Group #34045660 — subgroups

Subgroups — #34045660

Order 19072.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#18011903	C₂	1	1	yes	maximal, characteristic
2	#23749495	C₂	1	1	yes	maximal, characteristic
3	#23749388	C₂	1	4	yes	maximal
4	C₂×#97×C₁₄₉	C₂	1	1	yes	maximal, characteristic
5	#18011903	—	2	2	no	—
6	#18011903	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₁₄₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
10	C₂×#14558715	—	2	2	no	—
11	C₂²×C₈×C₁₄₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
13	#97×C₁₄₉	—	2	2	no	—
14	C₂²×Dic₁₄₉	—	2	2	no	—
15	C₂²×Dic₁₄₉	—	2	2	no	—
16	C₄⋊Dic₁₄₉	—	2	2	no	—
17	C₄⋊Dic₁₄₉	—	2	4	no	—
18	C₄⋊Dic₁₄₉	—	4	4	no	—
19	C₄⋊Dic₁₄₉	C₂×C₄	1	4	yes	—
20	#14558715	—	2	8	no	—
21	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
22	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₄₉	—	2	2	no	—
24	C₂×C₈×C₁₄₉	—	2	2	no	—
25	C₂×C₈×C₁₄₉	Q₈	1	4	yes	—
26	C₄²×C₁₄₉	C₂×C₄	1	2	yes	—
27	C₄²×C₁₄₉	D₄	1	2	yes	—
28	C₂×Dic₁₄₉	—	2	4	no	—
29	C₂×Dic₁₄₉	—	4	2	no	—
30	C₂×Dic₁₄₉	—	4	2	no	—
31	C₂×Dic₁₄₉	—	4	2	no	—
32	C₂×Dic₁₄₉	—	4	4	no	—
33	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₄₉	—	2	2	no	—
35	C₂×C₄×C₁₄₉	—	2	2	no	—
36	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₄₉	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₄₉	#41	1	2	yes	—
41	C₈×C₁₄₉	—	2	4	no	—
42	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₁₄₉	#124	1	1	yes	characteristic
44	C₂²×C₁₄₉	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₁₄₉	#137	1	2	yes	—
47	C₄×C₁₄₉	—	2	4	no	—
48	C₄×C₁₄₉	C₄×Q₈	1	2	yes	—
49	C₄×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
50	Dic₁₄₉	—	4	4	no	—
51	Dic₁₄₉	—	8	2	no	—
52	C₂×C₁₄₉	#391	1	1	yes	characteristic
53	C₂×C₁₄₉	#417	1	2	yes	—
54	C₂×C₁₄₉	#490	1	1	yes	characteristic
55	C₂×C₁₄₉	#491	1	1	yes	characteristic
56	C₂×C₁₄₉	#501	1	2	yes	—
57	C₁₄₉	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	149	1	no	maximal, Sylow
59	C₂×#125	—	149	1	no	—
60	C₂×#97	D₁₄₉	1	1	yes	characteristic
61	#384	—	149	1	no	—
62	#340	—	149	4	no	—
63	C₂²×C₈	C₂×D₁₄₉	1	2	yes	—
64	#125	—	298	2	no	—
65	#125	—	149	4	no	—
66	C₂×C₄⋊C₄	—	298	2	no	—
67	C₂×C₄⋊C₄	—	298	2	no	—
68	C₂×C₄⋊C₄	—	149	2	no	—
69	C₂×C₄⋊C₄	—	149	2	no	—
70	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
71	#97	—	2	2	no	—
72	C₂²×C₄	—	298	2	no	—
73	C₂²×C₄	—	298	2	no	—
74	C₂²×C₄	D₅₉₆	1	2	yes	—
75	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Dic₂₉₈	1	4	yes	—
79	C₄⋊C₄	—	298	2	no	—
80	C₄⋊C₄	—	298	8	no	—
81	C₄⋊C₄	—	298	4	no	—
82	C₄⋊C₄	—	596	4	no	—
83	C₄⋊C₄	—	149	4	no	—
84	C₄²	C₄×D₁₄₉	1	2	yes	—
85	C₄²	C₂²⋊D₁₄₉	1	2	yes	—
86	C₂³	C₂×D₅₉₆	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	298	4	no	—
90	C₂×C₄	—	596	2	no	—
91	C₂×C₄	—	596	2	no	—
92	C₂×C₄	—	596	4	no	—
93	C₂×C₄	—	596	2	no	—
94	C₂×C₄	#14558713	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₂₉₈	1	2	yes	—
98	C₂×C₄	#14558736	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×#14558713	1	1	yes	characteristic
101	C₂²	#18011899	1	1	yes	characteristic
102	C₂²	#18011911	1	1	yes	characteristic
103	C₂²	#18011939	1	2	yes	—
104	C₂²	#18011944	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	596	4	no	—
107	C₄	—	1192	2	no	—
108	C₄	C₄×Dic₂₉₈	1	2	yes	—
109	C₄	#18011890	1	2	yes	—
110	C₂	#23749531	1	1	yes	characteristic
111	C₂	#23749627	1	2	yes	—
112	C₂	#23749990	1	1	yes	characteristic
113	C₂	#23749994	1	1	yes	characteristic
114	C₂	#23750034	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011903

