Group #24923819 — subgroups

Subgroups — #24923819

Order 10688.

310 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×#14581950	C₂	1	2	yes	maximal, complemented
2	#19128832	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128829	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128820	C₂	1	2	yes	maximal, complemented
5	#19128806	C₂	1	4	yes	maximal, complemented
6	C₄×D₆₆₈	C₂	1	4	yes	maximal, complemented
7	C₂×C₄²×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14581950	—	2	8	no	—
9	C₂×Dic₃₃₄	C₂²	1	2	yes	—
10	C₂×D₆₆₈	C₂²	1	2	yes	—
11	C₂×D₆₆₈	C₂²	1	4	yes	—
12	C₂×C₂²⋊D₁₆₇	C₂²	1	4	yes	—
13	C₂×C₄×D₁₆₇	C₂²	1	4	yes	—
14	C₄⋊Dic₁₆₇	C₂²	1	4	yes	—
15	(C₂×D₁₆₇)⋊C₄	C₂²	1	8	yes	—
16	C₂²×C₄×C₁₆₇	C₂²	1	2	yes	—
17	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
18	C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
19	C₄²×C₁₆₇	C₂²	1	2	yes	—
20	C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₆₇	C₂³	1	4	yes	—
22	Dic₃₃₄	—	2	4	no	—
23	D₆₆₈	—	2	4	no	—
24	D₆₆₈	—	2	8	no	—
25	C₂²⋊D₁₆₇	—	2	8	no	—
26	C₂×Dic₁₆₇	C₂³	1	4	yes	—
27	C₄×D₁₆₇	—	2	8	no	—
28	C₂³×C₁₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₆₇	—	2	4	no	—
30	C₂×C₄×C₁₆₇	D₄	1	4	yes	complemented
31	C₂×C₄×C₁₆₇	C₂³	1	2	yes	—
32	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₁₆₇	C₂³	1	2	yes	—
34	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
35	C₂²×C₁₆₇	—	2	2	no	—
36	C₂²×C₁₆₇	C₂×D₄	1	2	yes	—
37	C₂²×C₁₆₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₆₇	—	2	4	no	—
39	C₂×D₁₆₇	—	2	8	no	—
40	C₄×C₁₆₇	—	2	2	no	—
41	C₄×C₁₆₇	C₂×D₄	1	2	yes	—
42	C₄×C₁₆₇	C₂×D₄	1	2	yes	—
43	C₄×C₁₆₇	#41	1	4	yes	complemented
44	Dic₁₆₇	—	2	4	no	—
45	C₂×C₁₆₇	—	2	2	no	—
46	C₂×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
47	C₂×C₁₆₇	C₂×#41	1	2	yes	—
48	D₁₆₇	—	4	4	no	—
49	C₁₆₇	#532	1	1	yes	characteristic, complemented, Sylow
50	#532	—	167	1	no	maximal, Sylow
51	C₂×#41	—	167	2	no	—
52	C₄⋊Q₈	—	167	1	no	—
53	#127	—	167	1	no	—
54	#124	—	167	2	no	—
55	#121	—	167	4	no	—
56	C₄×D₄	—	167	4	no	—
57	C₂×C₄²	D₁₆₇	1	1	yes	characteristic, complemented
58	#41	—	334	8	no	—
59	C₂×Q₈	—	167	2	no	—
60	C₂×D₄	—	167	4	no	—
61	C₂×D₄	—	167	4	no	—
62	C₂×D₄	—	167	2	no	—
63	C₂²×C₄	—	167	4	no	—
64	C₂²×C₄	C₂×D₁₆₇	1	2	yes	—
65	C₂²×C₄	C₂×D₁₆₇	1	1	yes	characteristic
66	C₄⋊C₄	—	167	4	no	—
67	C₂²⋊C₄	—	167	8	no	—
68	C₄²	C₂×D₁₆₇	1	1	yes	characteristic
69	C₄²	C₂×D₁₆₇	1	1	yes	characteristic
70	C₄²	C₂×D₁₆₇	1	2	yes	—
71	C₂³	—	167	4	no	—
72	C₂³	C₂²×D₁₆₇	1	1	yes	characteristic
73	Q₈	—	334	4	no	—
74	D₄	—	334	8	no	—
75	D₄	—	334	4	no	—
76	D₄	—	334	8	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	334	8	no	—
79	C₂×C₄	—	167	4	no	—
80	C₂×C₄	D₆₆₈	1	4	yes	complemented
81	C₂×C₄	C₂²×D₁₆₇	1	2	yes	—
82	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
83	C₂×C₄	C₂²×D₁₆₇	1	2	yes	—
84	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
85	C₂²	—	2	2	no	—
86	C₂²	—	334	4	no	—
87	C₂²	—	334	8	no	—
88	C₂²	C₂×D₆₆₈	1	2	yes	—
89	C₂²	C₂³×D₁₆₇	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	334	4	no	—
92	C₄	C₂×D₆₆₈	1	2	yes	—
93	C₄	C₂×D₆₆₈	1	2	yes	—
94	C₄	#14581950	1	4	yes	complemented
95	C₂	—	2	2	no	—
96	C₂	—	668	4	no	—
97	C₂	C₂²×D₆₆₈	1	1	yes	characteristic
98	C₂	C₂×#14581950	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581950

