Group #30674324 — subgroups

Subgroups — #30674324

Order 16768.

472 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 213 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	C₂×#16862238	C₂	1	2	yes	maximal
2	#22553210	C₂	1	8	yes	maximal
3	#22553182	C₂	1	2	yes	maximal
4	#22553178	C₂	1	2	yes	maximal
5	C₄³×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#16862238	—	2	8	no	—
7	C₂×C₄⋊Dic₁₃₁	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₁₃₁	C₂²	1	8	yes	—
9	C₂×Dic₁₃₁⋊C₄	C₂²	1	8	yes	—
10	C₁₃₁⋊#95	C₂²	1	8	yes	—
11	C₂×C₄²×C₁₃₁	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃₁	C₂³	1	8	yes	—
16	C₄⋊Dic₁₃₁	—	2	8	no	—
17	C₄⋊Dic₁₃₁	—	2	16	no	—
18	Dic₁₃₁⋊C₄	—	2	16	no	—
19	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
20	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
21	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
23	C₄²×C₁₃₁	—	2	8	no	—
24	C₄²×C₁₃₁	D₄	1	4	yes	—
25	C₄²×C₁₃₁	Q₈	1	4	yes	—
26	C₄²×C₁₃₁	Q₈	1	4	yes	—
27	C₂×Dic₁₃₁	—	2	8	no	—
28	C₂×Dic₁₃₁	—	2	16	no	—
29	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₁₃₁	—	2	4	no	—
31	C₂×C₄×C₁₃₁	—	2	4	no	—
32	C₂×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
35	C₂×C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
36	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₃₁	#41	1	8	yes	—
39	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
40	C₂²×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
41	C₂²×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
42	C₂²×C₁₃₁	C₂×#41	1	4	yes	—
43	C₄×C₁₃₁	—	2	4	no	—
44	C₄×C₁₃₁	—	2	4	no	—
45	C₄×C₁₃₁	#125	1	8	yes	complemented
46	C₄×C₁₃₁	C₄⋊Q₈	1	4	yes	—
47	Dic₁₃₁	—	4	8	no	—
48	C₂×C₁₃₁	C₂×#125	1	2	yes	—
49	C₂×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₁₃₁	#532	1	2	yes	—
51	C₂×C₁₃₁	#533	1	2	yes	—
52	C₁₃₁	#2859	1	1	yes	characteristic, complemented, Sylow
53	#2859	—	131	1	no	maximal, Sylow
54	C₂×#125	—	131	2	no	—
55	#389	—	131	8	no	—
56	#384	—	131	2	no	—
57	#382	—	131	2	no	—
58	C₄³	D₁₃₁	1	1	yes	characteristic
59	#125	—	262	8	no	—
60	C₂×C₄⋊C₄	—	131	4	no	—
61	C₂×C₄⋊C₄	—	131	8	no	—
62	C₂×C₄⋊C₄	—	131	8	no	—
63	C₂×C₄²	C₂×D₁₃₁	1	4	yes	—
64	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
67	#95	—	131	8	no	—
68	C₂²×C₄	—	131	8	no	—
69	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
71	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
73	C₄⋊C₄	—	262	16	no	—
74	C₄⋊C₄	—	262	8	no	—
75	C₄⋊C₄	—	262	16	no	—
76	C₄²	—	2	8	no	—
77	C₄²	D₅₂₄	1	4	yes	—
78	C₄²	Dic₂₆₂	1	4	yes	—
79	C₄²	Dic₂₆₂	1	4	yes	—
80	C₂³	C₂³×D₁₃₁	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	262	8	no	—
84	C₂×C₄	—	262	16	no	—
85	C₂×C₄	C₂×D₅₂₄	1	4	yes	—
86	C₂×C₄	C₂×D₅₂₄	1	2	yes	—
87	C₂×C₄	C₂×Dic₂₆₂	1	4	yes	—
88	C₂×C₄	C₂×Dic₂₆₂	1	4	yes	—
89	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
90	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
91	C₂×C₄	#14523513	1	8	yes	—
92	C₂²	C₂²×D₅₂₄	1	1	yes	characteristic
93	C₂²	C₂²×Dic₂₆₂	1	1	yes	characteristic
94	C₂²	C₂²×Dic₂₆₂	1	1	yes	characteristic
95	C₂²	C₂×#14523513	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	2	4	no	—
98	C₄	—	524	8	no	—
99	C₄	#16862238	1	8	yes	complemented
100	C₄	#16862246	1	4	yes	—
101	C₂	C₂×#16862238	1	2	yes	—
102	C₂	C₂×#16862246	1	1	yes	characteristic
103	C₂	#22553862	1	2	yes	—
104	C₂	#22553869	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16862238

