Group #34046768 — subgroups

Subgroups — #34046768

Order 19072.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#23750063	C₂	1	2	yes	maximal, complemented
2	#23750002	C₂	1	1	yes	maximal, characteristic, complemented
3	#23749743	C₂	1	1	yes	maximal, characteristic, complemented
4	#23749439	C₂	1	2	yes	maximal, complemented
5	#364×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#18011933	C₂²	1	1	yes	characteristic
7	C₂×D₁₁₉₂	—	2	1	no	—
8	C₂×D₁₁₉₂	C₂²	1	2	yes	—
9	#18011917	—	2	1	no	—
10	#18011908	—	2	1	no	—
11	#18011856	—	2	2	no	—
12	D₂₃₈₄	—	2	2	no	—
13	#18011847	C₂²	1	1	yes	characteristic
14	#18011839	—	2	1	no	—
15	M₃₂×C₁₄₉	C₂²	1	2	yes	—
16	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
17	#14558736	—	2	1	no	—
18	C₂×D₅₉₆	—	2	2	no	—
19	C₂×D₅₉₆	—	4	2	no	—
20	Dic₅₉₆	D₄	1	1	yes	characteristic, complemented
21	#14558725	—	2	1	no	—
22	D₁₁₉₂	—	2	2	no	—
23	D₁₁₉₂	—	2	2	no	—
24	D₁₁₉₂	—	4	1	no	—
25	D₁₁₉₂	—	4	1	no	—
26	D₁₁₉₂	D₄	1	1	yes	characteristic, complemented
27	#14558720	—	2	1	no	—
28	#14558719	—	4	1	no	—
29	C₈×D₁₄₉	—	4	1	no	—
30	C₂×C₈×C₁₄₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
32	C₄²×C₁₄₉	D₄	1	1	yes	characteristic
33	C₁₆×C₁₄₉	—	2	2	no	—
34	C₂²×D₁₄₉	—	4	2	no	—
35	Dic₂₉₈	—	2	1	no	—
36	D₅₉₆	—	2	1	no	—
37	D₅₉₆	—	4	2	no	—
38	D₅₉₆	—	4	2	no	—
39	D₅₉₆	—	8	4	no	—
40	C₂²⋊D₁₄₉	—	4	1	no	—
41	C₄×D₁₄₉	—	4	1	no	—
42	#11924746	—	4	1	no	—
43	C₂×C₄×C₁₄₉	—	2	1	no	—
44	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₄₉	D₈	1	1	yes	characteristic
46	C₈×C₁₄₉	D₈	1	1	yes	characteristic
47	C₈×C₁₄₉	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₄₉	#41	1	1	yes	characteristic
49	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₄₉	—	4	1	no	—
51	C₂×D₁₄₉	—	8	4	no	—
52	C₂×D₁₄₉	—	8	2	no	—
53	C₄×C₁₄₉	—	2	2	no	—
54	C₄×C₁₄₉	#121	1	1	yes	characteristic
55	C₄×C₁₄₉	C₂×D₈	1	1	yes	characteristic
56	Dic₁₄₉	—	4	1	no	—
57	C₂×C₁₄₉	—	2	1	no	—
58	C₂×C₁₄₉	#459	1	1	yes	characteristic
59	D₁₄₉	—	8	1	no	—
60	D₁₄₉	—	16	2	no	—
61	C₁₄₉	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	149	1	no	maximal, Sylow
63	#509	—	149	2	no	—
64	#493	—	149	1	no	—
65	#443	—	149	1	no	—
66	#364	D₁₄₉	1	1	yes	characteristic, complemented
67	#361	—	149	2	no	—
68	#135	—	149	1	no	—
69	C₂×D₈	—	298	1	no	—
70	C₂×D₈	—	149	2	no	—
71	#131	—	298	1	no	—
72	#127	—	298	1	no	—
73	SD₃₂	—	298	2	no	—
74	D₁₆	—	298	2	no	—
75	M₃₂	C₂×D₁₄₉	1	2	yes	—
76	#108	—	149	1	no	—
77	#104	—	298	1	no	—
78	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
79	#41	—	298	1	no	—
80	C₂×D₄	—	298	2	no	—
81	C₂×D₄	—	596	2	no	—
82	Q₁₆	—	149	1	no	—
83	SD₁₆	—	298	1	no	—
84	D₈	—	298	2	no	—
85	D₈	—	298	2	no	—
86	D₈	—	596	1	no	—
87	D₈	—	596	1	no	—
88	D₈	—	149	1	no	—
89	M₁₆	—	298	1	no	—
90	M₁₆	—	596	1	no	—
91	C₂×C₈	—	596	1	no	—
92	C₂×C₈	D₅₉₆	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
94	C₄²	D₅₉₆	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	596	2	no	—
97	Q₈	—	298	1	no	—
98	D₄	—	298	1	no	—
99	D₄	—	596	1	no	—
100	D₄	—	596	2	no	—
101	D₄	—	596	2	no	—
102	D₄	—	1192	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	596	1	no	—
105	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
106	C₈	—	596	1	no	—
107	C₈	D₁₁₉₂	1	1	yes	characteristic
108	C₈	D₁₁₉₂	1	1	yes	characteristic
109	C₈	D₄×D₁₄₉	1	1	yes	characteristic
110	C₈	#14558737	1	1	yes	characteristic
111	C₂²	—	596	1	no	—
112	C₂²	—	1192	4	no	—
113	C₂²	—	1192	2	no	—
114	C₂²	#18011939	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	596	1	no	—
117	C₄	#18011886	1	1	yes	characteristic
118	C₄	C₂×D₁₁₉₂	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2384	2	no	—
121	C₂	—	1192	1	no	—
122	C₂	#23749821	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750063

