Group #34046043 — subgroups

Subgroups — #34046043

Order 19072.

314 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 75 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	#23749497	C₂	1	3	yes	maximal
2	#23749392	C₂	1	3	yes	maximal
3	#543×C₁₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₄₉	C₂²	1	1	yes	characteristic
5	C₂×#14558715	—	2	3	no	—
6	C₂×C₂²⋊Dic₁₄₉	C₂²	1	3	yes	—
7	#18011818	—	2	3	no	—
8	C₂²⋊Q₈×C₁₄₉	—	2	3	no	—
9	C₂²⋊Q₈×C₁₄₉	—	2	3	no	—
10	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	3	yes	—
11	C₂²×Dic₁₄₉	—	2	3	no	—
12	C₂²×Dic₁₄₉	—	2	3	no	—
13	C₂²×Dic₁₄₉	—	4	2	no	—
14	#14558715	—	4	6	no	—
15	C₂²⋊Dic₁₄₉	—	2	3	no	—
16	C₂²⋊Dic₁₄₉	Q₈	1	6	yes	complemented
17	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
18	C₂×Q₈×C₁₄₉	—	2	1	no	—
19	C₂×Q₈×C₁₄₉	—	2	1	no	—
20	C₂²×C₄×C₁₄₉	D₄	1	3	yes	—
21	C₂²×C₄×C₁₄₉	D₄	1	3	yes	—
22	C₄⋊C₄×C₁₄₉	—	2	3	no	—
23	C₄⋊C₄×C₁₄₉	—	2	3	no	—
24	C₂²⋊C₄×C₁₄₉	—	2	6	no	—
25	C₂×Dic₁₄₉	—	2	6	no	—
26	C₂×Dic₁₄₉	—	2	6	no	—
27	C₂×Dic₁₄₉	—	4	3	no	—
28	C₂×Dic₁₄₉	—	4	3	no	—
29	C₂×Dic₁₄₉	—	4	3	no	—
30	C₂×Dic₁₄₉	—	4	1	no	—
31	C₂³×C₁₄₉	—	4	1	no	—
32	C₂³×C₁₄₉	—	4	1	no	—
33	C₂³×C₁₄₉	C₂×D₄	1	3	yes	—
34	C₂³×C₁₄₉	C₂×Q₈	1	3	yes	—
35	C₂³×C₁₄₉	#41	1	1	yes	characteristic
36	Q₈×C₁₄₉	—	8	1	no	—
37	Q₈×C₁₄₉	—	8	1	no	—
38	C₂×C₄×C₁₄₉	—	2	3	no	—
39	C₂×C₄×C₁₄₉	—	2	3	no	—
40	C₂×C₄×C₁₄₉	—	4	3	no	—
41	C₂×C₄×C₁₄₉	—	4	3	no	—
42	C₂²×C₁₄₉	—	2	3	no	—
43	C₂²×C₁₄₉	—	2	3	no	—
44	C₂²×C₁₄₉	—	4	3	no	—
45	C₂²×C₁₄₉	—	4	1	no	—
46	C₂²×C₁₄₉	#120	1	1	yes	characteristic
47	C₂²×C₁₄₉	C₂²⋊Q₈	1	3	yes	—
48	C₂²×C₁₄₉	C₄⋊Q₈	1	3	yes	—
49	C₄×C₁₄₉	—	4	3	no	—
50	C₄×C₁₄₉	—	4	3	no	—
51	Dic₁₄₉	—	4	6	no	—
52	Dic₁₄₉	—	4	2	no	—
53	C₂×C₁₄₉	—	2	3	no	—
54	C₂×C₁₄₉	—	2	3	no	—
55	C₂×C₁₄₉	#395	1	1	yes	characteristic
56	C₂×C₁₄₉	B(4,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₄₉	#458	1	1	yes	characteristic
58	C₁₄₉	#2062	1	1	yes	characteristic, complemented, Sylow
59	#2062	—	149	1	no	maximal, Sylow
60	#543	D₁₄₉	1	1	yes	characteristic
61	#385	—	149	3	no	—
62	#342	—	149	3	no	—
63	C₂³×C₄	—	149	1	no	—
64	C₂²⋊Q₈	—	2	3	no	—
65	C₂²⋊Q₈	—	2	3	no	—
66	C₂×C₄⋊C₄	—	298	3	no	—
67	C₂×C₂²⋊C₄	—	149	3	no	—
68	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	3	yes	—
69	#95	—	298	3	no	—
70	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
71	C₂×Q₈	—	2	1	no	—
72	C₂×Q₈	—	2	1	no	—
73	C₂²×C₄	—	298	3	no	—
74	C₂²×C₄	—	298	3	no	—
75	C₂²×C₄	—	596	2	no	—
76	C₂²×C₄	C₂²⋊D₁₄₉	1	3	yes	—
77	C₂²×C₄	C₂²⋊D₁₄₉	1	3	yes	—
78	C₄⋊C₄	—	2	3	no	—
79	C₄⋊C₄	—	2	3	no	—
80	C₄⋊C₄	—	596	6	no	—
81	C₂²⋊C₄	—	2	6	no	—
82	C₂²⋊C₄	—	298	3	no	—
83	C₂²⋊C₄	—	149	6	no	—
84	C₂³	—	4	1	no	—
85	C₂³	—	4	1	no	—
86	C₂³	C₂×C₂²⋊D₁₄₉	1	3	yes	—
87	C₂³	Q₈×D₁₄₉	1	3	yes	—
88	C₂³	#14558737	1	1	yes	characteristic
89	Q₈	—	8	1	no	—
90	Q₈	—	8	1	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	4	3	no	—
94	C₂×C₄	—	4	3	no	—
95	C₂×C₄	—	298	6	no	—
96	C₂×C₄	—	298	6	no	—
97	C₂×C₄	—	596	3	no	—
98	C₂×C₄	—	596	3	no	—
99	C₂×C₄	—	596	3	no	—
100	C₂×C₄	—	596	1	no	—
101	C₂²	—	2	3	no	—
102	C₂²	—	2	3	no	—
103	C₂²	—	4	3	no	—
104	C₂²	—	4	1	no	—
105	C₂²	#18011880	1	1	yes	characteristic
106	C₂²	#18011887	1	3	yes	—
107	C₂²	#18011909	1	3	yes	—
108	C₄	—	4	3	no	—
109	C₄	—	4	3	no	—
110	C₄	—	596	6	no	—
111	C₄	—	596	2	no	—
112	C₂	—	2	3	no	—
113	C₂	—	2	3	no	—
114	C₂	#23749544	1	1	yes	characteristic
115	C₂	#23749812	1	1	yes	characteristic
116	C₂	#23749815	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23749497

