Group #34045960 — subgroups

Subgroups — #34045960

Order 19072.

256 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×Dic₅₉₆	C₂	1	1	yes	maximal, characteristic
2	#23749651	C₂	1	1	yes	maximal, characteristic
3	#23749635	C₂	1	2	yes	maximal
4	C₂×#18011829	C₂	1	2	yes	maximal
5	#341×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
7	C₂×Dic₅₉₆	—	2	2	no	—
8	C₂×Dic₅₉₆	—	2	4	no	—
9	C₂×#14558720	C₂²	1	1	yes	characteristic
10	#18011870	C₂²	1	1	yes	characteristic
11	#18011840	—	2	2	no	—
12	#18011836	—	2	4	no	—
13	#18011829	—	2	4	no	—
14	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₈	—	2	2	no	—
17	C₂×Dic₂₉₈	—	2	2	no	—
18	C₂×Dic₂₉₈	—	2	2	no	—
19	C₂×Dic₂₉₈	—	2	2	no	—
20	C₂×Dic₂₉₈	—	2	2	no	—
21	C₂×Dic₂₉₈	D₄	1	4	yes	—
22	C₂²×Dic₁₄₉	—	2	2	no	—
23	Dic₅₉₆	—	2	8	no	—
24	#14558720	—	2	2	no	—
25	C₂×#11924746	—	2	1	no	—
26	C₄⋊Dic₁₄₉	D₄	1	2	yes	—
27	C₂²⋊Dic₁₄₉	—	2	1	no	—
28	C₄×Dic₁₄₉	—	2	1	no	—
29	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₄₉	—	2	2	no	—
31	M₁₆×C₁₄₉	—	2	2	no	—
32	C₂×C₈×C₁₄₉	—	2	2	no	—
33	C₂×C₈×C₁₄₉	—	2	2	no	—
34	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
35	Dic₂₉₈	—	2	4	no	—
36	Dic₂₉₈	—	2	4	no	—
37	Dic₂₉₈	—	4	2	no	—
38	Dic₂₉₈	—	4	2	no	—
39	C₂×Dic₁₄₉	—	2	2	no	—
40	C₂×Dic₁₄₉	—	2	4	no	—
41	C₂×Dic₁₄₉	—	4	2	no	—
42	C₂×Dic₁₄₉	—	4	2	no	—
43	#11924746	—	4	1	no	—
44	C₂³×C₁₄₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₄₉	#41	1	2	yes	—
49	C₈×C₁₄₉	—	2	2	no	—
50	C₈×C₁₄₉	—	4	2	no	—
51	C₂²×C₁₄₉	—	2	2	no	—
52	C₂²×C₁₄₉	#121	1	1	yes	characteristic
53	C₂²×C₁₄₉	#123	1	2	yes	—
54	C₄×C₁₄₉	#120	1	1	yes	characteristic
55	C₄×C₁₄₉	#121	1	2	yes	—
56	C₄×C₁₄₉	#124	1	1	yes	characteristic
57	Dic₁₄₉	—	4	4	no	—
58	Dic₁₄₉	—	4	2	no	—
59	C₂×C₁₄₉	—	2	2	no	—
60	C₂×C₁₄₉	#394	1	1	yes	characteristic
61	C₂×C₁₄₉	#473	1	2	yes	—
62	C₁₄₉	#2047	1	1	yes	characteristic, complemented, Sylow
63	#2047	—	149	1	no	maximal, Sylow
64	C₂²×Q₁₆	—	149	1	no	—
65	#424	—	149	1	no	—
66	#419	—	149	2	no	—
67	C₂×#101	—	149	2	no	—
68	#341	D₁₄₉	1	1	yes	characteristic
69	C₂²×Q₈	—	149	2	no	—
70	C₂×Q₁₆	—	298	4	no	—
71	C₂×Q₁₆	—	298	2	no	—
72	C₂×M₁₆	—	149	1	no	—
73	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
74	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
75	#117	—	149	1	no	—
76	C₄⋊C₈	—	298	2	no	—
77	#103	—	298	4	no	—
78	#101	—	298	4	no	—
79	C₂×Q₈	—	298	2	no	—
80	C₂×Q₈	—	298	2	no	—
81	C₂×Q₈	—	298	2	no	—
82	C₂×Q₈	—	298	2	no	—
83	C₂×Q₈	—	298	2	no	—
84	C₂×Q₈	—	149	4	no	—
85	C₂²×C₄	—	298	2	no	—
86	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
87	Q₁₆	—	298	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	298	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	298	1	no	—
94	C₂×C₈	D₅₉₆	1	2	yes	—
95	C₄⋊C₄	—	149	2	no	—
96	C₂²⋊C₄	—	298	1	no	—
97	C₄²	—	298	1	no	—
98	C₂³	#14558737	1	1	yes	characteristic
99	Q₈	—	298	4	no	—
100	Q₈	—	298	4	no	—
101	Q₈	—	596	2	no	—
102	Q₈	—	596	2	no	—
103	C₂×C₄	—	298	2	no	—
104	C₂×C₄	—	298	4	no	—
105	C₂×C₄	—	596	2	no	—
106	C₂×C₄	—	596	2	no	—
107	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
108	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
109	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
110	C₂×C₄	#14558736	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	596	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#18011886	1	1	yes	characteristic
116	C₂²	#18011896	1	2	yes	—
117	C₄	—	596	2	no	—
118	C₄	—	596	4	no	—
119	C₄	#18011881	1	1	yes	characteristic
120	C₄	#18011884	1	2	yes	—
121	C₄	#18011899	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#23749543	1	1	yes	characteristic
124	C₂	#23749901	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₉₆

