Group #34033954 — subgroups

Subgroups — #34033954

Order 18688.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22627150	C₂	1	2	yes	maximal, complemented
2	#22627000	C₂	1	1	yes	maximal, characteristic, complemented
3	#22626982	C₂	1	2	yes	maximal, complemented
4	#22626946	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626332	C₂	1	1	yes	maximal, characteristic
6	#17989709	—	2	2	no	—
7	#17989692	C₂²	1	2	yes	—
8	#17989688	—	2	2	no	—
9	#64408⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₈×#10596842	C₂²	1	1	yes	characteristic
11	#17989631	C₂²	1	2	yes	—
12	#17989515	C₄	1	1	yes	characteristic
13	#17989502	C₄	1	2	yes	—
14	C₁₆×#64408	C₂²	1	1	yes	characteristic
15	#17989020	C₄	1	1	yes	characteristic
16	#14556688	—	2	2	no	—
17	#14556688	—	4	2	no	—
18	#14556686	—	4	2	no	—
19	#14556686	D₄	1	1	yes	characteristic
20	#14556686	D₄	1	1	yes	characteristic
21	C₄×#10596842	—	2	1	no	—
22	#14556679	—	2	2	no	—
23	#14556664	C₂×C₄	1	1	yes	characteristic
24	#14556662	C₂×C₄	1	2	yes	—
25	#14556656	—	2	2	no	—
26	C₈×#64408	C₂³	1	1	yes	characteristic
27	#14556614	C₂×C₄	1	2	yes	—
28	#14556534	C₈	1	2	yes	complemented
29	D₁₁₆₈	C₈	1	1	yes	characteristic, complemented
30	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
31	Q₃₂×C₇₃	C₈	1	1	yes	characteristic, complemented
32	C₂×#10596842	—	2	2	no	—
33	#11924040	—	2	1	no	—
34	#11924040	—	4	2	no	—
35	#11924040	D₈	1	2	yes	—
36	#11924038	—	2	2	no	—
37	#11924038	—	4	2	no	—
38	#11924036	—	4	2	no	—
39	#11924036	C₂×D₄	1	1	yes	characteristic
40	C₄×#64408	#41	1	1	yes	characteristic
41	#11924031	—	2	1	no	—
42	#11924029	—	2	2	no	—
43	#11924015	—	2	2	no	—
44	D₅₈₄	C₂×C₈	1	2	yes	—
45	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
46	#11924003	C₂×C₈	1	1	yes	characteristic
47	Q₁₆×C₇₃	C₂×C₈	1	2	yes	—
48	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂×#64408	—	2	1	no	—
50	#10596842	—	4	2	no	—
51	Q₈×C₇₃	—	2	2	no	—
52	D₂₉₂	—	2	2	no	—
53	D₂₉₂	—	4	2	no	—
54	C₄×D₇₃	—	4	2	no	—
55	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
56	#10596833	—	4	2	no	—
57	#10596833	C₂×D₈	1	1	yes	characteristic
58	#10596833	#135	1	1	yes	characteristic
