Group #20261459 — subgroups

Subgroups — #20261459

Order 6208.

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

normal subgroups only

This group has 212 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	#15690805	C₂	1	4	yes	maximal, complemented
2	#15690799	C₂	1	2	yes	maximal
3	#15690790	C₂	1	2	yes	maximal, complemented
4	Q₈×Dic₉₇	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₃₈₈	C₂	1	1	yes	maximal, characteristic
7	#15690768	C₂	1	2	yes	maximal, complemented
8	C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic
10	C₂×D₃₈₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₉₇	C₂²	1	2	yes	—
12	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₉₇	C₂²	1	4	yes	—
15	#13095490	C₂²	1	2	yes	—
16	#13095488	C₂²	1	4	yes	—
17	#13095488	C₂²	1	2	yes	—
18	#13095488	C₂²	1	4	yes	—
19	C₄×Dic₉₇	C₂²	1	4	yes	—
20	C₄×Dic₉₇	C₂²	1	2	yes	—
21	C₂×Q₈×C₉₇	C₂²	1	2	yes	—
22	C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
24	C₄²×C₉₇	C₂²	1	1	yes	characteristic
25	C₂²×D₉₇	C₂³	1	2	yes	—
26	Q₈×C₉₇	—	2	4	no	—
27	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₉₇	C₂³	1	4	yes	—
29	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₉₇	C₂³	1	2	yes	—
32	C₂×Dic₉₇	C₂³	1	2	yes	—
33	C₂×Dic₉₇	C₂³	1	2	yes	—
34	D₃₈₈	—	2	2	no	—
35	C₄×D₉₇	—	2	4	no	—
36	Dic₁₉₄	—	2	2	no	—
37	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₉₇	—	2	2	no	—
39	C₂×D₉₇	—	2	2	no	—
40	C₂×D₉₇	—	2	2	no	—
41	C₄×C₉₇	—	2	4	no	—
42	C₄×C₉₇	—	2	1	no	—
43	C₄×C₉₇	#41	1	2	yes	—
44	C₄×C₉₇	#41	1	2	yes	—
45	Dic₉₇	—	2	2	no	—
46	Dic₉₇	—	2	2	no	—
47	Dic₉₇	—	2	2	no	—
48	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
49	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
50	C₂×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
51	D₉₇	—	4	2	no	—
52	C₉₇	#556	1	1	yes	characteristic, complemented, Sylow
53	#556	—	97	1	no	maximal, Sylow
54	C₄⋊Q₈	D₉₇	1	1	yes	characteristic, complemented
55	#126	—	97	4	no	—
56	#124	—	97	2	no	—
57	C₂²⋊Q₈	—	97	2	no	—
58	C₄×Q₈	—	97	2	no	—
59	C₄×Q₈	—	97	1	no	—
60	C₄×D₄	—	97	1	no	—
61	#117	—	97	2	no	—
62	C₂×Q₈	—	97	1	no	—
63	C₂×Q₈	C₂×D₉₇	1	2	yes	—
64	C₂×D₄	—	97	1	no	—
65	C₂²×C₄	—	97	2	no	—
66	C₄⋊C₄	—	97	2	no	—
67	C₄⋊C₄	—	97	1	no	—
68	C₄⋊C₄	—	97	1	no	—
69	C₄⋊C₄	—	97	4	no	—
70	C₄⋊C₄	C₂×D₉₇	1	2	yes	—
71	C₄⋊C₄	C₂×D₉₇	1	2	yes	—
72	C₂²⋊C₄	—	97	4	no	—
73	C₂²⋊C₄	—	97	2	no	—
74	C₂²⋊C₄	—	97	4	no	—
75	C₄²	—	97	4	no	—
76	C₄²	—	97	2	no	—
77	C₄²	C₂×D₉₇	1	1	yes	characteristic
78	C₂³	—	97	2	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	194	2	no	—
81	D₄	—	194	2	no	—
82	C₂×C₄	—	97	2	no	—
83	C₂×C₄	—	97	2	no	—
84	C₂×C₄	—	97	2	no	—
85	C₂×C₄	—	194	4	no	—
86	C₂×C₄	C₂²×D₉₇	1	4	yes	—
87	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
90	C₂²	—	194	2	no	—
91	C₂²	—	194	2	no	—
92	C₂²	—	194	2	no	—
93	C₂²	C₂³×D₉₇	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	2	1	no	—
96	C₄	—	194	2	no	—
97	C₄	—	194	2	no	—
98	C₄	—	194	2	no	—
99	C₄	#13095510	1	2	yes	—
100	C₄	#13095512	1	2	yes	—
101	C₂	—	388	2	no	—
102	C₂	C₂×#13095510	1	1	yes	characteristic
103	C₂	C₂×#13095512	1	1	yes	characteristic
104	C₂	#15690858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15690805

