Group #30760340 — subgroups

Subgroups — #30760340

Order 17152.

350 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22560342	C₂	1	3	yes	maximal
2	#22558090	C₂	1	3	yes	maximal
3	C₆₇×#2845	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523954	C₂²	1	3	yes	—
5	#16863266	C₂²	1	1	yes	characteristic
6	#16863158	—	2	6	no	—
7	#398×C₆₇	—	2	6	no	—
8	#382×C₆₇	C₂²	1	3	yes	—
9	C₂²×C₆₇⋊C₈	—	2	3	no	—
10	C₂×C₄⋊Dic₆₇	—	2	3	no	—
11	#14523954	—	2	3	no	—
12	#14523954	Q₈	1	6	yes	—
13	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
14	C₂×C₄⋊C₄×C₆₇	D₄	1	6	yes	—
15	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
16	#95×C₆₇	—	2	2	no	—
17	#95×C₆₇	—	2	6	no	—
18	C₂²×Dic₆₇	—	4	1	no	—
19	C₂×C₆₇⋊C₈	—	2	6	no	—
20	C₂×C₆₇⋊C₈	—	4	3	no	—
21	C₂×C₆₇⋊C₈	—	4	3	no	—
22	C₄⋊Dic₆₇	—	4	6	no	—
23	C₂²×C₄×C₆₇	—	2	6	no	—
24	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
25	C₄⋊C₄×C₆₇	—	4	6	no	—
26	C₄⋊C₄×C₆₇	—	8	6	no	—
27	C₄²×C₆₇	C₂×Q₈	1	3	yes	—
28	C₄²×C₆₇	#41	1	1	yes	characteristic
29	C₂³×C₆₇	#120	1	1	yes	characteristic
30	C₂×C₄×C₆₇	—	2	3	no	—
31	C₂×C₄×C₆₇	—	2	3	no	—
32	C₂×C₄×C₆₇	—	4	6	no	—
33	C₂×C₄×C₆₇	—	4	6	no	—
34	C₂×C₄×C₆₇	—	4	6	no	—
35	C₂×C₄×C₆₇	C₂²⋊Q₈	1	3	yes	—
36	C₂×C₄×C₆₇	C₄⋊Q₈	1	3	yes	—
37	C₂×Dic₆₇	—	8	3	no	—
38	C₆₇⋊C₈	—	4	6	no	—
39	C₂²×C₆₇	#395	1	1	yes	characteristic
40	C₂²×C₆₇	#454	1	3	yes	—
41	C₂²×C₆₇	#455	1	3	yes	—
42	C₄×C₆₇	—	2	3	no	—
43	C₄×C₆₇	—	2	3	no	—
44	C₄×C₆₇	—	8	6	no	—
45	Dic₆₇	—	16	1	no	—
46	#10586	—	67	1	no	maximal, Sylow
47	C₂×C₆₇	#1658	1	1	yes	characteristic
48	C₂×C₆₇	#1662	1	3	yes	—
49	C₂×C₆₇	#2061	1	3	yes	—
50	#2845	D₆₇	1	1	yes	characteristic
51	#1836	—	67	3	no	—
52	#1295	—	67	3	no	—
53	C₆₇	#10586	1	1	yes	characteristic, complemented, Sylow
54	C₂×#97	—	67	3	no	—
55	#398	—	2	6	no	—
56	#384	—	67	1	no	—
57	#382	C₂×D₆₇	1	3	yes	—
58	#340	—	134	6	no	—
59	C₂²×C₈	—	134	3	no	—
60	C₂×C₄⋊C₄	—	2	6	no	—
61	C₂×C₄⋊C₄	—	134	3	no	—
62	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	6	yes	—
63	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
64	#97	—	67	6	no	—
65	#97	—	134	3	no	—
66	#95	—	2	2	no	—
67	#95	—	2	6	no	—
68	C₂²×C₄	—	2	6	no	—
69	C₂²×C₄	—	268	1	no	—
70	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
71	C₂×C₈	—	268	3	no	—
72	C₂×C₈	—	268	3	no	—
73	C₂×C₈	—	134	6	no	—
74	C₄⋊C₄	—	4	6	no	—
75	C₄⋊C₄	—	8	6	no	—
76	C₄⋊C₄	—	268	6	no	—
77	C₄²	Q₈×D₆₇	1	3	yes	—
78	C₄²	#11761629	1	1	yes	characteristic
79	C₂³	#14524012	1	1	yes	characteristic
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	4	6	no	—
85	C₂×C₄	—	536	3	no	—
86	C₂×C₄	#14524019	1	3	yes	—
87	C₂×C₄	#14524041	1	3	yes	—
88	C₈	—	268	6	no	—
89	C₂²	#16863315	1	1	yes	characteristic
90	C₂²	#16863569	1	3	yes	—
91	C₂²	#16863574	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	1072	1	no	—
96	C₂	#22559511	1	1	yes	characteristic
97	C₂	#22559532	1	3	yes	—
98	C₂	#22561556	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560342

C₂

yes

maximal

#22558090

C₂

yes

maximal

C₆₇×#2845

C₂

yes

maximal, characteristic

C₂×#14523954

C₂²

yes

—

#16863266

C₂²

yes

characteristic

#16863158

—

#398×C₆₇

—

#382×C₆₇

C₂²

yes

—

C₂²×C₆₇⋊C₈

—

C₂×C₄⋊Dic₆₇

—

#14523954

—

#14523954

Q₈

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#395

yes

characteristic

C₂²×C₆₇

#454

yes

—

C₂²×C₆₇

#455

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#2061

yes

—

#2845

D₆₇

yes

characteristic

#1836

—

#1295

—

C₆₇

#10586

yes

characteristic, complemented, Sylow

C₂×#97

—

#398

—

#384

—

#382

C₂×D₆₇

yes

—

#340

—

134

—

C₂²×C₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#97

—

#97

—

134

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

268

—

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

C₂×C₈

—

268

—

C₂×C₈

—

268

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄²

Q₈×D₆₇

yes

—

C₄²

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

536

—

C₂×C₄

#14524019

yes

—

C₂×C₄

#14524041

yes

—

C₈

—

268

—

C₂²

#16863315

yes

characteristic

C₂²

#16863569

yes

—

C₂²

#16863574

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

1072

—

C₂

#22559511

yes

characteristic

C₂

#22559532

yes

—

C₂

#22561556

yes

—

	subgroup	G/H
center	C₂³	#14524012
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2845	C₂
socle, prime radical	C₂³×C₆₇	#120
hypercenter	#2845	D₆₇
hypocenter	C₆₇	#10586
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#14524012

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2845

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
53. C₆₇	#10586	semidirect

H ≅ G/N

type

53. C₆₇

#10586

semidirect

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

subgroup	index
#22560342	2
#22558090	2
C₆₇×#2845	2
#10586	67

subgroup

index

#22560342

#22558090

C₆₇×#2845

#10586

Subgroups — #30760340

Conjugacy classes of subgroups

Complements

Maximal subgroups