Group #30772224 — subgroups

Subgroups — #30772224

Order 17152.

274 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22560381	C₂	1	2	yes	maximal
2	#22558303	C₂	1	4	yes	maximal
3	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523979	—	2	2	no	—
5	C₂×#14523979	C₂²	1	4	yes	—
6	#16863420	—	2	2	no	—
7	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	#346×C₆₇	C₄	1	4	yes	—
10	C₂×#11761612	—	2	4	no	—
11	#14523979	—	2	8	no	—
12	#14523979	—	4	4	no	—
13	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
14	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₆₇	—	2	4	no	—
18	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
19	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₆₇	Q₈	1	2	yes	—
21	#11761612	—	4	4	no	—
22	#11761612	—	4	4	no	—
23	C₂×C₆₇⋊C₈	—	4	4	no	—
24	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
28	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₆₇	#41	1	4	yes	—
31	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
32	C₄²×C₆₇	D₈	1	2	yes	—
33	C₁₆×C₆₇	—	2	4	no	—
34	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₆₇	—	4	2	no	—
36	C₂×C₄×C₆₇	#107	1	4	yes	—
37	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	#124	1	2	yes	—
39	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
41	C₈×C₆₇	#106	1	4	yes	—
42	C₈×C₆₇	#117	1	2	yes	—
43	C₈×C₆₇	#125	1	2	yes	—
44	C₆₇⋊C₈	—	8	4	no	—
45	C₂²×C₆₇	—	2	2	no	—
46	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
47	C₂²×C₆₇	#487	1	2	yes	—
48	C₄×C₆₇	—	2	4	no	—
49	C₄×C₆₇	#382	1	1	yes	characteristic
50	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
51	C₄×C₆₇	#486	1	2	yes	—
52	#12892	—	67	1	no	maximal, Sylow
53	C₂×C₆₇	—	2	2	no	—
54	C₂×C₆₇	#1834	1	1	yes	characteristic
55	C₂×C₆₇	#2161	1	2	yes	—
56	C₂×#346	D₆₇	1	1	yes	characteristic
57	#1848	—	67	2	no	—
58	#1381	—	67	4	no	—
59	C₆₇	#12892	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
61	C₂×#108	—	67	4	no	—
62	C₂×#108	—	134	2	no	—
63	#423	—	134	2	no	—
64	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
65	#346	Dic₆₇	1	4	yes	—
66	C₂×M₁₆	—	134	4	no	—
67	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
68	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
69	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
72	#108	—	268	4	no	—
73	#108	—	134	8	no	—
74	C₄⋊C₈	—	268	4	no	—
75	C₄×C₈	Dic₁₃₄	1	2	yes	—
76	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
77	C₂²×C₄	Dic₂₆₈	1	2	yes	—
78	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
79	M₁₆	—	268	4	no	—
80	M₁₆	—	268	4	no	—
81	C₂×C₈	—	268	4	no	—
82	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
83	C₂×C₈	#11761617	1	4	yes	—
84	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
85	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
86	C₂×C₈	#11761628	1	4	yes	—
87	C₄²	C₄⋊Dic₆₇	1	2	yes	—
88	C₄²	D₅₃₆	1	2	yes	—
89	C₁₆	—	2	4	no	—
90	C₂³	C₂×Dic₂₆₈	1	1	yes	characteristic
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#14523977	1	4	yes	—
93	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
94	C₂×C₄	#14524031	1	2	yes	—
95	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
96	C₂×C₄	C₂×#11761617	1	2	yes	—
97	C₈	—	536	4	no	—
98	C₈	#14523975	1	4	yes	—
99	C₈	#14524002	1	2	yes	—
100	C₈	#14524035	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×#14523977	1	1	yes	characteristic
103	C₂²	#16863747	1	2	yes	—
104	C₄	—	2	4	no	—
105	C₄	#16863262	1	1	yes	characteristic
106	C₄	C₂×#14523975	1	1	yes	characteristic
107	C₄	#16863743	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	#22560337	1	1	yes	characteristic
110	C₂	#22562004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560381

C₂

yes

maximal

#22558303

C₂

yes

maximal

C₂×#346×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523979

—

C₂×#14523979

C₂²

yes

—

#16863420

—

C₂²×C₁₆×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#346×C₆₇

C₄

yes

—

C₂×#11761612

—

#14523979

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

Q₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

D₈

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

#106

yes

—

C₈×C₆₇

#117

yes

—

C₈×C₆₇

#125

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#107

yes

characteristic

C₂²×C₆₇

#487

yes

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#12892

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1834

yes

characteristic

C₂×C₆₇

#2161

yes

—

C₂×#346

D₆₇

yes

characteristic

#1848

—

#1381

—

C₆₇

#12892

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₆₇

yes

—

C₂×#108

—

C₂×#108

—

134

—

#423

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#346

Dic₆₇

yes

—

C₂×M₁₆

—

134

—

C₂²×C₈

D₂₆₈

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

D₂₆₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₆₇

yes

—

#108

—

268

—

#108

—

134

—

C₄⋊C₈

—

268

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₂²×C₄

Dic₂₆₈

yes

—

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

M₁₆

—

268

—

M₁₆

—

268

—

C₂×C₈

—

268

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₂×C₈

#11761617

yes

—

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

C₂×C₈

#11761628

yes

—

C₄²

C₄⋊Dic₆₇

yes

—

C₄²

D₅₃₆

yes

—

C₁₆

—

C₂³

C₂×Dic₂₆₈

yes

characteristic

C₂×C₄

—

C₂×C₄

#14523977

yes

—

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

#14524031

yes

—

C₂×C₄

C₂×D₅₃₆

yes

characteristic

C₂×C₄

C₂×#11761617

yes

—

C₈

—

536

—

C₈

#14523975

yes

—

C₈

#14524002

yes

—

100

C₈

#14524035

yes

—

101

C₂²

—

102

C₂²

C₂×#14523977

yes

characteristic

103

C₂²

#16863747

yes

—

104

C₄

—

105

C₄

#16863262

yes

characteristic

106

C₄

C₂×#14523975

yes

characteristic

107

C₄

#16863743

yes

—

108

C₂

—

109

C₂

#22560337

yes

characteristic

110

C₂

#22562004

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#346×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂³×C₆₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#346	D₆₇
hypocenter	C₆₇	#12892
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₅₃₆

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#346×C₆₇

C₂

socle

C₂²×C₆₇

C₂×#107

prime radical

C₂³×C₆₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
59. C₆₇	#12892	semidirect

H ≅ G/N

type

59. C₆₇

#12892

semidirect

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

subgroup	index
#22560381	2
#22558303	2
C₂×#346×C₆₇	2
#12892	67

subgroup

index

#22560381

#22558303

C₂×#346×C₆₇

#12892

Subgroups — #30772224

Conjugacy classes of subgroups

Complements

Maximal subgroups