Group #23749765 — subgroups

Subgroups — #23749765

Order 9536.

158 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 45 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×Q₈×D₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14558725	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14558723	C₂	1	1	yes	maximal, characteristic, complemented
4	#18011886	C₂	1	1	yes	maximal, characteristic
5	#18011834	C₂	1	1	yes	maximal, characteristic, complemented
6	#18011823	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₉₈	—	2	1	no	—
9	C₂×Dic₂₉₈	C₂²	1	1	yes	characteristic
10	Q₈×D₁₄₉	—	2	2	no	—
11	Q₈×D₁₄₉	—	2	2	no	—
12	C₂×D₅₉₆	—	2	1	no	—
13	C₂×D₅₉₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₄₉	—	2	1	no	—
15	C₂×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
16	#14558725	—	2	2	no	—
17	#14558723	—	2	2	no	—
18	C₂×#11924746	C₂²	1	1	yes	characteristic
19	#14558713	—	2	1	no	—
20	C₂×Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
23	C₂²×D₁₄₉	—	2	1	no	—
24	C₂²×D₁₄₉	D₄	1	1	yes	characteristic
25	Dic₂₉₈	—	2	4	no	—
26	Dic₂₉₈	—	2	1	no	—
27	Dic₂₉₈	D₄	1	2	yes	complemented
28	D₅₉₆	—	2	1	no	—
29	D₅₉₆	—	2	1	no	—
30	D₅₉₆	—	4	2	no	—
31	C₂×Dic₁₄₉	—	2	1	no	—
32	C₂×Dic₁₄₉	D₄	1	1	yes	characteristic
33	C₄×D₁₄₉	—	2	4	no	—
34	C₄×D₁₄₉	—	2	1	no	—
35	C₄×D₁₄₉	—	2	1	no	—
36	#11924746	—	2	1	no	—
37	Q₈×C₁₄₉	—	2	1	no	—
38	Q₈×C₁₄₉	D₄	1	2	yes	complemented
39	C₂×C₄×C₁₄₉	—	2	1	no	—
40	C₂×C₄×C₁₄₉	—	2	1	no	—
41	C₂×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
42	C₈×C₁₄₉	—	2	1	no	—
43	C₂²×C₁₄₉	C₂×D₄	1	1	yes	characteristic
44	C₂×D₁₄₉	—	2	1	no	—
45	C₂×D₁₄₉	—	2	1	no	—
46	C₂×D₁₄₉	—	4	1	no	—
47	C₂×D₁₄₉	—	4	1	no	—
48	C₂×D₁₄₉	—	4	1	no	—
49	C₂×D₁₄₉	SD₁₆	1	2	yes	complemented
50	C₄×C₁₄₉	—	2	2	no	—
51	C₄×C₁₄₉	—	4	1	no	—
52	C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
53	C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
54	Dic₁₄₉	—	2	2	no	—
55	Dic₁₄₉	—	2	1	no	—
56	C₂×C₁₄₉	#120	1	1	yes	characteristic
57	C₂×C₁₄₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₁₄₉	#137	1	1	yes	characteristic
59	D₁₄₉	—	2	2	no	—
60	D₁₄₉	—	8	1	no	—
61	C₁₄₉	#448	1	1	yes	characteristic, complemented, Sylow
62	#448	—	149	1	no	maximal, Sylow
63	C₂²×Q₈	—	149	1	no	—
64	C₂×SD₁₆	—	149	1	no	—
65	C₂×SD₁₆	—	149	1	no	—
66	#121	—	149	1	no	—
67	#103	—	149	1	no	—
68	#103	D₁₄₉	1	1	yes	characteristic, complemented
69	C₂²⋊C₈	—	149	1	no	—
70	C₂×Q₈	—	298	1	no	—
71	C₂×Q₈	—	298	2	no	—
72	C₂×Q₈	—	298	2	no	—
73	C₂×Q₈	—	149	1	no	—
74	C₂×Q₈	C₂×D₁₄₉	1	1	yes	characteristic
75	C₂×D₄	—	298	1	no	—
76	C₂×D₄	—	149	1	no	—
77	C₂²×C₄	—	298	1	no	—
78	C₂²×C₄	—	149	1	no	—
79	SD₁₆	—	298	2	no	—
80	SD₁₆	—	298	2	no	—
81	C₂×C₈	—	149	1	no	—
82	C₂×C₈	C₂×D₁₄₉	1	1	yes	characteristic
83	C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
84	C₂²⋊C₄	—	298	1	no	—
85	C₂³	—	298	1	no	—
86	C₂³	—	149	1	no	—
87	Q₈	—	2	1	no	—
88	Q₈	—	298	4	no	—
89	Q₈	—	298	1	no	—
90	Q₈	—	149	2	no	—
91	Q₈	D₅₉₆	1	2	yes	complemented
92	D₄	—	298	1	no	—
93	D₄	—	298	1	no	—
94	D₄	—	596	2	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	298	4	no	—
98	C₂×C₄	—	298	1	no	—
99	C₂×C₄	—	298	1	no	—
100	C₂×C₄	—	298	1	no	—
101	C₂×C₄	—	149	1	no	—
102	C₂×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	298	1	no	—
105	C₂²	—	298	1	no	—
106	C₂²	—	298	1	no	—
107	C₂²	—	596	1	no	—
108	C₂²	—	596	1	no	—
109	C₂²	—	596	1	no	—
110	C₂²	—	149	2	no	—
111	C₂²	D₄×D₁₄₉	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	4	1	no	—
114	C₄	—	298	2	no	—
115	C₄	—	298	1	no	—
116	C₄	D₄×D₁₄₉	1	1	yes	characteristic
117	C₄	C₂×D₅₉₆	1	1	yes	characteristic
118	C₂	—	298	2	no	—
119	C₂	—	1192	1	no	—
120	C₂	#18011881	1	1	yes	characteristic
121	C₂	SD₁₆×D₁₄₉	1	1	yes	characteristic
122	C₂	#18011943	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂×#14558725

