Group #30746253 — subgroups

Subgroups — #30746253

Order 17152.

574 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 291 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	C₄×C₈×Dic₆₇	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	C₂×C₈×Dic₆₇	C₄	1	6	yes	—
6	C₂×C₈×Dic₆₇	C₄	1	6	yes	—
7	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
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	C₈×Dic₆₇	—	2	12	no	—
15	C₈×Dic₆₇	C₈	1	24	yes	complemented
16	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
17	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₆₇	—	2	6	no	—
19	C₂²×Dic₆₇	C₄²	1	1	yes	characteristic
20	C₂²×Dic₆₇	C₄²	1	3	yes	—
21	C₂×C₆₇⋊C₈	—	2	6	no	—
22	C₂×C₆₇⋊C₈	—	2	6	no	—
23	C₂×C₆₇⋊C₈	C₂×C₈	1	12	yes	—
24	C₄×Dic₆₇	C₂×C₈	1	12	yes	—
25	C₄×Dic₆₇	M₁₆	1	12	yes	—
26	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
27	C₂×C₈×C₆₇	—	2	6	no	—
28	C₂×C₈×C₆₇	—	2	6	no	—
29	C₂×C₈×C₆₇	C₂×C₈	1	12	yes	—
30	C₄²×C₆₇	#41	1	3	yes	—
31	C₄²×C₆₇	#41	1	1	yes	characteristic
32	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
33	C₂×C₄×C₆₇	#117	1	3	yes	—
34	C₂×C₄×C₆₇	#117	1	3	yes	—
35	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
36	C₂×C₄×C₆₇	C₂×M₁₆	1	6	yes	—
37	C₂×Dic₆₇	C₄×C₈	1	6	yes	—
38	C₂×Dic₆₇	C₄×C₈	1	6	yes	—
39	C₂×Dic₆₇	#97	1	12	yes	—
40	C₈×C₆₇	—	2	12	no	—
41	C₆₇⋊C₈	—	2	12	no	—
42	C₂²×C₆₇	#376	1	1	yes	characteristic
43	C₂²×C₆₇	C₂×C₄×C₈	1	3	yes	—
44	C₂²×C₆₇	C₂×#97	1	3	yes	—
45	C₄×C₆₇	#431	1	6	yes	—
46	C₄×C₆₇	#431	1	6	yes	—
47	Dic₆₇	C₈²	1	4	yes	—
48	Dic₆₇	#322	1	12	yes	—
49	#8131	—	67	1	no	maximal, Sylow
50	C₂×C₆₇	C₂×C₈²	1	1	yes	characteristic
51	C₂×C₆₇	C₂×#322	1	3	yes	—
52	C₂×C₆₇	#1744	1	3	yes	—
53	C₄²×C₈	—	67	3	no	—
54	#1273	—	67	3	no	—
55	#1273	D₆₇	1	1	yes	characteristic
56	C₆₇	#8131	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄×C₈	—	67	6	no	—
58	C₂×C₄×C₈	—	67	6	no	—
59	C₂×C₄×C₈	—	67	3	no	—
60	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
61	C₄³	—	67	1	no	—
62	C₂²×C₈	—	67	6	no	—
63	C₂²×C₈	C₄×D₆₇	1	6	yes	—
64	C₂×C₄²	—	67	3	no	—
65	C₂×C₄²	—	67	3	no	—
66	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
67	C₄×C₈	—	2	6	no	—
68	C₄×C₈	—	67	24	no	—
69	C₄×C₈	—	134	12	no	—
70	C₄×C₈	—	134	3	no	—
71	C₄×C₈	—	134	3	no	—
72	C₂²×C₄	—	67	1	no	—
73	C₂²×C₄	—	67	3	no	—
74	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
75	C₂×C₈	—	2	6	no	—
76	C₂×C₈	—	2	6	no	—
77	C₂×C₈	—	67	12	no	—
78	C₂×C₈	—	134	6	no	—
79	C₂×C₈	—	134	6	no	—
80	C₂×C₈	C₈×D₆₇	1	12	yes	—
81	C₄²	—	67	12	no	—
82	C₄²	—	67	12	no	—
83	C₄²	#11761627	1	3	yes	—
84	C₄²	#11761629	1	1	yes	characteristic
85	C₂³	C₄²×D₆₇	1	1	yes	characteristic
86	C₂×C₄	—	67	6	no	—
87	C₂×C₄	—	67	6	no	—
88	C₂×C₄	—	67	12	no	—
89	C₂×C₄	#14523999	1	3	yes	—
90	C₂×C₄	#14524001	1	3	yes	—
91	C₂×C₄	C₂×C₈×D₆₇	1	6	yes	—
92	C₂×C₄	M₁₆×D₆₇	1	6	yes	—
93	C₈	—	2	12	no	—
94	C₈	—	134	12	no	—
95	C₂²	#16863241	1	1	yes	characteristic
96	C₂²	C₄×C₈×D₆₇	1	3	yes	—
97	C₂²	#97×D₆₇	1	3	yes	—
98	C₄	—	67	4	no	—
99	C₄	—	67	12	no	—
100	C₄	#16863446	1	6	yes	—
101	C₄	#16863448	1	6	yes	—
102	C₂	C₈²×D₆₇	1	1	yes	characteristic
103	C₂	#322×D₆₇	1	3	yes	—
104	C₂	#22559953	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×Dic₆₇

