Group #34000024 — subgroups

Subgroups — #34000024

Order 18688.

148 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 48 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	#22624953	C₂	1	1	yes	maximal, characteristic, complemented
2	#22624843	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608816	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924031	—	2	1	no	—
5	C₄×#11924031	—	2	1	no	—
6	#17988665	—	2	1	no	—
7	#17988340	C₄	1	1	yes	characteristic
8	D₄⋊#10596831	—	2	1	no	—
9	C₈×#10596831	C₂²	1	1	yes	characteristic
10	C₄×D₈×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924031	—	2	2	no	—
12	C₂×#11924031	—	4	1	no	—
13	C₂×#11924012	—	4	1	no	—
14	C₄⋊#10596831	—	2	1	no	—
15	C₂²⋊#10596831	—	4	1	no	—
16	C₄×#10596831	D₄	1	1	yes	characteristic
17	C₄×#10596831	D₄	1	1	yes	characteristic
18	C₂×D₈×C₇₃	C₈	1	1	yes	characteristic
19	C₄×D₄×C₇₃	—	2	1	no	—
20	#107×C₇₃	C₈	1	1	yes	characteristic
21	#102×C₇₃	—	2	1	no	—
22	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	#11924031	—	2	4	no	—
24	#11924031	—	8	1	no	—
25	#11924012	—	8	2	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	4	1	no	—
28	C₂×#10596831	D₈	1	1	yes	characteristic
29	C₂×#10596831	SD₁₆	1	1	yes	characteristic
30	C₂×D₄×C₇₃	—	2	1	no	—
31	C₂²×C₄×C₇₃	—	4	1	no	—
32	D₈×C₇₃	—	4	1	no	—
33	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₂²⋊C₄×C₇₃	—	4	1	no	—
37	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂³×C₇₃	—	4	1	no	—
39	D₄×C₇₃	—	4	1	no	—
40	D₄×C₇₃	—	4	1	no	—
41	C₂×C₄×C₇₃	—	4	1	no	—
42	C₂×C₄×C₇₃	—	8	1	no	—
43	C₂×C₄×C₇₃	—	8	1	no	—
44	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
46	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
47	C₈×C₇₃	—	2	1	no	—
48	C₈×C₇₃	—	2	1	no	—
49	#10596831	—	2	2	no	—
50	#10596831	—	2	1	no	—
51	#10596831	—	2	1	no	—
52	#10596831	—	8	1	no	—
53	C₂²×C₇₃	—	8	1	no	—
54	C₂²×C₇₃	—	8	1	no	—
55	C₂²×C₇₃	—	8	1	no	—
56	C₂²×C₇₃	D₄⋊C₈	1	1	yes	characteristic
57	C₄×C₇₃	—	2	1	no	—
58	C₄×C₇₃	—	8	1	no	—
59	C₄×C₇₃	#349	1	1	yes	characteristic
60	C₄×C₇₃	#350	1	1	yes	characteristic
61	C₄×C₇₃	#360	1	1	yes	characteristic
62	C₄×C₇₃	#361	1	1	yes	characteristic
63	#7074	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	—	16	1	no	—
65	C₂×C₇₃	#1330	1	1	yes	characteristic
66	C₂×C₇₃	#1333	1	1	yes	characteristic
67	C₂×C₇₃	#1335	1	1	yes	characteristic
68	#1589	—	73	1	no	—
69	#1367	—	73	1	no	—
70	#1311	—	73	1	no	—
71	C₇₃	#7074	1	1	yes	characteristic, complemented, Sylow
72	C₄×D₈	#64408	1	1	yes	characteristic
73	#436	—	146	1	no	—
74	C₄⋊C₁₆	—	146	1	no	—
75	C₄×C₁₆	—	146	1	no	—
76	#335	—	73	1	no	—
77	D₄⋊C₈	—	146	1	no	—
78	C₈²	—	73	1	no	—
79	C₂×D₈	#10596833	1	1	yes	characteristic
80	C₂×M₁₆	—	292	1	no	—
81	C₄×D₄	—	2	1	no	—
82	C₂×C₁₆	—	292	1	no	—
83	C₂×C₁₆	—	146	2	no	—
84	#107	#10596833	1	1	yes	characteristic
85	C₄⋊C₈	—	146	1	no	—
86	#102	—	2	1	no	—
87	C₂²⋊C₈	—	292	1	no	—
88	C₄×C₈	—	73	1	no	—
89	C₄×C₈	—	73	1	no	—
90	C₄×C₈	C₂×#64408	1	1	yes	characteristic
91	C₂×D₄	—	2	1	no	—
92	C₂²×C₄	—	4	1	no	—
93	D₈	—	4	1	no	—
94	M₁₆	—	584	2	no	—
95	C₂×C₈	—	292	1	no	—
96	C₂×C₈	—	73	1	no	—
97	C₂×C₈	—	73	1	no	—
98	C₂×C₈	—	146	1	no	—
99	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
100	C₂×C₈	#11924037	1	1	yes	characteristic
101	C₄⋊C₄	—	2	1	no	—
102	C₂²⋊C₄	—	4	1	no	—
103	C₄²	C₂²⋊#64408	1	1	yes	characteristic
104	C₁₆	—	584	1	no	—
105	C₁₆	—	146	4	no	—
106	C₂³	—	4	1	no	—
107	D₄	—	4	1	no	—
108	D₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	8	1	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
113	C₂×C₄	#14556653	1	1	yes	characteristic
114	C₂×C₄	#14556655	1	1	yes	characteristic
115	C₈	—	2	1	no	—
116	C₈	—	2	1	no	—
117	C₈	—	584	1	no	—
118	C₈	—	146	2	no	—
119	C₈	—	146	1	no	—
120	C₈	—	146	1	no	—
121	C₂²	—	8	1	no	—
122	C₂²	—	8	1	no	—
123	C₂²	—	8	1	no	—
124	C₂²	D₄⋊#10596833	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	8	1	no	—
127	C₄	#17989483	1	1	yes	characteristic
128	C₄	#17989485	1	1	yes	characteristic
129	C₄	#17989503	1	1	yes	characteristic
130	C₄	#17989505	1	1	yes	characteristic
131	C₂	—	16	1	no	—
132	C₂	#22624883	1	1	yes	characteristic
133	C₂	#22624889	1	1	yes	characteristic
134	C₂	#22624893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22624953

