Group #30746265 — subgroups

Subgroups — #30746265

Order 17152.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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	#22559954	C₂	1	3	yes	maximal
2	C₆₇⋊#1273	C₂	1	3	yes	maximal
3	C₆₇×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
5	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
6	#16863150	C₄	1	6	yes	—
7	#16863150	C₄	1	6	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
9	C₂²×C₆₇⋊C₈	C₂×C₄	1	6	yes	—
10	C₂×C₄×Dic₆₇	C₂×C₄	1	3	yes	—
11	C₂×C₄×Dic₆₇	C₂×C₄	1	3	yes	—
12	C₄×C₆₇⋊C₈	—	2	3	no	—
13	C₄×C₆₇⋊C₈	—	2	3	no	—
14	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
15	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₆₇	—	2	6	no	—
17	C₂²×Dic₆₇	C₄²	1	1	yes	characteristic
18	C₂²×Dic₆₇	C₄²	1	3	yes	—
19	C₂×C₆₇⋊C₈	—	2	6	no	—
20	C₂×C₆₇⋊C₈	—	2	6	no	—
21	C₂×C₆₇⋊C₈	—	2	6	no	—
22	C₄×Dic₆₇	—	2	6	no	—
23	C₄×Dic₆₇	—	2	6	no	—
24	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
25	C₂×C₈×C₆₇	—	2	6	no	—
26	C₂×C₈×C₆₇	—	2	6	no	—
27	C₂×C₈×C₆₇	—	2	6	no	—
28	C₄²×C₆₇	#41	1	3	yes	—
29	C₄²×C₆₇	#41	1	1	yes	characteristic
30	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₄×C₆₇	#117	1	3	yes	—
32	C₂×C₄×C₆₇	#117	1	3	yes	—
33	C₂×C₄×C₆₇	#131	1	6	yes	—
34	C₂×C₄×C₆₇	#131	1	6	yes	—
35	C₂×Dic₆₇	—	2	3	no	—
36	C₂×Dic₆₇	—	2	3	no	—
37	C₂×Dic₆₇	—	2	6	no	—
38	C₈×C₆₇	—	4	6	no	—
39	C₆₇⋊C₈	—	4	6	no	—
40	C₂²×C₆₇	#376	1	1	yes	characteristic
41	C₂²×C₆₇	#405	1	3	yes	—
42	C₂²×C₆₇	#405	1	3	yes	—
43	C₄×C₆₇	#433	1	6	yes	—
44	C₄×C₆₇	#433	1	6	yes	—
45	Dic₆₇	—	4	1	no	—
46	Dic₆₇	—	4	3	no	—
47	#8135	—	67	1	no	maximal, Sylow
48	C₂×C₆₇	#1454	1	1	yes	characteristic
49	C₂×C₆₇	#1454	1	3	yes	—
50	C₂×C₆₇	#1746	1	3	yes	—
51	#1744	—	67	3	no	—
52	#1273	—	67	3	no	—
53	#1273	D₆₇	1	1	yes	characteristic
54	C₆₇	#8135	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₄×C₈	—	67	3	no	—
56	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
57	C₄³	—	67	1	no	—
58	#336	—	67	6	no	—
59	#336	—	67	6	no	—
60	C₂²×C₈	—	67	6	no	—
61	C₂²×C₈	C₄×D₆₇	1	6	yes	—
62	C₂×C₄²	—	67	3	no	—
63	C₂×C₄²	—	67	3	no	—
64	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
65	C₄×C₈	—	2	6	no	—
66	C₄×C₈	—	134	3	no	—
67	C₄×C₈	—	134	3	no	—
68	C₂²×C₄	—	67	1	no	—
69	C₂²×C₄	—	67	3	no	—
70	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	—	2	6	no	—
73	C₂×C₈	—	2	6	no	—
74	C₂×C₈	—	134	6	no	—
75	C₂×C₈	—	134	6	no	—
76	C₂×C₈	—	134	6	no	—
77	C₄²	—	134	6	no	—
78	C₄²	—	134	6	no	—
79	C₄²	#11761627	1	3	yes	—
80	C₄²	#11761629	1	1	yes	characteristic
81	C₂³	C₄²×D₆₇	1	1	yes	characteristic
82	C₂×C₄	—	134	3	no	—
83	C₂×C₄	—	134	3	no	—
84	C₂×C₄	—	134	6	no	—
85	C₂×C₄	#14523999	1	3	yes	—
86	C₂×C₄	#14524001	1	3	yes	—
87	C₂×C₄	#14524049	1	6	yes	—
88	C₂×C₄	#14524051	1	6	yes	—
89	C₈	—	4	6	no	—
90	C₈	—	268	6	no	—
91	C₂²	#16863241	1	1	yes	characteristic
92	C₂²	#16863350	1	3	yes	—
93	C₂²	#16863353	1	3	yes	—
94	C₄	—	268	1	no	—
95	C₄	—	268	3	no	—
96	C₄	#16863454	1	6	yes	—
97	C₄	#16863456	1	6	yes	—
98	C₂	#22558508	1	1	yes	characteristic
99	C₂	#22558510	1	3	yes	—
100	C₂	#22559959	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559954

C₂

yes

maximal

C₆₇⋊#1273

C₂

yes

maximal

C₆₇×#1273

C₂

yes

maximal, characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

C₄²×Dic₆₇

C₂²

yes

characteristic

#16863150

C₄

yes

—

#16863150

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

—

C₂²×Dic₆₇

C₄²

yes

characteristic

C₂²×Dic₆₇

C₄²

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

#41

yes

characteristic

C₂³×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

#405

yes

—

C₂²×C₆₇

#405

yes

—

C₄×C₆₇

#433

yes

—

C₄×C₆₇

#433

yes

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#1746

yes

—

#1744

—

#1273

—

#1273

D₆₇

yes

characteristic

C₆₇

#8135

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₄³

—

#336

—

#336

—

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄×C₈

—

C₄×C₈

—

134

—

C₄×C₈

—

134

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×D₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₄²

—

134

—

C₄²

—

134

—

C₄²

#11761627

yes

—

C₄²

#11761629

yes

characteristic

C₂³

C₄²×D₆₇

yes

characteristic

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

#14523999

yes

—

C₂×C₄

#14524001

yes

—

C₂×C₄

#14524049

yes

—

C₂×C₄

#14524051

yes

—

C₈

—

C₈

—

268

—

C₂²

#16863241

yes

characteristic

C₂²

#16863350

yes

—

C₂²

#16863353

yes

—

C₄

—

268

—

C₄

—

268

—

C₄

#16863454

yes

—

C₄

#16863456

yes

—

C₂

#22558508

yes

characteristic

C₂

#22558510

yes

—

100

C₂

#22559959

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂×C₄²
Fitting	C₆₇×#1273	C₂
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1273	D₆₇
hypocenter	C₆₇	#8135
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₆₇

commutator, socle, prime radical

C₂³×C₆₇

C₂×C₄²

Fitting

C₆₇×#1273

C₂

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
54. C₆₇	#8135	semidirect

H ≅ G/N

type

54. C₆₇

#8135

semidirect

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

subgroup	index
#22559954	2
C₆₇⋊#1273	2
C₆₇×#1273	2
#8135	67

subgroup

index

#22559954

C₆₇⋊#1273

C₆₇×#1273

#8135

Subgroups — #30746265

Conjugacy classes of subgroups

Complements

Maximal subgroups