59	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
60	#10596831	#131	1	1	yes	characteristic
61	C₂×D₇₃	—	4	2	no	—
62	C₂×D₇₃	#443	1	1	yes	characteristic
63	#64408	—	2	2	no	—
64	C₄×C₇₃	—	4	2	no	—
65	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
66	Dic₇₃	C₄×D₈	1	1	yes	characteristic
67	#12047	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	C₈×D₈	1	1	yes	characteristic
69	D₇₃	—	2	1	no	—
70	D₇₃	—	8	2	no	—
71	#2178	—	73	1	no	—
72	#2170	—	73	2	no	—
73	#1373	—	73	1	no	—
74	#1335	—	73	2	no	—
75	#1311	—	73	1	no	—
76	C₇₃	#12047	1	1	yes	characteristic, complemented, Sylow
77	#508	—	73	1	no	—
78	#504	—	146	2	no	—
79	#443	—	73	2	no	—
80	#368	—	73	1	no	—
81	#364	—	73	2	no	—
82	#359	—	73	2	no	—
83	#350	—	146	2	no	—
84	C₄×C₁₆	—	73	1	no	—
85	C₄×C₁₆	—	73	1	no	—
86	C₈²	—	73	1	no	—
87	#135	—	73	2	no	—
88	#131	—	146	2	no	—
89	Q₃₂	#10596842	1	1	yes	characteristic, complemented
90	SD₃₂	—	73	2	no	—
91	D₁₆	—	73	1	no	—
92	M₃₂	—	292	2	no	—
93	M₃₂	—	146	2	no	—
94	C₂×C₁₆	—	292	2	no	—
95	C₂×C₁₆	—	73	1	no	—
96	C₂×C₁₆	—	73	1	no	—
97	C₂×C₁₆	—	73	1	no	—
98	C₂×C₁₆	—	73	1	no	—
99	#108	—	73	2	no	—
100	#104	—	146	2	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	146	1	no	—
103	#41	—	146	2	no	—
104	Q₁₆	C₂×#10596842	1	2	yes	—
105	SD₁₆	—	146	2	no	—
106	D₈	—	73	2	no	—
107	M₁₆	—	292	2	no	—
108	M₁₆	—	146	2	no	—
109	C₂×C₈	—	292	2	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	146	2	no	—
113	C₄²	—	73	1	no	—
114	C₁₆	—	292	2	no	—
115	C₁₆	—	73	2	no	—
116	C₁₆	—	73	1	no	—
117	C₁₆	—	146	1	no	—
118	C₁₆	—	146	1	no	—
119	C₁₆	C₂×#10596842	1	1	yes	characteristic
120	Q₈	—	2	2	no	—
121	D₄	—	292	2	no	—
122	D₄	—	146	2	no	—
123	C₂×C₄	—	292	2	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	146	1	no	—
126	C₈	—	292	2	no	—
127	C₈	—	292	2	no	—
128	C₈	—	73	1	no	—
129	C₈	—	73	1	no	—
130	C₈	—	73	1	no	—
131	C₈	C₂²×#10596842	1	1	yes	characteristic
132	C₂²	—	292	2	no	—
133	C₂²	—	73	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	73	1	no	—
136	C₄	—	146	2	no	—
137	C₄	D₄×#10596842	1	1	yes	characteristic
138	C₂	—	584	2	no	—
139	C₂	—	146	1	no	—
140	C₂	D₈×#10596842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627150

