Group #23749504 — subgroups

Subgroups — #23749504

Order 9536.

286 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 87 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×C₂²⋊D₁₄₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14558713	C₂	1	2	yes	maximal, complemented
4	#18011818	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₄₉	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₁₄₉	—	2	2	no	—
8	C₂×C₂²⋊D₁₄₉	—	2	2	no	—
9	C₂×C₂²⋊D₁₄₉	C₄	1	4	yes	complemented
10	C₂²×Dic₁₄₉	C₂²	1	2	yes	—
11	C₂²×Dic₁₄₉	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₁₄₉	—	2	2	no	—
13	#14558713	—	2	4	no	—
14	C₂⁴×C₁₄₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₄₉	—	2	2	no	—
16	C₂²×C₄×C₁₄₉	—	2	4	no	—
17	C₂²×C₄×C₁₄₉	C₂²	1	2	yes	—
18	C₂²×D₁₄₉	—	2	1	no	—
19	C₂²×D₁₄₉	—	2	1	no	—
20	C₂²×D₁₄₉	—	2	2	no	—
21	C₂²×D₁₄₉	—	2	2	no	—
22	C₂²×D₁₄₉	C₂×C₄	1	2	yes	complemented
23	C₂²⋊D₁₄₉	—	2	8	no	—
24	C₂×Dic₁₄₉	—	2	1	no	—
25	C₂×Dic₁₄₉	—	2	1	no	—
26	C₂×Dic₁₄₉	—	2	2	no	—
27	C₂×Dic₁₄₉	—	2	2	no	—
28	C₂×Dic₁₄₉	—	2	2	no	—
29	C₂×Dic₁₄₉	C₂×C₄	1	2	yes	complemented
30	C₂³×C₁₄₉	—	2	2	no	—
31	C₂³×C₁₄₉	—	2	2	no	—
32	C₂³×C₁₄₉	C₂×C₄	1	2	yes	—
33	C₂³×C₁₄₉	D₄	1	2	yes	—
34	C₂³×C₁₄₉	D₄	1	2	yes	—
35	C₂³×C₁₄₉	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₄₉	—	2	8	no	—
37	C₂×C₄×C₁₄₉	—	2	2	no	—
38	C₂×C₄×C₁₄₉	—	2	2	no	—
39	C₂×C₄×C₁₄₉	D₄	1	4	yes	complemented
40	C₂²×C₁₄₉	—	2	4	no	—
41	C₂²×C₁₄₉	—	2	2	no	—
42	C₂²×C₁₄₉	—	2	4	no	—
43	C₂²×C₁₄₉	—	2	2	no	—
44	C₂²×C₁₄₉	C₂²⋊C₄	1	4	yes	complemented
45	C₂²×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₁₄₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₁₄₉	C₂×D₄	1	2	yes	—
48	C₂²×C₁₄₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₁₄₉	#41	1	2	yes	—
50	C₂×D₁₄₉	—	2	2	no	—
51	C₂×D₁₄₉	—	2	4	no	—
52	C₂×D₁₄₉	—	4	1	no	—
53	C₂×D₁₄₉	—	4	2	no	—
54	C₂×D₁₄₉	—	4	1	no	—
55	C₄×C₁₄₉	—	2	4	no	—
56	Dic₁₄₉	—	2	2	no	—
57	Dic₁₄₉	—	4	2	no	—
58	C₂×C₁₄₉	—	2	4	no	—
59	C₂×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₁₄₉	C₄×D₄	1	2	yes	—
