Group #34052960 — subgroups

Subgroups — #34052960

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	#23749564	C₂	1	3	yes	maximal
2	#23749558	C₂	1	6	yes	maximal
3	#23749545	C₂	1	3	yes	maximal
4	#23749493	C₂	1	2	yes	maximal
5	#384×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₄₉	C₂²	1	3	yes	—
8	C₂×#14558715	C₂²	1	6	yes	—
9	#18011818	C₂²	1	6	yes	—
10	#18011818	C₂²	1	6	yes	—
11	#18011818	C₂²	1	3	yes	—
12	#18011818	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	3	yes	—
15	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₈	—	4	3	no	—
17	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
18	C₂²×Dic₁₄₉	C₂³	1	6	yes	—
19	C₄⋊Dic₁₄₉	—	4	3	no	—
20	#14558715	—	4	6	no	—
21	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₄₉	C₂³	1	3	yes	—
23	C₂²×C₄×C₁₄₉	C₂³	1	3	yes	—
24	C₄⋊C₄×C₁₄₉	—	4	3	no	—
25	C₄⋊C₄×C₁₄₉	—	4	3	no	—
26	C₄²×C₁₄₉	Q₈	1	4	yes	complemented
27	Dic₂₉₈	—	4	4	no	—
28	C₂×Dic₁₄₉	—	2	6	no	—
29	C₂×Dic₁₄₉	—	2	6	no	—
30	C₂×Dic₁₄₉	—	2	12	no	—
31	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₁₄₉	—	2	3	no	—
33	C₂×C₄×C₁₄₉	—	2	3	no	—
34	C₂×C₄×C₁₄₉	—	2	3	no	—
35	C₂×C₄×C₁₄₉	—	2	6	no	—
36	C₂×C₄×C₁₄₉	—	2	3	no	—
37	C₂×C₄×C₁₄₉	C₂×Q₈	1	6	yes	—
38	C₂²×C₁₄₉	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₁₄₉	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₁₄₉	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₁₄₉	—	2	6	no	—
42	C₄×C₁₄₉	—	4	3	no	—
43	C₄×C₁₄₉	—	4	1	no	—
44	Dic₁₄₉	—	4	2	no	—
45	Dic₁₄₉	—	4	6	no	—
46	C₂×C₁₄₉	#544	1	3	yes	—
47	C₂×C₁₄₉	#560	1	1	yes	characteristic
48	C₂×C₁₄₉	#563	1	3	yes	—
49	C₁₄₉	#2843	1	1	yes	characteristic, complemented, Sylow
50	#2843	—	149	1	no	maximal, Sylow
51	#400	—	149	3	no	—
52	#398	—	149	6	no	—
53	#395	—	149	3	no	—
54	#384	D₁₄₉	1	1	yes	characteristic
55	#383	—	149	2	no	—
56	C₂²×Q₈	—	149	1	no	—
57	C₂×C₄⋊C₄	—	149	6	no	—
58	C₂×C₄⋊C₄	—	149	3	no	—
59	C₂×C₄⋊C₄	C₂×D₁₄₉	1	3	yes	—
60	C₂×C₄⋊C₄	C₂×D₁₄₉	1	3	yes	—
61	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
62	#95	—	149	6	no	—
63	#95	—	149	6	no	—
64	#95	—	149	3	no	—
65	#95	—	149	3	no	—
66	C₂×Q₈	—	596	3	no	—
67	C₂²×C₄	—	149	6	no	—
68	C₂²×C₄	—	149	2	no	—
69	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₁₄₉	1	3	yes	—
71	C₂²×C₄	C₂²×D₁₄₉	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	596	6	no	—
75	C₄⋊C₄	—	596	3	no	—
76	C₄²	Dic₂₉₈	1	4	yes	complemented
77	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
78	Q₈	—	596	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	298	6	no	—
85	C₂×C₄	—	298	6	no	—
86	C₂×C₄	—	298	12	no	—
87	C₂×C₄	C₂×Dic₂₉₈	1	6	yes	—
88	C₂²	C₂²×Dic₂₉₈	1	1	yes	characteristic
89	C₂²	#18011956	1	3	yes	—
90	C₂²	#18011959	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	596	2	no	—
95	C₄	—	596	6	no	—
96	C₂	#23750256	1	3	yes	—
97	C₂	#23750362	1	1	yes	characteristic
98	C₂	#23750374	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23749564

C₂

yes

maximal

#23749558

C₂

yes

maximal

#23749545

C₂

yes

maximal

#23749493

C₂

yes

maximal

#384×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×#14558715

C₂²

yes

—

#18011818

C₂²

yes

—

#18011818

C₂²

yes

—

#18011818

C₂²

yes

—

#18011818

C₂²

yes

—

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

C₂²×Dic₁₄₉

C₂³

yes

—

C₂²×Dic₁₄₉

C₂³

yes

—

C₄⋊Dic₁₄₉

—

#14558715

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂²×C₄×C₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

Q₈

yes

complemented

Dic₂₉₈

—

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

—

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

—

#395

—

149

—

#384

D₁₄₉

yes

characteristic

#383

—

149

—

C₂²×Q₈

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#95

—

149

—

#95

—

149

—

#95

—

149

—

#95

—

149

—

C₂×Q₈

—

596

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂²×C₄

C₂²×D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

596

—

C₄⋊C₄

—

596

—

C₄²

Dic₂₉₈

yes

complemented

C₂³

C₂³×D₁₄₉

yes

characteristic

Q₈

—

596

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

C₂×Dic₂₉₈

yes

—

C₂²

C₂²×Dic₂₉₈

yes

characteristic

C₂²

#18011956

yes

—

C₂²

#18011959

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

596

—

C₄

—

596

—

C₂

#23750256

yes

—

C₂

#23750362

yes

characteristic

C₂

#23750374

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₄₉
commutator, socle, prime radical, cosocle	C₂³×C₁₄₉	C₂⁴
Fitting	#384×C₁₄₉	C₂
hypercenter	#384	D₁₄₉
hypocenter	C₁₄₉	#2843
LCS	G ▷ C₂³×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₄₉

commutator, socle, prime radical, cosocle

C₂³×C₁₄₉

C₂⁴

Fitting

#384×C₁₄₉

C₂

D₁₄₉

C₁₄₉

G ▷ C₂³×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₁₄₉	Q₈	semidirect
49. C₁₄₉	#2843	semidirect
76. C₄²	Dic₂₉₈	semidirect

H ≅ G/N

type

26. C₄²×C₁₄₉

Q₈

semidirect

49. C₁₄₉

#2843

semidirect

76. C₄²

Dic₂₉₈

semidirect

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

subgroup	index
#23749564	2
#23749558	2
#23749545	2
#23749493	2
#384×C₁₄₉	2
#2843	149

subgroup

index

#384×C₁₄₉

149

Subgroups — #34052960

Conjugacy classes of subgroups

Complements

Maximal subgroups