Group #34000848 — subgroups

Subgroups — #34000848

Order 18688.

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

normal subgroups only

This group has 190 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	#22624873	C₂	1	2	yes	maximal
2	#22610087	C₂	1	1	yes	maximal, characteristic
3	#17989432	—	2	2	no	—
4	C₂×#14556522	—	2	2	no	—
5	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
6	#17988457	C₂²	1	1	yes	characteristic
7	#384×C₇₃	C₄	1	1	yes	characteristic
8	C₂²⋊#10596833	—	4	2	no	—
9	C₂²×#10596831	—	2	1	no	—
10	C₂×C₄⋊Dic₇₃	—	4	1	no	—
11	C₂×C₄⋊Dic₇₃	D₄	1	2	yes	—
12	#14556522	—	4	4	no	—
13	C₄×#10596831	—	2	1	no	—
14	C₄×#10596831	C₈	1	2	yes	complemented
15	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
16	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
17	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#10596833	—	8	2	no	—
19	C₂²×Dic₇₃	—	2	2	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₄⋊Dic₇₃	—	4	2	no	—
24	C₄⋊Dic₇₃	—	8	2	no	—
25	C₂²×C₄×C₇₃	—	2	1	no	—
26	C₂²×C₄×C₇₃	—	4	1	no	—
27	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₇₃	—	4	2	no	—
29	C₄⋊C₄×C₇₃	—	8	2	no	—
30	C₄²×C₇₃	—	2	1	no	—
31	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
33	C₂³×C₇₃	#99	1	1	yes	characteristic
34	C₂×C₄×C₇₃	—	2	2	no	—
35	C₂×C₄×C₇₃	—	2	2	no	—
36	C₂×C₄×C₇₃	—	2	1	no	—
37	C₂×C₄×C₇₃	—	2	1	no	—
38	C₂×C₄×C₇₃	—	2	1	no	—
39	C₂×C₄×C₇₃	—	2	1	no	—
40	C₂×C₄×C₇₃	—	8	2	no	—
41	C₂×C₄×C₇₃	—	8	1	no	—
42	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×C₄×C₇₃	#101	1	1	yes	characteristic
44	C₂×Dic₇₃	—	4	2	no	—
45	C₂×Dic₇₃	—	8	2	no	—
46	#10596833	—	16	2	no	—
47	#10596831	—	2	4	no	—
48	C₂²×C₇₃	—	2	1	no	—
49	C₂²×C₇₃	—	2	1	no	—
50	C₂²×C₇₃	#324	1	1	yes	characteristic
51	C₂²×C₇₃	#353	1	1	yes	characteristic
52	C₂²×C₇₃	#356	1	1	yes	characteristic
53	C₄×C₇₃	—	2	2	no	—
54	C₄×C₇₃	—	2	2	no	—
55	C₄×C₇₃	—	2	1	no	—
56	C₄×C₇₃	—	2	1	no	—
57	C₄×C₇₃	—	16	1	no	—
58	Dic₇₃	—	8	2	no	—
59	#7371	—	73	1	no	maximal, Sylow
60	C₂×C₇₃	—	2	1	no	—
61	C₂×C₇₃	—	2	1	no	—
62	C₂×C₇₃	#1325	1	1	yes	characteristic
63	C₂×C₇₃	#1409	1	1	yes	characteristic
64	C₂×C₇₃	#1410	1	1	yes	characteristic
65	#1844	—	73	1	no	—
66	#1325	—	73	2	no	—
67	C₇₃	#7371	1	1	yes	characteristic, complemented, Sylow
68	C₂×#106	—	146	2	no	—
69	C₂×C₄×C₈	—	73	1	no	—
70	#384	—	73	1	no	—
71	#384	#64408	1	1	yes	characteristic
72	#324	—	146	2	no	—
73	C₂²×C₈	—	146	1	no	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄⋊C₄	—	292	1	no	—
77	C₂×C₄⋊C₄	—	73	2	no	—
78	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
79	#106	—	292	4	no	—
80	C₂²⋊C₈	—	292	2	no	—
81	C₄×C₈	—	73	2	no	—
82	C₄×C₈	—	146	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	4	1	no	—
85	C₂²×C₄	—	146	2	no	—
86	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
87	C₂×C₈	—	584	2	no	—
88	C₂×C₈	—	146	2	no	—
89	C₂×C₈	—	146	2	no	—
90	C₂×C₈	—	146	2	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	8	2	no	—
93	C₄⋊C₄	—	292	2	no	—
94	C₄⋊C₄	—	584	2	no	—
95	C₄²	—	2	1	no	—
96	C₄²	#11924036	1	1	yes	characteristic
97	C₄²	#11924038	1	1	yes	characteristic
98	C₂³	#14556649	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	—	8	1	no	—
107	C₂×C₄	—	292	2	no	—
108	C₂×C₄	—	584	2	no	—
109	C₂×C₄	#14556648	1	1	yes	characteristic
110	C₂×C₄	#14556652	1	1	yes	characteristic
111	C₈	—	1168	2	no	—
112	C₈	—	146	4	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	#17989433	1	1	yes	characteristic
116	C₂²	#17989491	1	1	yes	characteristic
117	C₂²	#17989497	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	16	1	no	—
123	C₄	—	584	2	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#22624874	1	1	yes	characteristic
127	C₂	#22625064	1	1	yes	characteristic
128	C₂	#22625066	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22624873