C₂

yes

maximal, characteristic

#23749495

C₂

yes

maximal, characteristic

#23749388

C₂

yes

maximal

C₂×#97×C₁₄₉

C₂

yes

maximal, characteristic

#18011903

—

#18011903

C₄

yes

complemented

C₂×C₄⋊Dic₁₄₉

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×#14558715

—

C₂²×C₈×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

#97×C₁₄₉

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

#14558715

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

Q₈

yes

—

C₄²×C₁₄₉

C₂×C₄

yes

—

C₄²×C₁₄₉

D₄

yes

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×Q₈

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₄₉

#124

yes

characteristic

C₂²×C₁₄₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₄₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₄₉

#137

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₄×Q₈

yes

—

C₄×C₁₄₉

C₂²⋊Q₈

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

#391

yes

characteristic

C₂×C₁₄₉

#417

yes

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

—

C₁₄₉

#1992

yes

characteristic, complemented, Sylow

#1992

—

149

maximal, Sylow

C₂×#125

—

149

—

C₂×#97

D₁₄₉

yes

characteristic

#384

—

149

—

#340

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

—

#125

—

298

—

#125

—

149

—

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#97

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

D₅₉₆

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₂₉₈

yes

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

596

—

C₄⋊C₄

—

149

—

C₄²

C₄×D₁₄₉

yes

—

C₄²

C₂²⋊D₁₄₉

yes

—

C₂³

C₂×D₅₉₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

298

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

#14558713

yes

—

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

C₂×C₄

C₂×Dic₂₉₈

yes

—

C₂×C₄

#14558736

yes

—

C₈

—

100

C₂²

C₂×#14558713

yes

characteristic

101

C₂²

#18011899

yes

characteristic

102

C₂²

#18011911

yes

characteristic

103

C₂²

#18011939

yes

—

104

C₂²

#18011944

yes

—

105

C₄

—

106

C₄

—

596

—

107

C₄

—

1192

—

108

C₄

C₄×Dic₂₉₈

yes

—

109

C₄

#18011890

yes

—

110

C₂

#23749531

yes

characteristic

111

C₂

#23749627

yes

—

112

C₂

#23749990

yes

characteristic

113

C₂

#23749994

yes

characteristic

114

C₂

#23750034

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₉₆
commutator	C₂×C₄×C₁₄₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	C₂×#97×C₁₄₉	C₂
socle, prime radical	C₂³×C₁₄₉	C₂×D₄
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	C₂×#97	D₁₄₉
hypocenter	C₁₄₉	#1992
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₉₆

commutator

C₂×C₄×C₁₄₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

C₂×#97×C₁₄₉

C₂

socle, prime radical

C₂³×C₁₄₉

C₂×D₄

cosocle

C₂²×C₄×C₁₄₉

C₂³

hypercenter

C₂×#97

D₁₄₉

hypocenter

C₁₄₉

#1992

LCS

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
6. #18011903	C₄	semidirect
57. C₁₄₉	#1992	semidirect

H ≅ G/N

type

6. #18011903

C₄

semidirect

57. C₁₄₉

#1992

semidirect

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

subgroup	index
C₂×#18011903	2
#23749495	2
#23749388	2
C₂×#97×C₁₄₉	2
#1992	149

subgroup

index

C₂×#97×C₁₄₉

149

Subgroups — #34045660

Conjugacy classes of subgroups

Complements

Maximal subgroups