Group #23749539 — subgroups

Subgroups — #23749539

Order 9536.

238 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 65 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×#14558715	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Dic₁₄₉	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₁₄₉	C₂	1	2	yes	maximal, complemented
4	#18011818	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₉	C₂²	1	2	yes	—
7	C₂²×Dic₁₄₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₄₉	C₂²	1	1	yes	characteristic
9	#14558715	—	2	2	no	—
10	C₂²⋊Dic₁₄₉	—	2	4	no	—
11	C₂²⋊Dic₁₄₉	—	2	4	no	—
12	C₂⁴×C₁₄₉	C₂²	1	2	yes	—
13	C₂×D₄×C₁₄₉	—	2	4	no	—
14	C₂×D₄×C₁₄₉	—	2	2	no	—
15	C₂²×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₄₉	—	2	1	no	—
17	C₂×Dic₁₄₉	—	2	1	no	—
18	C₂×Dic₁₄₉	—	2	2	no	—
19	C₂×Dic₁₄₉	—	2	2	no	—
20	C₂×Dic₁₄₉	—	2	2	no	—
21	C₂×Dic₁₄₉	—	2	1	no	—
22	C₂×Dic₁₄₉	—	2	2	no	—
23	C₂×Dic₁₄₉	D₄	1	2	yes	complemented
24	C₂³×C₁₄₉	—	2	2	no	—
25	C₂³×C₁₄₉	—	2	2	no	—
26	C₂³×C₁₄₉	—	2	2	no	—
27	C₂³×C₁₄₉	—	2	2	no	—
28	C₂³×C₁₄₉	—	2	2	no	—
29	C₂³×C₁₄₉	—	2	2	no	—
30	C₂³×C₁₄₉	D₄	1	4	yes	—
31	C₂³×C₁₄₉	C₂³	1	1	yes	characteristic
32	D₄×C₁₄₉	—	2	8	no	—
33	C₂×C₄×C₁₄₉	—	2	2	no	—
34	C₂×C₄×C₁₄₉	D₄	1	2	yes	—
35	C₂²×C₁₄₉	—	2	4	no	—
36	C₂²×C₁₄₉	—	2	4	no	—
37	C₂²×C₁₄₉	—	2	4	no	—
38	C₂²×C₁₄₉	—	4	2	no	—
39	C₂²×C₁₄₉	—	4	2	no	—
40	C₂²×C₁₄₉	—	4	2	no	—
41	C₂²×C₁₄₉	—	4	2	no	—
42	C₂²×C₁₄₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₁₄₉	C₂×D₄	1	2	yes	—
44	C₂²×C₁₄₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₁₄₉	#41	1	1	yes	characteristic
46	C₂²×C₁₄₉	#41	1	2	yes	—
47	C₄×C₁₄₉	—	2	2	no	—
48	Dic₁₄₉	—	2	2	no	—
49	Dic₁₄₉	—	4	1	no	—
50	Dic₁₄₉	—	4	2	no	—
51	C₂×C₁₄₉	—	4	4	no	—
52	C₂×C₁₄₉	#120	1	1	yes	characteristic
53	C₂×C₁₄₉	#121	1	1	yes	characteristic
54	C₂×C₁₄₉	#121	1	2	yes	—
55	C₂×C₁₄₉	#123	1	2	yes	—
56	C₂×C₁₄₉	#124	1	1	yes	characteristic
57	C₁₄₉	#394	1	1	yes	characteristic, complemented, Sylow
58	#394	—	149	1	no	maximal, Sylow
59	C₂²×D₄	D₁₄₉	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	149	1	no	—
61	C₂×C₂²⋊C₄	—	149	2	no	—
62	C₂×C₂²⋊C₄	—	149	2	no	—
63	#95	—	149	1	no	—
64	C₂⁴	C₂×D₁₄₉	1	2	yes	—
65	C₂×D₄	—	2	4	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂²×C₄	—	149	2	no	—
68	C₂²×C₄	—	149	1	no	—
69	C₂²×C₄	—	149	1	no	—
70	C₂²×C₄	C₂×D₁₄₉	1	1	yes	characteristic
71	C₄⋊C₄	—	298	2	no	—
72	C₂²⋊C₄	—	298	4	no	—
73	C₂²⋊C₄	—	298	4	no	—
74	C₂³	—	2	2	no	—
75	C₂³	—	2	2	no	—
76	C₂³	—	2	2	no	—
77	C₂³	—	2	2	no	—
78	C₂³	—	2	2	no	—
79	C₂³	—	2	2	no	—
80	C₂³	C₂²⋊D₁₄₉	1	4	yes	—
81	C₂³	C₂²×D₁₄₉	1	1	yes	characteristic
82	D₄	—	2	8	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	298	1	no	—
85	C₂×C₄	—	298	1	no	—
86	C₂×C₄	—	298	2	no	—
87	C₂×C₄	—	298	2	no	—
88	C₂×C₄	—	298	2	no	—
89	C₂×C₄	—	298	1	no	—
90	C₂×C₄	—	298	2	no	—
91	C₂×C₄	—	149	2	no	—
92	C₂×C₄	C₂²⋊D₁₄₉	1	2	yes	—
93	C₂²	—	2	4	no	—
94	C₂²	—	2	4	no	—
95	C₂²	—	2	4	no	—
96	C₂²	—	4	2	no	—
97	C₂²	—	4	2	no	—
98	C₂²	—	4	2	no	—
99	C₂²	—	4	2	no	—
100	C₂²	D₄×D₁₄₉	1	1	yes	characteristic
101	C₂²	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
102	C₂²	C₂×C₂²⋊D₁₄₉	1	2	yes	—
103	C₂²	#14558735	1	1	yes	characteristic
104	C₂²	#14558735	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	298	2	no	—
107	C₄	—	596	1	no	—
108	C₄	—	596	2	no	—
109	C₂	—	4	4	no	—
110	C₂	#18011880	1	1	yes	characteristic
111	C₂	#18011882	1	1	yes	characteristic
112	C₂	#18011883	1	2	yes	—
113	C₂	#18011892	1	2	yes	—
114	C₂	#18011898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14558715

