Group #16835026 — subgroups

Subgroups — #16835026

Order 3712.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#14272186	C₂	1	2	yes	maximal
2	#14272023	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₂₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×D₂₉	C₄	1	2	yes	—
5	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	#11745306	C₂²	1	2	yes	—
7	C₄⋊C₄×D₂₉	—	2	2	no	—
8	#11745250	—	2	2	no	—
9	C₄²×D₂₉	C₂²	1	1	yes	characteristic
10	#11745231	C₄	1	1	yes	characteristic
11	#11745220	C₂²	1	4	yes	—
12	#11745219	—	2	2	no	—
13	#11745218	—	2	2	no	—
14	C₂×Q₈×C₂₉	C₂×C₄	1	2	yes	—
15	Q₈×D₂₉	—	4	4	no	—
16	C₂×C₄×D₂₉	—	2	2	no	—
17	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₂₉	D₄	1	2	yes	—
19	C₂×Dic₅₈	—	2	2	no	—
20	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
21	C₂×#6599	—	2	4	no	—
22	C₄⋊C₄×C₂₉	—	2	2	no	—
23	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
24	C₄⋊Dic₂₉	—	2	2	no	—
25	#90779	—	2	4	no	—
26	C₄×Dic₂₉	D₄	1	2	yes	—
27	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
28	C₂²×D₂₉	C₂²⋊C₄	1	1	yes	characteristic
29	Q₈×C₂₉	—	4	2	no	—
30	C₂×C₄×C₂₉	—	2	2	no	—
31	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₂₉	—	2	2	no	—
34	C₂×Dic₂₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₂₉	C₂×D₄	1	2	yes	—
36	C₄×D₂₉	—	2	2	no	—
37	C₄×D₂₉	—	2	4	no	—
38	C₄×D₂₉	—	4	4	no	—
39	Dic₅₈	—	4	4	no	—
40	Dic₅₈	—	4	2	no	—
41	#6599	—	4	4	no	—
42	#1558	—	29	1	no	maximal, Sylow
43	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×D₂₉	—	2	2	no	—
45	C₂×D₂₉	#100	1	2	yes	—
46	C₄×C₂₉	—	2	1	no	—
47	C₄×C₂₉	—	2	2	no	—
48	C₄×C₂₉	—	4	2	no	—
49	Dic₂₉	—	2	1	no	—
50	Dic₂₉	—	4	2	no	—
51	Dic₂₉	#137	1	4	yes	—
52	C₂×C₄⋊Q₈	—	29	1	no	—
53	#432	—	29	2	no	—
54	#333	—	29	4	no	—
55	C₂×C₂₉	C₂×#100	1	1	yes	characteristic
56	C₂×C₂₉	#419	1	2	yes	—
57	D₂₉	—	2	2	no	—
58	C₂²×Q₈	—	29	2	no	—
59	C₄⋊Q₈	—	29	2	no	—
60	C₄⋊Q₈	—	29	1	no	—
61	C₄⋊Q₈	—	58	2	no	—
62	C₄⋊Q₈	#1181	1	1	yes	characteristic
63	C₂×C₄⋊C₄	—	58	2	no	—
64	C₂×C₄²	—	29	1	no	—
65	C₄⋊C₈	—	29	4	no	—
66	C₂²⋊C₈	—	58	2	no	—
67	#97	—	58	2	no	—
68	C₂₉	#1558	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₈	—	29	2	no	—
70	C₂×Q₈	—	58	2	no	—
71	C₂×Q₈	—	116	4	no	—
72	C₂×Q₈	C₂×#1181	1	2	yes	—
73	C₂²×C₄	—	29	2	no	—
74	C₂²×C₄	—	29	1	no	—
75	C₂²×C₄	—	58	2	no	—
76	C₂×C₈	—	58	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	58	4	no	—
79	C₄⋊C₄	—	58	2	no	—
80	C₄²	—	29	2	no	—
81	C₄²	—	29	1	no	—
82	C₄²	C₂×#1181	1	1	yes	characteristic
83	C₂³	—	29	1	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	116	2	no	—
86	Q₈	—	116	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	29	2	no	—
89	C₂×C₄	—	29	1	no	—
90	C₂×C₄	—	58	4	no	—
91	C₂×C₄	—	58	2	no	—
92	C₂×C₄	—	58	2	no	—
93	C₂×C₄	—	116	4	no	—
94	C₂×C₄	C₂²⋊#1181	1	2	yes	—
95	C₂×C₄	C₂²×#1181	1	1	yes	characteristic
96	C₈	—	116	4	no	—
97	C₂²	—	29	2	no	—
98	C₂²	—	58	2	no	—
99	C₂²	C₂×C₂²⋊#1181	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	29	4	no	—
104	C₄	—	58	1	no	—
105	C₄	—	116	2	no	—
106	C₂	—	58	2	no	—
107	C₂	D₂₉⋊#100	1	1	yes	characteristic
108	C₂	#14272156	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14272186

