Group #34000808 — subgroups

Subgroups — #34000808

Order 18688.

204 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 40 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	#22624871	C₂	1	2	yes	maximal
2	#22610751	C₂	1	1	yes	maximal, characteristic
3	#17989432	—	2	2	no	—
4	C₂×#14556591	C₄	1	1	yes	characteristic
5	C₂×#14556516	—	2	2	no	—
6	#17988453	C₂²	1	1	yes	characteristic
7	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
8	C₂²⋊#10596833	—	4	2	no	—
9	C₂²×Dic₁₄₆	—	2	1	no	—
10	#14556591	—	2	1	no	—
11	#14556591	—	4	1	no	—
12	#14556591	C₈	1	2	yes	complemented
13	C₂×C₄⋊Dic₇₃	—	4	1	no	—
14	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
15	#14556516	—	4	4	no	—
16	#14556498	—	4	1	no	—
17	C₂²×C₈×C₇₃	—	2	1	no	—
18	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
19	#97×C₇₃	—	4	1	no	—
20	C₂×#10596833	—	8	2	no	—
21	C₂×Dic₁₄₆	—	2	2	no	—
22	C₂×Dic₁₄₆	—	4	2	no	—
23	C₂×Dic₁₄₆	—	4	2	no	—
24	C₂×Dic₁₄₆	—	4	1	no	—
25	C₂²×Dic₇₃	—	2	2	no	—
26	C₂²×Dic₇₃	—	4	1	no	—
27	C₄⋊Dic₇₃	—	4	2	no	—
28	C₄⋊Dic₇₃	—	4	2	no	—
29	C₄⋊Dic₇₃	—	4	2	no	—
30	C₂²×C₄×C₇₃	—	2	1	no	—
31	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₃	—	2	2	no	—
33	C₂×C₈×C₇₃	—	4	1	no	—
34	C₂×C₈×C₇₃	—	4	1	no	—
35	C₄²×C₇₃	—	2	1	no	—
36	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
38	C₂³×C₇₃	#99	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	2	1	no	—
40	C₂×C₄×C₇₃	—	2	1	no	—
41	C₂×C₄×C₇₃	—	2	1	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	4	2	no	—
44	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
46	C₂×Dic₇₃	—	4	2	no	—
47	C₂×Dic₇₃	—	4	2	no	—
48	C₂×Dic₇₃	—	8	1	no	—
49	C₂×Dic₇₃	—	8	1	no	—
50	C₂×Dic₇₃	—	8	2	no	—
51	Dic₁₄₆	—	4	4	no	—
52	Dic₁₄₆	—	8	2	no	—
53	#10596833	—	16	2	no	—
54	C₈×C₇₃	—	4	2	no	—
55	C₂²×C₇₃	—	2	2	no	—
56	C₂²×C₇₃	#323	1	1	yes	characteristic
57	C₂²×C₇₃	#353	1	1	yes	characteristic
58	C₂²×C₇₃	#355	1	1	yes	characteristic
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	—	2	1	no	—
61	C₄×C₇₃	—	4	1	no	—
62	C₄×C₇₃	—	4	1	no	—
63	Dic₇₃	—	8	2	no	—
64	Dic₇₃	—	8	2	no	—
65	#7353	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	—	2	1	no	—
67	C₂×C₇₃	—	2	1	no	—
68	C₂×C₇₃	#1318	1	1	yes	characteristic
69	C₂×C₇₃	#1407	1	2	yes	—
70	#1960	—	73	1	no	—
71	#1324	—	73	2	no	—
72	C₇₃	#7353	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₄⋊Q₈	—	73	1	no	—
74	C₂×#103	—	146	2	no	—
75	C₂×#97	#64408	1	1	yes	characteristic
76	#382	—	73	1	no	—
77	#324	—	146	2	no	—
78	C₂²×Q₈	—	146	1	no	—
79	C₂²×C₈	—	2	1	no	—
80	C₄⋊Q₈	—	292	1	no	—
81	C₄⋊Q₈	—	73	2	no	—
82	C₄⋊Q₈	—	146	1	no	—
83	C₂×C₄⋊C₄	—	292	1	no	—
84	C₂×C₄⋊C₄	—	73	2	no	—
85	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
86	#103	—	292	4	no	—
87	C₂²⋊C₈	—	292	2	no	—
88	#97	—	4	1	no	—
89	#95	—	292	1	no	—
90	C₂×Q₈	—	292	2	no	—
91	C₂×Q₈	—	292	2	no	—
92	C₂×Q₈	—	292	1	no	—
93	C₂×Q₈	—	146	2	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	292	1	no	—
96	C₂²×C₄	—	146	2	no	—
97	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	584	2	no	—
102	C₄⋊C₄	—	292	2	no	—
103	C₄⋊C₄	—	292	2	no	—
104	C₄⋊C₄	—	292	2	no	—
105	C₄²	—	2	1	no	—
106	C₄²	#11924036	1	1	yes	characteristic
107	C₄²	#11924038	1	1	yes	characteristic
108	C₂³	#14556650	1	1	yes	characteristic
109	Q₈	—	292	4	no	—
110	Q₈	—	584	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	292	2	no	—
117	C₂×C₄	—	292	2	no	—
118	C₂×C₄	—	584	1	no	—
119	C₂×C₄	—	584	1	no	—
120	C₂×C₄	—	584	2	no	—
121	C₂×C₄	#14556648	1	1	yes	characteristic
122	C₂×C₄	#14556651	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	1168	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	#17989431	1	1	yes	characteristic
127	C₂²	#17989492	1	1	yes	characteristic
128	C₂²	#17989496	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	584	2	no	—
134	C₄	—	584	2	no	—
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	#22624860	1	1	yes	characteristic
138	C₂	#22625061	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22624871

