Group #22562429 — subgroups

Subgroups — #22562429

Order 8576.

124 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16863654	C₂	1	1	yes	maximal, characteristic
2	#16863445	C₂	1	1	yes	maximal, characteristic
3	#16863219	C₂	1	2	yes	maximal, complemented
4	#16863213	C₂	1	2	yes	maximal, complemented
5	#349×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×#11761612	—	2	1	no	—
8	#14524022	—	2	1	no	—
9	#14523979	—	2	1	no	—
10	#14523979	C₂²	1	2	yes	—
11	#14523975	C₂²	1	1	yes	characteristic
12	#14523968	—	2	1	no	—
13	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	M₃₂×C₆₇	C₂²	1	2	yes	—
15	C₂×Dic₁₃₄	—	2	1	no	—
16	Dic₂₆₈	—	2	2	no	—
17	#11761612	—	2	2	no	—
18	#11761612	—	4	1	no	—
19	C₂×C₆₇⋊C₈	—	4	1	no	—
20	C₄⋊Dic₆₇	—	2	1	no	—
21	Dic₆₇⋊C₄	—	4	1	no	—
22	C₂²⋊Dic₆₇	—	4	1	no	—
23	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
24	M₁₆×C₆₇	—	2	2	no	—
25	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₁₆×C₆₇	—	2	2	no	—
28	C₂³×C₆₇	SD₁₆	1	1	yes	characteristic
29	C₂×C₄×C₆₇	—	2	1	no	—
30	C₂×C₄×C₆₇	—	2	1	no	—
31	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₆₇	—	4	1	no	—
34	C₂×Dic₆₇	—	4	1	no	—
35	Dic₁₃₄	—	4	1	no	—
36	Dic₁₃₄	—	4	1	no	—
37	C₈×C₆₇	—	2	1	no	—
38	C₈×C₆₇	#41	1	2	yes	—
39	C₆₇⋊C₈	—	4	2	no	—
40	C₂²×C₆₇	—	2	1	no	—
41	C₂²×C₆₇	—	4	1	no	—
42	C₂²×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
43	C₄×C₆₇	—	2	1	no	—
44	C₄×C₆₇	#123	1	1	yes	characteristic
45	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
46	Dic₆₇	—	8	1	no	—
47	Dic₆₇	—	8	1	no	—
48	C₂×C₆₇	—	2	1	no	—
49	C₂×C₆₇	—	4	1	no	—
50	C₂×C₆₇	#481	1	1	yes	characteristic
51	#2239	—	67	1	no	maximal, Sylow
52	C₆₇	#2239	1	1	yes	characteristic, complemented, Sylow
53	#470	—	67	1	no	—
54	#430	—	67	1	no	—
55	#365	—	67	2	no	—
56	#362	—	67	2	no	—
57	#349	D₆₇	1	1	yes	characteristic
58	C₂×Q₁₆	—	67	1	no	—
59	C₂×M₁₆	—	134	1	no	—
60	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
61	C₂²⋊Q₈	—	134	1	no	—
62	M₃₂	C₂×D₆₇	1	2	yes	—
63	#108	—	67	2	no	—
64	#108	—	134	1	no	—
65	#106	—	67	1	no	—
66	#103	—	134	1	no	—
67	C₂×Q₈	—	134	1	no	—
68	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
69	Q₁₆	—	134	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	268	1	no	—
72	M₁₆	—	134	2	no	—
73	C₂×C₈	—	268	1	no	—
74	C₂×C₈	D₂₆₈	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
76	C₄⋊C₄	—	268	1	no	—
77	C₄⋊C₄	—	134	1	no	—
78	C₂²⋊C₄	—	268	1	no	—
79	C₁₆	—	2	2	no	—
80	C₂³	#11761617	1	1	yes	characteristic
81	Q₈	—	268	1	no	—
82	Q₈	—	268	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	268	1	no	—
86	C₂×C₄	—	268	1	no	—
87	C₂×C₄	#11761617	1	1	yes	characteristic
88	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	268	2	no	—
91	C₈	#11761627	1	2	yes	—
92	C₂²	—	2	1	no	—
93	C₂²	—	4	1	no	—
94	C₂²	C₂×#11761617	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	536	1	no	—
97	C₄	—	536	1	no	—
98	C₄	#14524027	1	1	yes	characteristic
99	C₄	#14524071	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	4	1	no	—
102	C₂	#16863715	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863654

C₂

yes

maximal, characteristic

#16863445

C₂

yes

maximal, characteristic

#16863219

C₂

yes

maximal, complemented

#16863213

C₂

yes

maximal, complemented

#349×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

—

—

C₂²

yes

—

#14523975

C₂²

yes

characteristic

#14523968

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

Dic₂₆₈

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

#41

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×M₁₆

—

134

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²⋊Q₈

—

134

—

M₃₂

C₂×D₆₇

yes

—

#108

—

#108

—

134

—

#106

—

#103

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

D₂₆₈

yes

characteristic

Q₁₆

—

134

—

M₁₆

—

M₁₆

—

268

—

M₁₆

—

134

—

C₂×C₈

—

268

—

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

268

—

C₁₆

—

C₂³

#11761617

yes

characteristic

Q₈

—

268

—

Q₈

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

#11761617

yes

characteristic

C₂×C₄

C₂×D₂₆₈

yes

characteristic

C₈

—

C₈

—

268

—

C₈

#11761627

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#11761617

yes

characteristic

C₄

—

C₄

—

536

—

C₄

—

536

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

—

102

C₂

#16863715

yes

characteristic

	subgroup	G/H
center	C₂	#16863715
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#349×C₆₇	C₂
socle	C₂×C₆₇	#481
hypercenter	#349	D₆₇
hypocenter	C₆₇	#2239
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

subgroup

G/H

center

C₂

#16863715

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#349×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

N	H ≅ G/N	type
3. #16863219	C₂	semidirect
4. #16863213	C₂	semidirect
52. C₆₇	#2239	semidirect

H ≅ G/N

type

3. #16863219

C₂

semidirect

4. #16863213

C₂

semidirect

52. C₆₇

#2239

semidirect

Subgroups — #22562429

Conjugacy classes of subgroups

Complements