C₂

yes

maximal, complemented

#19128832

C₂

yes

maximal, characteristic, complemented

#19128829

C₂

yes

maximal, characteristic, complemented

#19128820

C₂

yes

maximal, complemented

#19128806

C₂

yes

maximal, complemented

C₄×D₆₆₈

C₂

yes

maximal, complemented

C₂×C₄²×C₁₆₇

C₂

yes

maximal, characteristic, complemented

#14581950

—

C₂×Dic₃₃₄

C₂²

yes

—

C₂×D₆₆₈

C₂²

yes

—

C₂×D₆₆₈

C₂²

yes

—

C₂×C₂²⋊D₁₆₇

C₂²

yes

—

C₂×C₄×D₁₆₇

C₂²

yes

—

C₄⋊Dic₁₆₇

C₂²

yes

—

(C₂×D₁₆₇)⋊C₄

C₂²

yes

—

C₂²×C₄×C₁₆₇

C₂²

yes

—

C₂²×C₄×C₁₆₇

C₂²

yes

characteristic

C₄²×C₁₆₇

C₂²

yes

characteristic

C₄²×C₁₆₇

C₂²

yes

—

C₄²×C₁₆₇

C₂²

yes

characteristic

C₂²×D₁₆₇

C₂³

yes

—

Dic₃₃₄

—

D₆₆₈

—

D₆₆₈

—

C₂²⋊D₁₆₇

—

C₂×Dic₁₆₇

C₂³

yes

—

C₄×D₁₆₇

—

C₂³×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

D₄

yes

complemented

C₂×C₄×C₁₆₇

C₂³

yes

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂×D₄

yes

—

C₂²×C₁₆₇

C₂⁴

yes

characteristic

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×D₄

yes

—

C₄×C₁₆₇

C₂×D₄

yes

—

C₄×C₁₆₇

#41

yes

complemented

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

C₂²×D₄

yes

characteristic

C₂×C₁₆₇

C₂×#41

yes

—

D₁₆₇

—

C₁₆₇

#532

yes

characteristic, complemented, Sylow

#532

—

167

maximal, Sylow

C₂×#41

—

167

—

C₄⋊Q₈

—

167

—

#127

—

167

—

#124

—

167

—

#121

—

167

—

C₄×D₄

—

167

—

C₂×C₄²

D₁₆₇

yes

characteristic, complemented

#41

—

334

—

C₂×Q₈

—

167

—

C₂×D₄

—

167

—

C₂×D₄

—

167

—

C₂×D₄

—

167

—

C₂²×C₄

—

167

—

C₂²×C₄

C₂×D₁₆₇

yes

—

C₂²×C₄

C₂×D₁₆₇

yes

characteristic

C₄⋊C₄

—

167

—

C₂²⋊C₄

—

167

—

C₄²

C₂×D₁₆₇

yes

characteristic

C₄²

C₂×D₁₆₇

yes

characteristic

C₄²

C₂×D₁₆₇

yes

—

C₂³

—

167

—

C₂³

C₂²×D₁₆₇

yes

characteristic

Q₈

—

334

—

D₄

—

334

—

D₄

—

334

—

D₄

—

334

—

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

167

—

C₂×C₄

D₆₆₈

yes

complemented

C₂×C₄

C₂²×D₁₆₇

yes

—

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₂×C₄

C₂²×D₁₆₇

yes

—

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₂²

—

C₂²

—

334

—

C₂²

—

334

—

C₂²

C₂×D₆₆₈

yes

—

C₂²

C₂³×D₁₆₇

yes

characteristic

C₄

—

C₄

—

334

—

C₄

C₂×D₆₆₈

yes

—

C₄

C₂×D₆₆₈

yes

—

C₄

#14581950

yes

complemented

C₂

—

C₂

—

668

—

C₂

C₂²×D₆₆₈

yes

characteristic

C₂

C₂×#14581950

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₆₇
commutator, socle, cosocle	C₂²×C₁₆₇	C₂⁴
Frattini	C₂²	C₂³×D₁₆₇
Fitting	C₂×C₄²×C₁₆₇	C₂
hypercenter	C₂×C₄²	D₁₆₇
hypocenter	C₁₆₇	#532
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂×C₄

C₂²×D₁₆₇

commutator, socle, cosocle

C₂²×C₁₆₇

C₂⁴

Frattini

C₂²

C₂³×D₁₆₇

Fitting

C₂×C₄²×C₁₆₇

C₂

hypercenter

C₂×C₄²

D₁₆₇

C₁₆₇

G ▷ C₂²×C₁₆₇ ▷ C₁₆₇

C₁ ◁ C₂×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
1. C₂×#14581950	C₂	semidirect
2. #19128832	C₂	semidirect
3. #19128829	C₂	semidirect
4. #19128820	C₂	semidirect
5. #19128806	C₂	semidirect
6. C₄×D₆₆₈	C₂	semidirect
7. C₂×C₄²×C₁₆₇	C₂	semidirect
30. C₂×C₄×C₁₆₇	D₄	semidirect
43. C₄×C₁₆₇	#41	semidirect
49. C₁₆₇	#532	semidirect
57. C₂×C₄²	D₁₆₇	semidirect
80. C₂×C₄	D₆₆₈	semidirect
94. C₄	#14581950	semidirect

H ≅ G/N

type

1. C₂×#14581950

C₂

semidirect

2. #19128832

C₂

semidirect

3. #19128829

C₂

semidirect

4. #19128820

C₂

semidirect

5. #19128806

C₂

semidirect

6. C₄×D₆₆₈

C₂

semidirect

7. C₂×C₄²×C₁₆₇

C₂

semidirect

30. C₂×C₄×C₁₆₇

D₄

semidirect

43. C₄×C₁₆₇

#41

semidirect

49. C₁₆₇

#532

semidirect

57. C₂×C₄²

D₁₆₇

semidirect

80. C₂×C₄

D₆₆₈

semidirect

94. C₄

#14581950

semidirect

Subgroups — #24923819

Conjugacy classes of subgroups

Complements