C₂

yes

maximal

C₆₇⋊#1273

C₂

yes

maximal

C₆₇×#1273

C₂

yes

maximal, characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×C₈×Dic₆₇

C₄

yes

—

C₂×C₈×Dic₆₇

C₄

yes

—

C₄²×Dic₆₇

C₂²

yes

characteristic

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

—

C₈×Dic₆₇

—

C₈×Dic₆₇

C₈

yes

complemented

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

C₂×C₈

yes

—

C₄×Dic₆₇

C₂×C₈

yes

—

C₄×Dic₆₇

M₁₆

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₈

yes

—

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

C₂²×C₈

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×Dic₆₇

C₄×C₈

yes

—

C₂×Dic₆₇

C₄×C₈

yes

—

C₂×Dic₆₇

#97

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

C₂×C₄×C₈

yes

—

C₂²×C₆₇

C₂×#97

yes

—

C₄×C₆₇

#431

yes

—

C₄×C₆₇

#431

yes

—

Dic₆₇

C₈²

yes

—

Dic₆₇

#322

yes

—

#8131

—

maximal, Sylow

C₂×C₆₇

C₂×C₈²

yes

characteristic

C₂×C₆₇

C₂×#322

yes

—

C₂×C₆₇

#1744

yes

—

C₄²×C₈

—

#1273

—

#1273

D₆₇

yes

characteristic

C₆₇

#8131

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₄³

—

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

134

—

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

C₈×D₆₇

yes

—

C₄²

—

C₄²

—

C₄²

#11761627

yes

—

C₄²

#11761629

yes

characteristic

C₂³

C₄²×D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#14523999

yes

—

C₂×C₄

#14524001

yes

—

C₂×C₄

C₂×C₈×D₆₇

yes

—

C₂×C₄

M₁₆×D₆₇

yes

—

C₈

—

C₈

—

134

—

C₂²

#16863241

yes

characteristic

C₂²

C₄×C₈×D₆₇

yes

—

C₂²

#97×D₆₇

yes

—

C₄

—

C₄

—

100

C₄

#16863446

yes

—

101

C₄

#16863448

yes

—

102

C₂

C₈²×D₆₇

yes

characteristic

103

C₂

#322×D₆₇

yes

—

104

C₂

#22559953

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₈²
Fitting	C₆₇×#1273	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1273	D₆₇
hypocenter	C₆₇	#8131
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₆₇

commutator

C₂×C₆₇

C₂×C₈²

Fitting

C₆₇×#1273

C₂

socle, prime radical

C₂³×C₆₇

C₂×C₄²

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
15. C₈×Dic₆₇	C₈	semidirect
56. C₆₇	#8131	semidirect

H ≅ G/N

type

15. C₈×Dic₆₇

C₈

semidirect

56. C₆₇

#8131

semidirect

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

subgroup	index
C₄×C₈×Dic₆₇	2
C₆₇⋊#1273	2
C₆₇×#1273	2
#8131	67

subgroup

index

C₄×C₈×Dic₆₇

C₆₇⋊#1273

C₆₇×#1273

#8131

Subgroups — #30746253

Conjugacy classes of subgroups

Complements

Maximal subgroups