Group #34011952 — subgroups

Subgroups — #34011952

Order 18688.

214 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 78 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	#22625524	C₂	1	2	yes	maximal
2	#22624955	C₂	1	1	yes	maximal, characteristic
3	#22624891	C₂	1	2	yes	maximal
4	#22624847	C₂	1	1	yes	maximal, characteristic
5	#22611802	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596833	—	2	2	no	—
7	#17989476	C₂²	1	1	yes	characteristic
8	#17989448	C₂²	1	2	yes	—
9	Q₈⋊#10596833	—	2	2	no	—
10	#17989428	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
12	#17988415	C₄	1	1	yes	characteristic
13	#17988391	C₄	1	2	yes	—
14	#17988362	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924031	—	2	1	no	—
17	C₄⋊#10596833	—	2	2	no	—
18	C₄⋊#10596833	—	4	2	no	—
19	C₄×#10596833	—	4	2	no	—
20	C₄×#10596833	D₄	1	2	yes	—
21	Q₈×Dic₇₃	—	2	2	no	—
22	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
23	#14556525	C₂×C₄	1	2	yes	—
24	#14556515	—	2	2	no	—
25	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
26	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
27	Q₃₂×C₇₃	C₈	1	4	yes	complemented
28	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#10596833	—	2	2	no	—
30	C₂×#10596833	—	4	2	no	—
31	#11924031	—	4	1	no	—
32	C₂×#10596831	#41	1	1	yes	characteristic
33	C₄⋊Dic₇₃	—	2	2	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	C₄×Dic₇₃	—	4	2	no	—
36	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
37	#11924003	—	2	1	no	—
38	C₂×Q₈×C₇₃	—	2	2	no	—
39	Q₁₆×C₇₃	—	2	2	no	—
40	Q₁₆×C₇₃	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
43	Q₈×C₇₃	—	2	4	no	—
44	Q₈×C₇₃	—	4	2	no	—
45	C₂×C₄×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
47	C₂×Dic₇₃	—	4	2	no	—
48	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
49	#10596833	—	4	2	no	—
50	#10596833	—	8	2	no	—
51	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₇₃	#131	1	1	yes	characteristic
53	#10596831	—	2	1	no	—
54	C₂²×C₇₃	#442	1	1	yes	characteristic
55	C₄×C₇₃	—	4	4	no	—
56	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
57	C₄×C₇₃	#444	1	1	yes	characteristic
58	Dic₇₃	—	2	2	no	—
59	Dic₇₃	—	8	2	no	—
60	#12066	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	#1580	1	1	yes	characteristic
62	C₂×C₇₃	#2180	1	2	yes	—
63	#2176	—	73	1	no	—
64	#1579	—	73	2	no	—
65	#1368	—	73	1	no	—
66	#1334	—	73	2	no	—
67	#1313	—	73	1	no	—
68	C₇₃	#12066	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	#64408	1	1	yes	characteristic
70	C₈×Q₈	—	146	2	no	—
71	C₄×Q₁₆	—	73	2	no	—
72	#367	—	73	1	no	—
73	#358	—	73	2	no	—
74	#346	—	73	1	no	—
75	#346	—	73	1	no	—
76	#334	—	73	2	no	—
77	Q₈⋊C₈	—	146	2	no	—
78	#322	—	73	1	no	—
79	C₂×Q₁₆	C₂×#64408	1	2	yes	—
80	C₄×Q₈	—	146	2	no	—
81	Q₃₂	#10596833	1	4	yes	complemented
82	C₂×C₁₆	—	73	1	no	—
83	C₂×C₁₆	—	146	1	no	—
84	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
85	#107	—	73	2	no	—
86	C₄⋊C₈	—	292	2	no	—
87	C₄⋊C₈	—	146	2	no	—
88	#103	—	146	2	no	—
89	C₄×C₈	—	292	2	no	—
90	C₄×C₈	—	73	1	no	—
91	C₄×C₈	—	73	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	C₂×#10596833	1	4	yes	—
95	C₂×C₈	—	292	2	no	—
96	C₂×C₈	—	73	1	no	—
97	C₂×C₈	—	146	2	no	—
98	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
99	C₄⋊C₄	—	292	2	no	—
100	C₄⋊C₄	—	146	2	no	—
101	C₄²	—	292	2	no	—
102	C₄²	—	73	1	no	—
103	C₁₆	—	292	1	no	—
104	C₁₆	—	146	1	no	—
105	C₁₆	C₂×#10596833	1	2	yes	—
106	Q₈	—	2	4	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	292	2	no	—
110	C₂×C₄	—	73	2	no	—
111	C₂×C₄	D₄×#64408	1	1	yes	characteristic
112	C₈	—	292	2	no	—
113	C₈	—	584	2	no	—
114	C₈	—	146	1	no	—
115	C₈	C₂²×#10596833	1	1	yes	characteristic
116	C₈	#14556678	1	1	yes	characteristic
117	C₂²	#17989628	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	584	2	no	—
120	C₄	—	146	2	no	—
121	C₄	D₄×#10596833	1	1	yes	characteristic
122	C₄	#17989634	1	1	yes	characteristic
123	C₂	#22625527	1	1	yes	characteristic
124	C₂	#22626338	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625524

