Group #22561973 — subgroups

Subgroups — #22561973

Order 8576.

152 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 75 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	M₃₂×D₆₇	C₂	1	2	yes	maximal
2	#16863350	C₂	1	1	yes	maximal, characteristic
3	#16863217	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863216	C₂	1	2	yes	maximal, complemented
5	#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761611	C₄	1	1	yes	characteristic
7	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
8	#14524000	C₄	1	1	yes	characteristic
9	#14523983	C₂²	1	2	yes	—
10	#14523982	—	2	2	no	—
11	C₁₆×D₆₇	—	2	2	no	—
12	#14523954	C₄	1	1	yes	characteristic
13	#14523953	C₄	1	1	yes	characteristic
14	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
15	M₃₂×C₆₇	C₂²	1	2	yes	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
18	#11761611	C₈	1	4	yes	—
19	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₈×D₆₇	D₄	1	2	yes	—
22	C₈×D₆₇	Q₈	1	2	yes	—
23	Dic₆₇⋊C₄	—	2	1	no	—
24	(C₂×D₆₇)⋊C₄	—	2	1	no	—
25	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
26	C₆₇⋊C₁₆	—	2	2	no	—
27	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₆₇	—	2	2	no	—
31	C₂²×D₆₇	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₆₇	—	2	1	no	—
33	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₆₇	—	2	1	no	—
35	C₂×Dic₆₇	M₁₆	1	1	yes	characteristic
36	C₄×D₆₇	C₄⋊C₄	1	2	yes	—
37	C₄×D₆₇	C₂×C₈	1	2	yes	—
38	C₈×C₆₇	C₂×C₈	1	2	yes	—
39	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
42	C₆₇⋊C₈	C₂×C₈	1	2	yes	—
43	C₂²×C₆₇	C₂×M₁₆	1	1	yes	characteristic
44	C₂×D₆₇	—	4	1	no	—
45	C₂×D₆₇	C₄⋊C₈	1	2	yes	—
46	C₄×C₆₇	—	4	1	no	—
47	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
49	Dic₆₇	—	4	1	no	—
50	Dic₆₇	C₄⋊C₈	1	2	yes	—
51	C₂×C₆₇	—	2	1	no	—
52	C₂×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
53	D₆₇	—	2	2	no	—
54	#2153	—	67	1	no	maximal, Sylow
55	C₆₇	#2153	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₃₂	—	67	2	no	—
57	#405	—	67	1	no	—
58	#364	—	67	2	no	—
59	#364	—	67	1	no	—
60	#364	D₆₇	1	1	yes	characteristic, complemented
61	C₂×M₁₆	—	67	1	no	—
62	C₂²×C₈	—	67	1	no	—
63	#117	—	67	1	no	—
64	M₃₂	—	67	2	no	—
65	M₃₂	—	134	2	no	—
66	M₃₂	C₂×D₆₇	1	2	yes	—
67	C₂×C₁₆	—	134	2	no	—
68	#97	—	67	1	no	—
69	#97	—	67	1	no	—
70	C₄×C₈	—	67	1	no	—
71	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
72	C₂²×C₄	—	67	1	no	—
73	M₁₆	—	67	4	no	—
74	C₂×C₈	—	67	1	no	—
75	C₂×C₈	—	67	1	no	—
76	C₂×C₈	—	67	2	no	—
77	C₂×C₈	—	67	2	no	—
78	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
79	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
80	C₄⋊C₄	—	134	1	no	—
81	C₂²⋊C₄	—	134	1	no	—
82	C₄²	—	67	1	no	—
83	C₄²	C₄×D₆₇	1	1	yes	characteristic
84	C₁₆	—	2	2	no	—
85	C₁₆	—	134	2	no	—
86	C₂³	—	67	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	67	1	no	—
89	C₂×C₄	—	67	2	no	—
90	C₂×C₄	—	67	2	no	—
91	C₂×C₄	—	134	1	no	—
92	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
93	C₈	—	67	2	no	—
94	C₈	—	67	2	no	—
95	C₈	C₈×D₆₇	1	2	yes	—
96	C₈	D₄×D₆₇	1	1	yes	characteristic
97	C₈	Q₈×D₆₇	1	1	yes	characteristic
98	C₂²	—	268	1	no	—
99	C₂²	—	67	2	no	—
100	C₂²	M₁₆×D₆₇	1	1	yes	characteristic
101	C₄	—	4	1	no	—
102	C₄	—	268	1	no	—
103	C₄	—	67	2	no	—
104	C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
105	C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	134	2	no	—
108	C₂	C₄⋊C₈×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×D₆₇

