Group #34011968 — subgroups

Subgroups — #34011968

Order 18688.

204 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 66 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	#22626323	C₂	1	2	yes	maximal, complemented
2	#22624958	C₂	1	1	yes	maximal, characteristic, complemented
3	#22624896	C₂	1	2	yes	maximal, complemented
4	#22624847	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611816	C₂	1	1	yes	maximal, characteristic
6	#17989646	—	2	2	no	—
7	#17989509	C₂²	1	2	yes	—
8	#17989486	—	2	2	no	—
9	#17989476	C₂²	1	1	yes	characteristic
10	#17989428	C₂²	1	1	yes	characteristic
11	#17989019	C₄	1	1	yes	characteristic
12	#17988703	C₂²	1	2	yes	—
13	#17988393	C₄	1	2	yes	—
14	#17988362	C₂²	1	1	yes	characteristic
15	#368×C₇₃	C₄	1	1	yes	characteristic
16	#14556666	—	4	2	no	—
17	#14556665	—	2	2	no	—
18	#14556664	—	4	2	no	—
19	C₂×#11924031	D₄	1	2	yes	—
20	C₄×#10596833	—	2	1	no	—
21	#14556613	C₂×C₄	1	2	yes	—
22	#14556599	—	2	2	no	—
23	Dic₅₈₄	C₈	1	1	yes	characteristic, complemented
24	#14556536	C₈	1	2	yes	complemented
25	D₁₁₆₈	C₈	1	1	yes	characteristic, complemented
26	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
27	#14556517	—	2	2	no	—
28	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
29	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
30	#108×C₇₃	C₂×C₄	1	2	yes	—
31	C₂×#10596833	—	4	1	no	—
32	#11924031	—	2	2	no	—
33	#11924031	—	4	2	no	—
34	#11924028	—	2	2	no	—
35	Dic₂₉₂	C₂×C₈	1	2	yes	—
36	#11924017	—	2	2	no	—
37	D₅₈₄	C₂×C₈	1	2	yes	—
38	#11924011	—	4	2	no	—
39	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
40	C₈×D₇₃	—	4	2	no	—
41	C₄×Dic₇₃	#41	1	1	yes	characteristic
42	#11924003	—	2	1	no	—
43	M₁₆×C₇₃	—	2	2	no	—
44	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
48	C₂²⋊D₇₃	—	4	2	no	—
49	C₂×Dic₇₃	—	2	1	no	—
50	D₂₉₂	—	2	2	no	—
51	C₄×D₇₃	—	4	2	no	—
52	Dic₁₄₆	—	2	2	no	—
53	#10596833	—	8	1	no	—
54	C₈×C₇₃	—	4	2	no	—
55	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₇₃	#131	1	1	yes	characteristic
57	#10596831	AGL(1,ℤ/8)	1	2	yes	—
58	C₂²×C₇₃	#444	1	1	yes	characteristic
59	C₂×D₇₃	—	4	2	no	—
60	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₇₃	#442	1	1	yes	characteristic
62	Dic₇₃	—	4	1	no	—
63	Dic₇₃	—	4	2	no	—
64	#12068	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	—	2	1	no	—
66	C₂×C₇₃	#1580	1	1	yes	characteristic
67	D₇₃	—	8	2	no	—
68	#2179	—	73	1	no	—
69	#2171	—	73	2	no	—
70	#1369	—	73	1	no	—
71	#1336	—	73	2	no	—
72	#1313	—	73	1	no	—
73	C₇₃	#12068	1	1	yes	characteristic, complemented, Sylow
74	#508	—	73	1	no	—
75	#504	—	146	2	no	—
76	#443	—	73	2	no	—
77	#368	#64408	1	1	yes	characteristic
78	#364	—	73	2	no	—
79	#359	—	73	2	no	—
80	#350	—	146	2	no	—
81	#346	—	73	1	no	—
82	#346	—	73	1	no	—
83	#322	—	73	1	no	—
84	#135	—	73	2	no	—
85	#131	—	146	2	no	—
86	Q₃₂	—	73	1	no	—
87	SD₃₂	—	73	2	no	—
88	D₁₆	—	73	1	no	—
89	M₃₂	—	292	2	no	—
90	M₃₂	—	146	2	no	—
91	C₂×C₁₆	—	292	2	no	—
92	C₂×C₁₆	—	73	2	no	—
93	C₂×C₁₆	—	73	1	no	—
94	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
95	#108	C₂×#64408	1	2	yes	—
96	#104	—	146	2	no	—
97	C₄×C₈	—	73	1	no	—
98	C₄×C₈	—	146	1	no	—
99	#41	—	146	2	no	—
100	Q₁₆	—	73	2	no	—
101	SD₁₆	—	146	2	no	—
102	D₈	—	73	2	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	292	2	no	—
105	C₂×C₈	—	292	2	no	—
106	C₂×C₈	—	292	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
109	C₄²	—	73	1	no	—
110	C₁₆	—	292	2	no	—
111	C₁₆	—	146	2	no	—
112	C₁₆	—	146	1	no	—
113	C₁₆	#11924036	1	1	yes	characteristic
114	C₁₆	#11924036	1	1	yes	characteristic
115	Q₈	—	146	2	no	—
116	D₄	—	292	2	no	—
117	D₄	—	146	2	no	—
118	C₂×C₄	—	292	2	no	—
119	C₂×C₄	—	146	1	no	—
120	C₂×C₄	D₄×#64408	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	584	1	no	—
123	C₈	—	73	2	no	—
124	C₈	C₂×#11924036	1	1	yes	characteristic
125	C₈	#14556679	1	1	yes	characteristic
126	C₂²	—	292	2	no	—
127	C₂²	#17989633	1	1	yes	characteristic
128	C₄	—	292	1	no	—
129	C₄	—	292	2	no	—
130	C₄	#17989615	1	1	yes	characteristic
131	C₄	#17989628	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	584	2	no	—
134	C₂	#22625529	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626323