C₂

yes

maximal, complemented

#23750002

C₂

yes

maximal, characteristic, complemented

#23749743

C₂

yes

maximal, characteristic, complemented

#23749439

C₂

yes

maximal, complemented

#364×C₁₄₉

C₂

yes

maximal, characteristic, complemented

#18011933

C₂²

yes

characteristic

C₂×D₁₁₉₂

—

C₂×D₁₁₉₂

C₂²

yes

—

#18011917

—

#18011908

—

#18011856

—

D₂₃₈₄

—

#18011847

C₂²

yes

characteristic

#18011839

—

M₃₂×C₁₄₉

C₂²

yes

—

C₄×C₈×C₁₄₉

C₂²

yes

characteristic

#14558736

—

C₂×D₅₉₆

—

C₂×D₅₉₆

—

Dic₅₉₆

D₄

yes

characteristic, complemented

#14558725

—

D₁₁₉₂

—

D₁₁₉₂

—

D₁₁₉₂

—

D₁₁₉₂

—

D₁₁₉₂

D₄

yes

characteristic, complemented

#14558720

—

#14558719

—

C₈×D₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

characteristic

C₂×C₈×C₁₄₉

C₂³

yes

characteristic

C₄²×C₁₄₉

D₄

yes

characteristic

C₁₆×C₁₄₉

—

C₂²×D₁₄₉

—

Dic₂₉₈

—

D₅₉₆

—

D₅₉₆

—

D₅₉₆

—

D₅₉₆

—

C₂²⋊D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₈×C₁₄₉

D₈

yes

characteristic

C₈×C₁₄₉

D₈

yes

characteristic

C₈×C₁₄₉

C₂×D₄

yes

C₈×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#121

yes

characteristic

C₄×C₁₄₉

C₂×D₈

yes

characteristic

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#459

yes

characteristic

D₁₄₉

—

D₁₄₉

—

C₁₄₉

#2213

yes

characteristic, complemented, Sylow

#2213

—

149

maximal, Sylow

#509

—

149

—

#493

—

149

—

#443

—

149

—

#364

D₁₄₉

yes

characteristic, complemented

#361

—

149

—

#135

—

149

—

C₂×D₈

—

298

—

C₂×D₈

—

149

—

#131

—

298

—

#127

—

298

—

SD₃₂

—

298

—

D₁₆

—

298

—

M₃₂

C₂×D₁₄₉

yes

—

#108

—

149

—

#104

—

298

—

C₄×C₈

C₂×D₁₄₉

yes

characteristic

#41

—

298

—

C₂×D₄

—

298

—

C₂×D₄

—

596

—

Q₁₆

—

149

—

SD₁₆

—

298

—

D₈

—

298

—

D₈

—

298

—

D₈

—

596

—

D₈

—

596

—

D₈

—

149

—

M₁₆

—

298

—

M₁₆

—

596

—

C₂×C₈

—

596

—

C₂×C₈

D₅₉₆

yes

characteristic

C₂×C₈

C₂²×D₁₄₉

yes

characteristic

