Group #34011870 — subgroups

Subgroups — #34011870

Order 18688.

222 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 82 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	Q₁₆×#10596833	C₂	1	2	yes	maximal
2	#22624964	C₂	1	1	yes	maximal, characteristic
3	#22624887	C₂	1	2	yes	maximal
4	#22624842	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596833	—	2	2	no	—
7	#17989473	C₂²	1	1	yes	characteristic
8	#17989450	C₂²	1	2	yes	—
9	Q₈⋊#10596833	—	2	2	no	—
10	C₈×#10596833	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
12	#17988413	C₄	1	1	yes	characteristic
13	#17988391	C₄	1	2	yes	—
14	C₁₆×Dic₇₃	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	1	yes	characteristic
21	C₄×#10596833	D₄	1	1	yes	characteristic
22	Q₈×Dic₇₃	—	2	2	no	—
23	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
24	#14556525	C₂×C₄	1	2	yes	—
25	#14556515	—	2	2	no	—
26	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
27	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
28	Q₃₂×C₇₃	C₈	1	4	yes	complemented
29	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#10596833	—	2	1	no	—
31	C₂×#10596833	—	4	2	no	—
32	C₂×#10596833	D₈	1	2	yes	—
33	#11924031	—	2	2	no	—
34	C₂×#10596831	#41	1	1	yes	characteristic
35	C₄⋊Dic₇₃	—	2	2	no	—
36	C₄⋊Dic₇₃	—	4	2	no	—
37	C₄×Dic₇₃	—	4	2	no	—
38	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
39	#11924003	—	2	1	no	—
40	C₂×Q₈×C₇₃	—	2	2	no	—
41	Q₁₆×C₇₃	—	2	2	no	—
42	Q₁₆×C₇₃	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₁₆×C₇₃	C₂×C₈	1	2	yes	—
45	Q₈×C₇₃	—	2	4	no	—
46	Q₈×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
49	C₂×Dic₇₃	—	4	2	no	—
50	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₇₃	#135	1	1	yes	characteristic
52	#10596833	—	2	2	no	—
53	#10596833	—	4	1	no	—
54	#10596833	—	8	2	no	—
55	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₇₃	#131	1	1	yes	characteristic
57	#10596831	—	2	1	no	—
58	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
59	C₄×C₇₃	—	4	4	no	—
60	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₇₃	#443	1	1	yes	characteristic
62	Dic₇₃	—	2	1	no	—
63	Dic₇₃	—	8	2	no	—
64	Dic₇₃	#510	1	2	yes	—
65	#12044	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	C₈×D₈	1	1	yes	characteristic
67	C₂×C₇₃	#2176	1	1	yes	characteristic
68	C₂×C₇₃	#2179	1	1	yes	characteristic
69	C₄×Q₃₂	—	73	1	no	—
70	C₈×Q₁₆	—	73	2	no	—
71	#1372	—	73	1	no	—
72	#1332	—	73	2	no	—
73	#1311	—	73	1	no	—
74	C₇₃	#12044	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	#64408	1	1	yes	characteristic
76	C₈×Q₈	—	146	2	no	—
77	C₄×Q₁₆	—	73	2	no	—
78	#366	—	73	1	no	—
79	#358	—	73	2	no	—
80	C₄×C₁₆	—	73	1	no	—
81	C₄×C₁₆	—	73	1	no	—
82	#335	—	73	2	no	—
83	Q₈⋊C₈	—	146	2	no	—
84	C₈²	—	73	1	no	—
85	C₂×Q₁₆	C₂×#64408	1	2	yes	—
86	C₄×Q₈	—	146	2	no	—
87	Q₃₂	#10596833	1	4	yes	complemented
88	C₂×C₁₆	—	73	1	no	—
89	C₂×C₁₆	—	146	1	no	—
90	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
91	#107	—	73	2	no	—
92	C₄⋊C₈	—	292	2	no	—
93	C₄⋊C₈	—	146	2	no	—
94	#103	—	146	2	no	—
95	C₄×C₈	—	292	2	no	—
96	C₄×C₈	—	73	1	no	—
97	C₄×C₈	—	73	1	no	—
98	C₄×C₈	—	73	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	C₂×#10596833	1	4	yes	—
102	C₂×C₈	—	292	2	no	—
103	C₂×C₈	—	73	2	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
107	C₄⋊C₄	—	292	2	no	—
108	C₄⋊C₄	—	146	2	no	—
109	C₄²	—	292	2	no	—
110	C₄²	—	73	1	no	—
111	C₁₆	—	146	2	no	—
112	C₁₆	—	146	1	no	—
113	C₁₆	C₂×#10596833	1	2	yes	—
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	292	2	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	D₄×#64408	1	1	yes	characteristic
121	C₈	—	292	1	no	—
122	C₈	—	584	2	no	—
123	C₈	—	146	2	no	—
124	C₈	—	146	1	no	—
125	C₈	C₂²×#10596833	1	1	yes	characteristic
126	C₈	#14556678	1	1	yes	characteristic
127	C₂²	D₈×#64408	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	584	2	no	—
130	C₄	—	73	2	no	—
131	C₄	—	146	1	no	—
132	C₄	D₄×#10596833	1	1	yes	characteristic
133	C₄	#17989629	1	1	yes	characteristic
134	C₂	D₈×#10596833	1	1	yes	characteristic
135	C₂	#22626328	1	1	yes	characteristic
136	C₂	#22626334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×#10596833

