Group #16817327 — subgroups

Subgroups — #16817327

Order 3712.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊#6597	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₂₉	C₄	1	2	yes	—
4	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
5	C₂²×#6597	—	2	2	no	—
6	C₄×C₈×C₂₉	—	2	2	no	—
7	C₄⋊#6597	—	2	2	no	—
8	C₄⋊#6597	C₄	1	4	yes	—
9	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₉	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₂₉	—	2	2	no	—
13	C₂×C₈×C₂₉	—	2	2	no	—
14	C₂×C₈×C₂₉	C₈	1	4	yes	complemented
15	C₄²×C₂₉	C₂×C₄	1	2	yes	—
16	C₄²×C₂₉	D₄	1	2	yes	—
17	C₂×#6597	—	2	4	no	—
18	C₂×#6597	—	4	2	no	—
19	C₂×#6597	—	4	2	no	—
20	C₂³×C₂₉	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₉	—	2	2	no	—
22	C₂×C₄×C₂₉	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₉	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₉	M₁₆	1	2	yes	—
28	C₂×C₄×C₂₉	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
30	C₂×C₄×C₂₉	Q₁₆	1	1	yes	characteristic
31	C₈×C₂₉	—	2	4	no	—
32	#6597	—	4	4	no	—
33	#1277	—	29	1	no	maximal, Sylow
34	C₂²×C₂₉	#95	1	1	yes	characteristic
35	C₂²×C₂₉	C₄⋊C₈	1	2	yes	—
36	C₂²×C₂₉	#106	1	1	yes	characteristic
37	C₂²×C₂₉	#107	1	1	yes	characteristic
38	C₂²×C₂₉	#108	1	2	yes	—
39	C₄×C₂₉	—	2	2	no	—
40	C₄×C₂₉	C₄×C₈	1	1	yes	characteristic
41	C₄×C₂₉	#97	1	1	yes	characteristic
42	C₄×C₂₉	C₂²⋊C₈	1	2	yes	—
43	C₄×C₂₉	#102	1	2	yes	—
44	C₄×C₂₉	#103	1	2	yes	—
45	C₂×C₄⋊C₈	—	29	2	no	—
46	C₂×C₄×C₈	D₂₉	1	1	yes	characteristic
47	C₂×C₂₉	#334	1	2	yes	—
48	C₂×C₂₉	#335	1	2	yes	—
49	C₂×C₂₉	#336	1	1	yes	characteristic
50	C₂×C₂₉	#340	1	1	yes	characteristic
51	C₂×C₂₉	#341	1	1	yes	characteristic
52	C₂²×C₈	—	58	2	no	—
53	C₂²×C₈	Dic₂₉	1	2	yes	—
54	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
55	C₄⋊C₈	—	29	4	no	—
56	C₄⋊C₈	—	58	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₂₉	#1277	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₄	Dic₅₈	1	1	yes	characteristic
60	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
61	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	58	4	no	—
65	C₂×C₈	—	116	2	no	—
66	C₂×C₈	—	116	2	no	—
67	C₂×C₈	#6597	1	4	yes	complemented
68	C₄²	C₄×D₂₉	1	2	yes	—
69	C₄²	C₂²⋊D₂₉	1	2	yes	—
70	C₂³	C₄⋊Dic₂₉	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	#90773	1	2	yes	—
73	C₂×C₄	D₂₃₂	1	1	yes	characteristic
74	C₂×C₄	Dic₁₁₆	1	1	yes	characteristic
75	C₂×C₄	C₂×#6597	1	2	yes	—
76	C₂×C₄	#90777	1	2	yes	—
77	C₂×C₄	C₄×Dic₂₉	1	1	yes	characteristic
78	C₂×C₄	#90779	1	2	yes	—
79	C₂×C₄	#90781	1	2	yes	—
80	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	116	4	no	—
83	C₂²	C₄⋊#6597	1	2	yes	—
84	C₂²	#11745165	1	1	yes	characteristic
85	C₂²	#11745166	1	1	yes	characteristic
86	C₂²	#11745167	1	2	yes	—
87	C₂²	#11745179	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	C₄×#6597	1	1	yes	characteristic
90	C₄	#11745151	1	1	yes	characteristic
91	C₄	#11745164	1	2	yes	—
92	C₄	#11745169	1	2	yes	—
93	C₄	C₂²⋊#6597	1	2	yes	—
94	C₂	#14270914	1	2	yes	—
95	C₂	#14270916	1	2	yes	—
96	C₂	#14270918	1	1	yes	characteristic
97	C₂	#14270926	1	1	yes	characteristic
98	C₂	#14270928	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#6597

