Group #34013320 — subgroups

Subgroups — #34013320

Order 18688.

382 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 152 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×#17989510	C₂	1	4	yes	maximal
2	#22626263	C₂	1	2	yes	maximal
3	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	C₂²	1	4	yes	—
5	#17989510	—	2	8	no	—
6	C₄⋊#11924031	—	2	4	no	—
7	C₂×C₄×#10596831	C₄	1	2	yes	—
8	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10596831	C₂²	1	2	yes	—
11	C₄³×C₇₃	C₄	1	1	yes	characteristic
12	#14556665	—	2	8	no	—
13	C₂×#11924031	—	2	4	no	—
14	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
15	C₂²×#10596831	C₂×C₄	1	2	yes	—
16	C₂²×#10596831	C₂³	1	1	yes	characteristic
17	C₄×#10596831	—	2	4	no	—
18	C₄×#10596831	C₈	1	2	yes	—
19	C₄×#10596831	C₈	1	2	yes	—
20	C₄×#10596831	D₄	1	2	yes	—
21	C₄×#10596831	D₄	1	4	yes	—
22	C₄×#10596831	Q₈	1	4	yes	—
23	C₄×#10596831	Q₈	1	2	yes	—
24	C₂×C₄²×C₇₃	—	2	2	no	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	#11924031	—	4	4	no	—
28	C₂×#10596831	—	2	2	no	—
29	C₂×#10596831	C₄⋊C₄	1	2	yes	—
30	C₂×#10596831	C₄⋊C₄	1	4	yes	—
31	C₂×#10596831	C₂×C₈	1	4	yes	—
32	C₂×#10596831	M₁₆	1	4	yes	—
33	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
34	C₂×#10596831	C₂×D₄	1	2	yes	—
35	C₂×#10596831	C₂×Q₈	1	2	yes	—
36	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	—	2	2	no	—
38	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
39	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₇₃	—	2	8	no	—
41	C₄²×C₇₃	—	2	2	no	—
42	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
43	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
44	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	8	no	—
48	C₂×C₄×C₇₃	—	2	4	no	—
49	C₂×C₄×C₇₃	—	2	4	no	—
50	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
51	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
56	#10596831	—	2	2	no	—
57	#10596831	C₄⋊C₈	1	8	yes	—
58	#10596831	#127	1	1	yes	characteristic
59	#10596831	C₄⋊Q₈	1	1	yes	characteristic
60	#10596831	C₄⋊Q₈	1	2	yes	—
61	C₂²×C₇₃	—	2	2	no	—
62	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²×C₇₃	#423	1	2	yes	—
64	C₄×C₇₃	—	2	8	no	—
65	C₄×C₇₃	—	2	2	no	—
66	C₄×C₇₃	#364	1	4	yes	complemented
67	C₄×C₇₃	#384	1	1	yes	characteristic
68	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
69	C₄×C₇₃	#423	1	1	yes	characteristic
70	#12900	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	—	2	2	no	—
72	C₂×C₇₃	#1842	1	1	yes	characteristic
73	C₂×C₇₃	C₂×#364	1	2	yes	—
74	C₂×#364	—	73	4	no	—
75	#2150	—	73	2	no	—
76	C₄²×C₈	—	73	1	no	—
77	C₇₃	#12900	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₃₂	—	73	4	no	—
79	C₂×C₄×C₈	—	73	2	no	—
80	C₂×C₄×C₈	—	73	1	no	—
81	C₂×C₄×C₈	—	73	1	no	—
82	C₂×C₄×C₈	—	73	2	no	—
83	C₄³	#64408	1	1	yes	characteristic
84	#364	—	146	8	no	—
85	C₄⋊C₁₆	—	146	4	no	—
86	C₂²×C₈	—	73	2	no	—
87	C₂²×C₈	—	73	1	no	—
88	C₂²×C₈	—	73	1	no	—
89	C₂×C₄²	—	2	2	no	—
90	C₂×C₄²	C₂×#64408	1	2	yes	—
91	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
92	M₃₂	—	146	8	no	—
93	C₂×C₁₆	—	146	4	no	—
94	C₄×C₈	—	73	2	no	—
95	C₄×C₈	—	73	4	no	—
96	C₄×C₈	—	73	4	no	—
97	C₄×C₈	—	73	2	no	—
98	C₄×C₈	—	73	2	no	—
99	C₄×C₈	—	73	2	no	—
100	C₄×C₈	—	146	4	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	#11924037	1	2	yes	—
103	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
104	C₂×C₈	—	73	4	no	—
105	C₂×C₈	—	73	4	no	—
106	C₂×C₈	—	73	4	no	—
107	C₂×C₈	—	73	2	no	—
108	C₂×C₈	—	73	2	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	146	2	no	—
113	C₄²	—	2	8	no	—
114	C₄²	—	2	2	no	—
115	C₄²	C₄⋊#64408	1	4	yes	—
116	C₄²	C₄⋊#64408	1	2	yes	—
117	C₄²	#11924036	1	1	yes	characteristic
118	C₄²	#11924036	1	1	yes	characteristic
119	C₁₆	—	292	4	no	—
120	C₂³	C₂×#11924037	1	1	yes	characteristic
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	#14556658	1	2	yes	—
125	C₂×C₄	#14556658	1	2	yes	—
126	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
128	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
129	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
130	C₈	—	73	1	no	—
131	C₈	—	73	2	no	—
132	C₈	—	73	1	no	—
133	C₈	—	73	8	no	—
134	C₈	—	146	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×#14556658	1	1	yes	characteristic
137	C₂²	#17989590	1	2	yes	—
138	C₄	—	2	8	no	—
139	C₄	—	2	2	no	—
140	C₄	#17989510	1	4	yes	complemented
141	C₄	#17989533	1	1	yes	characteristic
142	C₄	#17989588	1	2	yes	—
143	C₄	#17989592	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	#22625864	1	1	yes	characteristic
146	C₂	C₂×#17989510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989510

