Group #34044977 — subgroups

Subgroups — #34044977

Order 19072.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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	#23749653	C₂	1	2	yes	maximal, complemented
2	#23749599	C₂	1	1	yes	maximal, characteristic
3	#23749592	C₂	1	1	yes	maximal, characteristic
4	#23749396	C₂	1	2	yes	maximal, complemented
5	#568×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558720	C₂²	1	1	yes	characteristic
7	C₂²×#11924746	C₂²	1	1	yes	characteristic
8	#18011870	C₂²	1	2	yes	—
9	#18011840	—	2	4	no	—
10	#18011826	C₄	1	4	yes	—
11	C₂²⋊#11924746	—	2	2	no	—
12	C₂×#41×C₁₄₉	C₂²	1	1	yes	characteristic
13	#131×C₁₄₉	—	2	4	no	—
14	C₂×M₁₆×C₁₄₉	—	2	2	no	—
15	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
16	#14558720	—	2	2	no	—
17	C₂×#11924746	—	2	1	no	—
18	C₂×#11924746	—	2	2	no	—
19	C₂×#11924746	—	2	1	no	—
20	C₄⋊Dic₁₄₉	—	2	2	no	—
21	C₂²⋊Dic₁₄₉	C₂×C₄	1	4	yes	—
22	C₄×Dic₁₄₉	—	2	2	no	—
23	#41×C₁₄₉	—	2	2	no	—
24	#41×C₁₄₉	—	2	2	no	—
25	C₂×Q₈×C₁₄₉	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₁₄₉	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₁₄₉	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
29	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₄₉	—	2	4	no	—
31	M₁₆×C₁₄₉	—	2	2	no	—
32	M₁₆×C₁₄₉	—	2	2	no	—
33	C₂×C₈×C₁₄₉	—	2	4	no	—
34	C₂×C₈×C₁₄₉	D₄	1	4	yes	—
35	C₂×Dic₁₄₉	—	2	4	no	—
36	#11924746	—	4	1	no	—
37	#11924746	—	4	1	no	—
38	C₂³×C₁₄₉	C₄⋊C₄	1	2	yes	—
39	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₁₄₉	—	2	2	no	—
41	D₄×C₁₄₉	—	2	4	no	—
42	D₄×C₁₄₉	—	2	2	no	—
43	C₂×C₄×C₁₄₉	—	2	2	no	—
44	C₂×C₄×C₁₄₉	—	2	2	no	—
45	C₂×C₄×C₁₄₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₄₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₄₉	#41	1	2	yes	—
50	C₈×C₁₄₉	—	2	4	no	—
51	C₂²×C₁₄₉	—	2	2	no	—
52	C₂²×C₁₄₉	—	4	1	no	—
53	C₂²×C₁₄₉	—	4	2	no	—
54	C₂²×C₁₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₄₉	C₄×D₄	1	2	yes	—
56	C₄×C₁₄₉	—	2	2	no	—
57	C₄×C₁₄₉	#120	1	1	yes	characteristic
58	C₄×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₄₉	#123	1	1	yes	characteristic
60	Dic₁₄₉	—	4	4	no	—
61	C₂×C₁₄₉	—	2	2	no	—
62	C₂×C₁₄₉	—	2	1	no	—
63	C₂×C₁₄₉	—	4	2	no	—
64	C₂×C₁₄₉	#385	1	1	yes	characteristic
65	C₁₄₉	#1868	1	1	yes	characteristic, complemented, Sylow
66	#1868	—	149	1	no	maximal, Sylow
67	#568	D₁₄₉	1	1	yes	characteristic
68	#424	—	149	2	no	—
69	#410	—	149	1	no	—
70	#408	—	149	1	no	—
71	#344	—	149	2	no	—
72	C₂×#41	C₂×D₁₄₉	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	149	1	no	—
76	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
77	C₂²×C₈	—	149	1	no	—
78	#117	—	149	2	no	—
79	C₄⋊C₈	—	298	4	no	—
80	#99	—	149	4	no	—
81	C₂²⋊C₈	—	298	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₂₉₈	1	1	yes	characteristic
85	C₂×D₄	C₄×D₁₄₉	1	2	yes	—
86	C₂×D₄	Dic₂₉₈	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₁₄₉	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
89	M₁₆	—	2	4	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	298	2	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	298	1	no	—
95	C₂×C₈	—	298	1	no	—
96	C₂×C₈	—	298	2	no	—
97	C₂×C₈	C₂²⋊D₁₄₉	1	4	yes	—
98	C₄⋊C₄	—	298	2	no	—
99	C₂²⋊C₄	—	149	4	no	—
100	C₄²	—	298	2	no	—
101	C₂³	#14558715	1	2	yes	—
102	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	298	4	no	—
109	C₂×C₄	#14558715	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₂₉₈	1	1	yes	characteristic
113	C₂×C₄	#14558736	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	596	1	no	—
116	C₈	—	596	1	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×#14558715	1	1	yes	characteristic
121	C₂²	C₄×C₂²⋊D₁₄₉	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	596	4	no	—
124	C₄	#18011880	1	1	yes	characteristic
125	C₄	#18011890	1	2	yes	—
126	C₄	#18011894	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#23749498	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23749653

