Group #34034469 — subgroups

Subgroups — #34034469

Order 18688.

186 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22627165	C₂	1	2	yes	maximal, complemented
2	#22627031	C₂	1	1	yes	maximal, characteristic, complemented
3	#22627028	C₂	1	2	yes	maximal, complemented
4	#22627014	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626320	C₂	1	1	yes	maximal, characteristic
6	#17989696	—	2	2	no	—
7	#17989695	C₂²	1	2	yes	—
8	#17989694	—	2	2	no	—
9	C₂×#14556681	C₂²	1	1	yes	characteristic
10	C₂×#14556681	C₂²	1	1	yes	characteristic
11	#17989646	C₂²	1	2	yes	—
12	#17989509	C₄	1	1	yes	characteristic
13	#17989486	C₄	1	2	yes	—
14	#17989473	C₂²	1	1	yes	characteristic
15	#17988703	C₄	1	1	yes	characteristic
16	#14556681	—	2	2	no	—
17	#14556681	—	2	1	no	—
18	#14556681	D₄	1	1	yes	characteristic
19	#14556681	D₄	1	1	yes	characteristic
20	#14556666	—	2	2	no	—
21	#14556665	C₂×C₄	1	2	yes	—
22	#14556664	—	2	2	no	—
23	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924031	C₂³	1	1	yes	characteristic
25	#14556613	C₈	1	1	yes	characteristic
26	#14556599	C₂×C₄	1	2	yes	—
27	#14556517	C₈	1	2	yes	—
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	#108×C₇₃	C₈	1	1	yes	characteristic
30	#11924031	—	2	2	no	—
31	#11924031	—	2	1	no	—
32	#11924031	C₂×D₄	1	1	yes	characteristic
33	#11924031	#41	1	1	yes	characteristic
34	#11924028	C₂×C₈	1	2	yes	—
35	Dic₂₉₂	C₁₆	1	1	yes	characteristic
36	#11924017	C₁₆	1	2	yes	—
37	D₅₈₄	C₁₆	1	1	yes	characteristic
38	#11924011	—	2	2	no	—
39	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
40	C₈×D₇₃	—	2	2	no	—
41	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
42	M₁₆×C₇₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
45	C₂²⋊D₇₃	—	2	2	no	—
46	C₂×Dic₇₃	—	2	1	no	—
47	D₂₉₂	C₂×C₁₆	1	2	yes	—
48	C₄×D₇₃	—	2	2	no	—
49	Dic₁₄₆	C₂×C₁₆	1	2	yes	—
50	C₈×C₇₃	—	2	2	no	—
51	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
53	#10596831	C₄×D₄	1	1	yes	characteristic
54	#10596831	#131	1	1	yes	characteristic
55	C₂²×C₇₃	#504	1	1	yes	characteristic
56	C₂×D₇₃	—	2	2	no	—
57	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
58	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
59	Dic₇₃	—	2	2	no	—
60	Dic₇₃	—	2	2	no	—
61	#13647	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	—	2	1	no	—
63	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
64	D₇₃	—	4	2	no	—
65	#2258	—	73	2	no	—
66	#2170	—	73	1	no	—
67	#1422	—	73	1	no	—
68	#1401	—	73	2	no	—
69	C₄×C₃₂	—	73	1	no	—
70	C₇₃	#13647	1	1	yes	characteristic, complemented, Sylow
71	#504	—	73	2	no	—
72	#443	—	73	1	no	—
73	M₆₄	—	73	2	no	—
74	M₆₄	—	146	2	no	—
75	C₂×C₃₂	—	73	1	no	—
76	C₂×C₃₂	—	73	1	no	—
77	C₂×C₃₂	—	146	2	no	—
78	#364	—	73	1	no	—
79	#350	—	73	2	no	—
80	C₄×C₁₆	—	73	1	no	—
81	#135	—	73	1	no	—
82	#131	—	73	2	no	—
83	M₃₂	—	73	2	no	—
84	M₃₂	—	146	2	no	—
85	C₂×C₁₆	—	73	1	no	—
86	C₂×C₁₆	—	73	1	no	—
87	C₂×C₁₆	—	146	2	no	—
88	#108	#10596842	1	1	yes	characteristic
89	#104	—	73	2	no	—
90	C₄×C₈	—	73	1	no	—
91	C₃₂	—	73	1	no	—
92	C₃₂	—	73	1	no	—
93	C₃₂	—	146	1	no	—
94	C₃₂	—	146	2	no	—
95	#41	—	73	2	no	—
96	Q₁₆	—	73	1	no	—
97	SD₁₆	—	73	2	no	—
98	D₈	—	73	1	no	—
99	M₁₆	—	146	2	no	—
100	M₁₆	C₂×#10596842	1	2	yes	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	146	2	no	—
103	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
104	C₄²	—	73	1	no	—
105	C₁₆	—	73	1	no	—
106	C₁₆	—	73	1	no	—
107	C₁₆	—	146	1	no	—
108	C₁₆	—	146	2	no	—
109	Q₈	—	73	2	no	—
110	D₄	—	73	2	no	—
111	D₄	—	146	2	no	—
112	C₂×C₄	—	146	1	no	—
113	C₂×C₄	—	146	2	no	—
114	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	—	73	1	no	—
117	C₈	—	73	1	no	—
118	C₈	#14556686	1	1	yes	characteristic
119	C₈	#14556686	1	1	yes	characteristic
120	C₂²	—	146	2	no	—
121	C₂²	#17989709	1	1	yes	characteristic
122	C₄	—	146	2	no	—
123	C₄	—	146	2	no	—
124	C₄	D₄×#10596842	1	1	yes	characteristic
125	C₄	C₂×#14556686	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	292	2	no	—
128	C₂	#22627140	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627165

