Group #34046412 — subgroups

Subgroups — #34046412

Order 19072.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	#23749663	C₂	1	1	yes	maximal, characteristic
3	#23749430	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₉₈	—	2	1	no	—
6	C₂×Dic₅₉₆	—	2	1	no	—
7	C₂×Dic₅₉₆	—	2	2	no	—
8	C₂×Dic₅₉₆	C₄	1	4	yes	complemented
9	C₂×Dic₅₉₆	C₂²	1	2	yes	—
10	#18011870	—	2	1	no	—
11	#18011845	C₂²	1	2	yes	—
12	#18011843	—	2	1	no	—
13	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
14	M₃₂×C₁₄₉	—	2	2	no	—
15	C₂×C₁₆×C₁₄₉	C₂²	1	2	yes	—
16	C₂×Dic₂₉₈	—	2	4	no	—
17	C₂×Dic₂₉₈	—	2	2	no	—
18	C₂×Dic₂₉₈	—	4	1	no	—
19	C₂×Dic₂₉₈	—	4	2	no	—
20	C₂²×Dic₁₄₉	—	4	1	no	—
21	Dic₅₉₆	—	2	4	no	—
22	Dic₅₉₆	—	2	2	no	—
23	Dic₅₉₆	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₁₄₉	—	2	2	no	—
25	C₂²⋊Dic₁₄₉	—	4	1	no	—
26	C₄×Dic₁₄₉	—	4	1	no	—
27	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₄₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₄₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
31	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
32	C₁₆×C₁₄₉	—	2	2	no	—
33	Dic₂₉₈	—	2	4	no	—
34	Dic₂₉₈	—	4	4	no	—
35	Dic₂₉₈	—	4	2	no	—
36	C₂×Dic₁₄₉	—	4	4	no	—
37	C₂×Dic₁₄₉	—	4	2	no	—
38	C₂×Dic₁₄₉	—	4	2	no	—
39	C₂³×C₁₄₉	D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₄₉	D₈	1	1	yes	characteristic
42	C₂×C₄×C₁₄₉	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₄₉	C₂²⋊C₄	1	2	yes	—
45	C₈×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₄₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₁₄₉	—	2	2	no	—
48	C₂²×C₁₄₉	#102	1	2	yes	—
49	C₂²×C₁₄₉	C₂×D₈	1	1	yes	characteristic
50	C₄×C₁₄₉	#102	1	2	yes	—
51	C₄×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₁₄₉	C₂×SD₁₆	1	1	yes	characteristic
53	Dic₁₄₉	—	4	4	no	—
54	Dic₁₄₉	—	8	2	no	—
55	C₂×C₁₄₉	—	2	2	no	—
56	C₂×C₁₄₉	C₂×#102	1	1	yes	characteristic
57	C₂×C₁₄₉	#510	1	2	yes	—
58	C₁₄₉	#2141	1	1	yes	characteristic, complemented, Sylow
59	#2141	—	149	1	no	maximal, Sylow
60	C₂²×Q₁₆	—	149	1	no	—
61	C₂×M₃₂	D₁₄₉	1	1	yes	characteristic
62	#427	—	149	1	no	—
63	#358	—	149	4	no	—
64	C₂²×Q₈	—	298	1	no	—
65	C₂×Q₁₆	—	298	2	no	—
66	C₂×Q₁₆	—	298	1	no	—
67	C₂×Q₁₆	—	149	4	no	—
68	C₂×Q₁₆	—	149	2	no	—
69	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
70	#117	—	298	1	no	—
71	M₃₂	—	2	2	no	—
72	C₂×C₁₆	C₂×D₁₄₉	1	2	yes	—
73	#107	—	149	2	no	—
74	#106	—	298	1	no	—
75	C₂×Q₈	—	298	2	no	—
76	C₂×Q₈	—	298	4	no	—
77	C₂×Q₈	—	596	1	no	—
78	C₂×Q₈	—	596	2	no	—
79	C₂²×C₄	—	596	1	no	—
80	C₂²×C₄	D₅₉₆	1	1	yes	characteristic
81	Q₁₆	—	298	2	no	—
82	Q₁₆	—	298	4	no	—
83	Q₁₆	—	149	4	no	—
84	C₂×C₈	C₄×D₁₄₉	1	2	yes	—
85	C₂×C₈	C₂²⋊D₁₄₉	1	2	yes	—
86	C₂×C₈	D₅₉₆	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
88	C₄⋊C₄	—	298	2	no	—
89	C₂²⋊C₄	—	596	1	no	—
90	C₄²	—	596	1	no	—
91	C₁₆	—	2	2	no	—
92	C₂³	D₁₁₉₂	1	1	yes	characteristic
93	Q₈	—	298	4	no	—
94	Q₈	—	596	2	no	—
95	Q₈	—	596	4	no	—
96	C₂×C₄	—	596	2	no	—
97	C₂×C₄	—	596	4	no	—
98	C₂×C₄	—	596	2	no	—
99	C₂×C₄	#14558713	1	2	yes	—
100	C₂×C₄	D₁₁₉₂	1	1	yes	characteristic
101	C₂×C₄	#14558725	1	2	yes	—
102	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
103	C₈	#14558713	1	2	yes	—
104	C₈	C₂×C₄×D₁₄₉	1	1	yes	characteristic
105	C₈	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#18011833	1	2	yes	—
108	C₂²	C₂×D₁₁₉₂	1	1	yes	characteristic
109	C₄	—	596	4	no	—
110	C₄	—	1192	2	no	—
111	C₄	#18011833	1	2	yes	—
112	C₄	C₂×#14558713	1	1	yes	characteristic
113	C₄	C₂×#14558725	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#18011833	1	1	yes	characteristic
116	C₂	#23750068	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₉₆

