Group #34034448 — subgroups

Subgroups — #34034448

Order 18688.

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

normal subgroups only

This group has 180 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⋊#14556681	C₂	1	2	yes	maximal, complemented
2	#22627024	C₂	1	2	yes	maximal
3	#22627017	C₂	1	1	yes	maximal, characteristic, complemented
4	#22627014	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626141	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556681	C₂²	1	2	yes	—
7	C₂×#14556681	C₂²	1	2	yes	—
8	C₂×#14556666	C₄	1	1	yes	characteristic
9	C₂×#14556664	C₂²	1	1	yes	characteristic
10	#17989476	C₄	1	1	yes	characteristic
11	#17989475	C₄	1	1	yes	characteristic
12	#17989473	C₂²	1	1	yes	characteristic
13	C₄×#11924031	C₂²	1	1	yes	characteristic
14	#17988544	C₄	1	1	yes	characteristic
15	#14556681	—	2	2	no	—
16	#14556681	—	2	2	no	—
17	#14556666	—	2	2	no	—
18	#14556664	—	2	2	no	—
19	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
22	C₂×#11924031	C₂³	1	1	yes	characteristic
23	C₂×#11924011	C₈	1	1	yes	characteristic
24	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
25	#14556548	C₈	1	1	yes	characteristic
26	#14556501	C₈	1	1	yes	characteristic
27	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	#97×C₇₃	C₈	1	1	yes	characteristic
30	#11924031	—	2	1	no	—
31	#11924031	—	2	1	no	—
32	#11924031	M₁₆	1	2	yes	—
33	#11924031	#41	1	2	yes	—
34	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
35	#11924011	—	2	2	no	—
36	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
37	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
38	C₈×D₇₃	—	2	1	no	—
39	C₈×D₇₃	—	2	1	no	—
40	#11924007	C₁₆	1	2	yes	—
41	#11924005	C₁₆	1	2	yes	—
42	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×D₇₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
50	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
51	C₄×D₇₃	—	2	1	no	—
52	C₄×D₇₃	—	2	1	no	—
53	C₈×C₇₃	—	2	1	no	—
54	C₈×C₇₃	M₃₂	1	2	yes	—
55	#10596831	—	2	1	no	—
56	#10596831	#117	1	1	yes	characteristic
57	#10596831	C₂×M₁₆	1	1	yes	characteristic
58	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
59	C₂×D₇₃	—	2	1	no	—
60	C₂×D₇₃	—	2	2	no	—
61	C₄×C₇₃	—	2	2	no	—
62	C₄×C₇₃	#431	1	1	yes	characteristic
63	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
64	Dic₇₃	—	2	1	no	—
65	Dic₇₃	—	2	2	no	—
66	#13641	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	#2162	1	1	yes	characteristic
68	C₂×C₇₃	#2258	1	2	yes	—
69	D₇₃	—	4	1	no	—
70	#2108	—	73	1	no	—
71	C₄⋊C₃₂	—	73	2	no	—
72	C₂²⋊C₃₂	—	73	2	no	—
73	#1397	—	73	1	no	—
74	C₄×C₃₂	—	73	1	no	—
75	C₇₃	#13641	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₃₂	—	73	1	no	—
77	C₂²×C₁₆	—	73	1	no	—
78	#405	—	73	1	no	—
79	C₂×C₃₂	—	73	2	no	—
80	C₂×C₃₂	—	73	2	no	—
81	#346	—	73	1	no	—
82	#346	—	73	1	no	—
83	C₄×C₁₆	—	73	1	no	—
84	C₄×C₁₆	—	73	1	no	—
85	C₂×M₁₆	—	73	1	no	—
86	C₂²×C₈	—	73	1	no	—
87	#117	—	73	1	no	—
88	M₃₂	—	146	2	no	—
89	C₂×C₁₆	—	73	1	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	73	1	no	—
93	C₂×C₁₆	—	146	2	no	—
94	#97	—	73	1	no	—
95	#97	#10596842	1	1	yes	characteristic
96	C₄×C₈	—	73	1	no	—
97	C₄×C₈	—	73	1	no	—
98	C₃₂	—	146	2	no	—
99	C₃₂	—	146	2	no	—
100	C₂²×C₄	—	73	1	no	—
101	M₁₆	—	146	2	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	146	1	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
107	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
108	C₄⋊C₄	—	73	2	no	—
109	C₂²⋊C₄	—	73	2	no	—
110	C₄²	—	73	1	no	—
111	C₄²	C₂×#10596842	1	1	yes	characteristic
112	C₁₆	—	73	2	no	—
113	C₁₆	—	73	2	no	—
114	C₁₆	—	146	1	no	—
115	C₁₆	—	146	1	no	—
116	C₂³	—	73	1	no	—
117	C₂×C₄	—	73	2	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	146	1	no	—
120	C₂×C₄	—	146	1	no	—
121	C₂×C₄	#14556686	1	2	yes	—
122	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	73	1	no	—
125	C₈	—	73	1	no	—
126	C₈	—	146	1	no	—
127	C₈	#14556688	1	2	yes	—
128	C₂²	—	146	1	no	—
129	C₂²	—	146	2	no	—
130	C₂²	C₂×#14556686	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	146	1	no	—
133	C₄	—	146	2	no	—
134	C₄	#17989700	1	1	yes	characteristic
135	C₄	C₂×#14556688	1	1	yes	characteristic
136	C₂	—	292	1	no	—
137	C₂	#22627131	1	1	yes	characteristic
138	C₂	#22627165	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#14556681