C₂

yes

maximal

#22624955

C₂

yes

maximal, characteristic

#22624891

C₂

yes

maximal

#22624847

C₂

yes

maximal, characteristic

#22611802

C₂

yes

maximal, characteristic

Q₈×#10596833

—

#17989476

C₂²

yes

characteristic

#17989448

C₂²

yes

—

Q₈⋊#10596833

—

#17989428

C₂²

yes

characteristic

Q₁₆×Dic₇₃

C₂²

yes

—

#17988415

C₄

yes

characteristic

#17988391

C₄

yes

—

#17988362

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

C₄

yes

—

—

—

—

D₄

yes

—

Q₈×Dic₇₃

—

C₂×#11924003

C₂×C₄

yes

characteristic

#14556525

C₂×C₄

yes

—

#14556515

—

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×Q₁₆×C₇₃

C₂×C₄

yes

—

Q₃₂×C₇₃

C₈

yes

complemented

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

—

—

yes

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

C₂×C₈

yes

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

AGL(1,ℤ/8)

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

C₂²×C₈

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

#442

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

#444

yes

characteristic

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

—

—

C₇₃

#12066

yes

characteristic, complemented, Sylow

C₂×Q₃₂

#64408

yes

characteristic

C₈×Q₈

—

146

—

C₄×Q₁₆

—

—

—

—

—

—

Q₈⋊C₈

—

146

—

#322

—

C₂×Q₁₆

C₂×#64408

yes

—

C₄×Q₈

—

146

—

Q₃₂

#10596833

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×#64408

yes

characteristic

#107

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

#103

—

146

—

C₄×C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×#10596833

yes

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₂²×#64408

yes

characteristic

C₄⋊C₄

—

292

—

100

C₄⋊C₄

—

146

—

101

C₄²

—

292

—

102

C₄²

—

103

C₁₆

—

292

—

104

C₁₆

—

146

—

105

C₁₆

C₂×#10596833

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

292

—

110

C₂×C₄

—

111

C₂×C₄

D₄×#64408

yes

characteristic

112

C₈

—

292

—

113

C₈

—

584

—

114

C₈

—

146

—

115

C₈

C₂²×#10596833

yes

characteristic

116

C₈

#14556678

yes

characteristic

117

C₂²

#17989628

yes

characteristic

118

C₄

—

119

C₄

—

584

—

120

C₄

—

146

—

121

C₄

D₄×#10596833

yes

characteristic

122

C₄

#17989634

yes

characteristic

123

C₂

#22625527

yes

characteristic

124

C₂

#22626338

yes

—

	subgroup	G/H
center	C₂²	#17989628
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×Q₃₂×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	#442
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂×Q₃₂	#64408
hypocenter	C₇₃	#12066
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

#17989628

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₂×Q₃₂×C₇₃

C₄

socle, prime radical

C₂²×C₇₃

#442

cosocle

C₈×Dic₇₃

C₂³

C₂×Q₃₂

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
27. Q₃₂×C₇₃	C₈	semidirect
68. C₇₃	#12066	semidirect
81. Q₃₂	#10596833	semidirect

H ≅ G/N

type

27. Q₃₂×C₇₃

C₈

semidirect

68. C₇₃

#12066

semidirect

81. Q₃₂

#10596833

semidirect

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

subgroup	index
#22625524	2
#22624955	2
#22624891	2
#22624847	2
#22611802	2
#12066	73

subgroup

index

Subgroups — #34011952

Conjugacy classes of subgroups

Complements

Maximal subgroups