C₂

yes

maximal

#14272023

C₂

yes

maximal, complemented

C₄⋊Q₈×D₂₉

C₂

yes

maximal, characteristic

C₂×Q₈×D₂₉

C₄

yes

—

C₂₉×C₄⋊Q₈

C₄

yes

characteristic

#11745306

C₂²

yes

—

C₄⋊C₄×D₂₉

—

#11745250

—

C₄²×D₂₉

C₂²

yes

characteristic

#11745231

C₄

yes

characteristic

#11745220

C₂²

yes

—

#11745219

—

#11745218

—

C₂×Q₈×C₂₉

C₂×C₄

yes

—

Q₈×D₂₉

—

C₂×C₄×D₂₉

—

C₂×C₄×D₂₉

C₂×C₄

yes

characteristic

C₂×C₄×D₂₉

D₄

yes

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×#6599

—

C₄⋊C₄×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄⋊Dic₂₉

—

#90779

—

C₄×Dic₂₉

D₄

yes

—

C₄×Dic₂₉

C₂³

yes

characteristic

C₂²×D₂₉

C₂²⋊C₄

yes

characteristic

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉

C₂×D₄

yes

—

C₄×D₂₉

—

C₄×D₂₉

—

C₄×D₂₉

—

Dic₅₈

—

Dic₅₈

—

#6599

—

#1558

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉

—

C₂×D₂₉

#100

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

Dic₂₉

—

Dic₂₉

—

Dic₂₉

#137

yes

—

C₂×C₄⋊Q₈

—

#432

—

#333

—

C₂×C₂₉

C₂×#100

yes

characteristic

C₂×C₂₉

#419

yes

—

D₂₉

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#1181

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂₉

#1558

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

116

—

C₂×Q₈

C₂×#1181

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#1181

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

116

—

Q₈

—

116

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

116

—

C₂×C₄

C₂²⋊#1181

yes

—

C₂×C₄

C₂²×#1181

yes

characteristic

C₈

—

116

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊#1181

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

116

—

106

C₂

—

107

C₂

D₂₉⋊#100

yes

characteristic

108

C₂

#14272156

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#1181
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1181
Fitting	C₂₉×C₄⋊Q₈	C₄
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₉	C₂²⋊C₄
cosocle	C₄×Dic₂₉	C₂³
hypercenter	C₄⋊Q₈	#1181
hypocenter	C₂₉	#1558
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#1181

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#1181

Fitting

C₂₉×C₄⋊Q₈

C₄

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₉

C₂²⋊C₄

cosocle

C₄×Dic₂₉

C₂³

C₄⋊Q₈

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #14272023	C₂	semidirect
68. C₂₉	#1558	semidirect

H ≅ G/N

type

2. #14272023

C₂

semidirect

68. C₂₉

#1558

semidirect

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

subgroup	index
#14272186	2
#14272023	2
C₄⋊Q₈×D₂₉	2
#1558	29

subgroup

index

#14272186

#14272023

C₄⋊Q₈×D₂₉

#1558

Subgroups — #16835026

Conjugacy classes of subgroups

Complements

Maximal subgroups