Group #34011878 — subgroups

Subgroups — #34011878

Order 18688.

216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 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	#22626320	C₂	1	2	yes	maximal, complemented
2	#22624966	C₂	1	1	yes	maximal, characteristic, complemented
3	#22624894	C₂	1	2	yes	maximal, complemented
4	C₁₆×#10596833	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611811	C₂	1	1	yes	maximal, characteristic
6	#17989646	—	2	2	no	—
7	#17989509	C₂²	1	2	yes	—
8	#17989486	—	2	2	no	—
9	#17989473	C₂²	1	1	yes	characteristic
10	C₈×#10596833	C₂²	1	1	yes	characteristic
11	#17989019	C₄	1	1	yes	characteristic
12	#17988703	C₂²	1	2	yes	—
13	#17988393	C₄	1	2	yes	—
14	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
15	#368×C₇₃	C₄	1	1	yes	characteristic
16	#14556666	—	4	2	no	—
17	#14556665	—	2	2	no	—
18	#14556664	—	4	2	no	—
19	C₂×#11924031	D₄	1	1	yes	characteristic
20	C₂×#11924031	D₄	1	1	yes	characteristic
21	C₄×#10596833	—	2	1	no	—
22	#14556613	C₂×C₄	1	2	yes	—
23	#14556599	—	2	2	no	—
24	Dic₅₈₄	C₈	1	1	yes	characteristic, complemented
25	#14556536	C₈	1	2	yes	complemented
26	D₁₁₆₈	C₈	1	1	yes	characteristic, complemented
27	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
28	#14556517	—	2	2	no	—
29	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
31	#108×C₇₃	C₂×C₄	1	2	yes	—
32	C₂×#10596833	—	2	2	no	—
33	#11924031	—	2	1	no	—
34	#11924031	—	4	2	no	—
35	#11924031	D₈	1	1	yes	characteristic
36	#11924031	D₈	1	1	yes	characteristic
37	#11924028	—	2	2	no	—
38	Dic₂₉₂	C₂×C₈	1	2	yes	—
39	#11924017	—	2	2	no	—
40	D₅₈₄	C₂×C₈	1	2	yes	—
41	#11924011	—	4	2	no	—
42	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
43	C₈×D₇₃	—	4	2	no	—
44	C₄×Dic₇₃	#41	1	1	yes	characteristic
45	#11924003	—	2	1	no	—
46	M₁₆×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
51	C₂²⋊D₇₃	—	4	2	no	—
52	C₂×Dic₇₃	—	2	1	no	—
53	D₂₉₂	—	2	2	no	—
54	C₄×D₇₃	—	4	2	no	—
55	Dic₁₄₆	—	2	2	no	—
56	#10596833	—	2	4	no	—
57	C₈×C₇₃	—	4	2	no	—
58	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₇₃	#131	1	1	yes	characteristic
60	#10596831	C₂×D₈	1	1	yes	characteristic
61	#10596831	#135	1	1	yes	characteristic
62	C₂²×C₇₃	#443	1	1	yes	characteristic
63	C₂×D₇₃	—	4	2	no	—
64	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
65	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
66	Dic₇₃	—	2	2	no	—
67	Dic₇₃	—	4	2	no	—
68	#12046	—	73	1	no	maximal, Sylow
69	C₂×C₇₃	—	2	1	no	—
70	C₂×C₇₃	C₈×D₈	1	1	yes	characteristic
71	D₇₃	—	8	2	no	—
72	#2178	—	73	1	no	—
73	#2170	—	73	2	no	—
74	#1373	—	73	1	no	—
75	#1335	—	73	2	no	—
76	C₈×C₁₆	—	73	1	no	—
77	C₇₃	#12046	1	1	yes	characteristic, complemented, Sylow
78	#508	—	73	1	no	—
79	#504	—	146	2	no	—
80	#443	—	73	2	no	—
81	#368	#64408	1	1	yes	characteristic
82	#364	—	73	2	no	—
83	#359	—	73	2	no	—
84	#350	—	146	2	no	—
85	C₄×C₁₆	—	73	1	no	—
86	C₄×C₁₆	—	73	1	no	—
87	C₈²	—	73	1	no	—
88	#135	—	73	2	no	—
89	#131	—	146	2	no	—
90	Q₃₂	—	73	1	no	—
91	SD₃₂	—	73	2	no	—
92	D₁₆	—	73	1	no	—
93	M₃₂	—	292	2	no	—
94	M₃₂	—	146	2	no	—
95	C₂×C₁₆	—	292	2	no	—
96	C₂×C₁₆	—	73	1	no	—
97	C₂×C₁₆	—	73	1	no	—
98	C₂×C₁₆	—	73	1	no	—
99	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
100	#108	C₂×#64408	1	2	yes	—
101	#104	—	146	2	no	—
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	—	146	1	no	—
104	#41	—	146	2	no	—
105	Q₁₆	—	73	2	no	—
106	SD₁₆	—	146	2	no	—
107	D₈	—	73	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	292	2	no	—
110	C₂×C₈	—	292	2	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	146	2	no	—
113	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
114	C₄²	—	73	1	no	—
115	C₁₆	—	292	2	no	—
116	C₁₆	—	73	1	no	—
117	C₁₆	—	73	1	no	—
118	C₁₆	—	146	1	no	—
119	C₁₆	—	146	1	no	—
120	C₁₆	#11924036	1	1	yes	characteristic
121	C₁₆	#11924036	1	1	yes	characteristic
122	Q₈	—	146	2	no	—
123	D₄	—	292	2	no	—
124	D₄	—	146	2	no	—
125	C₂×C₄	—	292	2	no	—
126	C₂×C₄	—	146	1	no	—
127	C₂×C₄	D₄×#64408	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	73	1	no	—
130	C₈	—	73	1	no	—
131	C₈	—	146	4	no	—
132	C₈	C₂×#11924036	1	1	yes	characteristic
133	C₈	#14556679	1	1	yes	characteristic
134	C₂²	—	292	2	no	—
135	C₂²	#17989631	1	1	yes	characteristic
136	C₄	—	292	2	no	—
137	C₄	—	146	2	no	—
138	C₄	#17989615	1	1	yes	characteristic
139	C₄	D₈×#64408	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	584	2	no	—
142	C₂	#22625512	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626320

