Group #34034466 — subgroups

Subgroups — #34034466

Order 18688.

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

normal subgroups only

This group has 202 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	C₂×#17989696	C₂	1	2	yes	maximal, complemented
2	C₄⋊#14556681	C₂	1	1	yes	maximal, characteristic, complemented
3	#22627024	C₂	1	2	yes	maximal, complemented
4	C₄×#14556681	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626310	C₂	1	1	yes	maximal, characteristic
6	#17989696	—	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	C₂×#14556664	C₂²	1	2	yes	—
11	C₄⋊#11924031	C₄	1	1	yes	characteristic
12	#17989482	C₄	1	2	yes	—
13	C₄×#11924031	C₂²	1	1	yes	characteristic
14	#17988652	C₄	1	1	yes	characteristic
15	#14556681	—	2	2	no	—
16	#14556681	—	2	1	no	—
17	#14556681	D₄	1	2	yes	complemented
18	#14556664	—	2	2	no	—
19	#14556664	—	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	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
24	C₄×D₂₉₂	C₈	1	1	yes	characteristic
25	#14556503	C₈	1	2	yes	—
26	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×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×D₂₉₂	C₁₆	1	1	yes	characteristic
33	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
34	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
35	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
36	C₈×D₇₃	—	2	2	no	—
37	C₈×D₇₃	—	2	2	no	—
38	C₄⋊Dic₇₃	C₁₆	1	1	yes	characteristic
39	#11924005	C₁₆	1	2	yes	—
40	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
41	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
43	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
47	D₂₉₂	—	2	2	no	—
48	C₄×D₇₃	—	2	2	no	—
49	C₄×D₇₃	—	2	2	no	—
50	C₈×C₇₃	—	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	C₂²×C₁₆	1	1	yes	characteristic
55	C₂×D₇₃	—	2	2	no	—
56	C₂×D₇₃	—	2	2	no	—
57	C₂×D₇₃	—	2	2	no	—
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	M₆₄	1	2	yes	complemented
60	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₇₃	#504	1	1	yes	characteristic
62	Dic₇₃	—	2	2	no	—
63	#13646	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
65	C₂×C₇₃	C₂×M₆₄	1	1	yes	characteristic
66	C₂×C₇₃	#2258	1	1	yes	characteristic
67	D₇₃	—	4	2	no	—
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	—	73	1	no	—
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	#10596842	1	1	yes	characteristic
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	—	73	1	no	—
97	C₂²×C₄	—	73	2	no	—
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	—	73	1	no	—
100	C₂×C₈	—	146	2	no	—
101	C₂×C₈	—	146	2	no	—
102	C₂×C₈	C₂×#10596842	1	2	yes	—
103	C₄⋊C₄	—	73	1	no	—
104	C₂²⋊C₄	—	73	2	no	—
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	—	73	2	no	—
111	D₄	—	146	2	no	—
112	C₂×C₄	—	73	2	no	—
113	C₂×C₄	—	146	2	no	—
114	C₂×C₄	—	146	2	no	—
115	C₂×C₄	#14556686	1	1	yes	characteristic
116	C₂×C₄	#14556686	1	1	yes	characteristic
117	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	73	1	no	—
120	C₈	—	73	1	no	—
121	C₈	—	146	1	no	—
122	C₂²	—	146	2	no	—
123	C₂²	—	146	2	no	—
124	C₂²	—	146	2	no	—
125	C₂²	C₂×#14556686	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	146	2	no	—
128	C₄	#17989696	1	2	yes	complemented
129	C₄	D₄×#10596842	1	1	yes	characteristic
130	C₄	#17989709	1	1	yes	characteristic
131	C₂	—	292	2	no	—
132	C₂	#22627140	1	1	yes	characteristic
133	C₂	C₂×#17989696	1	1	yes	characteristic
134	C₂	#22627165	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989696

C₂

yes

maximal, complemented

C₄⋊#14556681

C₂

yes

maximal, characteristic, complemented

#22627024

C₂

yes

maximal, complemented

C₄×#14556681

C₂

yes

maximal, characteristic, complemented

#22626310

C₂

yes

maximal, characteristic

#17989696

—

C₂×#14556681

C₂²

yes

—

C₂×#14556681

C₂²

yes

characteristic

C₂×#14556681

C₂²

yes

characteristic

C₂×#14556664

C₂²

yes

—

C₄⋊#11924031

C₄

yes

characteristic

#17989482

C₄

yes

—

C₄×#11924031

C₂²

yes

characteristic

#17988652

C₄

yes

—

—

D₄

yes

—

—

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂³

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄×D₂₉₂

C₈

yes

characteristic

#14556503

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

C₈

yes

—

—

C₂×D₄

yes

yes

C₂×D₂₉₂

C₁₆

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

C₁₆

yes

characteristic

#11924005

C₁₆

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

C₄×D₄

yes

yes

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

M₆₄

yes

complemented

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

#13646

—

maximal, Sylow

C₂×C₇₃

D₄×C₁₆

yes

characteristic

C₂×C₇₃

C₂×M₆₄

yes

characteristic

C₂×C₇₃

#2258

yes

characteristic

D₇₃

—

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

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₄⋊C₈

#10596842

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

146

—

C₃₂

—

146

—

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

146

—

101

C₂×C₈

—

146

—

102

C₂×C₈

C₂×#10596842

yes

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

C₂×#10596842

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

146

—

109

C₁₆

—

146

—

110

C₂³

—

111

D₄

—

146

—

112

C₂×C₄

—

113

C₂×C₄

—

146

—

114

C₂×C₄

—

146

—

115

C₂×C₄

#14556686

yes

characteristic

116

C₂×C₄

#14556686

yes

characteristic

117

C₂×C₄

C₂²×#10596842

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

146

—

122

C₂²

—

146

—

123

C₂²

—

146

—

124

C₂²

—

146

—

125

C₂²

C₂×#14556686

yes

characteristic

126

C₄

—

127

C₄

—

146

—

128

C₄

#17989696

yes

complemented

129

C₄

D₄×#10596842

yes

characteristic

130

C₄

#17989709

yes

characteristic

131

C₂

—

292

—

132

C₂

#22627140

yes

characteristic

133

C₂

C₂×#17989696

yes

characteristic

134

C₂

#22627165

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₄⋊C₈×C₇₃	C₈
prime radical	C₂×D₂₉₂	C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	C₄⋊C₈	#10596842
hypocenter	C₇₃	#13646
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₄⋊C₈×C₇₃

C₈

prime radical

C₂×D₂₉₂

C₁₆

C₂³

C₄⋊C₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
1. C₂×#17989696	C₂	semidirect
2. C₄⋊#14556681	C₂	semidirect
3. #22627024	C₂	semidirect
4. C₄×#14556681	C₂	semidirect
17. #14556681	D₄	semidirect
59. C₄×C₇₃	M₆₄	semidirect
73. C₇₃	#13646	semidirect
128. C₄	#17989696	semidirect

H ≅ G/N

type

1. C₂×#17989696

C₂

semidirect

2. C₄⋊#14556681

C₂

semidirect

3. #22627024

C₂

semidirect

4. C₄×#14556681

C₂

semidirect

17. #14556681

D₄

semidirect

59. C₄×C₇₃

M₆₄

semidirect

73. C₇₃

#13646

semidirect

128. C₄

#17989696

semidirect

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

subgroup	index
C₂×#17989696	2
C₄⋊#14556681	2
#22627024	2
C₄×#14556681	2
#22626310	2
#13646	73

subgroup

index

Subgroups — #34034466

Conjugacy classes of subgroups

Complements

Maximal subgroups