Group #34034464 — subgroups

Subgroups — #34034464

Order 18688.

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

normal subgroups only

This group has 202 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	C₂×#17989695	C₂	1	2	yes	maximal, complemented
2	C₄⋊#14556681	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#14556681	C₂	1	2	yes	maximal, complemented
4	C₄×#14556681	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#11924031	C₂	1	1	yes	maximal, characteristic
6	#17989695	—	2	4	no	—
7	C₂×#14556681	C₂²	1	2	yes	—
8	C₂×#14556681	C₂²	1	1	yes	characteristic
9	C₂×#14556681	C₂²	1	1	yes	characteristic
10	C₂²×#11924031	C₂²	1	2	yes	—
11	C₄⋊#11924031	C₄	1	1	yes	characteristic
12	C₂²⋊#11924031	C₄	1	2	yes	—
13	C₄×#11924031	C₂²	1	1	yes	characteristic
14	D₄×#10596831	C₄	1	1	yes	characteristic
15	#14556681	—	2	2	no	—
16	#14556681	—	2	1	no	—
17	#14556681	D₄	1	2	yes	complemented
18	C₂×#11924031	—	2	2	no	—
19	C₂×#11924031	—	2	2	no	—
20	C₂×#11924031	C₂×C₄	1	2	yes	—
21	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
22	C₂×#11924031	C₂³	1	1	yes	characteristic
23	C₂²×#10596831	C₂×C₄	1	2	yes	—
24	C₄⋊#10596831	C₈	1	1	yes	characteristic
25	C₂²⋊#10596831	C₈	1	2	yes	—
26	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₇₃	C₈	1	1	yes	characteristic
28	#11924031	—	2	2	no	—
29	#11924031	—	2	1	no	—
30	#11924031	C₂×D₄	1	1	yes	characteristic
31	#11924031	#41	1	1	yes	characteristic
32	C₂×#10596831	—	2	2	no	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
35	C₂×#10596831	C₂×C₈	1	2	yes	—
36	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
37	C₂×D₄×C₇₃	C₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
39	C₄⋊C₄×C₇₃	C₁₆	1	1	yes	characteristic
40	C₂²⋊C₄×C₇₃	C₁₆	1	2	yes	—
41	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂³×C₇₃	C₂×C₁₆	1	2	yes	—
43	D₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	2	no	—
45	C₂×C₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	#10596831	—	2	2	no	—
51	#10596831	—	2	1	no	—
52	#10596831	C₄×D₄	1	1	yes	characteristic
53	#10596831	#131	1	1	yes	characteristic
54	C₂²×C₇₃	—	2	2	no	—
55	C₂²×C₇₃	—	2	2	no	—
56	C₂²×C₇₃	—	2	2	no	—
57	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₇₃	—	2	2	no	—
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	M₆₄	1	2	yes	complemented
61	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₇₃	#504	1	1	yes	characteristic
63	#13646	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	—	4	2	no	—
65	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
66	C₂×C₇₃	C₂×M₆₄	1	1	yes	characteristic
67	C₂×C₇₃	#2258	1	1	yes	characteristic
68	C₂×M₆₄	—	73	2	no	—
69	D₄×C₁₆	—	73	1	no	—
70	C₄⋊C₃₂	—	73	1	no	—
71	C₂²⋊C₃₂	—	73	2	no	—
72	C₄×C₃₂	—	73	1	no	—
73	C₇₃	#13646	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₁₆	—	73	2	no	—
75	C₈×D₄	—	73	1	no	—
76	M₆₄	—	146	4	no	—
77	C₂×C₃₂	—	73	1	no	—
78	C₂×C₃₂	—	73	1	no	—
79	C₂×C₃₂	—	73	2	no	—
80	C₄⋊C₁₆	—	73	1	no	—
81	C₂²⋊C₁₆	—	73	2	no	—
82	C₄×C₁₆	—	73	1	no	—
83	C₂²×C₈	—	73	2	no	—
84	C₄×D₄	#10596842	1	1	yes	characteristic
85	C₂×C₁₆	—	73	1	no	—
86	C₂×C₁₆	—	73	1	no	—
87	C₂×C₁₆	—	73	2	no	—
88	C₂×C₁₆	—	146	2	no	—
89	C₂×C₁₆	—	146	2	no	—
90	C₄⋊C₈	—	73	1	no	—
91	C₂²⋊C₈	—	73	2	no	—
92	C₄×C₈	—	73	1	no	—
93	C₃₂	—	73	2	no	—
94	C₃₂	—	146	1	no	—
95	C₃₂	—	146	2	no	—
96	C₂×D₄	#11924040	1	1	yes	characteristic
97	C₂²×C₄	C₂×#10596842	1	2	yes	—
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	—	73	1	no	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	—	146	2	no	—
102	C₂×C₈	—	146	2	no	—
103	C₄⋊C₄	#11924040	1	1	yes	characteristic
104	C₂²⋊C₄	#11924040	1	2	yes	—
105	C₄²	C₂×#10596842	1	1	yes	characteristic
106	C₁₆	—	73	1	no	—
107	C₁₆	—	73	1	no	—
108	C₁₆	—	146	1	no	—
109	C₁₆	—	146	2	no	—
110	C₂³	C₂×#11924040	1	2	yes	—
111	D₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
115	C₂×C₄	C₂×#11924040	1	2	yes	—
116	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
117	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
118	C₈	—	73	1	no	—
119	C₈	—	73	1	no	—
120	C₈	—	146	2	no	—
121	C₈	—	146	1	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	C₂²×#11924040	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	#17989695	1	2	yes	complemented
129	C₄	D₄×#10596842	1	1	yes	characteristic
130	C₄	#17989708	1	1	yes	characteristic
131	C₂	—	4	2	no	—
132	C₂	D₄×#11924040	1	1	yes	characteristic
133	C₂	C₂×#17989695	1	1	yes	characteristic
134	C₂	#22627164	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989695