C₂

yes

maximal

C₂×C₄×C₈×C₂₉

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₉

C₄

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂²×#6597

—

C₄×C₈×C₂₉

—

C₄⋊#6597

—

C₄⋊#6597

C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

Q₈

yes

characteristic

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

C₈

yes

complemented

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

D₄

yes

—

C₂×#6597

—

C₂×#6597

—

C₂×#6597

—

C₂³×C₂₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₄²

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₉

C₂×C₈

yes

—

C₂×C₄×C₂₉

M₁₆

yes

—

C₂×C₄×C₂₉

D₈

yes

characteristic

C₂×C₄×C₂₉

SD₁₆

yes

—

C₂×C₄×C₂₉

Q₁₆

yes

characteristic

C₈×C₂₉

—

—

—

C₂²×C₂₉

yes

C₂²×C₂₉

C₄⋊C₈

yes

—

C₂²×C₂₉

yes

C₂²×C₂₉

yes

C₂²×C₂₉

yes

—

C₄×C₂₉

—

C₄×C₂₉

C₄×C₈

yes

characteristic

C₄×C₂₉

#97

yes

characteristic

C₄×C₂₉

C₂²⋊C₈

yes

—

C₄×C₂₉

#102

yes

—

C₄×C₂₉

#103

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₂₉

yes

characteristic

C₂×C₂₉

#334

yes

—

C₂×C₂₉

#335

yes

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×C₈

—

C₂²×C₈

Dic₂₉

yes

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₂₉

#1277

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₅₈

yes

characteristic

C₂²×C₄

D₁₁₆

yes

characteristic

C₂²×C₄

C₂×Dic₂₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

116

—

C₂×C₈

—

116

—

C₂×C₈

#6597

yes

complemented

C₄²

C₄×D₂₉

yes

—

C₄²

C₂²⋊D₂₉

yes

—

C₂³

C₄⋊Dic₂₉

yes

characteristic

C₂×C₄

—

C₂×C₄

#90773

yes

—

C₂×C₄

D₂₃₂

yes

characteristic

C₂×C₄

Dic₁₁₆

yes

characteristic

C₂×C₄

C₂×#6597

yes

—

C₂×C₄

#90777

yes

—

C₂×C₄

C₄×Dic₂₉

yes

characteristic

C₂×C₄

#90779

yes

—

C₂×C₄

#90781

yes

—

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

C₈

—

C₈

—

116

—

C₂²

C₄⋊#6597

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#11745179

yes

characteristic

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₄

#11745169

yes

—

C₄

C₂²⋊#6597

yes

—

C₂

#14270914

yes

—

C₂

#14270916

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	D₁₁₆
commutator	C₄×C₂₉	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₉
Fitting	C₂×C₄×C₈×C₂₉	C₂
socle, prime radical	C₂³×C₂₉	C₄⋊C₄
cosocle	C₂×C₄²×C₂₉	C₂²
hypercenter	C₂×C₄×C₈	D₂₉
hypocenter	C₂₉	#1277
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₁₁₆

commutator

C₄×C₂₉

C₄×C₈

Frattini

C₂×C₄²

C₂×D₂₉

Fitting

C₂×C₄×C₈×C₂₉

C₂

socle, prime radical

C₂³×C₂₉

C₄⋊C₄

cosocle

C₂×C₄²×C₂₉

C₂²

hypercenter

C₂×C₄×C₈

D₂₉

hypocenter

C₂₉

#1277

LCS

G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
14. C₂×C₈×C₂₉	C₈	semidirect
58. C₂₉	#1277	semidirect
67. C₂×C₈	#6597	semidirect

H ≅ G/N

type

14. C₂×C₈×C₂₉

C₈

semidirect

58. C₂₉

#1277

semidirect

67. C₂×C₈

#6597

semidirect

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

subgroup	index
C₂×C₄⋊#6597	2
C₂×C₄×C₈×C₂₉	2
#1277	29

subgroup

index

C₂×C₄⋊#6597

C₂×C₄×C₈×C₂₉

#1277

Subgroups — #16817327

Conjugacy classes of subgroups

Complements

Maximal subgroups