Group #24924004 — subgroups

Subgroups — #24924004

Order 10688.

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

normal subgroups only

This group has 236 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	#19128829	C₂	1	1	yes	maximal, characteristic
2	#19128821	C₂	1	2	yes	maximal, complemented
3	#19128820	C₂	1	2	yes	maximal, complemented
4	#19128817	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₃₃₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₆₆₈	C₂	1	2	yes	maximal, complemented
7	#19128794	C₂	1	2	yes	maximal, complemented
8	C₄×Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₃₃₄	C₂²	1	1	yes	characteristic
10	C₂×D₆₆₈	C₂²	1	2	yes	—
11	C₂×D₆₆₈	C₂²	1	2	yes	—
12	C₂×D₆₆₈	C₂²	1	2	yes	—
13	C₂×C₄×D₁₆₇	C₂²	1	4	yes	—
14	C₄⋊Dic₁₆₇	C₂²	1	1	yes	characteristic
15	Dic₁₆₇⋊C₄	C₂²	1	2	yes	—
16	(C₂×D₁₆₇)⋊C₄	C₂²	1	4	yes	—
17	(C₂×D₁₆₇)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₆₇)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₁₆₇	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
23	C₄²×C₁₆₇	C₂²	1	2	yes	—
24	C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
25	C₂²×D₁₆₇	C₂³	1	4	yes	—
26	Dic₃₃₄	—	2	2	no	—
27	D₆₆₈	—	2	4	no	—
28	D₆₆₈	—	2	4	no	—
29	D₆₆₈	—	4	2	no	—
30	C₂×Dic₁₆₇	C₂³	1	2	yes	—
31	C₂×Dic₁₆₇	C₂³	1	2	yes	—
32	C₄×D₁₆₇	—	4	4	no	—
33	Q₈×C₁₆₇	—	2	2	no	—
34	C₂×C₄×C₁₆₇	C₂³	1	2	yes	—
35	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₆₇	C₂³	1	2	yes	—
37	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₁₆₇	C₂⁴	1	1	yes	characteristic
40	C₂×D₁₆₇	—	2	4	no	—
41	C₂×D₁₆₇	—	2	4	no	—
42	C₂×D₁₆₇	—	2	4	no	—
43	C₄×C₁₆₇	—	2	2	no	—
44	C₄×C₁₆₇	—	2	2	no	—
45	C₄×C₁₆₇	—	2	1	no	—
46	C₄×C₁₆₇	#41	1	2	yes	—
47	C₄×C₁₆₇	#41	1	2	yes	—
48	Dic₁₆₇	—	2	2	no	—
49	Dic₁₆₇	—	2	2	no	—
50	C₂×C₁₆₇	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₆₇	C₂×#41	1	1	yes	characteristic
52	C₂×C₁₆₇	H(5,𝔽₂)	1	1	yes	characteristic
53	D₁₆₇	—	4	4	no	—
54	C₁₆₇	#559	1	1	yes	characteristic, complemented, Sylow
55	#559	—	167	1	no	maximal, Sylow
56	#127	—	167	1	no	—
57	#124	—	167	2	no	—
58	#124	—	167	2	no	—
59	#123	—	167	4	no	—
60	C₄×Q₈	—	167	1	no	—
61	C₄×Q₈	D₁₆₇	1	1	yes	characteristic, complemented
62	C₄×D₄	—	167	2	no	—
63	C₄×D₄	—	167	2	no	—
64	C₂×Q₈	—	167	1	no	—
65	C₂×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
66	C₂×D₄	—	167	2	no	—
67	C₂×D₄	—	167	2	no	—
68	C₂×D₄	—	167	2	no	—
69	C₂²×C₄	—	167	4	no	—
70	C₄⋊C₄	—	167	1	no	—
71	C₄⋊C₄	—	167	2	no	—
72	C₄⋊C₄	C₂×D₁₆₇	1	1	yes	characteristic
73	C₄⋊C₄	C₂×D₁₆₇	1	2	yes	—
74	C₂²⋊C₄	—	167	4	no	—
75	C₂²⋊C₄	—	167	4	no	—
76	C₂²⋊C₄	—	167	4	no	—
77	C₄²	—	167	2	no	—
78	C₄²	C₂×D₁₆₇	1	2	yes	—
79	C₄²	C₂×D₁₆₇	1	1	yes	characteristic
80	C₂³	—	167	4	no	—
81	Q₈	—	2	2	no	—
82	Q₈	—	334	2	no	—
83	D₄	—	334	4	no	—
84	D₄	—	334	4	no	—
85	D₄	—	668	2	no	—
86	C₂×C₄	—	668	4	no	—
87	C₂×C₄	—	167	2	no	—
88	C₂×C₄	—	167	2	no	—
89	C₂×C₄	C₂²×D₁₆₇	1	2	yes	—
90	C₂×C₄	C₂²×D₁₆₇	1	2	yes	—
91	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
92	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
94	C₂²	—	334	4	no	—
95	C₂²	—	334	4	no	—
96	C₂²	—	334	4	no	—
97	C₂²	C₂³×D₁₆₇	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	334	2	no	—
102	C₄	—	334	2	no	—
103	C₄	#14581950	1	2	yes	—
104	C₄	#14581951	1	2	yes	—
105	C₂	—	668	4	no	—
106	C₂	C₂×#14581950	1	1	yes	characteristic
107	C₂	C₂×#14581951	1	1	yes	characteristic
108	C₂	#19128878	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128829