C₂

yes

maximal

#16863350

C₂

yes

maximal, characteristic

#16863217

C₂

yes

maximal, characteristic, complemented

#16863216

C₂

yes

maximal, complemented

#364×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761611

C₄

yes

characteristic

C₂×C₈×D₆₇

C₂²

yes

characteristic

#14524000

C₄

yes

characteristic

#14523983

C₂²

yes

—

#14523982

—

C₁₆×D₆₇

—

#14523954

C₄

yes

characteristic

#14523953

C₄

yes

characteristic

C₈×Dic₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×D₆₇

C₂×C₄

yes

characteristic

#11761611

C₈

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₆₇

D₄

yes

—

C₈×D₆₇

Q₈

yes

—

Dic₆₇⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

C₂×C₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₁₆×C₆₇

—

C₂²×D₆₇

M₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

M₁₆

yes

characteristic

C₄×D₆₇

C₄⋊C₄

yes

—

C₄×D₆₇

C₂×C₈

yes

—

C₈×C₆₇

C₂×C₈

yes

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₆₇⋊C₈

C₄⋊C₄

yes

—

C₆₇⋊C₈

C₂×C₈

yes

—

C₂²×C₆₇

C₂×M₁₆

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

C₄⋊C₈

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₂²×C₈

yes

characteristic

Dic₆₇

—

Dic₆₇

C₄⋊C₈

yes

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄⋊C₈

yes

characteristic

D₆₇

—

#2153

—

maximal, Sylow

C₆₇

#2153

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

—

—

—

D₆₇

yes

characteristic, complemented

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

134

—

M₃₂

C₂×D₆₇

yes

—

C₂×C₁₆

—

134

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₄²

—

C₄²

C₄×D₆₇

yes

characteristic

C₁₆

—

C₁₆

—

134

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₈

—

C₈

—

C₈

C₈×D₆₇

yes

—

C₈

D₄×D₆₇

yes

characteristic

C₈

Q₈×D₆₇

yes

characteristic

C₂²

—

268

—

C₂²

—

100

C₂²

M₁₆×D₆₇

yes

characteristic

101

C₄

—

102

C₄

—

268

—

103

C₄

—

104

C₄

C₄⋊C₄×D₆₇

yes

characteristic

105

C₄

C₂×C₈×D₆₇

yes

characteristic

106

C₂

—

107

C₂

—

134

—

108

C₂

C₄⋊C₈×D₆₇

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#364×C₆₇	C₂
socle	C₂×C₆₇	C₂×C₄⋊C₈
hypercenter	#364	D₆₇
hypocenter	C₆₇	#2153
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×D₆₇

commutator

C₄×C₆₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#364×C₆₇

C₂

socle

C₂×C₆₇

C₂×C₄⋊C₈

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
3. #16863217	C₂	semidirect
4. #16863216	C₂	semidirect
5. #364×C₆₇	C₂	semidirect
55. C₆₇	#2153	semidirect
60. #364	D₆₇	semidirect

H ≅ G/N

type

3. #16863217

C₂

semidirect

4. #16863216

C₂

semidirect

5. #364×C₆₇

C₂

semidirect

55. C₆₇

#2153

semidirect

60. #364

D₆₇

semidirect

Subgroups — #22561973

Conjugacy classes of subgroups

Complements