C₂

yes

maximal, complemented

#23749599

C₂

yes

maximal, characteristic

#23749592

C₂

yes

maximal, characteristic

#23749396

C₂

yes

maximal, complemented

#568×C₁₄₉

C₂

yes

maximal, characteristic

C₂×#14558720

C₂²

yes

characteristic

C₂²×#11924746

C₂²

yes

characteristic

#18011870

C₂²

yes

—

#18011840

—

#18011826

C₄

yes

—

C₂²⋊#11924746

—

C₂×#41×C₁₄₉

C₂²

yes

characteristic

#131×C₁₄₉

—

C₂×M₁₆×C₁₄₉

—

C₂×M₁₆×C₁₄₉

C₂²

yes

—

—

—

—

—

C₄⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

C₂×C₄

yes

—

C₄×Dic₁₄₉

—

#41×C₁₄₉

—

#41×C₁₄₉

—

C₂×Q₈×C₁₄₉

Q₈

yes

characteristic

C₂×D₄×C₁₄₉

C₂×C₄

yes

—

C₂×D₄×C₁₄₉

Q₈

yes

characteristic

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₂×Dic₁₄₉

—

#11924746

—

#11924746

—

C₂³×C₁₄₉

C₄⋊C₄

yes

—

C₂³×C₁₄₉

C₂²×C₄

yes

characteristic

Q₈×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₄₉

C₄×D₄

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#120

yes

characteristic

C₄×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

#123

yes

characteristic

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#385

yes

characteristic

C₁₄₉

#1868

yes

characteristic, complemented, Sylow

#1868

—

149

maximal, Sylow

#568

D₁₄₉

yes

characteristic

#424

—

149

—

#410

—

149

—

#408

—

149

—

#344

—

149

—

C₂×#41

C₂×D₁₄₉

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

C₂²×C₈

—

149

—

#117

—

149

—

C₄⋊C₈

—

298

—

#99

—

149

—

C₂²⋊C₈

—

298

—

#41

—

#41

—

C₂×Q₈

Dic₂₉₈

yes

characteristic

C₂×D₄

C₄×D₁₄₉

yes

—

C₂×D₄

Dic₂₉₈

yes

characteristic

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

298

—

C₂×C₈

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

C₂²⋊D₁₄₉

yes

—

C₄⋊C₄

—

298

—

C₂²⋊C₄

—

149

—

100

C₄²

—

298

—

101

C₂³

#14558715

yes

—

102

C₂³

C₂×C₄×D₁₄₉

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

298

—

109

C₂×C₄

#14558715

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

111

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

112

C₂×C₄

C₂×Dic₂₉₈

yes

characteristic

113

C₂×C₄

#14558736

yes

—

114

C₈

—

115

C₈

—

596

—

116

C₈

—

596

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#14558715

yes

characteristic

121

C₂²

C₄×C₂²⋊D₁₄₉

yes

—

122

C₄

—

123

C₄

—

596

—

124

C₄

#18011880

yes

characteristic

125

C₄

#18011890

yes

—

126

C₄

#18011894

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#23749498

yes

characteristic

	subgroup	G/H
center	C₄	#18011880
commutator	C₂³×C₁₄₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#568×C₁₄₉	C₂
socle	C₂×C₁₄₉	#385
prime radical	C₂×#41×C₁₄₉	C₂²
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	#568	D₁₄₉
hypocenter	C₁₄₉	#1868
LCS	G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#18011880

commutator

C₂³×C₁₄₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#568×C₁₄₉

C₂

socle

C₂×C₁₄₉

#385

prime radical

C₂×#41×C₁₄₉

C₂²

cosocle

C₂²×C₄×C₁₄₉

C₂³

D₁₄₉

C₁₄₉

G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #23749653	C₂	semidirect
4. #23749396	C₂	semidirect
65. C₁₄₉	#1868	semidirect

H ≅ G/N

type

1. #23749653

C₂

semidirect

4. #23749396

C₂

semidirect

65. C₁₄₉

#1868

semidirect

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

subgroup	index
#23749653	2
#23749599	2
#23749592	2
#23749396	2
#568×C₁₄₉	2
#1868	149

subgroup

index

#568×C₁₄₉

149

Subgroups — #34044977

Conjugacy classes of subgroups

Complements

Maximal subgroups