C₂

yes

maximal, characteristic

#19128821

C₂

yes

maximal, complemented

#19128820

C₂

yes

maximal, complemented

#19128817

C₂

yes

maximal, complemented

C₄×Dic₃₃₄

C₂

yes

maximal, characteristic, complemented

C₄×D₆₆₈

C₂

yes

maximal, complemented

#19128794

C₂

yes

maximal, complemented

C₄×Q₈×C₁₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₃₄

C₂²

yes

characteristic

C₂×D₆₆₈

C₂²

yes

—

C₂×D₆₆₈

C₂²

yes

—

C₂×D₆₆₈

C₂²

yes

—

C₂×C₄×D₁₆₇

C₂²

yes

—

C₄⋊Dic₁₆₇

C₂²

yes

characteristic

Dic₁₆₇⋊C₄

C₂²

yes

—

(C₂×D₁₆₇)⋊C₄

C₂²

yes

—

(C₂×D₁₆₇)⋊C₄

C₂²

yes

—

(C₂×D₁₆₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₆₇

C₂²

yes

—

C₂×Q₈×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₆₇

C₂²

yes

—

C₄²×C₁₆₇

C₂²

yes

—

C₄²×C₁₆₇

C₂²

yes

characteristic

C₂²×D₁₆₇

C₂³

yes

—

Dic₃₃₄

—

D₆₆₈

—

D₆₆₈

—

D₆₆₈

—

C₂×Dic₁₆₇

C₂³

yes

—

C₂×Dic₁₆₇

C₂³

yes

—

C₄×D₁₆₇

—

Q₈×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂³

yes

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

—

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₄×C₁₆₇

C₂³

yes

characteristic

C₂²×C₁₆₇

C₂⁴

yes

characteristic

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₂×D₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

#41

yes

—

C₄×C₁₆₇

#41

yes

—

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

C₂×#41

yes

characteristic

C₂×C₁₆₇

C₂×#41

yes

characteristic

C₂×C₁₆₇

H(5,𝔽₂)

yes

characteristic

D₁₆₇

—

C₁₆₇

#559

yes

characteristic, complemented, Sylow

#559

—

167

maximal, Sylow

#127

—

167

—

#124

—

167

—

#124

—

167

—

#123

—

167

—

C₄×Q₈

—

167

—

C₄×Q₈

D₁₆₇

yes

characteristic, complemented

C₄×D₄

—

167

—

C₄×D₄

—

167

—

C₂×Q₈

—

167

—

C₂×Q₈

C₂×D₁₆₇

yes

characteristic

C₂×D₄

—

167

—

C₂×D₄

—

167

—

C₂×D₄

—

167

—

C₂²×C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

—

167

—

C₄⋊C₄

C₂×D₁₆₇

yes

characteristic

C₄⋊C₄

C₂×D₁₆₇

yes

—

C₂²⋊C₄

—

167

—

C₂²⋊C₄

—

167

—

C₂²⋊C₄

—

167

—

C₄²

—

167

—

C₄²

C₂×D₁₆₇

yes

—

C₄²

C₂×D₁₆₇

yes

characteristic

C₂³

—

167

—

Q₈

—

Q₈

—

334

—

D₄

—

334

—

D₄

—

334

—

D₄

—

668

—

C₂×C₄

—

668

—

C₂×C₄

—

167

—

C₂×C₄

—

167

—

C₂×C₄

C₂²×D₁₆₇

yes

—

C₂×C₄

C₂²×D₁₆₇

yes

—

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₂×C₄

C₂²×D₁₆₇

yes

characteristic

C₂²

—

334

—

C₂²

—

334

—

C₂²

—

334

—

C₂²

C₂³×D₁₆₇

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

—

334

—

102

C₄

—

334

—

103

C₄

#14581950

yes

—

104

C₄

#14581951

yes

—

105

C₂

—

668

—

106

C₂

C₂×#14581950

yes

characteristic

107

C₂

C₂×#14581951

yes

characteristic

108

C₂

#19128878

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₆₇
commutator, socle, cosocle	C₂²×C₁₆₇	C₂⁴
Fitting	C₄×Q₈×C₁₆₇	C₂
prime radical	#19128829	C₂
hypercenter	C₄×Q₈	D₁₆₇
hypocenter	C₁₆₇	#559
LCS	G ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₆₇

commutator, socle, cosocle

C₂²×C₁₆₇

C₂⁴

Fitting

C₄×Q₈×C₁₆₇

C₂

prime radical

#19128829

C₂

hypercenter

C₄×Q₈

D₁₆₇

C₁₆₇

G ▷ C₂²×C₁₆₇ ▷ C₁₆₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
2. #19128821	C₂	semidirect
3. #19128820	C₂	semidirect
4. #19128817	C₂	semidirect
5. C₄×Dic₃₃₄	C₂	semidirect
6. C₄×D₆₆₈	C₂	semidirect
7. #19128794	C₂	semidirect
8. C₄×Q₈×C₁₆₇	C₂	semidirect
54. C₁₆₇	#559	semidirect
61. C₄×Q₈	D₁₆₇	semidirect

H ≅ G/N

type

2. #19128821

C₂

semidirect

3. #19128820

C₂

semidirect

4. #19128817

C₂

semidirect

5. C₄×Dic₃₃₄

C₂

semidirect

6. C₄×D₆₆₈

C₂

semidirect

7. #19128794

C₂

semidirect

8. C₄×Q₈×C₁₆₇

C₂

semidirect

54. C₁₆₇

#559

semidirect

61. C₄×Q₈

D₁₆₇

semidirect

Subgroups — #24924004

Conjugacy classes of subgroups

Complements