Group #34046201 — subgroups

Subgroups — #34046201

Order 19072.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#23749668	C₂	1	2	yes	maximal
2	C₂×#18011843	C₂	1	1	yes	maximal, characteristic
3	#23749393	C₂	1	2	yes	maximal
4	#23749390	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558720	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
8	#18011844	—	2	2	no	—
9	#18011843	—	2	2	no	—
10	#18011842	—	2	2	no	—
11	C₂×M₁₆×C₁₄₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
13	#117×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₄₉	—	2	2	no	—
15	C₂²×Dic₁₄₉	—	2	2	no	—
16	#14558720	—	2	2	no	—
17	#14558720	Q₈	1	4	yes	—
18	C₂×#11924746	—	2	2	no	—
19	C₄⋊Dic₁₄₉	—	2	2	no	—
20	C₄⋊Dic₁₄₉	—	2	2	no	—
21	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₄₉	—	2	2	no	—
23	C₂×C₈×C₁₄₉	—	2	2	no	—
24	C₂×C₈×C₁₄₉	—	2	1	no	—
25	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₄₉	D₄	1	2	yes	—
27	C₂²⋊C₄×C₁₄₉	—	2	1	no	—
28	C₄²×C₁₄₉	—	2	1	no	—
29	C₂×Dic₁₄₉	—	4	2	no	—
30	C₂×Dic₁₄₉	—	8	2	no	—
31	#11924746	—	2	4	no	—
32	C₂³×C₁₄₉	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₄₉	—	2	2	no	—
34	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₄₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₄₉	#41	1	2	yes	—
38	C₈×C₁₄₉	—	2	2	no	—
39	C₈×C₁₄₉	—	4	1	no	—
40	C₂²×C₁₄₉	—	2	2	no	—
41	C₂²×C₁₄₉	#121	1	1	yes	characteristic
42	C₂²×C₁₄₉	#125	1	2	yes	—
43	C₄×C₁₄₉	—	4	2	no	—
44	C₄×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₁₄₉	#123	1	1	yes	characteristic
46	C₄×C₁₄₉	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₁₄₉	—	8	2	no	—
48	C₂×C₁₄₉	—	2	2	no	—
49	C₂×C₁₄₉	#398	1	1	yes	characteristic
50	C₂×C₁₄₉	#471	1	2	yes	—
51	C₁₄₉	#2090	1	1	yes	characteristic, complemented, Sylow
52	#2090	—	149	1	no	maximal, Sylow
53	#428	—	149	2	no	—
54	C₂×#106	—	149	1	no	—
55	#424	D₁₄₉	1	1	yes	characteristic
56	#343	—	149	2	no	—
57	#341	—	149	1	no	—
58	C₂×M₁₆	—	149	2	no	—
59	C₂×M₁₆	C₂×D₁₄₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
61	#117	C₂×D₁₄₉	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	149	2	no	—
63	#107	—	298	2	no	—
64	#106	—	298	2	no	—
65	#106	—	298	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	298	2	no	—
68	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	298	2	no	—
71	M₁₆	—	149	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	298	2	no	—
75	C₂×C₈	C₂²⋊D₁₄₉	1	2	yes	—
76	C₄⋊C₄	—	298	2	no	—
77	C₄⋊C₄	—	298	2	no	—
78	C₄⋊C₄	D₅₉₆	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#14558736	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	596	2	no	—
84	C₂×C₄	—	1192	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₄₉	1	2	yes	—
88	C₂×C₄	#14558735	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	298	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#18011885	1	1	yes	characteristic
94	C₂²	#18011901	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	1192	2	no	—
97	C₄	#18011891	1	2	yes	—
98	C₄	#18011892	1	1	yes	characteristic
99	C₄	#18011909	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#23749555	1	1	yes	characteristic
102	C₂	#23749888	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23749668

C₂

yes

maximal

C₂×#18011843

C₂

yes

maximal, characteristic

#23749393

C₂

yes

maximal

#23749390

C₂

yes

maximal, characteristic

#424×C₁₄₉

C₂

yes

maximal, characteristic

C₂×#14558720

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

#18011844

—

#18011843

—

#18011842

—

C₂×M₁₆×C₁₄₉

C₂²

yes

characteristic

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

#117×C₁₄₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₄₉

—

C₂²×Dic₁₄₉

—

#14558720

—

#14558720

Q₈

yes

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₄⋊C₄×C₁₄₉

D₄

yes

—

C₂²⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

#41

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×Q₈

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#121

yes

characteristic

C₂²×C₁₄₉

#125

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

#123

yes

characteristic

C₄×C₁₄₉

C₄⋊Q₈

yes

characteristic

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#398

yes

characteristic

C₂×C₁₄₉

#471

yes

—

C₁₄₉

#2090

yes

characteristic, complemented, Sylow

#2090

—

149

maximal, Sylow

#428

—

149

—

C₂×#106

—

149

—

#424

D₁₄₉

yes

characteristic

#343

—

149

—

#341

—

149

—

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

characteristic

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

#117

C₂×D₁₄₉

yes

characteristic

C₂×C₄⋊C₄

—

149

—

#107

—

298

—

#106

—

298

—

#106

—

298

—

C₄⋊C₈

—

C₂²×C₄

—

298

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

M₁₆

—

M₁₆

—

298

—

M₁₆

—

149

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

298

—

C₂×C₈

C₂²⋊D₁₄₉

yes

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

D₅₉₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#14558736

yes

characteristic

C₂×C₄

—

C₂×C₄

—

596

—

C₂×C₄

—

1192

—

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

C₂×C₄

C₂×D₅₉₆

yes

characteristic

C₂×C₄

Q₈×D₁₄₉

yes

—

C₂×C₄

#14558735

yes

—

C₈

—

C₈

—

C₈

—

298

—

C₂²

—

C₂²

#18011885

yes

characteristic

C₂²

#18011901

yes

—

C₄

—

C₄

—

1192

—

C₄

#18011891

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#23749555

yes

characteristic

102

C₂

#23749888

yes

—

	subgroup	G/H
center	C₂²	#18011885
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#424×C₁₄₉	C₂
socle	C₂²×C₁₄₉	#121
prime radical	C₂³×C₁₄₉	#41
hypercenter	#424	D₁₄₉
hypocenter	C₁₄₉	#2090
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#18011885

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#424×C₁₄₉

C₂

C₂²×C₁₄₉

C₂³×C₁₄₉

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
51. C₁₄₉	#2090	semidirect

H ≅ G/N

type

51. C₁₄₉

#2090

semidirect

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

subgroup	index
#23749668	2
C₂×#18011843	2
#23749393	2
#23749390	2
#424×C₁₄₉	2
#2090	149

subgroup

index

#424×C₁₄₉

149

Subgroups — #34046201

Conjugacy classes of subgroups

Complements

Maximal subgroups