61	C₂×C₁₄₉	#120	1	1	yes	characteristic
62	C₂×C₁₄₉	#121	1	2	yes	—
63	C₂×C₁₄₉	#123	1	1	yes	characteristic
64	D₁₄₉	—	4	2	no	—
65	C₁₄₉	#386	1	1	yes	characteristic, complemented, Sylow
66	#386	—	149	1	no	maximal, Sylow
67	C₂²×D₄	—	149	1	no	—
68	C₂³×C₄	D₁₄₉	1	1	yes	characteristic, complemented
69	C₂×C₂²⋊C₄	—	149	1	no	—
70	C₂×C₂²⋊C₄	—	149	2	no	—
71	#95	—	149	2	no	—
72	C₂⁴	—	149	1	no	—
73	C₂⁴	C₂×D₁₄₉	1	1	yes	characteristic
74	C₂×D₄	—	298	2	no	—
75	C₂×D₄	—	298	2	no	—
76	C₂×D₄	—	149	4	no	—
77	C₂²×C₄	—	2	2	no	—
78	C₂²×C₄	—	2	4	no	—
79	C₂²×C₄	—	149	2	no	—
80	C₂²×C₄	—	149	1	no	—
81	C₂²×C₄	C₂×D₁₄₉	1	2	yes	—
82	C₂²⋊C₄	—	298	2	no	—
83	C₂²⋊C₄	—	298	4	no	—
84	C₂³	—	2	2	no	—
85	C₂³	—	2	2	no	—
86	C₂³	—	298	2	no	—
87	C₂³	—	298	1	no	—
88	C₂³	—	298	1	no	—
89	C₂³	—	298	2	no	—
90	C₂³	—	149	2	no	—
91	C₂³	C₄×D₁₄₉	1	2	yes	—
92	C₂³	C₂²⋊D₁₄₉	1	2	yes	—
93	C₂³	D₅₉₆	1	2	yes	—
94	C₂³	C₂²×D₁₄₉	1	1	yes	characteristic
95	D₄	—	298	8	no	—
96	C₂×C₄	—	2	8	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	298	1	no	—
100	C₂×C₄	—	298	1	no	—
101	C₂×C₄	—	298	2	no	—
102	C₂×C₄	—	298	2	no	—
103	C₂×C₄	—	298	2	no	—
104	C₂×C₄	—	149	2	no	—
105	C₂×C₄	C₂²⋊D₁₄₉	1	4	yes	complemented
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	2	4	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	298	2	no	—
111	C₂²	—	298	4	no	—
112	C₂²	—	596	1	no	—
113	C₂²	—	596	2	no	—
114	C₂²	—	596	1	no	—
115	C₂²	#14558713	1	4	yes	complemented
116	C₂²	C₂×C₄×D₁₄₉	1	1	yes	characteristic
117	C₂²	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
118	C₂²	C₂×C₂²⋊D₁₄₉	1	2	yes	—
119	C₂²	C₂×D₅₉₆	1	1	yes	characteristic
120	C₂²	#14558736	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	298	2	no	—
123	C₄	—	596	2	no	—
124	C₂	—	2	4	no	—
125	C₂	—	596	2	no	—
126	C₂	C₂×#14558713	1	1	yes	characteristic
127	C₂	C₄×C₂²⋊D₁₄₉	1	2	yes	—
128	C₂	#18011880	1	1	yes	characteristic
129	C₂	#18011885	1	2	yes	—
130	C₂	#18011894	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊D₁₄₉

