Group #34034027 — subgroups

Subgroups — #34034027

Order 18688.

204 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 59 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	#22627152	C₂	1	2	yes	maximal, complemented
2	#22626994	C₂	1	1	yes	maximal, characteristic, complemented
3	#22626984	C₂	1	2	yes	maximal, complemented
4	#22626951	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626334	C₂	1	1	yes	maximal, characteristic
6	#17989709	—	2	2	no	—
7	#17989691	C₂²	1	2	yes	—
8	#17989688	—	2	2	no	—
9	#17989682	C₂²	1	1	yes	characteristic
10	#17989671	C₂²	1	1	yes	characteristic
11	#17989631	C₂²	1	2	yes	—
12	#17989516	C₄	1	1	yes	characteristic
13	#17989502	C₄	1	2	yes	—
14	#17989474	C₂²	1	1	yes	characteristic
15	#17989020	C₄	1	1	yes	characteristic
16	#14556688	—	2	2	no	—
17	#14556688	—	4	2	no	—
18	#14556686	—	4	2	no	—
19	#14556686	D₄	1	2	yes	—
20	C₄×#10596842	—	2	1	no	—
21	#14556679	—	2	2	no	—
22	#14556664	C₂×C₄	1	1	yes	characteristic
23	#14556662	C₂×C₄	1	2	yes	—
24	#14556656	—	2	2	no	—
25	C₈×#64408	C₂³	1	1	yes	characteristic
26	#14556614	C₂×C₄	1	2	yes	—
27	#14556534	C₈	1	2	yes	complemented
28	D₁₁₆₈	C₈	1	1	yes	characteristic, complemented
29	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
30	Q₃₂×C₇₃	C₈	1	1	yes	characteristic, complemented
31	C₂×#10596842	—	4	1	no	—
32	#11924040	—	2	2	no	—
33	#11924040	—	4	2	no	—
34	#11924038	—	2	2	no	—
35	#11924038	—	4	2	no	—
36	#11924036	—	4	2	no	—
37	#11924036	C₂×D₄	1	1	yes	characteristic
38	C₄×#64408	#41	1	1	yes	characteristic
39	#11924031	—	2	1	no	—
40	#11924029	—	2	2	no	—
41	#11924015	—	2	2	no	—
42	D₅₈₄	C₂×C₈	1	2	yes	—
43	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
44	#11924003	C₂×C₈	1	1	yes	characteristic
45	Q₁₆×C₇₃	C₂×C₈	1	2	yes	—
46	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂×#64408	—	2	1	no	—
48	#10596842	—	8	1	no	—
49	Q₈×C₇₃	—	2	2	no	—
50	D₂₉₂	—	2	2	no	—
51	D₂₉₂	—	4	2	no	—
52	C₄×D₇₃	—	4	2	no	—
53	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
54	#10596833	—	4	2	no	—
55	#10596833	AGL(1,ℤ/8)	1	2	yes	—
56	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
57	#10596831	#131	1	1	yes	characteristic
58	C₂×D₇₃	—	4	2	no	—
59	C₂×D₇₃	#444	1	1	yes	characteristic
60	#64408	—	4	1	no	—
61	C₄×C₇₃	—	4	2	no	—
62	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
63	Dic₇₃	#442	1	1	yes	characteristic
64	#12068	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	#1580	1	1	yes	characteristic
66	D₇₃	—	2	1	no	—
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	—	73	1	no	—
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	#10596842	1	1	yes	characteristic, complemented
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	1	no	—
93	C₂×C₁₆	—	73	2	no	—
94	C₂×C₁₆	—	73	1	no	—
95	#108	—	73	2	no	—
96	#104	—	146	2	no	—
97	C₄×C₈	—	73	1	no	—
98	C₄×C₈	—	146	1	no	—
99	#41	—	146	2	no	—
100	Q₁₆	C₂×#10596842	1	2	yes	—
101	SD₁₆	—	146	2	no	—
102	D₈	—	73	2	no	—
103	M₁₆	—	292	2	no	—
104	M₁₆	—	146	2	no	—
105	C₂×C₈	—	292	2	no	—
106	C₂×C₈	—	292	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	—	73	1	no	—
109	C₄²	—	73	1	no	—
110	C₁₆	—	292	2	no	—
111	C₁₆	—	73	1	no	—
112	C₁₆	—	146	1	no	—
113	C₁₆	—	146	2	no	—
114	C₁₆	C₂×#10596842	1	1	yes	characteristic
115	Q₈	—	2	2	no	—
116	D₄	—	292	2	no	—
117	D₄	—	146	2	no	—
118	C₂×C₄	—	292	2	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	146	1	no	—
121	C₈	—	292	2	no	—
122	C₈	—	584	1	no	—
123	C₈	—	73	2	no	—
124	C₈	—	73	1	no	—
125	C₈	C₂²×#10596842	1	1	yes	characteristic
126	C₂²	—	292	2	no	—
127	C₂²	—	73	1	no	—
128	C₄	—	4	2	no	—
129	C₄	—	292	1	no	—
130	C₄	—	73	1	no	—
131	C₄	D₄×#10596842	1	1	yes	characteristic
132	C₂	—	584	2	no	—
133	C₂	—	146	1	no	—
134	C₂	#22627062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627152