C₂

yes

maximal, complemented

#22627000

C₂

yes

maximal, characteristic, complemented

#22626982

C₂

yes

maximal, complemented

#22626946

C₂

yes

maximal, characteristic, complemented

#22626332

C₂

yes

maximal, characteristic

#17989709

—

#17989692

C₂²

yes

—

#17989688

—

#64408⋊C₁₆

C₂²

yes

characteristic

C₈×#10596842

C₂²

yes

characteristic

#17989631

C₂²

yes

—

#17989515

C₄

yes

characteristic

#17989502

C₄

yes

—

C₁₆×#64408

C₂²

yes

characteristic

#17989020

C₄

yes

—

—

—

D₄

yes

characteristic

#14556686

D₄

yes

—

—

C₂×C₄

yes

characteristic

#14556662

C₂×C₄

yes

—

#14556656

—

C₈×#64408

C₂³

yes

characteristic

#14556614

C₂×C₄

yes

—

#14556534

C₈

yes

complemented

D₁₁₆₈

C₈

yes

characteristic, complemented

C₁₆×D₇₃

C₂×C₄

yes

characteristic

Q₃₂×C₇₃

C₈

yes

characteristic, complemented

—

—

—

D₈

yes

—

—

—

—

C₂×D₄

yes

yes

—

—

—

D₅₈₄

C₂×C₈

yes

—

C₈×D₇₃

C₂²×C₄

yes

characteristic

#11924003

C₂×C₈

yes

characteristic

Q₁₆×C₇₃

C₂×C₈

yes

—

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₂×#64408

—

#10596842

—

Q₈×C₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

C₄×D₄

yes

characteristic

#10596833

—

#10596833

C₂×D₈

yes

yes

C₈×C₇₃

C₂²×C₈

yes

yes

C₂×D₇₃

—

C₂×D₇₃

#443

yes

characteristic

#64408

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

Dic₇₃

C₄×D₈

yes

characteristic

#12047

—

maximal, Sylow

C₂×C₇₃

C₈×D₈

yes

characteristic

D₇₃

—

D₇₃

—

—

—

—

—

—

C₇₃

#12047

yes

characteristic, complemented, Sylow

#508

—

#504

—

146

—

—

—

—

—

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

#135

—

#131

—

146

—

Q₃₂

#10596842

yes

characteristic, complemented

SD₃₂

—

D₁₆

—

M₃₂

—

292

—

M₃₂

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

100

#104

—

146

—

101

C₄×C₈

—

102

C₄×C₈

—

146

—

103

#41

—

146

—

104

Q₁₆

C₂×#10596842

yes

—

105

SD₁₆

—

146

—

106

D₈

—

107

M₁₆

—

292

—

108

M₁₆

—

146

—

109

C₂×C₈

—

292

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

146

—

113

C₄²

—

114

C₁₆

—

292

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

146

—

118

C₁₆

—

146

—

119

C₁₆

C₂×#10596842

yes

characteristic

120

Q₈

—

121

D₄

—

292

—

122

D₄

—

146

—

123

C₂×C₄

—

292

—

124

C₂×C₄

—

125

C₂×C₄

—

146

—

126

C₈

—

292

—

127

C₈

—

292

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

C₂²×#10596842

yes

characteristic

132

C₂²

—

292

—

133

C₂²

—

134

C₄

—

135

C₄

—

136

C₄

—

146

—

137

C₄

D₄×#10596842

yes

characteristic

138

C₂

—

584

—

139

C₂

—

146

—

140

C₂

D₈×#10596842

yes

characteristic

	subgroup	G/H
center	C₂	D₈×#10596842
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₈	C₂²×#10596842
Fitting	Q₃₂×C₇₃	C₈
socle	C₂×C₇₃	C₈×D₈
prime radical	#17989020	C₄
cosocle	C₈×#64408	C₂³
hypercenter	Q₃₂	#10596842
hypocenter	C₇₃	#12047
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

D₈×#10596842

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₈

C₂²×#10596842

Fitting

Q₃₂×C₇₃

C₈

socle

C₂×C₇₃

C₈×D₈

C₄

C₂³

Q₃₂

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
1. #22627150	C₂	semidirect
2. #22627000	C₂	semidirect
3. #22626982	C₂	semidirect
4. #22626946	C₂	semidirect
28. #14556534	C₈	semidirect
29. D₁₁₆₈	C₈	semidirect
31. Q₃₂×C₇₃	C₈	semidirect
76. C₇₃	#12047	semidirect
89. Q₃₂	#10596842	semidirect

H ≅ G/N

type

1. #22627150

C₂

semidirect

2. #22627000

C₂

semidirect

3. #22626982

C₂

semidirect

4. #22626946

C₂

semidirect

28. #14556534

C₈

semidirect

29. D₁₁₆₈

C₈

semidirect

31. Q₃₂×C₇₃

C₈

semidirect

76. C₇₃

#12047

semidirect

89. Q₃₂

#10596842

semidirect

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

subgroup	index
#22627150	2
#22627000	2
#22626982	2
#22626946	2
#22626332	2
#12047	73

subgroup

index

Subgroups — #34033954

Conjugacy classes of subgroups

Complements

Maximal subgroups