C₂

yes

maximal

#23749392

C₂

yes

maximal

#543×C₁₄₉

C₂

yes

maximal, characteristic

C₂³×Dic₁₄₉

C₂²

yes

characteristic

C₂×#14558715

—

C₂×C₂²⋊Dic₁₄₉

C₂²

yes

—

#18011818

—

C₂²⋊Q₈×C₁₄₉

—

C₂²⋊Q₈×C₁₄₉

—

C₂×C₂²⋊C₄×C₁₄₉

C₂²

yes

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

#14558715

—

C₂²⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

Q₈

yes

complemented

C₂⁴×C₁₄₉

C₂³

yes

characteristic

C₂×Q₈×C₁₄₉

—

C₂×Q₈×C₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₂²⋊C₄×C₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

—

C₂³×C₁₄₉

C₂×Q₈

yes

—

C₂³×C₁₄₉

#41

yes

characteristic

Q₈×C₁₄₉

—

Q₈×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#120

yes

characteristic

C₂²×C₁₄₉

C₂²⋊Q₈

yes

—

C₂²×C₁₄₉

C₄⋊Q₈

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2062

—

149

maximal, Sylow

#543

D₁₄₉

yes

characteristic

#385

—

149

—

#342

—

149

—

C₂³×C₄

—

149

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

298

—

C₂×C₂²⋊C₄

—

149

—

C₂×C₂²⋊C₄

C₂×D₁₄₉

yes

—

#95

—

298

—

C₂⁴

C₂²×D₁₄₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

—

596

—

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

596

—

C₂²⋊C₄

—

C₂²⋊C₄

—

298

—

C₂²⋊C₄

—

149

—

C₂³

—

C₂³

—

C₂³

C₂×C₂²⋊D₁₄₉

yes

—

C₂³

Q₈×D₁₄₉

yes

—

C₂³

#14558737

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

C₂×C₄

—

596

—

100

C₂×C₄

—

596

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

#18011880

yes

characteristic

106

C₂²

#18011887

yes

—

107

C₂²

#18011909

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

596

—

111

C₄

—

596

—

112

C₂

—

113

C₂

—

114

C₂

#23749544

yes

characteristic

115

C₂

#23749812

yes

characteristic

116

C₂

#23749815

yes

characteristic

	subgroup	G/H
center	C₂²	#18011880
commutator, prime radical, cosocle	C₂⁴×C₁₄₉	C₂³
Frattini	C₂⁴	C₂²×D₁₄₉
Fitting	#543×C₁₄₉	C₂
socle	C₂²×C₁₄₉	#120
hypercenter	#543	D₁₄₉
hypocenter	C₁₄₉	#2062
LCS	G ▷ C₂⁴×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#18011880

commutator, prime radical, cosocle

C₂⁴×C₁₄₉

C₂³

Frattini

C₂⁴

C₂²×D₁₄₉

Fitting

#543×C₁₄₉

C₂

C₂²×C₁₄₉

D₁₄₉

C₁₄₉

G ▷ C₂⁴×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
16. C₂²⋊Dic₁₄₉	Q₈	semidirect
58. C₁₄₉	#2062	semidirect

H ≅ G/N

type

16. C₂²⋊Dic₁₄₉

Q₈

semidirect

58. C₁₄₉

#2062

semidirect

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

subgroup	index
#23749497	2
#23749392	2
#543×C₁₄₉	2
#2062	149

subgroup

index

#23749497

#23749392

#543×C₁₄₉

#2062

149

Subgroups — #34046043

Conjugacy classes of subgroups

Complements

Maximal subgroups