C₂

yes

maximal, complemented

#22624958

C₂

yes

maximal, characteristic, complemented

#22624896

C₂

yes

maximal, complemented

#22624847

C₂

yes

maximal, characteristic, complemented

#22611816

C₂

yes

maximal, characteristic

#17989646

—

#17989509

C₂²

yes

—

#17989486

—

#17989476

C₂²

yes

characteristic

#17989428

C₂²

yes

characteristic

#17989019

C₄

yes

characteristic

#17988703

C₂²

yes

—

#17988393

C₄

yes

—

#17988362

C₂²

yes

characteristic

#368×C₇₃

C₄

yes

—

—

—

D₄

yes

—

C₄×#10596833

—

#14556613

C₂×C₄

yes

—

#14556599

—

Dic₅₈₄

C₈

yes

characteristic, complemented

#14556536

C₈

yes

complemented

D₁₁₆₈

C₈

yes

characteristic, complemented

C₂×#11924003

C₂×C₄

yes

characteristic

#14556517

—

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

#108×C₇₃

C₂×C₄

yes

—

—

—

—

—

Dic₂₉₂

C₂×C₈

yes

—

#11924017

—

D₅₈₄

C₂×C₈

yes

—

#11924011

—

C₂×#10596831

C₂×D₄

yes

characteristic

C₈×D₇₃

—

C₄×Dic₇₃

#41

yes

characteristic

#11924003

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

C₂²×C₈

yes

C₈×C₇₃

yes

yes

—

C₂²×C₇₃

#444

yes

characteristic

C₂×D₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

#442

yes

characteristic

Dic₇₃

—

Dic₇₃

—

#12068

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1580

yes

characteristic

D₇₃

—

—

—

—

—

—

C₇₃

#12068

yes

characteristic, complemented, Sylow

#508

—

#504

—

146

—

—

yes

—

—

—

146

—

—

—

—

—

—

146

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

292

—

M₃₂

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

#104

—

146

—

C₄×C₈

—

C₄×C₈

—

146

—

#41

—

146

—

100

Q₁₆

—

101

SD₁₆

—

146

—

102

D₈

—

103

M₁₆

—

104

M₁₆

—

292

—

105

C₂×C₈

—

292

—

106

C₂×C₈

—

292

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×#64408

yes

characteristic

109

C₄²

—

110

C₁₆

—

292

—

111

C₁₆

—

146

—

112

C₁₆

—

146

—

113

C₁₆

#11924036

yes

characteristic

114

C₁₆

#11924036

yes

characteristic

115

Q₈

—

146

—

116

D₄

—

292

—

117

D₄

—

146

—

118

C₂×C₄

—

292

—

119

C₂×C₄

—

146

—

120

C₂×C₄

D₄×#64408

yes

characteristic

121

C₈

—

122

C₈

—

584

—

123

C₈

—

124

C₈

C₂×#11924036

yes

characteristic

125

C₈

#14556679

yes

characteristic

126

C₂²

—

292

—

127

C₂²

#17989633

yes

characteristic

128

C₄

—

292

—

129

C₄

—

292

—

130

C₄

#17989615

yes

characteristic

131

C₄

#17989628

yes

characteristic

132

C₂

—

133

C₂

—

584

—

134

C₂

#22625529

yes

characteristic

	subgroup	G/H
center	C₄	#17989628
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	#368×C₇₃	C₄
socle	C₂×C₇₃	#1580
prime radical	#17989019	C₄
cosocle	C₈×Dic₇₃	C₂³
hypercenter	#368	#64408
hypocenter	C₇₃	#12068
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

#17989628

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

#368×C₇₃

C₄

C₂×C₇₃

C₄

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
1. #22626323	C₂	semidirect
2. #22624958	C₂	semidirect
3. #22624896	C₂	semidirect
4. #22624847	C₂	semidirect
23. Dic₅₈₄	C₈	semidirect
24. #14556536	C₈	semidirect
25. D₁₁₆₈	C₈	semidirect
73. C₇₃	#12068	semidirect

H ≅ G/N

type

1. #22626323

C₂

semidirect

2. #22624958

C₂

semidirect

3. #22624896

C₂

semidirect

4. #22624847

C₂

semidirect

23. Dic₅₈₄

C₈

semidirect

24. #14556536

C₈

semidirect

25. D₁₁₆₈

C₈

semidirect

73. C₇₃

#12068

semidirect

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

subgroup	index
#22626323	2
#22624958	2
#22624896	2
#22624847	2
#22611816	2
#12068	73

subgroup

index

Subgroups — #34011968

Conjugacy classes of subgroups

Complements

Maximal subgroups