Group #34013283 — subgroups

Subgroups — #34013283

Order 18688.

342 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 128 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	#22626263	C₂	1	2	yes	maximal
2	#22624975	C₂	1	4	yes	maximal
3	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	C₂²	1	4	yes	—
5	C₄⋊#11924031	—	2	4	no	—
6	C₂×C₄×#10596831	C₄	1	2	yes	—
7	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	2	yes	—
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₄³×C₇₃	C₄	1	1	yes	characteristic
11	#14556665	—	2	8	no	—
12	C₂×#11924031	—	2	4	no	—
13	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
14	C₂²×#10596831	C₂×C₄	1	2	yes	—
15	C₂²×#10596831	C₂³	1	1	yes	characteristic
16	C₄×#10596831	—	2	4	no	—
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	Q₈	1	2	yes	—
22	C₂×C₄²×C₇₃	—	2	2	no	—
23	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
25	#11924031	—	4	4	no	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	—	2	2	no	—
28	C₂×#10596831	—	2	2	no	—
29	C₂×#10596831	C₄⋊C₄	1	2	yes	—
30	C₂×#10596831	C₂×C₈	1	4	yes	—
31	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
32	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
33	C₂×#10596831	#41	1	4	yes	—
34	C₂²×C₄×C₇₃	—	2	2	no	—
35	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
36	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₇₃	—	2	8	no	—
38	C₄²×C₇₃	—	2	2	no	—
39	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
40	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₇₃	M₁₆	1	4	yes	—
43	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₃	—	2	8	no	—
45	C₂×C₄×C₇₃	—	2	4	no	—
46	C₂×C₄×C₇₃	—	2	4	no	—
47	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
48	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#117	1	2	yes	—
51	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
53	#10596831	—	2	2	no	—
54	#10596831	—	2	4	no	—
55	#10596831	#124	1	2	yes	—
56	#10596831	#125	1	2	yes	—
57	C₂²×C₇₃	—	2	2	no	—
58	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₇₃	#432	1	2	yes	—
60	C₄×C₇₃	—	2	8	no	—
61	C₄×C₇₃	—	2	2	no	—
62	C₄×C₇₃	#364	1	4	yes	—
63	C₄×C₇₃	#382	1	1	yes	characteristic
64	C₄×C₇₃	#423	1	1	yes	characteristic
65	C₄×C₇₃	#431	1	2	yes	—
66	#12883	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	—	2	2	no	—
68	C₂×C₇₃	#1831	1	1	yes	characteristic
69	C₂×C₇₃	C₂×#364	1	2	yes	—
70	#2150	—	73	2	no	—
71	C₄²×C₈	—	73	1	no	—
72	#1376	—	73	4	no	—
73	C₇₃	#12883	1	1	yes	characteristic, complemented, Sylow
74	C₂×M₃₂	—	73	4	no	—
75	C₂×C₄×C₈	—	73	1	no	—
76	C₂×C₄×C₈	—	73	1	no	—
77	C₂×C₄×C₈	—	73	2	no	—
78	C₂×C₄×C₈	—	73	2	no	—
79	C₄³	#64408	1	1	yes	characteristic
80	C₄⋊C₁₆	—	146	4	no	—
81	C₂²×C₈	—	73	1	no	—
82	C₂²×C₈	—	73	2	no	—
83	C₂²×C₈	—	73	1	no	—
84	C₂×C₄²	—	2	2	no	—
85	C₂×C₄²	C₂×#64408	1	2	yes	—
86	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
87	M₃₂	—	146	8	no	—
88	C₂×C₁₆	—	146	4	no	—
89	C₄×C₈	—	73	2	no	—
90	C₄×C₈	—	73	2	no	—
91	C₄×C₈	—	73	2	no	—
92	C₄×C₈	—	73	2	no	—
93	C₄×C₈	—	146	4	no	—
94	C₄×C₈	—	146	4	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#11924037	1	2	yes	—
97	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	—	73	1	no	—
100	C₂×C₈	—	73	4	no	—
101	C₂×C₈	—	73	4	no	—
102	C₂×C₈	—	73	2	no	—
103	C₂×C₈	—	146	2	no	—
104	C₂×C₈	—	146	2	no	—
105	C₂×C₈	—	146	2	no	—
106	C₄²	—	2	8	no	—
107	C₄²	—	2	2	no	—
108	C₄²	C₄⋊#64408	1	2	yes	—
109	C₄²	#11924036	1	1	yes	characteristic
110	C₄²	#11924036	1	1	yes	characteristic
111	C₄²	#11924038	1	4	yes	—
112	C₁₆	—	292	4	no	—
113	C₂³	C₂×#11924037	1	1	yes	characteristic
114	C₂×C₄	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	#14556658	1	2	yes	—
118	C₂×C₄	#14556658	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
120	C₂×C₄	#14556670	1	2	yes	—
121	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
122	C₂×C₄	C₂×#11924038	1	2	yes	—
123	C₈	—	73	2	no	—
124	C₈	—	73	2	no	—
125	C₈	—	146	2	no	—
126	C₈	—	146	4	no	—
127	C₂²	—	2	2	no	—
128	C₂²	C₂×#14556658	1	1	yes	characteristic
129	C₂²	#17989608	1	2	yes	—
130	C₄	—	2	8	no	—
131	C₄	—	2	2	no	—
132	C₄	#17989510	1	4	yes	—
133	C₄	#17989531	1	1	yes	characteristic
134	C₄	#17989591	1	1	yes	characteristic
135	C₄	#17989606	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#22625848	1	1	yes	characteristic
138	C₂	C₂×#17989510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626263

