Group #20261442 — subgroups

Subgroups — #20261442

Order 6208.

228 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 115 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	#15690810	C₂	1	1	yes	maximal, characteristic, complemented
2	#15690801	C₂	1	2	yes	maximal, complemented
3	#15690789	C₂	1	2	yes	maximal, complemented
4	#15690787	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₉₇	C₂	1	2	yes	maximal
7	C₂×C₄⋊Dic₉₇	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₉₄	C₂²	1	2	yes	—
11	C₂²×Dic₉₇	C₂²	1	4	yes	—
12	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₉₇	C₂²	1	4	yes	—
14	C₄⋊Dic₉₇	C₂²	1	2	yes	—
15	C₄⋊Dic₉₇	C₂²	1	2	yes	—
16	#13095490	C₂²	1	2	yes	—
17	#13095490	C₂²	1	4	yes	—
18	C₂²⋊Dic₉₇	C₂²	1	4	yes	—
19	C₄×Dic₉₇	C₂²	1	2	yes	—
20	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₉₇	C₂²	1	2	yes	—
22	C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
24	C₄²×C₉₇	C₂²	1	1	yes	characteristic
25	C₂³×C₉₇	C₂³	1	2	yes	—
26	D₄×C₉₇	Q₈	1	4	yes	complemented
27	C₂×C₄×C₉₇	—	2	4	no	—
28	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂³	1	2	yes	—
30	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₉₇	—	4	4	no	—
33	C₂×Dic₉₇	C₂³	1	2	yes	—
34	C₂×Dic₉₇	C₂³	1	2	yes	—
35	C₂×Dic₉₇	C₂³	1	4	yes	—
36	Dic₁₉₄	—	2	2	no	—
37	Dic₁₉₄	—	4	2	no	—
38	C₂²×C₉₇	—	2	2	no	—
39	C₂²×C₉₇	—	2	2	no	—
40	C₂²×C₉₇	C₂×Q₈	1	4	yes	—
41	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
42	C₄×C₉₇	—	2	2	no	—
43	C₄×C₉₇	—	2	1	no	—
44	C₄×C₉₇	C₂×Q₈	1	2	yes	—
45	C₄×C₉₇	#41	1	2	yes	—
46	Dic₉₇	—	2	2	no	—
47	Dic₉₇	—	2	4	no	—
48	Dic₉₇	—	2	2	no	—
49	C₂×C₉₇	—	2	4	no	—
50	C₂×C₉₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
52	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
53	C₉₇	#554	1	1	yes	characteristic, complemented, Sylow
54	#554	—	97	1	no	maximal, Sylow
55	C₄⋊Q₈	—	97	1	no	—
56	#125	—	97	2	no	—
57	C₂²⋊Q₈	—	97	2	no	—
58	C₂²⋊Q₈	—	97	4	no	—
59	C₄×Q₈	—	97	1	no	—
60	C₄×D₄	—	97	2	no	—
61	C₄×D₄	D₉₇	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	97	2	no	—
63	C₂×Q₈	—	97	2	no	—
64	C₂×Q₈	—	97	1	no	—
65	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
66	C₂²×C₄	—	97	4	no	—
67	C₂²×C₄	C₂×D₉₇	1	2	yes	—
68	C₄⋊C₄	—	97	2	no	—
69	C₄⋊C₄	—	97	4	no	—
70	C₄⋊C₄	—	97	1	no	—
71	C₄⋊C₄	—	97	4	no	—
72	C₄⋊C₄	—	97	2	no	—
73	C₄⋊C₄	—	97	2	no	—
74	C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
75	C₂²⋊C₄	—	97	4	no	—
76	C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
77	C₄²	—	97	2	no	—
78	C₄²	C₂×D₉₇	1	1	yes	characteristic
79	C₂³	C₂²×D₉₇	1	2	yes	—
80	Q₈	—	388	2	no	—
81	Q₈	—	194	2	no	—
82	D₄	Dic₁₉₄	1	4	yes	complemented
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	97	2	no	—
85	C₂×C₄	—	97	2	no	—
86	C₂×C₄	—	97	4	no	—
87	C₂×C₄	—	388	4	no	—
88	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₉₇	1	2	yes	—
90	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	—	2	2	no	—
94	C₂²	C₂×Dic₁₉₄	1	4	yes	—
95	C₂²	C₂³×D₉₇	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	1	no	—
98	C₄	—	194	2	no	—
99	C₄	—	194	2	no	—
100	C₄	—	194	4	no	—
101	C₄	C₂×Dic₁₉₄	1	2	yes	—
102	C₄	#13095510	1	2	yes	—
103	C₂	—	2	4	no	—
104	C₂	C₂²×Dic₁₉₄	1	1	yes	characteristic
105	C₂	C₂×#13095510	1	1	yes	characteristic
106	C₂	#15690856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15690810