C₂

yes

maximal, characteristic, complemented

#22624843

C₂

yes

maximal, characteristic, complemented

#22608816

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

characteristic

D₄⋊#10596831

—

C₈×#10596831

C₂²

yes

characteristic

C₄×D₈×C₇₃

C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₂×D₈×C₇₃

C₈

yes

characteristic

C₄×D₄×C₇₃

—

#107×C₇₃

C₈

yes

characteristic

#102×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

—

D₈

yes

characteristic

C₂×#10596831

SD₁₆

yes

characteristic

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

—

—

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

D₄⋊C₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

C₇₃

#7074

yes

characteristic, complemented, Sylow

C₄×D₈

#64408

yes

characteristic

#436

—

146

—

C₄⋊C₁₆

—

146

—

C₄×C₁₆

—

146

—

#335

—

D₄⋊C₈

—

146

—

C₈²

—

C₂×D₈

#10596833

yes

characteristic

C₂×M₁₆

—

292

—

C₄×D₄

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

146

—

#107

#10596833

yes

characteristic

C₄⋊C₈

—

146

—

#102

—

C₂²⋊C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

D₈

—

M₁₆

—

584

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₂×#10596833

yes

characteristic

100

C₂×C₈

#11924037

yes

characteristic

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

C₂²⋊#64408

yes

characteristic

104

C₁₆

—

584

—

105

C₁₆

—

146

—

106

C₂³

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²⋊#10596833

yes

characteristic

113

C₂×C₄

#14556653

yes

characteristic

114

C₂×C₄

#14556655

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

584

—

118

C₈

—

146

—

119

C₈

—

146

—

120

C₈

—

146

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

D₄⋊#10596833

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

#17989483

yes

characteristic

128

C₄

#17989485

yes

characteristic

129

C₄

#17989503

yes

characteristic

130

C₄

#17989505

yes

characteristic

131

C₂

—

132

C₂

#22624883

yes

characteristic

133

C₂

#22624889

yes

characteristic

134

C₂

#22624893

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556653
commutator	C₂×C₈×C₇₃	C₂×C₈
Frattini	C₄×C₈	C₂×#64408
Fitting	C₄×D₈×C₇₃	C₄
socle	C₂²×C₇₃	D₄⋊C₈
prime radical	C₂×D₈×C₇₃	C₈
cosocle	C₈×#10596831	C₂²
hypercenter	C₄×D₈	#64408
hypocenter	C₇₃	#7074
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

#14556653

commutator

C₂×C₈×C₇₃

C₂×C₈

Frattini

C₄×C₈

C₂×#64408

Fitting

C₄×D₈×C₇₃

C₄

socle

C₂²×C₇₃

D₄⋊C₈

prime radical

C₂×D₈×C₇₃

C₈

C₂²

C₄×D₈

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

N	H ≅ G/N	type
1. #22624953	C₂	semidirect
2. #22624843	C₂	semidirect
71. C₇₃	#7074	semidirect

H ≅ G/N

type

1. #22624953

C₂

semidirect

2. #22624843

C₂

semidirect

71. C₇₃

#7074

semidirect

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

subgroup	index
#22624953	2
#22624843	2
#22608816	2
#7074	73

subgroup

index

Subgroups — #34000024

Conjugacy classes of subgroups

Complements

Maximal subgroups