C₂

yes

maximal, characteristic

#23749663

C₂

yes

maximal, characteristic

#23749430

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

—

C₂×Dic₅₉₆

—

C₂×Dic₅₉₆

—

C₂×Dic₅₉₆

C₄

yes

complemented

C₂×Dic₅₉₆

C₂²

yes

—

#18011870

—

#18011845

C₂²

yes

—

#18011843

—

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

M₃₂×C₁₄₉

—

C₂×C₁₆×C₁₄₉

C₂²

yes

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂²×Dic₁₄₉

—

Dic₅₉₆

—

Dic₅₉₆

—

Dic₅₉₆

C₂×C₄

yes

—

C₄⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

characteristic

C₂×C₈×C₁₄₉

C₂×C₄

yes

—

C₂×C₈×C₁₄₉

D₄

yes

characteristic

C₂×C₈×C₁₄₉

D₄

yes

—

C₂×C₈×C₁₄₉

C₂³

yes

characteristic

C₁₆×C₁₄₉

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂³×C₁₄₉

D₈

yes

characteristic

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

D₈

yes

characteristic

C₂×C₄×C₁₄₉

SD₁₆

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₈×C₁₄₉

C₂²⋊C₄

yes

—

C₈×C₁₄₉

C₂²×C₄

yes

characteristic

C₈×C₁₄₉

C₂×D₄

yes

characteristic

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#102

yes

—

C₂²×C₁₄₉

C₂×D₈

yes

characteristic

C₄×C₁₄₉

#102

yes

—

C₄×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₄₉

C₂×SD₁₆

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

C₂×#102

yes

characteristic

C₂×C₁₄₉

#510

yes

—

C₁₄₉

#2141

yes

characteristic, complemented, Sylow

#2141

—

149

maximal, Sylow

C₂²×Q₁₆

—

149

—

C₂×M₃₂

D₁₄₉

yes

characteristic

#427

—

149

—

#358

—

149

—

C₂²×Q₈

—

298

—

C₂×Q₁₆

—

298

—

C₂×Q₁₆

—

298

—

C₂×Q₁₆

—

149

—

C₂×Q₁₆

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

#117

—

298

—

M₃₂

—

C₂×C₁₆

C₂×D₁₄₉

yes

—

#107

—

149

—

#106

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

596

—

C₂×Q₈

—

596

—

C₂²×C₄

—

596

—

C₂²×C₄

D₅₉₆

yes

characteristic

Q₁₆

—

298

—

Q₁₆

—

298

—

Q₁₆

—

149

—

C₂×C₈

C₄×D₁₄₉

yes

—

C₂×C₈

C₂²⋊D₁₄₉

yes

—

C₂×C₈

D₅₉₆

yes

characteristic

C₂×C₈

C₂²×D₁₄₉

yes

characteristic

C₄⋊C₄

—

298

—

C₂²⋊C₄

—

596

—

C₄²

—

596

—

C₁₆

—

C₂³

D₁₁₉₂

yes

characteristic

Q₈

—

298

—

Q₈

—

596

—

Q₈

—

596

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

#14558713

yes

—

100

C₂×C₄

D₁₁₉₂

yes

characteristic

101

C₂×C₄

#14558725

yes

—

102

C₂×C₄

C₂×D₅₉₆

yes

characteristic

103

C₈

#14558713

yes

—

104

C₈

C₂×C₄×D₁₄₉

yes

characteristic

105

C₈

C₂×C₂²⋊D₁₄₉

yes

characteristic

106

C₂²

—

107

C₂²

#18011833

yes

—

108

C₂²

C₂×D₁₁₉₂

yes

characteristic

109

C₄

—

596

—

110

C₄

—

1192

—

111

C₄

#18011833

yes

—

112

C₄

C₂×#14558713

yes

characteristic

113

C₄

C₂×#14558725

yes

characteristic

114

C₂

—

115

C₂

C₂×#18011833

yes

characteristic

116

C₂

#23750068

yes

—

	subgroup	G/H
center	C₂²	C₂×D₁₁₉₂
commutator	C₈×C₁₄₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₄₉
Fitting	C₂×M₃₂×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂×D₈
prime radical	C₂³×C₁₄₉	D₈
cosocle	C₂×C₈×C₁₄₉	C₂³
hypercenter	C₂×M₃₂	D₁₄₉
hypocenter	C₁₄₉	#2141
LCS	G ▷ C₈×C₁₄₉ ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₁₁₉₂

commutator

C₈×C₁₄₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₄₉

Fitting

C₂×M₃₂×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂×D₈

prime radical

C₂³×C₁₄₉

D₈

cosocle

C₂×C₈×C₁₄₉

C₂³

hypercenter

C₂×M₃₂

D₁₄₉

hypocenter

C₁₄₉

#2141

LCS

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #23749430	C₂	semidirect
8. C₂×Dic₅₉₆	C₄	semidirect
58. C₁₄₉	#2141	semidirect

H ≅ G/N

type

3. #23749430

C₂

semidirect

8. C₂×Dic₅₉₆

C₄

semidirect

58. C₁₄₉

#2141

semidirect

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

subgroup	index
C₂²×Dic₅₉₆	2
#23749663	2
#23749430	2
C₂×M₃₂×C₁₄₉	2
#2141	149

subgroup

index

C₂²×Dic₅₉₆

#23749663

#23749430

C₂×M₃₂×C₁₄₉

#2141

149

Subgroups — #34046412

Conjugacy classes of subgroups

Complements

Maximal subgroups