C₂

yes

maximal, characteristic, complemented

#15690801

C₂

yes

maximal, complemented

#15690789

C₂

yes

maximal, complemented

#15690787

C₂

yes

maximal, complemented

C₄×Dic₁₉₄

C₂

yes

maximal, characteristic, complemented

D₄×Dic₉₇

C₂

yes

maximal

C₂×C₄⋊Dic₉₇

C₂

yes

maximal, complemented

C₄×D₄×C₉₇

C₂

yes

maximal, characteristic

C₂×Dic₁₉₄

C₂²

yes

characteristic

C₂×Dic₁₉₄

C₂²

yes

—

C₂²×Dic₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

—

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

C₂²⋊Dic₉₇

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

—

C₂×D₄×C₉₇

C₂²

yes

characteristic

C₂²×C₄×C₉₇

C₂²

yes

—

C₄⋊C₄×C₉₇

C₂²

yes

characteristic

C₂²⋊C₄×C₉₇

C₂²

yes

—

C₄²×C₉₇

C₂²

yes

characteristic

C₂³×C₉₇

C₂³

yes

—

D₄×C₉₇

Q₈

yes

complemented

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

Dic₁₉₄

—

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×Q₈

yes

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×Q₈

yes

—

C₄×C₉₇

#41

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂²×Q₈

yes

characteristic

C₂×C₉₇

C₂×#41

yes

C₂×C₉₇

yes

C₉₇

yes

characteristic, complemented, Sylow

#554

—

maximal, Sylow

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₉₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₉₇

yes

—

C₄²

—

C₄²

C₂×D₉₇

yes

characteristic

C₂³

C₂²×D₉₇

yes

—

Q₈

—

388

—

Q₈

—

194

—

D₄

Dic₁₉₄

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₄

C₂²×D₉₇

yes

—

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₁₉₄

yes

—

C₂²

C₂³×D₉₇

yes

characteristic

C₄

—

C₄

—

C₄

—

194

—

C₄

—

194

—

100

C₄

—

194

—

101

C₄

C₂×Dic₁₉₄

yes

—

102

C₄

#13095510

yes

—

103

C₂

—

104

C₂

C₂²×Dic₁₉₄

yes

characteristic

105

C₂

C₂×#13095510

yes

characteristic

106

C₂

#15690856

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₄×D₄×C₉₇	C₂
hypercenter	C₄×D₄	D₉₇
hypocenter	C₉₇	#554
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

C₄×D₄×C₉₇

C₂

hypercenter

C₄×D₄

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #15690810	C₂	semidirect
2. #15690801	C₂	semidirect
3. #15690789	C₂	semidirect
4. #15690787	C₂	semidirect
5. C₄×Dic₁₉₄	C₂	semidirect
7. C₂×C₄⋊Dic₉₇	C₂	semidirect
26. D₄×C₉₇	Q₈	semidirect
53. C₉₇	#554	semidirect
82. D₄	Dic₁₉₄	semidirect

H ≅ G/N

type

1. #15690810

C₂

semidirect

2. #15690801

C₂

semidirect

3. #15690789

C₂

semidirect

4. #15690787

C₂

semidirect

5. C₄×Dic₁₉₄

C₂

semidirect

7. C₂×C₄⋊Dic₉₇

C₂

semidirect

26. D₄×C₉₇

Q₈

semidirect

53. C₉₇

#554

semidirect

82. D₄

Dic₁₉₄

semidirect

Subgroups — #20261442

Conjugacy classes of subgroups

Complements