Group #34046742 — subgroups

Subgroups — #34046742

Order 19072.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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×Dic₁₁₉₂	C₂	1	2	yes	maximal
2	#23750005	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₅₉₆	C₂	1	1	yes	maximal, characteristic
4	#23749430	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₉₆	—	2	1	no	—
7	C₂×Dic₅₉₆	C₂²	1	2	yes	—
8	C₂×Dic₅₉₆	C₂²	1	1	yes	characteristic
9	#18011911	—	2	1	no	—
10	C₄×Dic₂₉₈	—	2	1	no	—
11	Dic₁₁₉₂	—	2	4	no	—
12	#18011845	C₂²	1	1	yes	characteristic
13	#18011836	—	2	1	no	—
14	C₂×C₁₆×C₁₄₉	C₂²	1	2	yes	—
15	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₈	—	2	1	no	—
17	C₂×Dic₂₉₈	—	2	2	no	—
18	Dic₅₉₆	—	2	2	no	—
19	Dic₅₉₆	—	2	1	no	—
20	Dic₅₉₆	—	2	2	no	—
21	Dic₅₉₆	—	4	2	no	—
22	Dic₅₉₆	D₄	1	2	yes	—
23	C₄⋊Dic₁₄₉	—	2	1	no	—
24	C₄⋊Dic₁₄₉	—	4	1	no	—
25	C₄⋊Dic₁₄₉	—	4	1	no	—
26	#14558715	—	4	1	no	—
27	C₄×Dic₁₄₉	—	4	1	no	—
28	C₂×C₈×C₁₄₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
30	C₄²×C₁₄₉	D₄	1	1	yes	characteristic
31	C₁₆×C₁₄₉	—	2	2	no	—
32	Dic₂₉₈	—	2	2	no	—
33	Dic₂₉₈	—	4	1	no	—
34	Dic₂₉₈	—	4	2	no	—
35	Dic₂₉₈	—	4	2	no	—
36	C₂×Dic₁₄₉	—	4	1	no	—
37	C₂×Dic₁₄₉	—	4	2	no	—
38	C₂×Dic₁₄₉	—	4	1	no	—
39	C₂×C₄×C₁₄₉	D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₄₉	D₈	1	1	yes	characteristic
41	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
42	C₈×C₁₄₉	—	2	1	no	—
43	C₈×C₁₄₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₄₉	#41	1	1	yes	characteristic
45	C₂²×C₁₄₉	C₂×D₈	1	1	yes	characteristic
46	C₄×C₁₄₉	—	2	1	no	—
47	C₄×C₁₄₉	Q₃₂	1	2	yes	complemented
48	C₄×C₁₄₉	#121	1	1	yes	characteristic
49	C₄×C₁₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₁₄₉	—	4	2	no	—
51	Dic₁₄₉	—	8	2	no	—
52	Dic₁₄₉	—	8	1	no	—
53	C₂×C₁₄₉	#459	1	1	yes	characteristic
54	C₂×C₁₄₉	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₁₄₉	#510	1	1	yes	characteristic
56	C₁₄₉	#2209	1	1	yes	characteristic, complemented, Sylow
57	#2209	—	149	1	no	maximal, Sylow
58	C₂×Q₃₂	—	149	2	no	—
59	#494	—	149	1	no	—
60	C₄×Q₁₆	—	149	1	no	—
61	C₄⋊C₁₆	D₁₄₉	1	1	yes	characteristic
62	#358	—	149	2	no	—
63	C₂×Q₁₆	—	298	1	no	—
64	C₂×Q₁₆	—	149	2	no	—
65	C₂×Q₁₆	—	149	1	no	—
66	C₄⋊Q₈	—	298	1	no	—
67	C₄×Q₈	—	298	1	no	—
68	Q₃₂	—	298	4	no	—
69	C₂×C₁₆	C₂×D₁₄₉	1	2	yes	—
70	#107	—	149	1	no	—
71	#103	—	298	1	no	—
72	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
73	C₂×Q₈	—	298	1	no	—
74	C₂×Q₈	—	298	2	no	—
75	Q₁₆	—	298	2	no	—
76	Q₁₆	—	298	1	no	—
77	Q₁₆	—	298	2	no	—
78	Q₁₆	—	596	2	no	—
79	Q₁₆	—	149	2	no	—
80	C₂×C₈	D₅₉₆	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
82	C₄⋊C₄	—	298	1	no	—
83	C₄⋊C₄	—	596	1	no	—
84	C₄⋊C₄	—	596	1	no	—
85	C₄⋊C₄	—	596	1	no	—
86	C₄²	—	596	1	no	—
87	C₄²	D₅₉₆	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	298	2	no	—
90	Q₈	—	596	1	no	—
91	Q₈	—	596	2	no	—
92	Q₈	—	596	2	no	—
93	C₂×C₄	—	596	1	no	—
94	C₂×C₄	—	596	1	no	—
95	C₂×C₄	—	596	2	no	—
96	C₂×C₄	D₁₁₉₂	1	1	yes	characteristic
97	C₂×C₄	D₁₁₉₂	1	1	yes	characteristic
98	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₁₄₉	1	1	yes	characteristic
101	C₈	#14558737	1	1	yes	characteristic
102	C₂²	C₂×D₁₁₉₂	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	596	2	no	—
105	C₄	—	1192	1	no	—
106	C₄	—	1192	2	no	—
107	C₄	Dic₁₁₉₂	1	2	yes	complemented
108	C₄	#18011886	1	1	yes	characteristic
109	C₄	#18011939	1	1	yes	characteristic
110	C₂	#23749821	1	1	yes	characteristic
111	C₂	C₂×Dic₁₁₉₂	1	1	yes	characteristic
112	C₂	#23750068	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₁₉₂