C₂

yes

maximal, complemented

#22624966

C₂

yes

maximal, characteristic, complemented

#22624894

C₂

yes

maximal, complemented

C₁₆×#10596833

C₂

yes

maximal, characteristic, complemented

#22611811

C₂

yes

maximal, characteristic

#17989646

—

#17989509

C₂²

yes

—

#17989486

—

#17989473

C₂²

yes

characteristic

C₈×#10596833

C₂²

yes

characteristic

#17989019

C₄

yes

characteristic

#17988703

C₂²

yes

—

#17988393

C₄

yes

—

C₁₆×Dic₇₃

C₂²

yes

characteristic

#368×C₇₃

C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11924031

D₄

yes

characteristic

C₄×#10596833

—

#14556613

C₂×C₄

yes

—

#14556599

—

Dic₅₈₄

C₈

yes

characteristic, complemented

#14556536

C₈

yes

complemented

D₁₁₆₈

C₈

yes

characteristic, complemented

C₂×#11924003

C₂×C₄

yes

characteristic

#14556517

—

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

#108×C₇₃

C₂×C₄

yes

—

—

—

—

D₈

yes

characteristic

#11924031

D₈

yes

characteristic

#11924028

—

Dic₂₉₂

C₂×C₈

yes

—

#11924017

—

D₅₈₄

C₂×C₈

yes

—

#11924011

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₈×D₇₃

—

C₄×Dic₇₃

#41

yes

characteristic

#11924003

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

C₂²×C₈

yes

C₈×C₇₃

yes

C₂×D₈

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

C₄×D₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

#12046

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₈×D₈

yes

characteristic

D₇₃

—

—

—

—

—

C₈×C₁₆

—

C₇₃

#12046

yes

characteristic, complemented, Sylow

#508

—

#504

—

146

—

—

yes

—

—

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

#135

—

#131

—

146

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

292

—

M₃₂

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

100

yes

—

101

#104

—

146

—

102

C₄×C₈

—

103

C₄×C₈

—

146

—

104

#41

—

146

—

105

Q₁₆

—

106

SD₁₆

—

146

—

107

D₈

—

108

M₁₆

—

109

M₁₆

—

292

—

110

C₂×C₈

—

292

—

111

C₂×C₈

—

112

C₂×C₈

—

146

—

113

C₂×C₈

C₂²×#64408

yes

characteristic

114

C₄²

—

115

C₁₆

—

292

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

146

—

119

C₁₆

—

146

—

120

C₁₆

#11924036

yes

characteristic

121

C₁₆

#11924036

yes

characteristic

122

Q₈

—

146

—

123

D₄

—

292

—

124

D₄

—

146

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

146

—

127

C₂×C₄

D₄×#64408

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

146

—

132

C₈

C₂×#11924036

yes

characteristic

133

C₈

#14556679

yes

characteristic

134

C₂²

—

292

—

135

C₂²

#17989631

yes

characteristic

136

C₄

—

292

—

137

C₄

—

146

—

138

C₄

#17989615

yes

characteristic

139

C₄

D₈×#64408

yes

characteristic

140

C₂

—

141

C₂

—

584

—

142

C₂

#22625512

yes

characteristic

	subgroup	G/H
center	C₄	D₈×#64408
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	#368×C₇₃	C₄
socle	C₂×C₇₃	C₈×D₈
prime radical	#17989019	C₄
cosocle	C₈×Dic₇₃	C₂³
hypercenter	#368	#64408
hypocenter	C₇₃	#12046
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

D₈×#64408

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

#368×C₇₃

C₄

socle

C₂×C₇₃

C₈×D₈

prime radical

#17989019

C₄

cosocle

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
1. #22626320	C₂	semidirect
2. #22624966	C₂	semidirect
3. #22624894	C₂	semidirect
4. C₁₆×#10596833	C₂	semidirect
24. Dic₅₈₄	C₈	semidirect
25. #14556536	C₈	semidirect
26. D₁₁₆₈	C₈	semidirect
77. C₇₃	#12046	semidirect

H ≅ G/N

type

1. #22626320

C₂

semidirect

2. #22624966

C₂

semidirect

3. #22624894

C₂

semidirect

4. C₁₆×#10596833

C₂

semidirect

24. Dic₅₈₄

C₈

semidirect

25. #14556536

C₈

semidirect

26. D₁₁₆₈

C₈

semidirect

77. C₇₃

#12046

semidirect

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

subgroup	index
#22626320	2
#22624966	2
#22624894	2
C₁₆×#10596833	2
#22611811	2
#12046	73

subgroup

index

C₁₆×#10596833

Subgroups — #34011878

Conjugacy classes of subgroups

Complements

Maximal subgroups