Group #30749547 — subgroups

Subgroups — #30749547

Order 17152.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22558972	C₂	1	2	yes	maximal
2	#22558071	C₂	1	4	yes	maximal
3	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic
4	#16863453	C₂²	1	4	yes	—
5	#16863149	—	2	2	no	—
6	#16863147	—	2	2	no	—
7	#431×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	#322×C₆₇	C₄	1	4	yes	—
10	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
11	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
12	#14523954	—	2	4	no	—
13	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
14	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
15	C₄⋊C₈×C₆₇	—	2	2	no	—
16	C₂²⋊C₈×C₆₇	—	2	2	no	—
17	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
18	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₆₇	D₄	1	2	yes	—
20	C₂×C₆₇⋊C₈	—	4	4	no	—
21	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₆₇	#41	1	2	yes	—
23	C₂×C₈×C₆₇	—	2	4	no	—
24	C₂×C₈×C₆₇	—	4	2	no	—
25	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
26	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
27	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
28	C₄²×C₆₇	#41	1	2	yes	—
29	C₂³×C₆₇	#125	1	1	yes	characteristic
30	C₂×C₄×C₆₇	—	2	2	no	—
31	C₂×C₄×C₆₇	—	4	2	no	—
32	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	#117	1	2	yes	—
34	C₂×C₄×C₆₇	#124	1	2	yes	—
35	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
36	C₈×C₆₇	—	2	4	no	—
37	C₈×C₆₇	—	4	4	no	—
38	C₆₇⋊C₈	—	8	4	no	—
39	C₂²×C₆₇	—	2	2	no	—
40	C₂²×C₆₇	—	2	1	no	—
41	C₂²×C₆₇	#382	1	1	yes	characteristic
42	C₄×C₆₇	—	2	2	no	—
43	C₄×C₆₇	—	2	2	no	—
44	C₄×C₆₇	—	2	1	no	—
45	C₄×C₆₇	#430	1	2	yes	—
46	#8799	—	67	1	no	maximal, Sylow
47	C₂×C₆₇	—	4	1	no	—
48	C₂×C₆₇	#1833	1	2	yes	—
49	C₂×C₆₇	#1838	1	1	yes	characteristic
50	#1568	—	67	2	no	—
51	#1452	D₆₇	1	1	yes	characteristic
52	#1287	—	67	4	no	—
53	C₆₇	#8799	1	1	yes	characteristic, complemented, Sylow
54	#432	—	67	4	no	—
55	#431	C₂×D₆₇	1	2	yes	—
56	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
57	#335	—	134	2	no	—
58	#334	—	134	2	no	—
59	#322	Dic₆₇	1	4	yes	—
60	C₂²×C₈	Dic₁₃₄	1	2	yes	—
61	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	—	134	4	no	—
64	C₂²⋊C₈	—	2	2	no	—
65	C₂²⋊C₈	—	134	4	no	—
66	#97	—	134	4	no	—
67	C₄×C₈	D₂₆₈	1	2	yes	—
68	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
69	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
70	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
71	C₂²×C₄	#11761628	1	2	yes	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	4	2	no	—
74	C₂×C₈	—	268	4	no	—
75	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
76	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
77	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
78	C₄²	#11761628	1	2	yes	—
79	C₂³	#14524035	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
83	C₂×C₄	#14524002	1	2	yes	—
84	C₂×C₄	#14524031	1	2	yes	—
85	C₂×C₄	#14524035	1	1	yes	characteristic
86	C₈	—	2	4	no	—
87	C₈	—	4	4	no	—
88	C₈	—	536	4	no	—
89	C₂²	—	2	2	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#16863262	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	2	1	no	—
95	C₄	#16863445	1	2	yes	—
96	C₂	—	4	1	no	—
97	C₂	#22560333	1	2	yes	—
98	C₂	#22560350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22558972

C₂

yes

maximal

#22558071

C₂

yes

maximal

C₆₇×#1452

C₂

yes

maximal, characteristic

#16863453

C₂²

yes

—

#16863149

—

#16863147

—

#431×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₂²×C₈×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#125

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

#125

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#382

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#430

yes

—

#8799

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1833

yes

—

C₂×C₆₇

yes

—

D₆₇

yes

characteristic

#1287

—

C₆₇

#8799

yes

characteristic, complemented, Sylow

#432

—

#431

C₂×D₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#335

—

134

—

#334

—

134

—

#322

Dic₆₇

yes

—

C₂²×C₈

Dic₁₃₄

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

C₂²⋊C₈

—

134

—

#97

—

134

—

C₄×C₈

D₂₆₈

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

C₂²×C₄

#11761628

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

268

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₄²

C₂²×Dic₆₇

yes

characteristic

C₄²

C₂×D₂₆₈

yes

characteristic

C₄²

#11761628

yes

—

C₂³

#14524035

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

#14524002

yes

—

C₂×C₄

#14524031

yes

—

C₂×C₄

#14524035

yes

characteristic

C₈

—

C₈

—

C₈

—

536

—

C₂²

—

C₂²

—

C₂²

#16863262

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#16863445

yes

—

C₂

—

C₂

#22560333

yes

—

C₂

#22560350

yes

characteristic

	subgroup	G/H
center	C₂²	#16863262
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1452	C₂
socle	C₂²×C₆₇	#382
prime radical	C₂³×C₆₇	#125
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1452	D₆₇
hypocenter	C₆₇	#8799
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16863262

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1452

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
53. C₆₇	#8799	semidirect

H ≅ G/N

type

53. C₆₇

#8799

semidirect

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

subgroup	index
#22558972	2
#22558071	2
C₆₇×#1452	2
#8799	67

subgroup

index

#22558972

#22558071

C₆₇×#1452

#8799

Subgroups — #30749547

Conjugacy classes of subgroups

Complements

Maximal subgroups