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₄₉

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₄₉

C₂

yes

maximal, complemented

#18011818

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₉

C₂²

yes

—

C₂²×Dic₁₄₉

C₂²

yes

characteristic

C₂²×Dic₁₄₉

C₂²

yes

characteristic

#14558715

—

C₂²⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

—

C₂⁴×C₁₄₉

C₂²

yes

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

D₄

yes

complemented

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

D₄

yes

—

C₂³×C₁₄₉

C₂³

yes

characteristic

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

D₄

yes

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×D₄

yes

characteristic

C₂²×C₁₄₉

C₂×D₄

yes

—

C₂²×C₁₄₉

C₂×D₄

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

—

C₄×C₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

—

C₂×C₁₄₉

#123

yes

—

C₂×C₁₄₉

#124

yes

characteristic

C₁₄₉

#394

yes

characteristic, complemented, Sylow

#394

—

149

maximal, Sylow

C₂²×D₄

D₁₄₉

yes

characteristic

C₂×C₄⋊C₄

—

149

—

C₂×C₂²⋊C₄

—

149

—

C₂×C₂²⋊C₄

—

149

—

#95

—

149

—

C₂⁴

C₂×D₁₄₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₂×D₁₄₉

yes

characteristic

C₄⋊C₄

—

298

—

C₂²⋊C₄

—

298

—

C₂²⋊C₄

—

298

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊D₁₄₉

yes

—

C₂³

C₂²×D₁₄₉

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

149

—

C₂×C₄

C₂²⋊D₁₄₉

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

D₄×D₁₄₉

yes

characteristic

101

C₂²

C₂×C₂²⋊D₁₄₉

yes

characteristic

102

C₂²

C₂×C₂²⋊D₁₄₉

yes

—

103

C₂²

#14558735

yes

characteristic

104

C₂²

#14558735

yes

—

105

C₄

—

106

C₄

—

298

—

107

C₄

—

596

—

108

C₄

—

596

—

109

C₂

—

110

C₂

#18011880

yes

characteristic

111

C₂

#18011882

yes

characteristic

112

C₂

#18011883

yes

—

113

C₂

#18011892

yes

—

114

C₂

#18011898

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₄₉
commutator, socle, cosocle	C₂³×C₁₄₉	C₂³
Fitting, prime radical	C₂²×D₄×C₁₄₉	C₂
hypercenter	C₂²×D₄	D₁₄₉
hypocenter	C₁₄₉	#394
LCS	G ▷ C₂³×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₄₉

commutator, socle, cosocle

C₂³×C₁₄₉

C₂³

Fitting, prime radical

C₂²×D₄×C₁₄₉

C₂

hypercenter

C₂²×D₄

D₁₄₉

C₁₄₉

G ▷ C₂³×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×#14558715	C₂	semidirect
2. C₂×C₂²⋊Dic₁₄₉	C₂	semidirect
3. C₂×C₂²⋊Dic₁₄₉	C₂	semidirect
4. #18011818	C₂	semidirect
23. C₂×Dic₁₄₉	D₄	semidirect
57. C₁₄₉	#394	semidirect

H ≅ G/N

type

1. C₂×#14558715

C₂

semidirect

2. C₂×C₂²⋊Dic₁₄₉

C₂

semidirect

3. C₂×C₂²⋊Dic₁₄₉

C₂

semidirect

4. #18011818

C₂

semidirect

23. C₂×Dic₁₄₉

D₄

semidirect

57. C₁₄₉

#394

semidirect

Subgroups — #23749539

Conjugacy classes of subgroups

Complements