C₂

yes

maximal, characteristic, complemented

C₂×#14558723

C₂

yes

maximal, characteristic, complemented

#18011886

C₂

yes

maximal, characteristic

#18011834

C₂

yes

maximal, characteristic, complemented

#18011823

C₂

yes

maximal, characteristic, complemented

#103×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

C₂²

yes

characteristic

Q₈×D₁₄₉

—

Q₈×D₁₄₉

—

C₂×D₅₉₆

—

C₂×D₅₉₆

C₂²

yes

characteristic

C₂×C₄×D₁₄₉

—

C₂×C₄×D₁₄₉

C₂²

yes

—

—

C₂²

yes

characteristic

#14558713

—

C₂×Q₈×C₁₄₉

C₂²

yes

characteristic

C₂×C₈×C₁₄₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₄₉

C₂²

yes

characteristic

C₂²×D₁₄₉

—

C₂²×D₁₄₉

D₄

yes

characteristic

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

D₄

yes

complemented

D₅₉₆

—

D₅₉₆

—

D₅₉₆

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

D₄

yes

characteristic

C₄×D₁₄₉

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

Q₈×C₁₄₉

—

Q₈×C₁₄₉

D₄

yes

complemented

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂³

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

C₂×D₄

yes

characteristic

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

SD₁₆

yes

complemented

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₄×C₁₄₉

C₂×D₄

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

#120

yes

characteristic

C₂×C₁₄₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₄₉

#137

yes

characteristic

D₁₄₉

—

D₁₄₉

—

C₁₄₉

#448

yes

characteristic, complemented, Sylow

#448

—

149

maximal, Sylow

C₂²×Q₈

—

149

—

C₂×SD₁₆

—

149

—

C₂×SD₁₆

—

149

—

#121

—

149

—

#103

—

149

—

#103

D₁₄₉

yes

characteristic, complemented

C₂²⋊C₈

—

149

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

C₂×Q₈

C₂×D₁₄₉

yes

characteristic

C₂×D₄

—

298

—

C₂×D₄

—

149

—

C₂²×C₄

—

298

—

C₂²×C₄

—

149

—

SD₁₆

—

298

—

SD₁₆

—

298

—

C₂×C₈

—

149

—

C₂×C₈

C₂×D₁₄₉

yes

characteristic

C₄⋊C₄

C₂×D₁₄₉

yes

characteristic

C₂²⋊C₄

—

298

—

C₂³

—

298

—

C₂³

—

149

—

Q₈

—

Q₈

—

298

—

Q₈

—

298

—

Q₈

—

149

—

Q₈

D₅₉₆

yes

complemented

D₄

—

298

—

D₄

—

298

—

D₄

—

596

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

100

C₂×C₄

—

298

—

101

C₂×C₄

—

149

—

102

C₂×C₄

C₂²×D₁₄₉

yes

characteristic

103

C₈

—

104

C₈

—

298

—

105

C₂²

—

298

—

106

C₂²

—

298

—

107

C₂²

—

596

—

108

C₂²

—

596

—

109

C₂²

—

596

—

110

C₂²

—

149

—

111

C₂²

D₄×D₁₄₉

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

298

—

115

C₄

—

298

—

116

C₄

D₄×D₁₄₉

yes

characteristic

117

C₄

C₂×D₅₉₆

yes

characteristic

118

C₂

—

298

—

119

C₂

—

1192

—

120

C₂

#18011881

yes

characteristic

121

C₂

SD₁₆×D₁₄₉

yes

characteristic

122

C₂

#18011943

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₄₉
commutator, cosocle	C₂×C₄×C₁₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₄₉
Fitting	#103×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂×D₄
prime radical	#18011886	C₂
hypercenter	#103	D₁₄₉
hypocenter	C₁₄₉	#448
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₄₉

commutator, cosocle

C₂×C₄×C₁₄₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₄₉

Fitting

#103×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂×D₄

C₂

D₁₄₉

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
1. C₂×Q₈×D₁₄₉	C₂	semidirect
2. C₂×#14558725	C₂	semidirect
3. C₂×#14558723	C₂	semidirect
5. #18011834	C₂	semidirect
6. #18011823	C₂	semidirect
7. #103×C₁₄₉	C₂	semidirect
27. Dic₂₉₈	D₄	semidirect
38. Q₈×C₁₄₉	D₄	semidirect
49. C₂×D₁₄₉	SD₁₆	semidirect
61. C₁₄₉	#448	semidirect
68. #103	D₁₄₉	semidirect
91. Q₈	D₅₉₆	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₄₉

C₂

semidirect

2. C₂×#14558725

C₂

semidirect

3. C₂×#14558723

C₂

semidirect

5. #18011834

C₂

semidirect

6. #18011823

C₂

semidirect

7. #103×C₁₄₉

C₂

semidirect

27. Dic₂₉₈

D₄

semidirect

38. Q₈×C₁₄₉

D₄

semidirect

49. C₂×D₁₄₉

SD₁₆

semidirect

61. C₁₄₉

#448

semidirect

68. #103

D₁₄₉

semidirect

91. Q₈

D₅₉₆

semidirect

Subgroups — #23749765

Conjugacy classes of subgroups

Complements