C₂

yes

maximal, complemented

C₄⋊#14556681

C₂

yes

maximal, characteristic, complemented

C₂²⋊#14556681

C₂

yes

maximal, complemented

C₄×#14556681

C₂

yes

maximal, characteristic, complemented

D₄×#11924031

C₂

yes

maximal, characteristic

#17989695

—

C₂×#14556681

C₂²

yes

—

C₂×#14556681

C₂²

yes

characteristic

C₂×#14556681

C₂²

yes

characteristic

C₂²×#11924031

C₂²

yes

—

C₄⋊#11924031

C₄

yes

characteristic

C₂²⋊#11924031

C₄

yes

—

C₄×#11924031

C₂²

yes

characteristic

D₄×#10596831

C₄

yes

—

—

D₄

yes

—

—

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂³

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

—

C₄⋊#10596831

C₈

yes

characteristic

C₂²⋊#10596831

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×D₄×C₇₃

C₈

yes

—

—

C₂×D₄

yes

yes

—

—

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×D₄×C₇₃

C₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₄⋊C₄×C₇₃

C₁₆

yes

characteristic

C₂²⋊C₄×C₇₃

C₁₆

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂³×C₇₃

C₂×C₁₆

yes

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

—

—

C₄×D₄

yes

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

M₆₄

yes

complemented

C₄×C₇₃

C₈×D₄

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

D₄×C₁₆

yes

characteristic

C₂×C₇₃

C₂×M₆₄

yes

characteristic

C₂×C₇₃

#2258

yes

characteristic

C₂×M₆₄

—

D₄×C₁₆

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₇₃

#13646

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₈×D₄

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₄×D₄

#10596842

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

146

—

C₃₂

—

146

—

C₂×D₄

#11924040

yes

characteristic

C₂²×C₄

C₂×#10596842

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

146

—

102

C₂×C₈

—

146

—

103

C₄⋊C₄

#11924040

yes

characteristic

104

C₂²⋊C₄

#11924040

yes

—

105

C₄²

C₂×#10596842

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

146

—

109

C₁₆

—

146

—

110

C₂³

C₂×#11924040

yes

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×#11924040

yes

characteristic

115

C₂×C₄

C₂×#11924040

yes

—

116

C₂×C₄

C₂×#11924040

yes

characteristic

117

C₂×C₄

C₂²×#10596842

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

146

—

121

C₈

—

146

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂²×#11924040

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

#17989695

yes

complemented

129

C₄

D₄×#10596842

yes

characteristic

130

C₄

#17989708

yes

characteristic

131

C₂

—

132

C₂

D₄×#11924040

yes

characteristic

133

C₂

C₂×#17989695

yes

characteristic

134

C₂

#22627164

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₄×D₄×C₇₃	C₈
prime radical	C₂×D₄×C₇₃	C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	C₄×D₄	#10596842
hypocenter	C₇₃	#13646
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₄×D₄×C₇₃

C₈

prime radical

C₂×D₄×C₇₃

C₁₆

C₂³

C₄×D₄

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#17989695	C₂	semidirect
2. C₄⋊#14556681	C₂	semidirect
3. C₂²⋊#14556681	C₂	semidirect
4. C₄×#14556681	C₂	semidirect
17. #14556681	D₄	semidirect
60. C₄×C₇₃	M₆₄	semidirect
73. C₇₃	#13646	semidirect
128. C₄	#17989695	semidirect

H ≅ G/N

type

1. C₂×#17989695

C₂

semidirect

2. C₄⋊#14556681

C₂

semidirect

3. C₂²⋊#14556681

C₂

semidirect

4. C₄×#14556681

C₂

semidirect

17. #14556681

D₄

semidirect

60. C₄×C₇₃

M₆₄

semidirect

73. C₇₃

#13646

semidirect

128. C₄

#17989695

semidirect

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

subgroup	index
C₂×#17989695	2
C₄⋊#14556681	2
C₂²⋊#14556681	2
C₄×#14556681	2
D₄×#11924031	2
#13646	73

subgroup

index

Subgroups — #34034464

Conjugacy classes of subgroups

Complements

Maximal subgroups