Group #34009757 — subgroups

Subgroups — #34009757

Order 18688.

246 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₂×#17988309	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	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	#17988309	C₄	1	2	yes	—
10	#17988309	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924031	C₂×C₄	1	8	yes	—
13	C₂²×#10596831	—	2	1	no	—
14	C₂²×#10596831	C₂×C₄	1	2	yes	—
15	C₄×#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	C₂²×C₈×C₇₃	—	2	1	no	—
21	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×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	—	4	2	no	—
28	C₂×#10596831	C₄²	1	2	yes	—
29	C₂×#10596831	C₂²×C₄	1	2	yes	—
30	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	—	4	2	no	—
35	C₄²×C₇₃	C₄²	1	1	yes	characteristic
36	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
37	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
38	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂³×C₇₃	#97	1	1	yes	characteristic
40	C₂×C₄×C₇₃	—	2	2	no	—
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
44	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
45	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
49	C₈×C₇₃	—	4	2	no	—
50	#10596831	—	2	2	no	—
51	#10596831	—	4	2	no	—
52	#10596831	#131	1	4	yes	—
53	C₂²×C₇₃	—	2	1	no	—
54	C₂²×C₇₃	—	2	1	no	—
55	C₂²×C₇₃	#347	1	2	yes	—
56	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
57	C₄×C₇₃	—	2	1	no	—
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	—	2	1	no	—
61	C₄×C₇₃	#347	1	2	yes	—
62	C₄×C₇₃	#405	1	2	yes	—
63	#11415	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	—	2	2	no	—
65	C₂×C₇₃	#1452	1	1	yes	characteristic
66	C₂×C₇₃	C₂×#347	1	1	yes	characteristic
67	C₂×C₇₃	C₂×#347	1	1	yes	characteristic
68	#2163	—	73	2	no	—
69	C₂×#322	—	73	1	no	—
70	#1313	—	73	4	no	—
71	C₇₃	#11415	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄×C₈	—	73	1	no	—
73	C₂×C₄×C₈	—	73	1	no	—
74	C₂×C₄×C₈	#64408	1	1	yes	characteristic
75	C₄⋊C₁₆	—	73	4	no	—
76	C₂²⋊C₁₆	—	73	4	no	—
77	#346	—	73	4	no	—
78	#322	—	73	2	no	—
79	#322	—	73	2	no	—
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	—	73	2	no	—
82	C₂²×C₈	—	146	1	no	—
83	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
84	C₂×C₁₆	—	73	8	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	73	2	no	—
87	C₄×C₈	—	73	1	no	—
88	C₄×C₈	—	73	1	no	—
89	C₄×C₈	—	73	2	no	—
90	C₄×C₈	—	146	1	no	—
91	C₄×C₈	C₂×#64408	1	2	yes	—
92	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
93	C₂²×C₄	#11924037	1	1	yes	characteristic
94	C₂²×C₄	#11924038	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	292	2	no	—
98	C₂×C₈	—	73	2	no	—
99	C₂×C₈	—	73	2	no	—
100	C₂×C₈	—	146	2	no	—
101	C₂×C₈	—	146	4	no	—
102	C₄²	C₄×#64408	1	1	yes	characteristic
103	C₄²	#11924037	1	1	yes	characteristic
104	C₄²	#11924038	1	1	yes	characteristic
105	C₄²	C₂²×#64408	1	1	yes	characteristic
106	C₁₆	—	146	8	no	—
107	C₂³	#14556646	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	#14556646	1	1	yes	characteristic
112	C₂×C₄	#14556646	1	1	yes	characteristic
113	C₂×C₄	#14556646	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
115	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
116	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
117	C₈	—	4	2	no	—
118	C₈	—	292	2	no	—
119	C₈	—	73	4	no	—
120	C₈	—	146	2	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	#17989480	1	2	yes	—
124	C₂²	C₂×#14556646	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	#17989479	1	2	yes	—
130	C₄	#17989550	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	#22625144	1	1	yes	characteristic
133	C₂	C₂×#17989479	1	1	yes	characteristic
134	C₂	C₂×#17989480	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626292

C₂

yes

maximal

#22624849

C₂

yes

maximal, complemented

C₂×#17988309

C₂

yes

maximal, characteristic

C₄⋊#11924031

C₄

yes

—

C₂²⋊#11924031

C₄

yes

—

#17989475

C₂²

yes

—

C₂×C₄×#10596831

C₄

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

characteristic

#17988309

C₄

yes

—

#17988309

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

—

C₂²×#10596831

—

C₂²×#10596831

C₂×C₄

yes

—

C₄×#10596831

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂³

yes

characteristic

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄×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

—

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

#347

yes

—

C₂²×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#347

yes

—

C₄×C₇₃

#405

yes

—

#11415

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

C₇₃

#11415

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#64408

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

#322

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₄×C₈

C₂×#64408

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

146

—

101

C₂×C₈

—

146

—

102

C₄²

C₄×#64408

yes

characteristic

103

C₄²

#11924037

yes

characteristic

104

C₄²

#11924038

yes

characteristic

105

C₄²

C₂²×#64408

yes

characteristic

106

C₁₆

—

146

—

107

C₂³

#14556646

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#14556646

yes

characteristic

112

C₂×C₄

#14556646

yes

characteristic

113

C₂×C₄

#14556646

yes

characteristic

114

C₂×C₄

C₂×C₄×#64408

yes

characteristic

115

C₂×C₄

C₂×#11924037

yes

characteristic

116

C₂×C₄

C₂×#11924038

yes

characteristic

117

C₈

—

118

C₈

—

292

—

119

C₈

—

120

C₈

—

146

—

121

C₂²

—

122

C₂²

—

123

C₂²

#17989480

yes

—

124

C₂²

C₂×#14556646

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#17989479

yes

—

130

C₄

#17989550

yes

—

131

C₂

—

132

C₂

#22625144

yes

characteristic

133

C₂

C₂×#17989479

yes

characteristic

134

C₂

C₂×#17989480

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄²	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×C₇₃	#97
cosocle	C₄×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#11415
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×C₄×#64408

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₄²

C₂²×#64408

Fitting

C₂×C₄×C₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×#97

C₂³×C₇₃

C₂³

C₂×C₄×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #22624849	C₂	semidirect
71. C₇₃	#11415	semidirect

H ≅ G/N

type

2. #22624849

C₂

semidirect

71. C₇₃

#11415

semidirect

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

subgroup	index
#22626292	2
#22624849	2
C₂×#17988309	2
#11415	73

subgroup

index

Subgroups — #34009757

Conjugacy classes of subgroups

Complements

Maximal subgroups