Group #34034470 — subgroups

Subgroups — #34034470

Order 18688.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22627164	C₂	1	2	yes	maximal, complemented
2	#22627031	C₂	1	1	yes	maximal, characteristic, complemented
3	#22627027	C₂	1	2	yes	maximal, complemented
4	#22627017	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626318	C₂	1	1	yes	maximal, characteristic
6	#17989695	—	2	2	no	—
7	#17989695	C₂²	1	2	yes	—
8	C₂×#14556681	—	2	2	no	—
9	C₂×#14556681	C₂²	1	1	yes	characteristic
10	C₂×#14556681	C₂²	1	1	yes	characteristic
11	#17989645	C₂²	1	2	yes	—
12	#17989509	C₄	1	1	yes	characteristic
13	#17989485	C₄	1	2	yes	—
14	#17989473	C₂²	1	1	yes	characteristic
15	#17988704	C₄	1	1	yes	characteristic
16	#14556681	—	2	2	no	—
17	#14556681	—	2	1	no	—
18	#14556681	D₄	1	2	yes	—
19	#14556665	—	2	2	no	—
20	#14556665	C₂×C₄	1	2	yes	—
21	C₂×#11924031	—	2	2	no	—
22	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924031	C₂³	1	1	yes	characteristic
24	#14556598	C₂×C₄	1	2	yes	—
25	#14556527	C₈	1	1	yes	characteristic
26	#14556518	C₈	1	2	yes	—
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	#135×C₇₃	C₈	1	1	yes	characteristic
29	#11924031	—	2	2	no	—
30	#11924031	—	2	1	no	—
31	#11924031	C₂×D₄	1	1	yes	characteristic
32	#11924031	#41	1	1	yes	characteristic
33	#11924012	—	2	2	no	—
34	#11924012	C₂×C₈	1	2	yes	—
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
37	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
38	#41×C₇₃	C₂×C₈	1	2	yes	—
39	Q₁₆×C₇₃	C₁₆	1	1	yes	characteristic
40	SD₁₆×C₇₃	C₁₆	1	2	yes	—
41	D₈×C₇₃	C₁₆	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
43	Q₈×C₇₃	C₂×C₁₆	1	2	yes	—
44	D₄×C₇₃	—	2	2	no	—
45	D₄×C₇₃	C₂×C₁₆	1	2	yes	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×Dic₇₃	—	2	1	no	—
49	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
50	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
51	#10596831	—	2	2	no	—
52	#10596831	C₄×D₄	1	1	yes	characteristic
53	#10596831	#131	1	1	yes	characteristic
54	C₂²×C₇₃	—	2	2	no	—
55	C₂²×C₇₃	#504	1	1	yes	characteristic
56	C₄×C₇₃	—	2	2	no	—
57	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
58	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
59	Dic₇₃	—	4	1	no	—
60	#13648	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	—	2	1	no	—
62	C₂×C₇₃	—	4	2	no	—
63	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
64	#2258	—	73	2	no	—
65	#2170	—	73	1	no	—
66	#1422	—	73	1	no	—
67	#1401	—	73	2	no	—
68	#1397	—	73	1	no	—
69	C₇₃	#13648	1	1	yes	characteristic, complemented, Sylow
70	#504	—	73	2	no	—
71	#443	—	73	1	no	—
72	M₆₄	—	73	2	no	—
73	M₆₄	—	146	2	no	—
74	C₂×C₃₂	—	73	1	no	—
75	C₂×C₃₂	—	73	1	no	—
76	C₂×C₃₂	—	146	2	no	—
77	#364	—	73	1	no	—
78	#350	—	73	2	no	—
79	C₄×C₁₆	—	73	1	no	—
80	#135	#10596842	1	1	yes	characteristic
81	#131	—	73	2	no	—
82	M₃₂	—	73	2	no	—
83	M₃₂	—	146	2	no	—
84	C₂×C₁₆	—	73	1	no	—
85	C₂×C₁₆	—	73	1	no	—
86	C₂×C₁₆	—	146	2	no	—
87	#108	—	73	1	no	—
88	#104	—	73	2	no	—
89	C₄×C₈	—	73	1	no	—
90	C₃₂	—	73	2	no	—
91	C₃₂	—	146	1	no	—
92	C₃₂	—	146	2	no	—
93	#41	C₂×#10596842	1	2	yes	—
94	Q₁₆	#11924040	1	1	yes	characteristic
95	SD₁₆	#11924040	1	2	yes	—
96	D₈	#11924040	1	1	yes	characteristic
97	M₁₆	—	73	2	no	—
98	M₁₆	—	146	2	no	—
99	C₂×C₈	—	73	1	no	—
100	C₂×C₈	—	146	2	no	—
101	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
102	C₄²	—	73	1	no	—
103	C₁₆	—	73	1	no	—
104	C₁₆	—	73	1	no	—
105	C₁₆	—	146	1	no	—
106	C₁₆	—	146	2	no	—
107	Q₈	C₂×#11924040	1	2	yes	—
108	D₄	—	2	2	no	—
109	D₄	C₂×#11924040	1	2	yes	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	146	1	no	—
112	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
113	C₈	—	73	1	no	—
114	C₈	—	73	1	no	—
115	C₈	—	146	2	no	—
116	C₈	C₂×#11924040	1	1	yes	characteristic
117	C₈	C₂×#11924040	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#17989708	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	292	1	no	—
122	C₄	D₄×#10596842	1	1	yes	characteristic
123	C₄	C₂²×#11924040	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	2	no	—
126	C₂	D₄×#11924040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627164

