Group #34046245 — subgroups

Subgroups — #34046245

Order 19072.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#23749674	C₂	1	1	yes	maximal, characteristic, complemented
2	#23749652	C₂	1	2	yes	maximal, complemented
3	#23749643	C₂	1	2	yes	maximal
4	#23749593	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14558736	C₂²	1	1	yes	characteristic
7	C₂×#14558720	C₂²	1	2	yes	—
8	C₂²×#11924746	C₂²	1	1	yes	characteristic
9	#18011847	—	2	2	no	—
10	C₄⋊#11924746	—	2	4	no	—
11	#18011839	—	2	4	no	—
12	#18011823	—	2	2	no	—
13	C₂×M₁₆×C₁₄₉	C₂²	1	1	yes	characteristic
14	#117×C₁₄₉	C₂²	1	2	yes	—
15	#14558736	—	2	2	no	—
16	#14558736	—	2	2	no	—
17	C₂×Dic₂₉₈	Q₈	1	1	yes	characteristic
18	C₂×D₅₉₆	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₄₉	—	2	1	no	—
20	C₂×C₄×D₁₄₉	—	2	1	no	—
21	#14558720	—	2	2	no	—
22	#14558720	—	2	2	no	—
23	C₂×#11924746	—	2	1	no	—
24	C₂×#11924746	—	2	2	no	—
25	C₂×#11924746	—	2	2	no	—
26	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₄₉	—	2	2	no	—
28	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₄₉	—	2	2	no	—
30	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
31	C₄²×C₁₄₉	D₄	1	4	yes	—
32	C₂²×D₁₄₉	—	2	1	no	—
33	Dic₂₉₈	—	2	2	no	—
34	D₅₉₆	—	2	2	no	—
35	C₂²⋊D₁₄₉	—	4	2	no	—
36	C₂×Dic₁₄₉	—	2	1	no	—
37	C₄×D₁₄₉	—	4	1	no	—
38	C₄×D₁₄₉	—	4	1	no	—
39	#11924746	—	4	2	no	—
40	#11924746	—	4	1	no	—
41	C₂³×C₁₄₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₄₉	—	2	4	no	—
43	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₄₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₄₉	#41	1	2	yes	—
47	C₈×C₁₄₉	—	2	2	no	—
48	C₂²×C₁₄₉	—	4	1	no	—
49	C₂²×C₁₄₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₄₉	#123	1	2	yes	—
51	C₂×D₁₄₉	—	4	2	no	—
52	C₂×D₁₄₉	—	4	1	no	—
53	C₄×C₁₄₉	—	4	4	no	—
54	C₄×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₄₉	#123	1	1	yes	characteristic
56	C₄×C₁₄₉	#127	1	1	yes	characteristic
57	Dic₁₄₉	—	4	1	no	—
58	C₂×C₁₄₉	—	2	2	no	—
59	C₂×C₁₄₉	—	2	1	no	—
60	C₂×C₁₄₉	#399	1	1	yes	characteristic
61	D₁₄₉	—	8	1	no	—
62	C₁₄₉	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	149	1	no	maximal, Sylow
64	#430	—	149	1	no	—
65	#424	—	149	2	no	—
66	#421	—	149	2	no	—
67	#408	—	149	1	no	—
68	H(3,ℤ/4)	D₁₄₉	1	1	yes	characteristic, complemented
69	C₂×#41	—	149	1	no	—
70	C₂×M₁₆	—	149	2	no	—
71	C₂×M₁₆	C₂×D₁₄₉	1	1	yes	characteristic
72	C₂²×C₈	—	149	1	no	—
73	#117	C₂×D₁₄₉	1	2	yes	—
74	#108	—	298	2	no	—
75	C₄⋊C₈	—	298	4	no	—
76	#104	—	298	4	no	—
77	C₂²⋊C₈	—	298	2	no	—
78	#41	—	298	2	no	—
79	#41	—	298	2	no	—
80	C₂×Q₈	—	149	1	no	—
81	C₂×D₄	—	298	1	no	—
82	C₂×D₄	—	149	1	no	—
83	C₂²×C₄	—	298	1	no	—
84	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	298	2	no	—
87	M₁₆	—	298	2	no	—
88	C₂×C₈	—	298	1	no	—
89	C₂×C₈	—	298	2	no	—
90	C₂×C₈	—	298	2	no	—
91	C₂×C₈	D₅₉₆	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₅₉₆	1	4	yes	—
95	C₂³	—	298	1	no	—
96	C₂³	#14558735	1	1	yes	characteristic
97	Q₈	—	298	2	no	—
98	D₄	—	298	2	no	—
99	D₄	—	596	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	298	1	no	—
102	C₂×C₄	—	596	1	no	—
103	C₂×C₄	—	596	1	no	—
104	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
105	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₄₉	1	1	yes	characteristic
107	C₂×C₄	#14558735	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	596	1	no	—
110	C₈	—	596	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	596	2	no	—
113	C₂²	—	596	1	no	—
114	C₂²	#18011891	1	1	yes	characteristic
115	C₂²	#18011895	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	596	1	no	—
118	C₄	#18011891	1	2	yes	—
119	C₄	#18011895	1	1	yes	characteristic
120	C₄	#18011908	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1192	1	no	—
124	C₂	#23749562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23749674

