Group #34009758 — subgroups

Subgroups — #34009758

Order 18688.

228 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 100 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	#22626292	C₂	1	2	yes	maximal
2	#22624849	C₂	1	4	yes	maximal, complemented
3	#22608215	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924031	C₄	1	4	yes	—
5	C₂²⋊#11924031	C₄	1	4	yes	—
6	#17989475	C₂²	1	4	yes	—
7	#17988640	C₄	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	#17988309	C₂²	1	2	yes	—
10	C₈×#10596831	C₄	1	2	yes	—
11	#431×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924031	C₂×C₄	1	8	yes	—
13	C₂²×#10596831	C₂×C₄	1	2	yes	—
14	C₄⋊#10596831	—	2	1	no	—
15	C₂²⋊#10596831	—	2	1	no	—
16	C₄×#10596831	C₂×C₄	1	2	yes	—
17	C₄×#10596831	C₂×C₄	1	2	yes	—
18	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₄×#10596831	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₈×C₇₃	—	2	1	no	—
22	C₂²⋊C₈×C₇₃	—	2	1	no	—
23	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
24	#11924031	—	2	8	no	—
25	C₂×#10596831	—	2	4	no	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	C₄²	1	2	yes	—
28	C₂×#10596831	C₂²×C₄	1	2	yes	—
29	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₇₃	—	2	2	no	—
33	C₄²×C₇₃	C₄²	1	1	yes	characteristic
34	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
35	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
36	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂³×C₇₃	#97	1	1	yes	characteristic
38	C₂×C₄×C₇₃	—	2	2	no	—
39	C₂×C₄×C₇₃	—	4	1	no	—
40	C₂×C₄×C₇₃	—	4	1	no	—
41	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
42	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
43	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
47	C₈×C₇₃	—	4	2	no	—
48	#10596831	—	2	2	no	—
49	#10596831	—	4	2	no	—
50	#10596831	#131	1	4	yes	—
51	C₂²×C₇₃	—	2	1	no	—
52	C₂²×C₇₃	—	2	1	no	—
53	C₂²×C₇₃	—	2	1	no	—
54	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
55	C₄×C₇₃	—	2	1	no	—
56	C₄×C₇₃	—	2	1	no	—
57	C₄×C₇₃	—	2	1	no	—
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	#405	1	2	yes	—
61	#11416	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	—	4	1	no	—
63	C₂×C₇₃	#1452	1	1	yes	characteristic
64	C₂×C₇₃	#2110	1	1	yes	characteristic
65	C₂×C₇₃	#2110	1	1	yes	characteristic
66	#2163	—	73	2	no	—
67	#1450	—	73	1	no	—
68	#1313	—	73	4	no	—
69	C₇₃	#11416	1	1	yes	characteristic, complemented, Sylow
70	#431	—	73	1	no	—
71	#431	#64408	1	1	yes	characteristic
72	C₂×C₄×C₈	—	73	1	no	—
73	C₄⋊C₁₆	—	73	4	no	—
74	C₂²⋊C₁₆	—	73	4	no	—
75	#346	—	73	4	no	—
76	#322	—	73	2	no	—
77	C₈²	—	73	2	no	—
78	C₂²×C₈	—	73	2	no	—
79	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
80	C₂×C₁₆	—	73	8	no	—
81	C₄⋊C₈	—	2	1	no	—
82	C₄⋊C₈	—	146	1	no	—
83	C₂²⋊C₈	—	2	1	no	—
84	C₂²⋊C₈	—	146	1	no	—
85	C₄×C₈	—	73	2	no	—
86	C₄×C₈	—	73	1	no	—
87	C₄×C₈	—	73	1	no	—
88	C₄×C₈	—	73	2	no	—
89	C₄×C₈	C₂×#64408	1	2	yes	—
90	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
91	C₂²×C₄	#11924037	1	1	yes	characteristic
92	C₂²×C₄	#11924038	1	1	yes	characteristic
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	73	2	no	—
95	C₂×C₈	—	73	2	no	—
96	C₂×C₈	—	146	2	no	—
97	C₂×C₈	—	146	4	no	—
98	C₄²	C₄×#64408	1	1	yes	characteristic
99	C₄²	#11924037	1	1	yes	characteristic
100	C₄²	#11924038	1	1	yes	characteristic
101	C₄²	C₂²×#64408	1	1	yes	characteristic
102	C₁₆	—	146	8	no	—
103	C₂³	#14556646	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	#14556646	1	1	yes	characteristic
108	C₂×C₄	#14556646	1	1	yes	characteristic
109	C₂×C₄	#14556646	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
111	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
112	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	292	2	no	—
115	C₈	—	73	4	no	—
116	C₈	—	146	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×#14556646	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	#17989550	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	#22625144	1	1	yes	characteristic
129	C₂	#22626152	1	1	yes	characteristic
130	C₂	#22626154	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626292

C₂

yes

maximal

#22624849

C₂

yes

maximal, complemented

#22608215

C₂

yes

maximal, characteristic

C₄⋊#11924031

C₄

yes

—

C₂²⋊#11924031

C₄

yes

—

#17989475

C₂²

yes

—

#17988640

C₄

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

characteristic

#17988309

C₂²

yes

—

C₈×#10596831

C₄

yes

—

#431×C₇₃

C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

—

C₄⋊#10596831

—

C₂²⋊#10596831

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂³

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

C₄²

yes

—

C₂×#10596831

C₂²×C₄

yes

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₄²

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂³×C₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₈×C₇₃

—

—

—

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#405

yes

—

#11416

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

C₇₃

#11416

yes

characteristic, complemented, Sylow

—

yes

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₈²

—

C₂²×C₈

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄²

yes

C₄²

yes

100

C₄²

#11924038

yes

characteristic

101

C₄²

C₂²×#64408

yes

characteristic

102

C₁₆

—

146

—

103

C₂³

#14556646

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#14556646

yes

characteristic

108

C₂×C₄

#14556646

yes

characteristic

109

C₂×C₄

#14556646

yes

characteristic

110

C₂×C₄

C₂×C₄×#64408

yes

characteristic

111

C₂×C₄

C₂×#11924037

yes

characteristic

112

C₂×C₄

C₂×#11924038

yes

characteristic

113

C₈

—

114

C₈

—

292

—

115

C₈

—

116

C₈

—

146

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#14556646

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#17989550

yes

—

127

C₂

—

128

C₂

#22625144

yes

characteristic

129

C₂

#22626152

yes

characteristic

130

C₂

#22626154

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄²	C₂²×#64408
Fitting	#431×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×C₇₃	#97
cosocle	C₄×#10596831	C₂³
hypercenter	#431	#64408
hypocenter	C₇₃	#11416
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₄×#64408

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₄²

C₂²×#64408

Fitting

#431×C₇₃

C₄

socle

C₂²×C₇₃

C₂×#97

C₂³×C₇₃

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #22624849	C₂	semidirect
69. C₇₃	#11416	semidirect

H ≅ G/N

type

2. #22624849

C₂

semidirect

69. C₇₃

#11416

semidirect

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

subgroup	index
#22626292	2
#22624849	2
#22608215	2
#11416	73

subgroup

index

Subgroups — #34009758

Conjugacy classes of subgroups

Complements

Maximal subgroups