Group #34052958 — subgroups

Subgroups — #34052958

Order 19072.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#23749567	C₂	1	3	yes	maximal
2	#23749559	C₂	1	3	yes	maximal
3	#23749557	C₂	1	3	yes	maximal
4	#23749546	C₂	1	3	yes	maximal
5	#23749495	C₂	1	1	yes	maximal, characteristic
6	#23749493	C₂	1	1	yes	maximal, characteristic
7	#383×C₁₄₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₄₉	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₁₄₉	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₁₄₉	C₂²	1	3	yes	—
12	C₂×#14558715	C₂²	1	3	yes	—
13	C₂×#14558715	C₂²	1	3	yes	—
14	#18011818	C₂²	1	3	yes	—
15	#18011818	C₂²	1	3	yes	—
16	#18011818	C₂²	1	3	yes	—
17	#18011818	C₂²	1	3	yes	—
18	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
19	#95×C₁₄₉	C₂²	1	3	yes	—
20	#95×C₁₄₉	C₂²	1	3	yes	—
21	C₂×Dic₂₉₈	—	4	3	no	—
22	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₄₉	C₂³	1	3	yes	—
25	C₂²×Dic₁₄₉	C₂³	1	3	yes	—
26	C₄⋊Dic₁₄₉	—	4	3	no	—
27	C₄⋊Dic₁₄₉	—	4	3	no	—
28	C₄⋊Dic₁₄₉	—	4	3	no	—
29	#14558715	—	4	3	no	—
30	#14558715	—	4	3	no	—
31	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₄₉	C₂³	1	3	yes	—
33	C₂²×C₄×C₁₄₉	C₂³	1	3	yes	—
34	C₄²×C₁₄₉	Q₈	1	4	yes	complemented
35	Dic₂₉₈	—	4	4	no	—
36	C₂×Dic₁₄₉	—	2	3	no	—
37	C₂×Dic₁₄₉	—	2	3	no	—
38	C₂×Dic₁₄₉	—	2	3	no	—
39	C₂×Dic₁₄₉	—	2	3	no	—
40	C₂×Dic₁₄₉	—	2	3	no	—
41	C₂×Dic₁₄₉	—	2	3	no	—
42	C₂×Dic₁₄₉	—	2	6	no	—
43	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₁₄₉	—	2	3	no	—
45	C₂×C₄×C₁₄₉	—	2	3	no	—
46	C₂×C₄×C₁₄₉	—	2	3	no	—
47	C₂×C₄×C₁₄₉	—	2	3	no	—
48	C₂×C₄×C₁₄₉	—	2	6	no	—
49	C₂×C₄×C₁₄₉	C₂×Q₈	1	6	yes	—
50	C₂²×C₁₄₉	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₄₉	H(5,𝔽₂)	1	3	yes	—
52	C₂²×C₁₄₉	2₋¹⁺⁴	1	3	yes	—
53	C₄×C₁₄₉	—	2	6	no	—
54	C₄×C₁₄₉	—	4	1	no	—
55	C₄×C₁₄₉	—	4	3	no	—
56	Dic₁₄₉	—	4	3	no	—
57	Dic₁₄₉	—	4	1	no	—
58	Dic₁₄₉	—	4	1	no	—
59	Dic₁₄₉	—	4	3	no	—
60	C₂×C₁₄₉	#544	1	3	yes	—
61	C₂×C₁₄₉	#560	1	1	yes	characteristic
62	C₂×C₁₄₉	#563	1	3	yes	—
63	C₁₄₉	#2843	1	1	yes	characteristic, complemented, Sylow
64	#2843	—	149	1	no	maximal, Sylow
65	#400	—	149	3	no	—
66	#398	—	149	3	no	—
67	#398	—	149	3	no	—
68	#395	—	149	3	no	—
69	#384	—	149	1	no	—
70	#383	—	149	1	no	—
71	#383	D₁₄₉	1	1	yes	characteristic
72	C₂²×Q₈	—	149	1	no	—
73	C₂×C₄⋊C₄	—	149	3	no	—
74	C₂×C₄⋊C₄	—	149	3	no	—
75	C₂×C₄⋊C₄	—	149	3	no	—
76	C₂×C₄⋊C₄	—	149	3	no	—
77	C₂×C₄⋊C₄	—	149	3	no	—
78	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
79	#95	—	149	3	no	—
80	#95	—	149	3	no	—
81	#95	—	149	3	no	—
82	#95	—	149	3	no	—
83	#95	C₂×D₁₄₉	1	3	yes	—
84	#95	C₂×D₁₄₉	1	3	yes	—
85	C₂×Q₈	—	596	3	no	—
86	C₂²×C₄	—	149	3	no	—
87	C₂²×C₄	—	149	1	no	—
88	C₂²×C₄	—	149	1	no	—
89	C₂²×C₄	—	149	3	no	—
90	C₂²×C₄	C₂²×D₁₄₉	1	3	yes	—
91	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₁₄₉	1	3	yes	—
93	C₄⋊C₄	—	596	3	no	—
94	C₄⋊C₄	—	596	3	no	—
95	C₄⋊C₄	—	596	3	no	—
96	C₄⋊C₄	—	596	3	no	—
97	C₄⋊C₄	—	596	3	no	—
98	C₄²	Dic₂₉₈	1	4	yes	complemented
99	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
100	Q₈	—	596	4	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	298	3	no	—
107	C₂×C₄	—	298	3	no	—
108	C₂×C₄	—	298	3	no	—
109	C₂×C₄	—	298	3	no	—
110	C₂×C₄	—	298	3	no	—
111	C₂×C₄	—	298	3	no	—
112	C₂×C₄	—	298	6	no	—
113	C₂×C₄	C₂×Dic₂₉₈	1	6	yes	—
114	C₂²	C₂²×Dic₂₉₈	1	1	yes	characteristic
115	C₂²	#18011957	1	3	yes	—
116	C₂²	#18011958	1	3	yes	—
117	C₄	—	2	6	no	—
118	C₄	—	4	1	no	—
119	C₄	—	4	3	no	—
120	C₄	—	596	1	no	—
121	C₄	—	596	1	no	—
122	C₄	—	596	3	no	—
123	C₄	—	596	3	no	—
124	C₂	#23750253	1	3	yes	—
125	C₂	#23750366	1	1	yes	characteristic
126	C₂	#23750377	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23749567