C₂

yes

maximal

#22624964

C₂

yes

maximal, characteristic

#22624887

C₂

yes

maximal

#22624842

C₂

yes

maximal, characteristic

Q₃₂×Dic₇₃

C₂

yes

maximal, characteristic

Q₈×#10596833

—

#17989473

C₂²

yes

characteristic

#17989450

C₂²

yes

—

Q₈⋊#10596833

—

C₈×#10596833

C₂²

yes

characteristic

Q₁₆×Dic₇₃

C₂²

yes

—

#17988413

C₄

yes

characteristic

#17988391

C₄

yes

—

C₁₆×Dic₇₃

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

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

—

—

D₈

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

C₂×D₈

yes

C₂×Dic₇₃

yes

—

—

—

C₈×C₇₃

C₂²×C₈

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

#443

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#510

yes

—

#12044

—

maximal, Sylow

C₂×C₇₃

C₈×D₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×Q₃₂

—

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

—

C₇₃

#12044

yes

characteristic, complemented, Sylow

C₂×Q₃₂

#64408

yes

characteristic

C₈×Q₈

—

146

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

146

—

C₈²

—

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

—

C₄×C₈

—

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

C₂×#10596833

yes

—

102

C₂×C₈

—

292

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₂²×#64408

yes

characteristic

107

C₄⋊C₄

—

292

—

108

C₄⋊C₄

—

146

—

109

C₄²

—

292

—

110

C₄²

—

111

C₁₆

—

146

—

112

C₁₆

—

146

—

113

C₁₆

C₂×#10596833

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

292

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×#64408

yes

characteristic

121

C₈

—

292

—

122

C₈

—

584

—

123

C₈

—

146

—

124

C₈

—

146

—

125

C₈

C₂²×#10596833

yes

characteristic

126

C₈

#14556678

yes

characteristic

127

C₂²

D₈×#64408

yes

characteristic

128

C₄

—

129

C₄

—

584

—

130

C₄

—

131

C₄

—

146

—

132

C₄

D₄×#10596833

yes

characteristic

133

C₄

#17989629

yes

characteristic

134

C₂

D₈×#10596833

yes

characteristic

135

C₂

#22626328

yes

characteristic

136

C₂

#22626334

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#64408
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×Q₃₂×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₄×D₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂×Q₃₂	#64408
hypocenter	C₇₃	#12044
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×#64408

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₂×Q₃₂×C₇₃

C₄

socle, prime radical

C₂²×C₇₃

C₄×D₈

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
28. Q₃₂×C₇₃	C₈	semidirect
74. C₇₃	#12044	semidirect
87. Q₃₂	#10596833	semidirect

H ≅ G/N

type

28. Q₃₂×C₇₃

C₈

semidirect

74. C₇₃

#12044

semidirect

87. Q₃₂

#10596833

semidirect

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

subgroup	index
Q₁₆×#10596833	2
#22624964	2
#22624887	2
#22624842	2
Q₃₂×Dic₇₃	2
#12044	73

subgroup

index

Q₁₆×#10596833

Q₃₂×Dic₇₃

Subgroups — #34011870

Conjugacy classes of subgroups

Complements

Maximal subgroups