C₄²

D₅₉₆

yes

characteristic

C₁₆

—

C₂³

—

596

—

Q₈

—

298

—

D₄

—

298

—

D₄

—

596

—

100

D₄

—

596

—

101

D₄

—

596

—

102

D₄

—

1192

—

103

C₂×C₄

—

104

C₂×C₄

—

596

—

105

C₂×C₄

C₂×D₅₉₆

yes

characteristic

106

C₈

—

596

—

107

C₈

D₁₁₉₂

yes

characteristic

108

C₈

D₁₁₉₂

yes

characteristic

109

C₈

D₄×D₁₄₉

yes

characteristic

110

C₈

#14558737

yes

characteristic

111

C₂²

—

596

—

112

C₂²

—

1192

—

113

C₂²

—

1192

—

114

C₂²

#18011939

yes

characteristic

115

C₄

—

116

C₄

—

596

—

117

C₄

#18011886

yes

characteristic

118

C₄

C₂×D₁₁₉₂

yes

characteristic

119

C₂

—

120

C₂

—

2384

—

121

C₂

—

1192

—

122

C₂

#23749821

yes

characteristic

	subgroup	G/H
center	C₂	#23749821
commutator, cosocle	C₂×C₈×C₁₄₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₄₉
Fitting	#364×C₁₄₉	C₂
socle	C₂×C₁₄₉	#459
hypercenter	#364	D₁₄₉
hypocenter	C₁₄₉	#2213
LCS	G ▷ C₂×C₈×C₁₄₉ ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#23749821

commutator, cosocle

C₂×C₈×C₁₄₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₄₉

Fitting

#364×C₁₄₉

C₂

C₂×C₁₄₉

D₁₄₉

C₁₄₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #23750063	C₂	semidirect
2. #23750002	C₂	semidirect
3. #23749743	C₂	semidirect
4. #23749439	C₂	semidirect
5. #364×C₁₄₉	C₂	semidirect
20. Dic₅₉₆	D₄	semidirect
26. D₁₁₉₂	D₄	semidirect
61. C₁₄₉	#2213	semidirect
66. #364	D₁₄₉	semidirect

H ≅ G/N

type

1. #23750063

C₂

semidirect

2. #23750002

C₂

semidirect

3. #23749743

C₂

semidirect

4. #23749439

C₂

semidirect

5. #364×C₁₄₉

C₂

semidirect

20. Dic₅₉₆

D₄

semidirect

26. D₁₁₉₂

D₄

semidirect

61. C₁₄₉

#2213

semidirect

66. #364

D₁₄₉

semidirect

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

subgroup	index
#23750063	2
#23750002	2
#23749743	2
#23749439	2
#364×C₁₄₉	2
#2213	149

subgroup

index

#364×C₁₄₉

149

Subgroups — #34046768

Conjugacy classes of subgroups

Complements

Maximal subgroups