Group #34068250 — subgroups

Subgroups — #34068250

Order 19328.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#23752591	C₂	1	1	yes	maximal, characteristic
2	#23752507	C₂	1	2	yes	maximal
3	#23751961	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₁₅₁⋊C₈)	C₂	1	2	yes	maximal
5	Q₈²×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558994	—	2	2	no	—
7	#18012739	C₂²	1	1	yes	characteristic
8	#18012672	—	2	2	no	—
9	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	2	yes	—
10	#18012663	—	2	2	no	—
11	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₅₁	—	2	2	no	—
13	C₄⋊Q₈×C₁₅₁	—	2	2	no	—
14	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₅₁	—	2	2	no	—
16	C₄×Q₈×C₁₅₁	C₂²	1	2	yes	—
17	C₂×Dic₃₀₂	—	2	1	no	—
18	#14558994	—	4	4	no	—
19	C₂×C₁₅₁⋊C₈	—	2	1	no	—
20	C₂×C₁₅₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₅₁	—	2	2	no	—
22	C₂×Q₈×C₁₅₁	—	2	2	no	—
23	C₂×Q₈×C₁₅₁	D₄	1	2	yes	—
24	C₂×Q₈×C₁₅₁	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₅₁	—	2	4	no	—
26	C₄⋊C₄×C₁₅₁	—	2	2	no	—
27	C₄⋊C₄×C₁₅₁	—	2	2	no	—
28	C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
29	C₄²×C₁₅₁	—	2	2	no	—
30	C₄²×C₁₅₁	—	2	2	no	—
31	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
32	Dic₃₀₂	—	4	2	no	—
33	C₂×Dic₁₅₁	—	4	1	no	—
34	C₁₅₁⋊C₈	—	2	2	no	—
35	C₁₅₁⋊C₈	—	4	2	no	—
36	Q₈×C₁₅₁	—	2	2	no	—
37	Q₈×C₁₅₁	—	4	2	no	—
38	Q₈×C₁₅₁	—	8	2	no	—
39	Q₈×C₁₅₁	Q₁₆	1	4	yes	complemented
40	C₂×C₄×C₁₅₁	—	2	2	no	—
41	C₂×C₄×C₁₅₁	—	2	2	no	—
42	C₂×C₄×C₁₅₁	—	2	2	no	—
43	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
45	C₂²×C₁₅₁	#120	1	1	yes	characteristic
46	C₄×C₁₅₁	—	2	4	no	—
47	C₄×C₁₅₁	—	2	1	no	—
48	C₄×C₁₅₁	—	4	2	no	—
49	C₄×C₁₅₁	—	4	2	no	—
50	C₄×C₁₅₁	C₂×Q₁₆	1	2	yes	—
51	C₄×C₁₅₁	#137	1	2	yes	—
52	Dic₁₅₁	—	8	1	no	—
53	C₂×C₁₅₁	#451	1	2	yes	—
54	C₂×C₁₅₁	#453	1	1	yes	characteristic
55	C₁₅₁	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	151	1	no	maximal, Sylow
57	Q₈²	D₁₅₁	1	1	yes	characteristic
58	#494	—	151	1	no	—
59	#477	—	151	2	no	—
60	#333	—	151	1	no	—
61	Q₈⋊C₈	—	151	2	no	—
62	C₂×Q₁₆	—	302	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	151	1	no	—
66	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₁₅₁	1	2	yes	—
69	#107	—	302	2	no	—
70	C₄⋊C₈	—	151	2	no	—
71	#103	—	302	2	no	—
72	C₄×C₈	—	151	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	302	1	no	—
75	C₂×Q₈	C₂²⋊D₁₅₁	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₁₅₁	1	2	yes	—
77	Q₁₆	—	604	4	no	—
78	C₂×C₈	—	302	1	no	—
79	C₂×C₈	—	302	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	302	2	no	—
84	C₄⋊C₄	C₂²⋊D₁₅₁	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	604	2	no	—
92	Q₈	#14558994	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	604	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
99	C₈	—	302	2	no	—
100	C₈	—	604	2	no	—
101	C₂²	#18012708	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	1208	1	no	—
107	C₄	C₂×#14558994	1	2	yes	—
108	C₄	#18012770	1	2	yes	—
109	C₂	#23752368	1	2	yes	—
110	C₂	#23752380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752591

C₂

yes

maximal, characteristic

#23752507

C₂

yes

maximal

#23751961

C₂

yes

maximal, characteristic

Q₈⋊(C₁₅₁⋊C₈)

C₂

yes

maximal

Q₈²×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558994

—

#18012739

C₂²

yes

characteristic

#18012672

—

C₄⋊(C₁₅₁⋊C₈)

C₂²

yes

—

#18012663

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

C₄×Q₈×C₁₅₁

—

C₄×Q₈×C₁₅₁

C₂²

yes

—

C₂×Dic₃₀₂

—

#14558994

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

D₄

yes

—

C₂×Q₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

Q₁₆

yes

complemented

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂²×C₁₅₁

#120

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×Q₁₆

yes

—

C₄×C₁₅₁

#137

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

#451

yes

—

C₂×C₁₅₁

#453

yes

characteristic

C₁₅₁

#1633

yes

characteristic, complemented, Sylow

#1633

—

151

maximal, Sylow

Q₈²

D₁₅₁

yes

characteristic

#494

—

151

—

#477

—

151

—

#333

—

151

—

Q₈⋊C₈

—

151

—

C₂×Q₁₆

—

302

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₁₅₁

yes

—

#107

—

302

—

C₄⋊C₈

—

151

—

#103

—

302

—

C₄×C₈

—

151

—

C₂×Q₈

—

C₂×Q₈

—

302

—

C₂×Q₈

C₂²⋊D₁₅₁

yes

—

C₂×Q₈

C₂²⋊D₁₅₁

yes

—

Q₁₆

—

604

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

C₂²⋊D₁₅₁

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₅₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

604

—

Q₈

#14558994

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

604

—

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

C₈

—

302

—

100

C₈

—

604

—

101

C₂²

#18012708

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

1208

—

107

C₄

C₂×#14558994

yes

—

108

C₄

#18012770

yes

—

109

C₂

#23752368

yes

—

110

C₂

#23752380

yes

characteristic

	subgroup	G/H
center	C₂²	#18012708
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	Q₈²×C₁₅₁	C₂
socle, prime radical	C₂²×C₁₅₁	#120
hypercenter	Q₈²	D₁₅₁
hypocenter	C₁₅₁	#1633
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#18012708

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

Q₈²×C₁₅₁

C₂

socle, prime radical

C₂²×C₁₅₁

#120

hypercenter

Q₈²

D₁₅₁

hypocenter

C₁₅₁

#1633

LCS

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
39. Q₈×C₁₅₁	Q₁₆	semidirect
55. C₁₅₁	#1633	semidirect
92. Q₈	#14558994	semidirect

H ≅ G/N

type

39. Q₈×C₁₅₁

Q₁₆

semidirect

55. C₁₅₁

#1633

semidirect

92. Q₈

#14558994

semidirect

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

subgroup	index
#23752591	2
#23752507	2
#23751961	2
Q₈⋊(C₁₅₁⋊C₈)	2
Q₈²×C₁₅₁	2
#1633	151

subgroup

index

#23752591

#23752507

#23751961

Q₈⋊(C₁₅₁⋊C₈)

Q₈²×C₁₅₁

#1633

151

Subgroups — #34068250

Conjugacy classes of subgroups

Complements

Maximal subgroups