Group #34032784 — subgroups

Subgroups — #34032784

Order 18688.

146 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 47 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	#22626995	C₂	1	1	yes	maximal, characteristic
2	C₈×#11924040	C₂	1	1	yes	maximal, characteristic
3	#22625519	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924040	—	2	1	no	—
5	C₄×#11924040	—	2	1	no	—
6	#17989636	—	2	1	no	—
7	#17989450	C₄	1	1	yes	characteristic
8	#17989437	—	2	1	no	—
9	C₈×#10596833	C₂²	1	1	yes	characteristic
10	#17988681	C₄	1	1	yes	characteristic
11	C₂×#11924040	—	2	2	no	—
12	C₂×#11924040	—	4	1	no	—
13	#14556658	—	4	1	no	—
14	C₄⋊#10596833	—	2	1	no	—
15	C₄×#10596833	—	4	1	no	—
16	C₄×#10596833	D₄	1	1	yes	characteristic
17	C₄×#10596833	D₄	1	1	yes	characteristic
18	C₂×Dic₂₉₂	C₈	1	1	yes	characteristic
19	#14556556	—	2	1	no	—
20	#14556514	—	2	1	no	—
21	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
22	#107×C₇₃	C₈	1	1	yes	characteristic
23	#11924040	—	2	4	no	—
24	#11924040	—	8	1	no	—
25	C₂×#10596833	—	2	1	no	—
26	C₂×#10596833	—	4	1	no	—
27	C₂×#10596833	D₈	1	1	yes	characteristic
28	C₂×#10596833	SD₁₆	1	1	yes	characteristic
29	C₂×Dic₁₄₆	—	2	1	no	—
30	Dic₂₉₂	—	2	1	no	—
31	Dic₂₉₂	C₁₆	1	2	yes	complemented
32	C₂×#10596831	M₁₆	1	1	yes	characteristic
33	#11924007	—	4	1	no	—
34	C₄×Dic₇₃	—	4	1	no	—
35	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₂×C₄×C₇₃	—	4	1	no	—
39	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×Dic₇₃	—	4	1	no	—
41	C₂×Dic₇₃	#102	1	1	yes	characteristic
42	C₂×Dic₇₃	#104	1	1	yes	characteristic
43	Dic₁₄₆	—	2	2	no	—
44	Dic₁₄₆	—	4	1	no	—
45	#10596833	—	2	2	no	—
46	#10596833	—	2	1	no	—
47	#10596833	—	8	1	no	—
48	#10596833	SD₃₂	1	1	yes	characteristic
49	#10596833	Q₃₂	1	1	yes	characteristic
50	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₇₃	M₃₂	1	1	yes	characteristic
52	#10596831	—	2	1	no	—
53	C₂²×C₇₃	D₄⋊C₈	1	1	yes	characteristic
54	C₄×C₇₃	—	8	1	no	—
55	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₄×C₇₃	#350	1	1	yes	characteristic
57	Dic₇₃	—	2	1	no	—
58	Dic₇₃	—	4	2	no	—
59	Dic₇₃	#358	1	1	yes	characteristic
60	Dic₇₃	#359	1	1	yes	characteristic
61	#7069	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	D₄⋊C₁₆	1	1	yes	characteristic
63	C₂×C₇₃	#1332	1	1	yes	characteristic
64	C₂×C₇₃	#1335	1	1	yes	characteristic
65	C₈×Q₁₆	—	73	1	no	—
66	#1370	—	73	1	no	—
67	C₈×C₁₆	—	73	1	no	—
68	C₇₃	#7069	1	1	yes	characteristic, complemented, Sylow
69	C₈×Q₈	—	146	1	no	—
70	C₄×Q₁₆	—	73	1	no	—
71	C₄⋊C₁₆	—	146	1	no	—
72	C₄×C₁₆	—	146	1	no	—
73	#335	—	73	1	no	—
74	Q₈⋊C₈	—	146	1	no	—
75	C₈²	—	73	1	no	—
76	C₂×Q₁₆	—	73	1	no	—
77	C₄×Q₈	—	146	1	no	—
78	C₂×C₁₆	—	292	1	no	—
79	C₂×C₁₆	—	146	2	no	—
80	#107	#10596842	1	1	yes	characteristic
81	C₄⋊C₈	—	292	1	no	—
82	C₄⋊C₈	—	146	1	no	—
83	#103	—	146	1	no	—
84	C₄×C₈	—	292	1	no	—
85	C₄×C₈	—	73	1	no	—
86	C₄×C₈	—	73	1	no	—
87	C₄×C₈	—	73	1	no	—
88	C₂×Q₈	—	146	1	no	—
89	Q₁₆	—	73	2	no	—
90	Q₁₆	—	146	1	no	—
91	C₂×C₈	—	292	1	no	—
92	C₂×C₈	—	73	1	no	—
93	C₂×C₈	—	73	1	no	—
94	C₂×C₈	—	73	1	no	—
95	C₂×C₈	—	146	1	no	—
96	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	292	1	no	—
99	C₄²	—	292	1	no	—
100	C₄²	—	73	1	no	—
101	C₁₆	—	584	1	no	—
102	C₁₆	—	146	4	no	—
103	Q₈	—	292	1	no	—
104	Q₈	—	146	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	292	1	no	—
107	C₂×C₄	—	73	1	no	—
108	C₂×C₄	—	73	1	no	—
109	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
110	C₈	—	584	1	no	—
111	C₈	—	73	1	no	—
112	C₈	—	73	1	no	—
113	C₈	—	146	1	no	—
114	C₈	—	146	2	no	—
115	C₈	—	146	1	no	—
116	C₈	#14556686	1	1	yes	characteristic
117	C₈	#14556688	1	1	yes	characteristic
118	C₂²	D₄⋊#10596842	1	1	yes	characteristic
119	C₄	—	8	1	no	—
120	C₄	—	292	2	no	—
121	C₄	—	73	1	no	—
122	C₄	—	73	1	no	—
123	C₄	—	146	1	no	—
124	C₄	#17989684	1	1	yes	characteristic
125	C₄	#17989688	1	1	yes	characteristic
126	C₂	#22626974	1	1	yes	characteristic
127	C₂	#22626978	1	1	yes	characteristic
128	C₂	#22626981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626995