C₂

yes

maximal

#23749559

C₂

yes

maximal

#23749557

C₂

yes

maximal

#23749546

C₂

yes

maximal

#23749495

C₂

yes

maximal, characteristic

#23749493

C₂

yes

maximal, characteristic

#383×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×#14558715

C₂²

yes

—

C₂×#14558715

C₂²

yes

—

#18011818

C₂²

yes

—

#18011818

C₂²

yes

—

#18011818

C₂²

yes

—

#18011818

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

#95×C₁₄₉

C₂²

yes

—

#95×C₁₄₉

C₂²

yes

—

C₂×Dic₂₉₈

—

C₂²×Dic₁₄₉

C₂³

yes

characteristic

C₂²×Dic₁₄₉

C₂³

yes

characteristic

C₂²×Dic₁₄₉

C₂³

yes

—

C₂²×Dic₁₄₉

C₂³

yes

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

#14558715

—

#14558715

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂²×C₄×C₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

—

C₄²×C₁₄₉

Q₈

yes

complemented

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂³×C₁₄₉

C₂⁴

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×Q₈

yes

—

C₂²×C₁₄₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₄₉

H(5,𝔽₂)

yes

—

C₂²×C₁₄₉

2₋¹⁺⁴

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

#544

yes

—

C₂×C₁₄₉

#560

yes

characteristic

C₂×C₁₄₉

#563

yes

—

C₁₄₉

#2843

yes

characteristic, complemented, Sylow

#2843

—

149

maximal, Sylow

#400

—

149

—

#398

—

149

—

#398

—

149

—

#395

—

149

—

#384

—

149

—

#383

—

149

—

#383

D₁₄₉

yes

characteristic

C₂²×Q₈

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#95

—

149

—

#95

—

149

—

#95

—

149

—

#95

—

149

—

#95

C₂×D₁₄₉

yes

—

#95

C₂×D₁₄₉

yes

—

C₂×Q₈

—

596

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₂²×D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂²×C₄

C₂²×D₁₄₉

yes

—

C₄⋊C₄

—

596

—

C₄⋊C₄

—

596

—

C₄⋊C₄

—

596

—

C₄⋊C₄

—

596

—

C₄⋊C₄

—

596

—

C₄²

Dic₂₉₈

yes

complemented

C₂³

C₂³×D₁₄₉

yes

characteristic

100

Q₈

—

596

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

298

—

107

C₂×C₄

—

298

—

108

C₂×C₄

—

298

—

109

C₂×C₄

—

298

—

110

C₂×C₄

—

298

—

111

C₂×C₄

—

298

—

112

C₂×C₄

—

298

—

113

C₂×C₄

C₂×Dic₂₉₈

yes

—

114

C₂²

C₂²×Dic₂₉₈

yes

characteristic

115

C₂²

#18011957

yes

—

116

C₂²

#18011958

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

596

—

121

C₄

—

596

—

122

C₄

—

596

—

123

C₄

—

596

—

124

C₂

#23750253

yes

—

125

C₂

#23750366

yes

characteristic

126

C₂

#23750377

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₄₉
commutator, socle, prime radical, cosocle	C₂³×C₁₄₉	C₂⁴
Fitting	#383×C₁₄₉	C₂
hypercenter	#383	D₁₄₉
hypocenter	C₁₄₉	#2843
LCS	G ▷ C₂³×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₄₉

commutator, socle, prime radical, cosocle

C₂³×C₁₄₉

C₂⁴

Fitting

#383×C₁₄₉

C₂

D₁₄₉

C₁₄₉

G ▷ C₂³×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
34. C₄²×C₁₄₉	Q₈	semidirect
63. C₁₄₉	#2843	semidirect
98. C₄²	Dic₂₉₈	semidirect

H ≅ G/N

type

34. C₄²×C₁₄₉

Q₈

semidirect

63. C₁₄₉

#2843

semidirect

98. C₄²

Dic₂₉₈

semidirect

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

subgroup	index
#23749567	2
#23749559	2
#23749557	2
#23749546	2
#23749495	2
#23749493	2
#383×C₁₄₉	2
#2843	149

subgroup

index

#383×C₁₄₉

149

Subgroups — #34052958

Conjugacy classes of subgroups

Complements

Maximal subgroups