C₂

yes

maximal, characteristic

#23749651

C₂

yes

maximal, characteristic

#23749635

C₂

yes

maximal

C₂×#18011829

C₂

yes

maximal

#341×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

—

C₂×Dic₅₉₆

—

C₂×Dic₅₉₆

—

C₂×#14558720

C₂²

yes

characteristic

#18011870

C₂²

yes

—

—

—

C₂×M₁₆×C₁₄₉

C₂²

yes

—

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

D₄

yes

—

C₂²×Dic₁₄₉

—

Dic₅₉₆

—

#14558720

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

D₄

yes

—

C₂²⋊Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

#41

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#121

yes

characteristic

C₂²×C₁₄₉

#123

yes

—

C₄×C₁₄₉

#120

yes

characteristic

C₄×C₁₄₉

#121

yes

—

C₄×C₁₄₉

#124

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#394

yes

characteristic

C₂×C₁₄₉

#473

yes

—

C₁₄₉

#2047

yes

characteristic, complemented, Sylow

#2047

—

149

maximal, Sylow

C₂²×Q₁₆

—

149

—

#424

—

149

—

#419

—

149

—

C₂×#101

—

149

—

#341

D₁₄₉

yes

characteristic

C₂²×Q₈

—

149

—

C₂×Q₁₆

—

298

—

C₂×Q₁₆

—

298

—

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

#117

—

149

—

C₄⋊C₈

—

298

—

#103

—

298

—

#101

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

C₂²×C₄

—

298

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

Q₁₆

—

298

—

M₁₆

—

M₁₆

—

M₁₆

—

298

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

298

—

C₂×C₈

D₅₉₆

yes

—

C₄⋊C₄

—

149

—

C₂²⋊C₄

—

298

—

C₄²

—

298

—

C₂³

#14558737

yes

characteristic

Q₈

—

298

—

100

Q₈

—

298

—

101

Q₈

—

596

—

102

Q₈

—

596

—

103

C₂×C₄

—

298

—

104

C₂×C₄

—

298

—

105

C₂×C₄

—

596

—

106

C₂×C₄

—

596

—

107

C₂×C₄

D₄×D₁₄₉

yes

—

108

C₂×C₄

D₄×D₁₄₉

yes

characteristic

109

C₂×C₄

C₂×D₅₉₆

yes

characteristic

110

C₂×C₄

#14558736

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

596

—

114

C₂²

—

115

C₂²

#18011886

yes

characteristic

116

C₂²

#18011896

yes

—

117

C₄

—

596

—

118

C₄

—

596

—

119

C₄

#18011881

yes

characteristic

120

C₄

#18011884

yes

—

121

C₄

#18011899

yes

characteristic

122

C₂

—

123

C₂

#23749543

yes

characteristic

124

C₂

#23749901

yes

—

	subgroup	G/H
center	C₂²	#18011886
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#341×C₁₄₉	C₂
socle	C₂²×C₁₄₉	#121
prime radical	C₂³×C₁₄₉	#41
hypercenter	#341	D₁₄₉
hypocenter	C₁₄₉	#2047
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#18011886

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#341×C₁₄₉

C₂

C₂²×C₁₄₉

C₂³×C₁₄₉

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
62. C₁₄₉	#2047	semidirect

H ≅ G/N

type

62. C₁₄₉

#2047

semidirect

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

subgroup	index
C₂²×Dic₅₉₆	2
#23749651	2
#23749635	2
C₂×#18011829	2
#341×C₁₄₉	2
#2047	149

subgroup

index

C₂²×Dic₅₉₆

#341×C₁₄₉

149

Subgroups — #34045960

Conjugacy classes of subgroups

Complements

Maximal subgroups