Group #34046708 — subgroups

Subgroups — #34046708

Order 19072.

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

normal subgroups only

This group has 248 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	#23749797	C₂	1	3	yes	maximal, complemented
2	#23749422	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#18011959	C₂²	1	1	yes	characteristic, complemented
5	#18011937	—	2	3	no	—
6	#18011839	—	2	3	no	—
7	#18011830	C₂²	1	3	yes	complemented
8	#124×C₁₄₉	—	2	3	no	—
9	#124×C₁₄₉	C₂²	1	3	yes	—
10	#120×C₁₄₉	—	2	3	no	—
11	#14558737	—	4	1	no	—
12	#14558736	—	2	3	no	—
13	Q₈×D₁₄₉	—	4	1	no	—
14	#14558724	—	4	3	no	—
15	#14558721	—	4	3	no	—
16	#14558720	—	2	3	no	—
17	C₂⁴×C₁₄₉	—	2	1	no	—
18	C₂×Q₈×C₁₄₉	—	2	1	no	—
19	C₂×Q₈×C₁₄₉	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₁₄₉	—	2	3	no	—
21	C₂×D₄×C₁₄₉	D₄	1	3	yes	—
22	C₂²⋊C₄×C₁₄₉	—	2	3	no	—
23	C₂²⋊C₄×C₁₄₉	—	2	6	no	—
24	C₄²×C₁₄₉	D₄	1	3	yes	—
25	Dic₂₉₈	—	2	3	no	—
26	D₅₉₆	—	2	3	no	—
27	C₂²⋊D₁₄₉	—	4	1	no	—
28	C₄×D₁₄₉	—	4	3	no	—
29	#11924746	—	4	3	no	—
30	C₂³×C₁₄₉	—	2	3	no	—
31	C₂³×C₁₄₉	—	8	2	no	—
32	C₂³×C₁₄₉	—	8	1	no	—
33	Q₈×C₁₄₉	—	4	1	no	—
34	Q₈×C₁₄₉	—	8	1	no	—
35	D₄×C₁₄₉	—	4	3	no	—
36	D₄×C₁₄₉	—	8	3	no	—
37	C₂×C₄×C₁₄₉	—	2	3	no	—
38	C₂×C₄×C₁₄₉	C₂×D₄	1	3	yes	—
39	C₂²×C₁₄₉	—	4	6	no	—
40	C₂²×C₁₄₉	—	4	3	no	—
41	C₂²×C₁₄₉	—	8	4	no	—
42	C₂²×C₁₄₉	#120	1	1	yes	characteristic
43	C₂×D₁₄₉	—	4	1	no	—
44	C₄×C₁₄₉	—	2	3	no	—
45	C₄×C₁₄₉	—	4	3	no	—
46	Dic₁₄₉	—	4	1	no	—
47	C₂×C₁₄₉	—	2	1	no	—
48	C₂×C₁₄₉	—	8	3	no	—
49	C₂×C₁₄₉	B(4,𝔽₂)	1	1	yes	characteristic
50	D₁₄₉	—	8	1	no	—
51	C₁₄₉	#2202	1	1	yes	characteristic, complemented, Sylow
52	#2202	—	149	1	no	maximal, Sylow
53	H(3,𝔽₄)	D₁₄₉	1	1	yes	characteristic, complemented
54	#454	—	149	3	no	—
55	#355	—	149	3	no	—
56	2₋¹⁺⁴	—	149	1	no	—
57	AGL(1,ℤ/8)	—	298	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₁₄₉	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	298	3	no	—
62	#101	—	149	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	298	3	no	—
65	#41	—	596	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	596	1	no	—
68	C₂×Q₈	C₂²×D₁₄₉	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₁₄₉	1	3	yes	—
71	SD₁₆	—	596	3	no	—
72	D₈	—	596	3	no	—
73	M₁₆	—	298	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₁₄₉	1	3	yes	—
77	C₂³	—	2	3	no	—
78	C₂³	—	8	2	no	—
79	C₂³	—	8	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	8	1	no	—
82	Q₈	—	298	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	298	3	no	—
86	D₄	—	596	1	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	596	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	3	yes	—
90	C₈	—	596	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	596	1	no	—
95	C₂²	#18011880	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	596	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	1192	1	no	—
102	C₂	#23749812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23749797