C₂

yes

maximal, characteristic

C₈×#11924040

C₂

yes

maximal, characteristic

#22625519

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

characteristic

#17989437

—

C₈×#10596833

C₂²

yes

characteristic

#17988681

C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₂×Dic₂₉₂

C₈

yes

characteristic

#14556556

—

#14556514

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

#107×C₇₃

C₈

yes

—

—

—

—

D₈

yes

characteristic

C₂×#10596833

SD₁₆

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

C₁₆

yes

complemented

C₂×#10596831

M₁₆

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

Dic₁₄₆

—

Dic₁₄₆

—

—

—

—

SD₃₂

yes

characteristic

#10596833

Q₃₂

yes

characteristic

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

M₃₂

yes

characteristic

#10596831

—

C₂²×C₇₃

D₄⋊C₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₃

#350

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

D₄⋊C₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₈×Q₁₆

—

#1370

—

C₈×C₁₆

—

C₇₃

#7069

yes

characteristic, complemented, Sylow

C₈×Q₈

—

146

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

146

—

C₄×C₁₆

—

146

—

#335

—

Q₈⋊C₈

—

146

—

C₈²

—

C₂×Q₁₆

—

C₄×Q₈

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

146

—

#107

#10596842

yes

characteristic

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

#103

—

146

—

C₄×C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

146

—

Q₁₆

—

Q₁₆

—

146

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₂×#10596842

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄²

—

292

—

100

C₄²

—

101

C₁₆

—

584

—

102

C₁₆

—

146

—

103

Q₈

—

292

—

104

Q₈

—

146

—

105

C₂×C₄

—

106

C₂×C₄

—

292

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊#10596842

yes

characteristic

110

C₈

—

584

—

111

C₈

—

112

C₈

—

113

C₈

—

146

—

114

C₈

—

146

—

115

C₈

—

146

—

116

C₈

#14556686

yes

characteristic

117

C₈

#14556688

yes

characteristic

118

C₂²

D₄⋊#10596842

yes

characteristic

119

C₄

—

120

C₄

—

292

—

121

C₄

—

122

C₄

—

123

C₄

—

146

—

124

C₄

#17989684

yes

characteristic

125

C₄

#17989688

yes

characteristic

126

C₂

#22626974

yes

characteristic

127

C₂

#22626978

yes

characteristic

128

C₂

#22626981

yes

characteristic

	subgroup	G/H
center	C₂²	D₄⋊#10596842
commutator	C₈×C₇₃	C₂×C₁₆
Frattini	C₂×C₈	C₂×#10596842
Fitting	#107×C₇₃	C₈
socle, prime radical	C₂²×C₇₃	D₄⋊C₈
cosocle	C₈×#10596833	C₂²
hypercenter	#107	#10596842
hypocenter	C₇₃	#7069
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

subgroup

G/H

center

C₂²

D₄⋊#10596842

commutator

C₈×C₇₃

C₂×C₁₆

Frattini

C₂×C₈

C₂×#10596842

Fitting

#107×C₇₃

C₈

socle, prime radical

C₂²×C₇₃

D₄⋊C₈

C₂²

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

N	H ≅ G/N	type
31. Dic₂₉₂	C₁₆	semidirect
68. C₇₃	#7069	semidirect

H ≅ G/N

type

31. Dic₂₉₂

C₁₆

semidirect

68. C₇₃

#7069

semidirect

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

subgroup	index
#22626995	2
C₈×#11924040	2
#22625519	2
#7069	73

subgroup

index

Subgroups — #34032784

Conjugacy classes of subgroups

Complements

Maximal subgroups