C₂

yes

maximal, complemented

#22627031

C₂

yes

maximal, characteristic, complemented

#22627027

C₂

yes

maximal, complemented

#22627017

C₂

yes

maximal, characteristic, complemented

#22626318

C₂

yes

maximal, characteristic

#17989695

—

#17989695

C₂²

yes

—

C₂×#14556681

—

C₂×#14556681

C₂²

yes

characteristic

C₂×#14556681

C₂²

yes

characteristic

#17989645

C₂²

yes

—

#17989509

C₄

yes

characteristic

#17989485

C₄

yes

—

#17989473

C₂²

yes

characteristic

#17988704

C₄

yes

—

—

D₄

yes

—

#14556665

—

#14556665

C₂×C₄

yes

—

C₂×#11924031

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂³

yes

characteristic

#14556598

C₂×C₄

yes

—

#14556527

C₈

yes

characteristic

#14556518

C₈

yes

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

#135×C₇₃

C₈

yes

—

—

C₂×D₄

yes

yes

—

C₂×C₈

yes

—

C₂×#10596831

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

#41×C₇₃

C₂×C₈

yes

—

Q₁₆×C₇₃

C₁₆

yes

characteristic

SD₁₆×C₇₃

C₁₆

yes

—

D₈×C₇₃

C₁₆

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

Q₈×C₇₃

C₂×C₁₆

yes

—

D₄×C₇₃

—

D₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

C₂×C₁₆

yes

characteristic

#10596831

—

#10596831

C₄×D₄

yes

yes

C₂²×C₇₃

—

C₂²×C₇₃

#504

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

Dic₇₃

—

#13648

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

D₄×C₁₆

yes

—

—

—

—

—

C₇₃

#13648

yes

characteristic, complemented, Sylow

#504

—

#443

—

M₆₄

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

#364

—

#350

—

C₄×C₁₆

—

yes

—

M₃₂

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#108

—

#104

—

C₄×C₈

—

C₃₂

—

C₃₂

—

146

—

C₃₂

—

146

—

#41

C₂×#10596842

yes

—

Q₁₆

#11924040

yes

characteristic

SD₁₆

#11924040

yes

—

D₈

#11924040

yes

characteristic

M₁₆

—

M₁₆

—

146

—

C₂×C₈

—

100

C₂×C₈

—

146

—

101

C₂×C₈

C₂×#10596842

yes

characteristic

102

C₄²

—

103

C₁₆

—

104

C₁₆

—

105

C₁₆

—

146

—

106

C₁₆

—

146

—

107

Q₈

C₂×#11924040

yes

—

108

D₄

—

109

D₄

C₂×#11924040

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

146

—

112

C₂×C₄

C₂²×#10596842

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

146

—

116

C₈

C₂×#11924040

yes

characteristic

117

C₈

C₂×#11924040

yes

characteristic

118

C₂²

—

119

C₂²

#17989708

yes

characteristic

120

C₄

—

121

C₄

—

292

—

122

C₄

D₄×#10596842

yes

characteristic

123

C₄

C₂²×#11924040

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

D₄×#11924040

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#10596842
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#10596842
Fitting, prime radical	#135×C₇₃	C₈
socle	C₂×C₇₃	D₄×C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	#135	#10596842
hypocenter	C₇₃	#13648
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₄

D₄×#10596842

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#10596842

Fitting, prime radical

#135×C₇₃

C₈

socle

C₂×C₇₃

D₄×C₁₆

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #22627164	C₂	semidirect
2. #22627031	C₂	semidirect
3. #22627027	C₂	semidirect
4. #22627017	C₂	semidirect
69. C₇₃	#13648	semidirect

H ≅ G/N

type

1. #22627164

C₂

semidirect

2. #22627031

C₂

semidirect

3. #22627027

C₂

semidirect

4. #22627017

C₂

semidirect

69. C₇₃

#13648

semidirect

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

subgroup	index
#22627164	2
#22627031	2
#22627027	2
#22627017	2
#22626318	2
#13648	73

subgroup

index

Subgroups — #34034470

Conjugacy classes of subgroups

Complements

Maximal subgroups