C₂

yes

maximal

#23750005

C₂

yes

maximal, characteristic

C₄×Dic₅₉₆

C₂

yes

maximal, characteristic

#23749430

C₂

yes

maximal

C₄⋊C₁₆×C₁₄₉

C₂

yes

maximal, characteristic

C₂×Dic₅₉₆

—

C₂×Dic₅₉₆

C₂²

yes

—

C₂×Dic₅₉₆

C₂²

yes

characteristic

#18011911

—

C₄×Dic₂₉₈

—

Dic₁₁₉₂

—

#18011845

C₂²

yes

characteristic

#18011836

—

C₂×C₁₆×C₁₄₉

C₂²

yes

—

C₄×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

Dic₅₉₆

—

Dic₅₉₆

—

Dic₅₉₆

—

Dic₅₉₆

—

Dic₅₉₆

D₄

yes

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

#14558715

—

C₄×Dic₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

characteristic

C₂×C₈×C₁₄₉

C₂³

yes

characteristic

C₄²×C₁₄₉

D₄

yes

characteristic

C₁₆×C₁₄₉

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×C₄×C₁₄₉

D₈

yes

characteristic

C₂×C₄×C₁₄₉

D₈

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₈×C₁₄₉

—

C₈×C₁₄₉

C₂×D₄

yes

characteristic

C₈×C₁₄₉

#41

yes

characteristic

C₂²×C₁₄₉

C₂×D₈

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

Q₃₂

yes

C₄×C₁₄₉

yes

C₄×C₁₄₉

yes

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

#459

yes

characteristic

C₂×C₁₄₉

C₂×Q₃₂

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2209

—

149

maximal, Sylow

C₂×Q₃₂

—

149

—

#494

—

149

—

C₄×Q₁₆

—

149

—

C₄⋊C₁₆

D₁₄₉

yes

characteristic

#358

—

149

—

C₂×Q₁₆

—

298

—

C₂×Q₁₆

—

149

—

C₂×Q₁₆

—

149

—

C₄⋊Q₈

—

298

—

C₄×Q₈

—

298

—

Q₃₂

—

298

—

C₂×C₁₆

C₂×D₁₄₉

yes

—

#107

—

149

—

#103

—

298

—

C₄×C₈

C₂×D₁₄₉

yes

characteristic

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

Q₁₆

—

298

—

Q₁₆

—

298

—

Q₁₆

—

298

—

Q₁₆

—

596

—

Q₁₆

—

149

—

C₂×C₈

D₅₉₆

yes

characteristic

C₂×C₈

C₂²×D₁₄₉

yes

characteristic

C₄⋊C₄

—

298

—

C₄⋊C₄

—

596

—

C₄⋊C₄

—

596

—

C₄⋊C₄

—

596

—

C₄²

—

596

—

C₄²

D₅₉₆

yes

characteristic

C₁₆

—

Q₈

—

298

—

Q₈

—

596

—

Q₈

—

596

—

Q₈

—

596

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

D₁₁₉₂

yes

characteristic

C₂×C₄

D₁₁₉₂

yes

characteristic

C₂×C₄

C₂×D₅₉₆

yes

characteristic

C₈

—

100

C₈

D₄×D₁₄₉

yes

characteristic

101

C₈

#14558737

yes

characteristic

102

C₂²

C₂×D₁₁₉₂

yes

characteristic

103

C₄

—

104

C₄

—

596

—

105

C₄

—

1192

—

106

C₄

—

1192

—

107

C₄

Dic₁₁₉₂

yes

complemented

108

C₄

#18011886

yes

characteristic

109

C₄

#18011939

yes

characteristic

110

C₂

#23749821

yes

characteristic

111

C₂

C₂×Dic₁₁₉₂

yes

characteristic

112

C₂

#23750068

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₁₉₂
commutator, cosocle	C₂×C₈×C₁₄₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₄₉
Fitting	C₄⋊C₁₆×C₁₄₉	C₂
socle, prime radical	C₂²×C₁₄₉	C₂×D₈
hypercenter	C₄⋊C₁₆	D₁₄₉
hypocenter	C₁₄₉	#2209
LCS	G ▷ C₂×C₈×C₁₄₉ ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₁₁₉₂

commutator, cosocle

C₂×C₈×C₁₄₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₄₉

Fitting

C₄⋊C₁₆×C₁₄₉

C₂

socle, prime radical

C₂²×C₁₄₉

C₂×D₈

hypercenter

C₄⋊C₁₆

D₁₄₉

hypocenter

C₁₄₉

#2209

LCS

G ▷ C₂×C₈×C₁₄₉ ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₁₄₉	Q₃₂	semidirect
56. C₁₄₉	#2209	semidirect
107. C₄	Dic₁₁₉₂	semidirect

H ≅ G/N

type

47. C₄×C₁₄₉

Q₃₂

semidirect

56. C₁₄₉

#2209

semidirect

107. C₄

Dic₁₁₉₂

semidirect

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

subgroup	index
C₂×Dic₁₁₉₂	2
#23750005	2
C₄×Dic₅₉₆	2
#23749430	2
C₄⋊C₁₆×C₁₄₉	2
#2209	149

subgroup

index

C₂×Dic₁₁₉₂

#23750005

C₄×Dic₅₉₆

#23749430

C₄⋊C₁₆×C₁₄₉

#2209

149

Subgroups — #34046742

Conjugacy classes of subgroups

Complements

Maximal subgroups