C₂

yes

maximal, complemented

#22627024

C₂

yes

maximal

#22627017

C₂

yes

maximal, characteristic, complemented

#22627014

C₂

yes

maximal, characteristic, complemented

#22626141

C₂

yes

maximal, characteristic

C₂×#14556681

C₂²

yes

—

C₂×#14556681

C₂²

yes

—

C₂×#14556666

C₄

yes

characteristic

C₂×#14556664

C₂²

yes

characteristic

#17989476

C₄

yes

characteristic

#17989475

C₄

yes

characteristic

#17989473

C₂²

yes

characteristic

C₄×#11924031

C₂²

yes

characteristic

#17988544

C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂³

yes

characteristic

C₂×#11924011

C₈

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

#14556548

C₈

yes

characteristic

#14556501

C₈

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

#97×C₇₃

C₈

yes

—

—

M₁₆

yes

—

#11924031

#41

yes

—

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

#11924011

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

#11924007

C₁₆

yes

—

#11924005

C₁₆

yes

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₂×Dic₇₃

C₂×C₁₆

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

M₃₂

yes

—

—

yes

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#431

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2108

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

#1397

—

C₄×C₃₂

—

C₇₃

#13641

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₂×C₃₂

—

C₂×C₃₂

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

—

yes

C₄×C₈

—

C₄×C₈

—

C₃₂

—

146

—

C₃₂

—

146

—

100

C₂²×C₄

—

101

M₁₆

—

146

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

146

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₂×#10596842

yes

characteristic

107

C₂×C₈

C₂×#10596842

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₂×#10596842

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

146

—

115

C₁₆

—

146

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

146

—

120

C₂×C₄

—

146

—

121

C₂×C₄

#14556686

yes

—

122

C₂×C₄

C₂²×#10596842

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

146

—

127

C₈

#14556688

yes

—

128

C₂²

—

146

—

129

C₂²

—

146

—

130

C₂²

C₂×#14556686

yes

characteristic

131

C₄

—

132

C₄

—

146

—

133

C₄

—

146

—

134

C₄

#17989700

yes

characteristic

135

C₄

C₂×#14556688

yes

characteristic

136

C₂

—

292

—

137

C₂

#22627131

yes

characteristic

138

C₂

#22627165

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	#97×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂×C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	#97	#10596842
hypocenter	C₇₃	#13641
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

#97×C₇₃

C₈

prime radical

C₂²×D₇₃

C₂×C₁₆

C₂³

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
1. C₄⋊#14556681	C₂	semidirect
3. #22627017	C₂	semidirect
4. #22627014	C₂	semidirect
75. C₇₃	#13641	semidirect

H ≅ G/N

type

1. C₄⋊#14556681

C₂

semidirect

3. #22627017

C₂

semidirect

4. #22627014

C₂

semidirect

75. C₇₃

#13641

semidirect

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

subgroup	index
C₄⋊#14556681	2
#22627024	2
#22627017	2
#22627014	2
#22626141	2
#13641	73

subgroup

index

Subgroups — #34034448

Conjugacy classes of subgroups

Complements

Maximal subgroups