C₂

yes

maximal, characteristic, complemented

#23749652

C₂

yes

maximal, complemented

#23749643

C₂

yes

maximal

#23749593

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂×#14558736

C₂²

yes

characteristic

C₂×#14558720

C₂²

yes

—

C₂²×#11924746

C₂²

yes

—

—

—

—

C₂×M₁₆×C₁₄₉

C₂²

yes

characteristic

#117×C₁₄₉

C₂²

yes

—

#14558736

—

#14558736

—

C₂×Dic₂₉₈

Q₈

yes

characteristic

C₂×D₅₉₆

Q₈

yes

characteristic

C₂×C₂²⋊D₁₄₉

—

C₂×C₄×D₁₄₉

—

—

—

—

—

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₄⋊C₄×C₁₄₉

—

C₂²⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

D₄

yes

—

C₂²×D₁₄₉

—

Dic₂₉₈

—

D₅₉₆

—

C₂²⋊D₁₄₉

—

C₂×Dic₁₄₉

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

#11924746

—

C₂³×C₁₄₉

#41

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₄₉

#123

yes

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

yes

C₄×C₁₄₉

yes

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#399

yes

characteristic

D₁₄₉

—

C₁₄₉

#2098

yes

characteristic, complemented, Sylow

#2098

—

149

maximal, Sylow

#430

—

149

—

#424

—

149

—

#421

—

149

—

#408

—

149

—

H(3,ℤ/4)

D₁₄₉

yes

characteristic, complemented

C₂×#41

—

149

—

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

characteristic

C₂²×C₈

—

149

—

#117

C₂×D₁₄₉

yes

—

#108

—

298

—

C₄⋊C₈

—

298

—

#104

—

298

—

C₂²⋊C₈

—

298

—

#41

—

298

—

#41

—

298

—

C₂×Q₈

—

149

—

C₂×D₄

—

298

—

C₂×D₄

—

149

—

C₂²×C₄

—

298

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

M₁₆

—

M₁₆

—

298

—

M₁₆

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

D₅₉₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₅₉₆

yes

—

C₂³

—

298

—

C₂³

#14558735

yes

characteristic

Q₈

—

298

—

D₄

—

298

—

D₄

—

596

—

100

C₂×C₄

—

101

C₂×C₄

—

298

—

102

C₂×C₄

—

596

—

103

C₂×C₄

—

596

—

104

C₂×C₄

C₂×D₅₉₆

yes

—

105

C₂×C₄

C₂×D₅₉₆

yes

characteristic

106

C₂×C₄

Q₈×D₁₄₉

yes

characteristic

107

C₂×C₄

#14558735

yes

—

108

C₈

—

109

C₈

—

596

—

110

C₈

—

596

—

111

C₂²

—

112

C₂²

—

596

—

113

C₂²

—

596

—

114

C₂²

#18011891

yes

characteristic

115

C₂²

#18011895

yes

—

116

C₄

—

117

C₄

—

596

—

118

C₄

#18011891

yes

—

119

C₄

#18011895

yes

characteristic

120

C₄

#18011908

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1192

—

124

C₂

#23749562

yes

characteristic

	subgroup	G/H
center	C₄	#18011908
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	H(3,ℤ/4)×C₁₄₉	C₂
socle	C₂×C₁₄₉	#399
prime radical	C₂×#14558736	C₂²
hypercenter	H(3,ℤ/4)	D₁₄₉
hypocenter	C₁₄₉	#2098
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#18011908

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

H(3,ℤ/4)×C₁₄₉

C₂

C₂×C₁₄₉

C₂²

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #23749674	C₂	semidirect
2. #23749652	C₂	semidirect
5. H(3,ℤ/4)×C₁₄₉	C₂	semidirect
62. C₁₄₉	#2098	semidirect
68. H(3,ℤ/4)	D₁₄₉	semidirect

H ≅ G/N

type

1. #23749674

C₂

semidirect

2. #23749652

C₂

semidirect

5. H(3,ℤ/4)×C₁₄₉

C₂

semidirect

62. C₁₄₉

#2098

semidirect

68. H(3,ℤ/4)

D₁₄₉

semidirect

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

subgroup	index
#23749674	2
#23749652	2
#23749643	2
#23749593	2
H(3,ℤ/4)×C₁₄₉	2
#2098	149

subgroup

index

H(3,ℤ/4)×C₁₄₉

149

Subgroups — #34046245

Conjugacy classes of subgroups

Complements

Maximal subgroups