C₂

yes

maximal, complemented

#23749422

C₂

yes

maximal, complemented

H(3,𝔽₄)×C₁₄₉

C₂

yes

maximal, characteristic, complemented

#18011959

C₂²

yes

characteristic, complemented

#18011937

—

#18011839

—

#18011830

C₂²

yes

complemented

#124×C₁₄₉

—

#124×C₁₄₉

C₂²

yes

—

#120×C₁₄₉

—

#14558737

—

#14558736

—

Q₈×D₁₄₉

—

#14558724

—

#14558721

—

#14558720

—

C₂⁴×C₁₄₉

—

C₂×Q₈×C₁₄₉

—

C₂×Q₈×C₁₄₉

C₂³

yes

characteristic

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

D₄

yes

—

C₂²⋊C₄×C₁₄₉

—

C₂²⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

D₄

yes

—

Dic₂₉₈

—

D₅₉₆

—

C₂²⋊D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

Q₈×C₁₄₉

—

Q₈×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#120

yes

characteristic

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

B(4,𝔽₂)

yes

characteristic

D₁₄₉

—

C₁₄₉

#2202

yes

characteristic, complemented, Sylow

#2202

—

149

maximal, Sylow

H(3,𝔽₄)

D₁₄₉

yes

characteristic, complemented

#454

—

149

—

#355

—

149

—

2₋¹⁺⁴

—

149

—

AGL(1,ℤ/8)

—

298

—

#124

—

#124

C₂×D₁₄₉

yes

—

#120

—

#104

—

298

—

#101

—

149

—

C₂⁴

—

#41

—

298

—

#41

—

596

—

C₂×Q₈

—

C₂×Q₈

—

596

—

C₂×Q₈

C₂²×D₁₄₉

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₄₉

yes

—

SD₁₆

—

596

—

D₈

—

596

—

M₁₆

—

298

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₁₄₉

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

298

—

D₄

—

D₄

—

D₄

—

298

—

D₄

—

596

—

C₂×C₄

—

C₂×C₄

—

596

—

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

C₈

—

596

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

596

—

C₂²

#18011880

yes

characteristic

C₄

—

C₄

—

C₄

—

596

—

C₂

—

100

C₂

—

101

C₂

—

1192

—

102

C₂

#23749812

yes

characteristic

	subgroup	G/H
center	C₂	#23749812
commutator, cosocle	C₂×Q₈×C₁₄₉	C₂³
Frattini	C₂×Q₈	C₂²×D₁₄₉
Fitting	H(3,𝔽₄)×C₁₄₉	C₂
socle	C₂×C₁₄₉	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₁₄₉
hypocenter	C₁₄₉	#2202
LCS	G ▷ C₂×Q₈×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#23749812

commutator, cosocle

C₂×Q₈×C₁₄₉

C₂³

Frattini

C₂×Q₈

C₂²×D₁₄₉

Fitting

H(3,𝔽₄)×C₁₄₉

C₂

C₂×C₁₄₉

D₁₄₉

C₁₄₉

G ▷ C₂×Q₈×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #23749797	C₂	semidirect
2. #23749422	C₂	semidirect
3. H(3,𝔽₄)×C₁₄₉	C₂	semidirect
4. #18011959	C₂²	semidirect
7. #18011830	C₂²	semidirect
51. C₁₄₉	#2202	semidirect
53. H(3,𝔽₄)	D₁₄₉	semidirect

H ≅ G/N

type

1. #23749797

C₂

semidirect

2. #23749422

C₂

semidirect

3. H(3,𝔽₄)×C₁₄₉

C₂

semidirect

4. #18011959

C₂²

semidirect

7. #18011830

C₂²

semidirect

51. C₁₄₉

#2202

semidirect

53. H(3,𝔽₄)

D₁₄₉

semidirect

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

subgroup	index
#23749797	2
#23749422	2
H(3,𝔽₄)×C₁₄₉	2
#2202	149

subgroup

index

#23749797

#23749422

H(3,𝔽₄)×C₁₄₉

#2202

149

Subgroups — #34046708

Conjugacy classes of subgroups

Complements

Maximal subgroups