C₂

yes

maximal

#22610087

C₂

yes

maximal, characteristic

#17989432

—

C₂×#14556522

—

C₂×C₄×#10596831

C₄

yes

characteristic

#17988457

C₂²

yes

characteristic

#384×C₇₃

C₄

yes

characteristic

C₂²⋊#10596833

—

C₂²×#10596831

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

D₄

yes

—

#14556522

—

C₄×#10596831

—

C₄×#10596831

C₈

yes

complemented

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×#10596833

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×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

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

#101

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596833

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

#7371

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

C₇₃

#7371

yes

characteristic, complemented, Sylow

C₂×#106

—

146

—

C₂×C₄×C₈

—

—

yes

—

146

—

C₂²×C₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

292

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×#64408

yes

characteristic

#106

—

292

—

C₂²⋊C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²⋊#64408

yes

characteristic

C₂×C₈

—

584

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

584

—

C₄²

—

C₄²

yes

C₄²

yes

C₂³

yes

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

292

—

108

C₂×C₄

—

584

—

109

C₂×C₄

#14556648

yes

characteristic

110

C₂×C₄

#14556652

yes

characteristic

111

C₈

—

1168

—

112

C₈

—

146

—

113

C₂²

—

114

C₂²

—

115

C₂²

#17989433

yes

characteristic

116

C₂²

#17989491

yes

characteristic

117

C₂²

#17989497

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

584

—

124

C₂

—

125

C₂

—

126

C₂

#22624874

yes

characteristic

127

C₂

#22625064

yes

characteristic

128

C₂

#22625066

yes

characteristic

	subgroup	G/H
center	C₂²	#17989491
commutator	C₄²×C₇₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	#384×C₇₃	C₄
socle	C₂²×C₇₃	#353
prime radical	C₂³×C₇₃	#99
cosocle	#17988457	C₂²
hypercenter	#384	#64408
hypocenter	C₇₃	#7371
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂²

#17989491

commutator

C₄²×C₇₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#64408

Fitting

#384×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₂²

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
14. C₄×#10596831	C₈	semidirect
67. C₇₃	#7371	semidirect

H ≅ G/N

type

14. C₄×#10596831

C₈

semidirect

67. C₇₃

#7371

semidirect

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

subgroup	index
#22624873	2
#22610087	2
#7371	73

subgroup

index

#22624873

#22610087

#7371

Subgroups — #34000848

Conjugacy classes of subgroups

Complements

Maximal subgroups