C₂

yes

maximal

#22610751

C₂

yes

maximal, characteristic

#17989432

—

C₂×#14556591

C₄

yes

characteristic

C₂×#14556516

—

#17988453

C₂²

yes

characteristic

C₂×#97×C₇₃

C₄

yes

characteristic

C₂²⋊#10596833

—

C₂²×Dic₁₄₆

—

#14556591

—

#14556591

—

#14556591

C₈

yes

complemented

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

D₄

yes

—

#14556516

—

#14556498

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

#97×C₇₃

—

C₂×#10596833

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

#99

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

#100

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

#7353

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1318

yes

characteristic

C₂×C₇₃

#1407

yes

—

#1960

—

#1324

—

C₇₃

#7353

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#103

—

146

—

C₂×#97

yes

—

—

146

—

C₂²×Q₈

—

146

—

C₂²×C₈

—

C₄⋊Q₈

—

292

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₂×C₄⋊C₄

—

292

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×#64408

yes

characteristic

#103

—

292

—

C₂²⋊C₈

—

292

—

#97

—

#95

—

292

—

C₂×Q₈

—

292

—

C₂×Q₈

—

292

—

C₂×Q₈

—

292

—

C₂×Q₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

292

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²⋊#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

584

—

102

C₄⋊C₄

—

292

—

103

C₄⋊C₄

—

292

—

104

C₄⋊C₄

—

292

—

105

C₄²

—

106

C₄²

#11924036

yes

characteristic

107

C₄²

#11924038

yes

characteristic

108

C₂³

#14556650

yes

characteristic

109

Q₈

—

292

—

110

Q₈

—

584

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

292

—

117

C₂×C₄

—

292

—

118

C₂×C₄

—

584

—

119

C₂×C₄

—

584

—

120

C₂×C₄

—

584

—

121

C₂×C₄

#14556648

yes

characteristic

122

C₂×C₄

#14556651

yes

characteristic

123

C₈

—

124

C₈

—

1168

—

125

C₂²

—

126

C₂²

#17989431

yes

characteristic

127

C₂²

#17989492

yes

characteristic

128

C₂²

#17989496

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

584

—

134

C₄

—

584

—

135

C₂

—

136

C₂

—

137

C₂

#22624860

yes

characteristic

138

C₂

#22625061

yes

—

	subgroup	G/H
center	C₂²	#17989492
commutator	C₄²×C₇₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×#97×C₇₃	C₄
socle	C₂²×C₇₃	#353
prime radical	C₂³×C₇₃	#99
cosocle	#17988453	C₂²
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#7353
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂²

#17989492

commutator

C₄²×C₇₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#64408

Fitting

C₂×#97×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₂²

C₂×#97

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
12. #14556591	C₈	semidirect
72. C₇₃	#7353	semidirect

H ≅ G/N

type

12. #14556591

C₈

semidirect

72. C₇₃

#7353

semidirect

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

subgroup	index
#22624871	2
#22610751	2
#7353	73

subgroup

index

#22624871

#22610751

#7353

Subgroups — #34000808

Conjugacy classes of subgroups

Complements

Maximal subgroups