes

characteristic

120

C₂×C₄

#13063347

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

344

—

125

C₈

C₂×C₈×D₄₃

yes

characteristic

126

C₈

#13063331

yes

characteristic

127

C₂²

C₄×D₃₄₄

yes

characteristic

128

C₄

—

129

C₄

—

344

—

130

C₄

—

172

—

131

C₄

C₈×D₁₇₂

yes

characteristic

132

C₄

#14583017

yes

characteristic

133

C₄

#14583342

yes

—

134

C₂

C₈×D₃₄₄

yes

characteristic

135

C₂

#19136467

yes

characteristic

136

C₂

#19136482

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₄₄
commutator	C₈×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1311	C₂
socle, prime radical	C₂²×C₄₃	C₄×D₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1311	D₄₃
hypocenter	C₄₃	#12044
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

C₂×D₃₄₄

commutator

C₈×C₄₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1311

C₂

socle, prime radical

C₂²×C₄₃

C₄×D₈

cosocle

C₄×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

N	H ≅ G/N	type
18. Dic₃₄₄	C₈	semidirect
84. C₄₃	#12044	semidirect

H ≅ G/N

type

18. Dic₃₄₄

C₈

semidirect

84. C₄₃

#12044

semidirect

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

subgroup	index
C₄×Dic₃₄₄	2
C₈×Dic₁₇₂	2
#19132669	2
#19132560	2
C₄₃×#1311	2
#12044	43

subgroup

index

C₄×Dic₃₄₄

C₈×Dic₁₇₂

#19132669

#19132560

C₄₃×#1311

#12044

Subgroups — #25008344

Conjugacy classes of subgroups

Complements

Maximal subgroups