Group #34013323 — subgroups

Subgroups — #34013323

Order 18688.

342 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 144 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	#22609618	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×#14556502	C₄	1	2	yes	—
8	C₂×#14556502	C₄	1	1	yes	characteristic
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10596831	C₂²	1	2	yes	—
11	#376×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	#14556502	—	2	4	no	—
18	#14556502	C₈	1	4	yes	—
19	C₄×#10596831	D₄	1	2	yes	—
20	C₄×#10596831	D₄	1	4	yes	—
21	C₄×#10596831	Q₈	1	4	yes	—
22	C₄×#10596831	Q₈	1	2	yes	—
23	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
25	#95×C₇₃	—	2	2	no	—
26	#11924031	—	4	4	no	—
27	C₂×#10596831	—	2	2	no	—
28	C₂×#10596831	C₄⋊C₄	1	2	yes	—
29	C₂×#10596831	C₄⋊C₄	1	4	yes	—
30	C₂×#10596831	C₂×C₈	1	4	yes	—
31	C₂×#10596831	M₁₆	1	4	yes	—
32	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
33	C₂×#10596831	C₂×D₄	1	2	yes	—
34	C₂×#10596831	C₂×Q₈	1	2	yes	—
35	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	—	2	2	no	—
37	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
38	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₇₃	—	2	2	no	—
40	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
41	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
42	C₄²×C₇₃	C₂×C₈	1	2	yes	—
43	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₃	—	2	4	no	—
45	C₂×C₄×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	—	4	4	no	—
47	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
48	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
52	#10596831	—	2	2	no	—
53	#10596831	C₄⋊C₈	1	8	yes	—
54	#10596831	#127	1	1	yes	characteristic
55	#10596831	C₄⋊Q₈	1	1	yes	characteristic
56	#10596831	C₄⋊Q₈	1	2	yes	—
57	C₂²×C₇₃	—	2	2	no	—
58	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₇₃	#423	1	2	yes	—
60	C₄×C₇₃	—	2	4	no	—
61	C₄×C₇₃	—	4	4	no	—
62	C₄×C₇₃	#384	1	1	yes	characteristic
63	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
64	C₄×C₇₃	#423	1	1	yes	characteristic
65	#12901	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	—	2	2	no	—
67	C₂×C₇₃	#1842	1	1	yes	characteristic
68	C₂×C₇₃	#2153	1	2	yes	—
69	C₂×#364	—	73	4	no	—
70	#2150	—	73	2	no	—
71	#1726	—	73	1	no	—
72	C₇₃	#12901	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₃₂	—	73	4	no	—
74	C₂×#97	—	73	1	no	—
75	C₂×#97	—	73	2	no	—
76	C₂×C₄×C₈	—	73	2	no	—
77	C₂×C₄×C₈	—	73	1	no	—
78	#376	#64408	1	1	yes	characteristic
79	#364	—	146	8	no	—
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	C₂×#64408	1	2	yes	—
85	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
86	M₃₂	—	146	8	no	—
87	C₂×C₁₆	—	146	4	no	—
88	#97	—	73	4	no	—
89	#97	—	146	4	no	—
90	C₄×C₈	—	73	2	no	—
91	C₄×C₈	—	73	4	no	—
92	C₄×C₈	—	73	4	no	—
93	C₄×C₈	—	73	2	no	—
94	#95	—	2	2	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	4	no	—
99	C₂×C₈	—	73	4	no	—
100	C₂×C₈	—	73	4	no	—
101	C₂×C₈	—	73	2	no	—
102	C₂×C₈	—	73	2	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	73	2	no	—
106	C₂×C₈	—	146	2	no	—
107	C₄²	—	2	2	no	—
108	C₄²	C₄⋊#64408	1	4	yes	—
109	C₄²	C₄⋊#64408	1	2	yes	—
110	C₄²	#11924036	1	2	yes	—
111	C₁₆	—	292	4	no	—
112	C₂³	C₂×#11924037	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	#14556658	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
119	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
120	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
121	C₈	—	73	1	no	—
122	C₈	—	73	2	no	—
123	C₈	—	73	1	no	—
124	C₈	—	73	8	no	—
125	C₈	—	146	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂×#14556658	1	1	yes	characteristic
128	C₂²	#17989590	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#17989533	1	1	yes	characteristic
132	C₄	#17989588	1	2	yes	—
133	C₄	#17989592	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#22625864	1	1	yes	characteristic
136	C₂	#22626274	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989510

C₂

yes

maximal

#22626263

C₂

yes

maximal

#22609618

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

#17989510

—

C₄⋊#11924031

—

C₂×#14556502

C₄

yes

—

C₂×#14556502

C₄

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

—

#376×C₇₃

C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

characteristic

#14556502

—

#14556502

C₈

yes

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

Q₈

yes

—

C₄×#10596831

Q₈

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

#95×C₇₃

—

#11924031

—

C₂×#10596831

—

C₂×#10596831

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

C₄⋊C₄

yes

—

C₄²×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂×C₈

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

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

#384

yes

characteristic

C₄×C₇₃

C₂×C₄⋊C₈

yes

—

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1842

yes

characteristic

C₂×C₇₃

#2153

yes

—

C₂×#364

—

#2150

—

#1726

—

C₇₃

#12901

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

yes

—

146

—

C₄⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#64408

yes

—

C₂×C₄²

C₂×#64408

yes

characteristic

M₃₂

—

146

—

C₂×C₁₆

—

146

—

#97

—

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

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

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

146

—

107

C₄²

—

108

C₄²

C₄⋊#64408

yes

—

109

C₄²

C₄⋊#64408

yes

—

110

C₄²

#11924036

yes

—

111

C₁₆

—

292

—

112

C₂³

C₂×#11924037

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#14556658

yes

—

117

C₂×C₄

C₂×C₄⋊#64408

yes

—

118

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

119

C₂×C₄

C₂×#11924036

yes

characteristic

120

C₂×C₄

D₇₃⋊M₁₆

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

146

—

126

C₂²

—

127

C₂²

C₂×#14556658

yes

characteristic

128

C₂²

#17989590

yes

—

129

C₄

—

130

C₄

—

131

C₄

#17989533

yes

characteristic

132

C₄

#17989588

yes

—

133

C₄

#17989592

yes

characteristic

134

C₂

—

135

C₂

#22625864

yes

characteristic

136

C₂

#22626274

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#376×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	#376	#64408
hypocenter	C₇₃	#12901
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#376×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄⋊C₈

prime radical

C₂³×C₇₃

C₂×M₁₆

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
72. C₇₃	#12901	semidirect

H ≅ G/N

type

72. C₇₃

#12901

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
#22609618	2
#12901	73

subgroup

index

Subgroups — #34013323

Conjugacy classes of subgroups

Complements

Maximal subgroups