C₂

yes

maximal, complemented

#22626994

C₂

yes

maximal, characteristic, complemented

#22626984

C₂

yes

maximal, complemented

#22626951

C₂

yes

maximal, characteristic, complemented

#22626334

C₂

yes

maximal, characteristic

#17989709

—

#17989691

C₂²

yes

—

#17989688

—

#17989682

C₂²

yes

characteristic

#17989671

C₂²

yes

characteristic

#17989631

C₂²

yes

—

#17989516

C₄

yes

characteristic

#17989502

C₄

yes

—

#17989474

C₂²

yes

characteristic

#17989020

C₄

yes

—

—

—

D₄

yes

—

C₄×#10596842

—

#14556679

—

#14556664

C₂×C₄

yes

characteristic

#14556662

C₂×C₄

yes

—

#14556656

—

C₈×#64408

C₂³

yes

characteristic

#14556614

C₂×C₄

yes

—

#14556534

C₈

yes

complemented

D₁₁₆₈

C₈

yes

characteristic, complemented

C₁₆×D₇₃

C₂×C₄

yes

characteristic

Q₃₂×C₇₃

C₈

yes

characteristic, complemented

—

—

—

—

—

—

C₂×D₄

yes

yes

—

—

—

D₅₈₄

C₂×C₈

yes

—

C₈×D₇₃

C₂²×C₄

yes

characteristic

#11924003

C₂×C₈

yes

characteristic

Q₁₆×C₇₃

C₂×C₈

yes

—

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₂×#64408

—

#10596842

—

Q₈×C₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

C₄×D₄

yes

—

yes

—

C₈×C₇₃

C₂²×C₈

yes

yes

C₂×D₇₃

—

C₂×D₇₃

#444

yes

characteristic

#64408

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

—

—

—

—

—

C₇₃

#12068

yes

characteristic, complemented, Sylow

#508

—

#504

—

146

—

—

—

—

—

—

146

—

—

—

—

—

—

146

—

Q₃₂

#10596842

yes

characteristic, complemented

SD₃₂

—

D₁₆

—

M₃₂

—

292

—

M₃₂

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#104

—

146

—

C₄×C₈

—

C₄×C₈

—

146

—

#41

—

146

—

100

Q₁₆

C₂×#10596842

yes

—

101

SD₁₆

—

146

—

102

D₈

—

103

M₁₆

—

292

—

104

M₁₆

—

146

—

105

C₂×C₈

—

292

—

106

C₂×C₈

—

292

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄²

—

110

C₁₆

—

292

—

111

C₁₆

—

112

C₁₆

—

146

—

113

C₁₆

—

146

—

114

C₁₆

C₂×#10596842

yes

characteristic

115

Q₈

—

116

D₄

—

292

—

117

D₄

—

146

—

118

C₂×C₄

—

292

—

119

C₂×C₄

—

120

C₂×C₄

—

146

—

121

C₈

—

292

—

122

C₈

—

584

—

123

C₈

—

124

C₈

—

125

C₈

C₂²×#10596842

yes

characteristic

126

C₂²

—

292

—

127

C₂²

—

128

C₄

—

129

C₄

—

292

—

130

C₄

—

131

C₄

D₄×#10596842

yes

characteristic

132

C₂

—

584

—

133

C₂

—

146

—

134

C₂

#22627062

yes

characteristic

	subgroup	G/H
center	C₂	#22627062
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₈	C₂²×#10596842
Fitting	Q₃₂×C₇₃	C₈
socle	C₂×C₇₃	#1580
prime radical	#17989020	C₄
cosocle	C₈×#64408	C₂³
hypercenter	Q₃₂	#10596842
hypocenter	C₇₃	#12068
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

#22627062

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₈

C₂²×#10596842

Fitting

Q₃₂×C₇₃

C₈

C₂×C₇₃

C₄

C₂³

Q₃₂

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
1. #22627152	C₂	semidirect
2. #22626994	C₂	semidirect
3. #22626984	C₂	semidirect
4. #22626951	C₂	semidirect
27. #14556534	C₈	semidirect
28. D₁₁₆₈	C₈	semidirect
30. Q₃₂×C₇₃	C₈	semidirect
73. C₇₃	#12068	semidirect
86. Q₃₂	#10596842	semidirect

H ≅ G/N

type

1. #22627152

C₂

semidirect

2. #22626994

C₂

semidirect

3. #22626984

C₂

semidirect

4. #22626951

C₂

semidirect

27. #14556534

C₈

semidirect

28. D₁₁₆₈

C₈

semidirect

30. Q₃₂×C₇₃

C₈

semidirect

73. C₇₃

#12068

semidirect

86. Q₃₂

#10596842

semidirect

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

subgroup	index
#22627152	2
#22626994	2
#22626984	2
#22626951	2
#22626334	2
#12068	73

subgroup

index

Subgroups — #34034027

Conjugacy classes of subgroups

Complements

Maximal subgroups