C₂

yes

maximal

#22624975

C₂

yes

maximal

C₄²×#10596831

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

C₄⋊#11924031

—

C₂×C₄×#10596831

C₄

yes

—

C₂×C₄×#10596831

C₄

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₄³×C₇₃

C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

—

—

C₈

yes

—

C₄×#10596831

C₈

yes

—

C₄×#10596831

D₄

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

C₂×C₈

yes

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂×#10596831

#41

yes

—

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

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

—

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

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

—

—

yes

—

#10596831

#125

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₃

#432

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#364

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#12883

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1831

yes

characteristic

C₂×C₇₃

C₂×#364

yes

—

#2150

—

C₄²×C₈

—

#1376

—

C₇₃

#12883

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

#64408

yes

characteristic

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

—

146

—

C₄×C₈

—

146

—

C₂²×C₄

—

C₂²×C₄

#11924037

yes

—

C₂²×C₄

C₂²×#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

146

—

104

C₂×C₈

—

146

—

105

C₂×C₈

—

146

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₄⋊#64408

yes

—

109

C₄²

#11924036

yes

characteristic

110

C₄²

#11924036

yes

characteristic

111

C₄²

#11924038

yes

—

112

C₁₆

—

292

—

113

C₂³

C₂×#11924037

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#14556658

yes

—

118

C₂×C₄

#14556658

yes

—

119

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

120

C₂×C₄

#14556670

yes

—

121

C₂×C₄

C₂×#11924036

yes

characteristic

122

C₂×C₄

C₂×#11924038

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

146

—

126

C₈

—

146

—

127

C₂²

—

128

C₂²

C₂×#14556658

yes

characteristic

129

C₂²

#17989608

yes

—

130

C₄

—

131

C₄

—

132

C₄

#17989510

yes

—

133

C₄

#17989531

yes

characteristic

134

C₄

#17989591

yes

characteristic

135

C₄

#17989606

yes

—

136

C₂

—

137

C₂

#22625848

yes

characteristic

138

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	#12883
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
73. C₇₃	#12883	semidirect

H ≅ G/N

type

73. C₇₃

#12883

semidirect

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

subgroup	index
#22626263	2
#22624975	2
C₄²×#10596831	2
#12883	73

subgroup

index

Subgroups — #34013283

Conjugacy classes of subgroups

Complements

Maximal subgroups