C₂

yes

maximal

#22626263

C₂

yes

maximal

C₄²×#10596831

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

#17989510

—

C₄⋊#11924031

—

C₂×C₄×#10596831

C₄

yes

—

C₂×C₄×#10596831

C₄

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

—

C₄³×C₇₃

C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

characteristic

C₄×#10596831

—

C₄×#10596831

C₈

yes

—

C₄×#10596831

C₈

yes

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

Q₈

yes

—

C₄×#10596831

Q₈

yes

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

—

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

—

C₂×#10596831

C₂×Q₈

yes

—

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

#10596831

—

#10596831

C₄⋊C₈

yes

—

yes

C₄⋊Q₈

yes

characteristic

#10596831

C₄⋊Q₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₃

#423

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×C₄⋊C₈

yes

—

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1842

yes

characteristic

C₂×C₇₃

C₂×#364

yes

—

C₂×#364

—

#2150

—

C₄²×C₈

—

C₇₃

#12900

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

#64408

yes

characteristic

#364

—

146

—

C₄⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₂×#64408

yes

—

C₂×C₄²

C₂×#64408

yes

characteristic

M₃₂

—

146

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

146

—

101

C₂²×C₄

—

102

C₂²×C₄

#11924037

yes

—

103

C₂²×C₄

C₂²×#64408

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

146

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₄⋊#64408

yes

—

116

C₄²

C₄⋊#64408

yes

—

117

C₄²

#11924036

yes

characteristic

118

C₄²

#11924036

yes

characteristic

119

C₁₆

—

292

—

120

C₂³

C₂×#11924037

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#14556658

yes

—

125

C₂×C₄

#14556658

yes

—

126

C₂×C₄

C₂×C₄⋊#64408

yes

—

127

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

128

C₂×C₄

C₂×#11924036

yes

characteristic

129

C₂×C₄

D₇₃⋊M₁₆

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

146

—

135

C₂²

—

136

C₂²

C₂×#14556658

yes

characteristic

137

C₂²

#17989590

yes

—

138

C₄

—

139

C₄

—

140

C₄

#17989510

yes

complemented

141

C₄

#17989533

yes

characteristic

142

C₄

#17989588

yes

—

143

C₄

#17989592

yes

characteristic

144

C₂

—

145

C₂

#22625864

yes

characteristic

146

C₂

C₂×#17989510

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₄³×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₄³	#64408
hypocenter	C₇₃	#12900
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₄³×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄⋊C₈

prime radical

C₂³×C₇₃

C₂×M₁₆

C₂³

C₄³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
66. C₄×C₇₃	#364	semidirect
77. C₇₃	#12900	semidirect
140. C₄	#17989510	semidirect

H ≅ G/N

type

66. C₄×C₇₃

#364

semidirect

77. C₇₃

#12900

semidirect

140. C₄

#17989510

semidirect

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

subgroup	index
C₂×#17989510	2
#22626263	2
C₄²×#10596831	2
#12900	73

subgroup

index

Subgroups — #34013320

Conjugacy classes of subgroups

Complements

Maximal subgroups