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂×#14558713

C₂

yes

maximal, complemented

#18011818

C₂

yes

maximal, complemented

C₂³×C₄×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₄₉

C₂²

yes

characteristic

C₂×C₂²⋊D₁₄₉

—

C₂×C₂²⋊D₁₄₉

—

C₂×C₂²⋊D₁₄₉

C₄

yes

complemented

C₂²×Dic₁₄₉

C₂²

yes

—

C₂²×Dic₁₄₉

C₂²

yes

characteristic

C₂²⋊Dic₁₄₉

—

#14558713

—

C₂⁴×C₁₄₉

C₂²

yes

characteristic

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

C₂²

yes

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

C₂×C₄

yes

complemented

C₂²⋊D₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

C₂×C₄

yes

complemented

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂×C₄

yes

—

C₂³×C₁₄₉

D₄

yes

—

C₂³×C₁₄₉

D₄

yes

—

C₂³×C₁₄₉

C₂³

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

D₄

yes

complemented

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂²⋊C₄

yes

complemented

C₂²×C₁₄₉

C₂²×C₄

yes

characteristic

C₂²×C₁₄₉

C₂×D₄

yes

characteristic

C₂²×C₁₄₉

C₂×D₄

yes

—

C₂²×C₁₄₉

C₂×D₄

yes

characteristic

C₂²×C₁₄₉

#41

yes

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₄₉

C₄×D₄

yes

—

C₂×C₁₄₉

#120

yes

characteristic

C₂×C₁₄₉

#121

yes

—

C₂×C₁₄₉

#123

yes

characteristic

D₁₄₉

—

C₁₄₉

#386

yes

characteristic, complemented, Sylow

#386

—

149

maximal, Sylow

C₂²×D₄

—

149

—

C₂³×C₄

D₁₄₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

149

—

C₂×C₂²⋊C₄

—

149

—

#95

—

149

—

C₂⁴

—

149

—

C₂⁴

C₂×D₁₄₉

yes

characteristic

C₂×D₄

—

298

—

C₂×D₄

—

298

—

C₂×D₄

—

149

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₂×D₁₄₉

yes

—

C₂²⋊C₄

—

298

—

C₂²⋊C₄

—

298

—

C₂³

—

C₂³

—

C₂³

—

298

—

C₂³

—

298

—

C₂³

—

298

—

C₂³

—

298

—

C₂³

—

149

—

C₂³

C₄×D₁₄₉

yes

—

C₂³

C₂²⋊D₁₄₉

yes

—

C₂³

D₅₉₆

yes

—

C₂³

C₂²×D₁₄₉

yes

characteristic

D₄

—

298

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

298

—

100

C₂×C₄

—

298

—

101

C₂×C₄

—

298

—

102

C₂×C₄

—

298

—

103

C₂×C₄

—

298

—

104

C₂×C₄

—

149

—

105

C₂×C₄

C₂²⋊D₁₄₉

yes

complemented

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

298

—

111

C₂²

—

298

—

112

C₂²

—

596

—

113

C₂²

—

596

—

114

C₂²

—

596

—

115

C₂²

#14558713

yes

complemented

116

C₂²

C₂×C₄×D₁₄₉

yes

characteristic

117

C₂²

C₂×C₂²⋊D₁₄₉

yes

characteristic

118

C₂²

C₂×C₂²⋊D₁₄₉

yes

—

119

C₂²

C₂×D₅₉₆

yes

characteristic

120

C₂²

#14558736

yes

—

121

C₄

—

122

C₄

—

298

—

123

C₄

—

596

—

124

C₂

—

125

C₂

—

596

—

126

C₂

C₂×#14558713

yes

characteristic

127

C₂

C₄×C₂²⋊D₁₄₉

yes

—

128

C₂

#18011880

yes

characteristic

129

C₂

#18011885

yes

—

130

C₂

#18011894

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₄₉
commutator	C₂²×C₁₄₉	C₂²×C₄
Fitting	C₂³×C₄×C₁₄₉	C₂
socle, cosocle	C₂³×C₁₄₉	C₂³
prime radical	C₂²×C₂²⋊D₁₄₉	C₂
hypercenter	C₂³×C₄	D₁₄₉
hypocenter	C₁₄₉	#386
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₄₉

commutator

C₂²×C₁₄₉

C₂²×C₄

Fitting

C₂³×C₄×C₁₄₉

C₂

socle, cosocle

C₂³×C₁₄₉

C₂³

prime radical

C₂²×C₂²⋊D₁₄₉

C₂

hypercenter

C₂³×C₄

D₁₄₉

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂³ ◁ C₂³×C₄

N	H ≅ G/N	type
2. C₂×C₂²⋊Dic₁₄₉	C₂	semidirect
3. C₂×#14558713	C₂	semidirect
4. #18011818	C₂	semidirect
5. C₂³×C₄×C₁₄₉	C₂	semidirect
9. C₂×C₂²⋊D₁₄₉	C₄	semidirect
22. C₂²×D₁₄₉	C₂×C₄	semidirect
29. C₂×Dic₁₄₉	C₂×C₄	semidirect
39. C₂×C₄×C₁₄₉	D₄	semidirect
44. C₂²×C₁₄₉	C₂²⋊C₄	semidirect
65. C₁₄₉	#386	semidirect
68. C₂³×C₄	D₁₄₉	semidirect
105. C₂×C₄	C₂²⋊D₁₄₉	semidirect
115. C₂²	#14558713	semidirect

H ≅ G/N

type

2. C₂×C₂²⋊Dic₁₄₉

C₂

semidirect

3. C₂×#14558713

C₂

semidirect

4. #18011818

C₂

semidirect

5. C₂³×C₄×C₁₄₉

C₂

semidirect

9. C₂×C₂²⋊D₁₄₉

C₄

semidirect

22. C₂²×D₁₄₉

C₂×C₄

semidirect

29. C₂×Dic₁₄₉

C₂×C₄

semidirect

39. C₂×C₄×C₁₄₉

D₄

semidirect

44. C₂²×C₁₄₉

C₂²⋊C₄

semidirect

65. C₁₄₉

#386

semidirect

68. C₂³×C₄

D₁₄₉

semidirect

105. C₂×C₄

C₂²⋊D₁₄₉

semidirect

115. C₂²

#14558713

semidirect

Subgroups — #23749504

Conjugacy classes of subgroups

Complements