C₂

yes

maximal

#22553210

C₂

yes

maximal

#22553182

C₂

yes

maximal

#22553178

C₂

yes

maximal

C₄³×C₁₃₁

C₂

yes

maximal, characteristic

#16862238

—

C₂×C₄⋊Dic₁₃₁

C₂²

yes

—

C₂×C₄⋊Dic₁₃₁

C₂²

yes

—

C₂×Dic₁₃₁⋊C₄

C₂²

yes

—

C₁₃₁⋊#95

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₂²×Dic₁₃₁

C₂³

yes

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

Dic₁₃₁⋊C₄

—

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂²×C₄×C₁₃₁

C₂³

yes

—

C₄²×C₁₃₁

—

C₄²×C₁₃₁

D₄

yes

—

C₄²×C₁₃₁

Q₈

yes

—

C₄²×C₁₃₁

Q₈

yes

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂³×C₁₃₁

C₂⁴

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×Q₈

yes

—

C₂×C₄×C₁₃₁

C₂×Q₈

yes

—

C₂×C₄×C₁₃₁

C₂×Q₈

yes

—

C₂×C₄×C₁₃₁

C₂×Q₈

yes

—

C₂×C₄×C₁₃₁

#41

yes

—

C₂²×C₁₃₁

C₂²×D₄

yes

characteristic

C₂²×C₁₃₁

C₂²×Q₈

yes

characteristic

C₂²×C₁₃₁

C₂²×Q₈

yes

characteristic

C₂²×C₁₃₁

C₂×#41

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#125

yes

complemented

C₄×C₁₃₁

C₄⋊Q₈

yes

—

Dic₁₃₁

—

C₂×C₁₃₁

C₂×#125

yes

—

C₂×C₁₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₁

#532

yes

—

C₂×C₁₃₁

#533

yes

—

C₁₃₁

#2859

yes

characteristic, complemented, Sylow

#2859

—

131

maximal, Sylow

C₂×#125

—

131

—

#389

—

131

—

#384

—

131

—

#382

—

131

—

C₄³

D₁₃₁

yes

characteristic

#125

—

262

—

C₂×C₄⋊C₄

—

131

—

C₂×C₄⋊C₄

—

131

—

C₂×C₄⋊C₄

—

131

—

C₂×C₄²

C₂×D₁₃₁

yes

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

#95

—

131

—

C₂²×C₄

—

131

—

C₂²×C₄

C₂²×D₁₃₁

yes

—

C₂²×C₄

C₂²×D₁₃₁

yes

—

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

C₂²×C₄

C₂²×D₁₃₁

yes

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

262

—

C₄²

—

C₄²

D₅₂₄

yes

—

C₄²

Dic₂₆₂

yes

—

C₄²

Dic₂₆₂

yes

—

C₂³

C₂³×D₁₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

C₂×D₅₂₄

yes

—

C₂×C₄

C₂×D₅₂₄

yes

—

C₂×C₄

C₂×Dic₂₆₂

yes

—

C₂×C₄

C₂×Dic₂₆₂

yes

—

C₂×C₄

C₂×Dic₂₆₂

yes

—

C₂×C₄

C₂×Dic₂₆₂

yes

—

C₂×C₄

#14523513

yes

—

C₂²

C₂²×D₅₂₄

yes

characteristic

C₂²

C₂²×Dic₂₆₂

yes

characteristic

C₂²

C₂²×Dic₂₆₂

yes

characteristic

C₂²

C₂×#14523513

yes

—

C₄

—

C₄

—

C₄

—

524

—

C₄

#16862238

yes

complemented

100

C₄

#16862246

yes

—

101

C₂

C₂×#16862238

yes

—

102

C₂

C₂×#16862246

yes

characteristic

103

C₂

#22553862

yes

—

104

C₂

#22553869

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₁
commutator, socle, prime radical, cosocle	C₂³×C₁₃₁	C₂⁴
Fitting	C₄³×C₁₃₁	C₂
hypercenter	C₄³	D₁₃₁
hypocenter	C₁₃₁	#2859
LCS	G ▷ C₂³×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃₁

commutator, socle, prime radical, cosocle

C₂³×C₁₃₁

C₂⁴

Fitting

C₄³×C₁₃₁

C₂

hypercenter

C₄³

D₁₃₁

C₁₃₁

G ▷ C₂³×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
45. C₄×C₁₃₁	#125	semidirect
52. C₁₃₁	#2859	semidirect
99. C₄	#16862238	semidirect

H ≅ G/N

type

45. C₄×C₁₃₁

#125

semidirect

52. C₁₃₁

#2859

semidirect

99. C₄

#16862238

semidirect

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

subgroup	index
C₂×#16862238	2
#22553210	2
#22553182	2
#22553178	2
C₄³×C₁₃₁	2
#2859	131

subgroup

index

C₄³×C₁₃₁

131

Subgroups — #30674324

Conjugacy classes of subgroups

Complements

Maximal subgroups