C₂

yes

maximal, complemented

#22627031

C₂

yes

maximal, characteristic, complemented

#22627028

C₂

yes

maximal, complemented

#22627014

C₂

yes

maximal, characteristic, complemented

#22626320

C₂

yes

maximal, characteristic

#17989696

—

#17989695

C₂²

yes

—

#17989694

—

C₂×#14556681

C₂²

yes

characteristic

C₂×#14556681

C₂²

yes

characteristic

#17989646

C₂²

yes

—

#17989509

C₄

yes

characteristic

#17989486

C₄

yes

—

#17989473

C₂²

yes

characteristic

#17988703

C₄

yes

—

—

D₄

yes

characteristic

#14556681

D₄

yes

characteristic

#14556666

—

#14556665

C₂×C₄

yes

—

#14556664

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂³

yes

characteristic

#14556613

C₈

yes

characteristic

#14556599

C₂×C₄

yes

—

#14556517

C₈

yes

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

#108×C₇₃

C₈

yes

—

—

C₂×D₄

yes

yes

C₂×C₈

yes

—

Dic₂₉₂

C₁₆

yes

characteristic

#11924017

C₁₆

yes

—

D₅₈₄

C₁₆

yes

characteristic

#11924011

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

M₁₆×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

C₂×C₁₆

yes

—

C₄×D₇₃

—

Dic₁₄₆

C₂×C₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

C₂×C₁₆

yes

characteristic

#10596831

C₄×D₄

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

#13647

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

D₄×C₁₆

yes

characteristic

D₇₃

—

—

—

—

—

C₄×C₃₂

—

C₇₃

#13647

yes

characteristic, complemented, Sylow

#504

—

#443

—

M₆₄

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

#364

—

#350

—

C₄×C₁₆

—

#135

—

#131

—

M₃₂

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

yes

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

—

146

—

C₃₂

—

146

—

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

146

—

100

M₁₆

C₂×#10596842

yes

—

101

C₂×C₈

—

102

C₂×C₈

—

146

—

103

C₂×C₈

C₂×#10596842

yes

characteristic

104

C₄²

—

105

C₁₆

—

106

C₁₆

—

107

C₁₆

—

146

—

108

C₁₆

—

146

—

109

Q₈

—

110

D₄

—

111

D₄

—

146

—

112

C₂×C₄

—

146

—

113

C₂×C₄

—

146

—

114

C₂×C₄

C₂²×#10596842

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

#14556686

yes

characteristic

119

C₈

#14556686

yes

characteristic

120

C₂²

—

146

—

121

C₂²

#17989709

yes

characteristic

122

C₄

—

146

—

123

C₄

—

146

—

124

C₄

D₄×#10596842

yes

characteristic

125

C₄

C₂×#14556686

yes

characteristic

126

C₂

—

127

C₂

—

292

—

128

C₂

#22627140

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#10596842
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#10596842
Fitting	#108×C₇₃	C₈
socle	C₂×C₇₃	D₄×C₁₆
prime radical	#14556613	C₈
cosocle	C₂×#11924031	C₂³
hypercenter	#108	#10596842
hypocenter	C₇₃	#13647
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₄

D₄×#10596842

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#10596842

Fitting

#108×C₇₃

C₈

socle

C₂×C₇₃

D₄×C₁₆

C₈

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
1. #22627165	C₂	semidirect
2. #22627031	C₂	semidirect
3. #22627028	C₂	semidirect
4. #22627014	C₂	semidirect
70. C₇₃	#13647	semidirect

H ≅ G/N

type

1. #22627165

C₂

semidirect

2. #22627031

C₂

semidirect

3. #22627028

C₂

semidirect

4. #22627014

C₂

semidirect

70. C₇₃

#13647

semidirect

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

subgroup	index
#22627165	2
#22627031	2
#22627028	2
#22627014	2
#22626320	2
#13647	73

subgroup

index

Subgroups — #34034469

Conjugacy classes of subgroups

Complements

Maximal subgroups