C₂

yes

maximal, complemented

#15690799

C₂

yes

maximal

#15690790

C₂

yes

maximal, complemented

Q₈×Dic₉₇

C₂

yes

maximal, complemented

C₄×Dic₁₉₄

C₂

yes

maximal, characteristic, complemented

C₄×D₃₈₈

C₂

yes

maximal, characteristic

#15690768

C₂

yes

maximal, complemented

C₄⋊Q₈×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₉₄

C₂²

yes

characteristic

C₂×D₃₈₈

C₂²

yes

characteristic

C₂×C₄×D₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

—

#13095490

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

—

C₂×Q₈×C₉₇

C₂²

yes

—

C₄⋊C₄×C₉₇

C₂²

yes

—

C₄⋊C₄×C₉₇

C₂²

yes

—

C₄²×C₉₇

C₂²

yes

characteristic

C₂²×D₉₇

C₂³

yes

—

Q₈×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

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

D₃₈₈

—

C₄×D₉₇

—

Dic₁₉₄

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#41

yes

—

C₄×C₉₇

#41

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

C₂×#41

yes

characteristic

C₂×C₉₇

C₂×#41

yes

characteristic

C₂×C₉₇

2₋¹⁺⁴

yes

characteristic

D₉₇

—

C₉₇

#556

yes

characteristic, complemented, Sylow

#556

—

maximal, Sylow

C₄⋊Q₈

D₉₇

yes

characteristic, complemented

#126

—

#124

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₉₇

yes

—

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

—

C₄⋊C₄

C₂×D₉₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₉₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

194

—

D₄

—

194

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

C₂²×D₉₇

yes

—

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₂²

—

194

—

C₂²

—

194

—

C₂²

—

194

—

C₂²

C₂³×D₉₇

yes

characteristic

C₄

—

C₄

—

C₄

—

194

—

C₄

—

194

—

C₄

—

194

—

C₄

#13095510

yes

—

100

C₄

#13095512

yes

—

101

C₂

—

388

—

102

C₂

C₂×#13095510

yes

characteristic

103

C₂

C₂×#13095512

yes

characteristic

104

C₂

#15690858

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₄⋊Q₈×C₉₇	C₂
hypercenter	C₄⋊Q₈	D₉₇
hypocenter	C₉₇	#556
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

C₄⋊Q₈×C₉₇

C₂

hypercenter

C₄⋊Q₈

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #15690805	C₂	semidirect
3. #15690790	C₂	semidirect
4. Q₈×Dic₉₇	C₂	semidirect
5. C₄×Dic₁₉₄	C₂	semidirect
7. #15690768	C₂	semidirect
8. C₄⋊Q₈×C₉₇	C₂	semidirect
52. C₉₇	#556	semidirect
54. C₄⋊Q₈	D₉₇	semidirect

H ≅ G/N

type

1. #15690805

C₂

semidirect

3. #15690790

C₂

semidirect

4. Q₈×Dic₉₇

C₂

semidirect

5. C₄×Dic₁₉₄

C₂

semidirect

7. #15690768

C₂

semidirect

8. C₄⋊Q₈×C₉₇

C₂

semidirect

52. C₉₇

#556

semidirect

54. C₄⋊Q₈

D₉₇

semidirect

Subgroups — #20261459

Conjugacy classes of subgroups

Complements