Group #20261400 — subgroups

Subgroups — #20261400

Order 6208.

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

normal subgroups only

This group has 280 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	C₂×Q₈×D₉₇	C₂	1	2	yes	maximal, complemented
2	#15690809	C₂	1	2	yes	maximal, complemented
3	#15690788	C₂	1	4	yes	maximal, complemented
4	#15690786	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	#15690772	C₂	1	2	yes	maximal
8	C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₁₉₄	C₂²	1	4	yes	—
10	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic
11	Q₈×D₉₇	—	2	8	no	—
12	C₂×D₃₈₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₉₇	C₂²	1	4	yes	—
14	C₂×C₄×D₉₇	C₂²	1	2	yes	—
15	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
17	#13095490	C₂²	1	2	yes	—
18	#13095490	C₂²	1	4	yes	—
19	#13095488	C₂²	1	4	yes	—
20	#13095488	C₂²	1	2	yes	—
21	C₄×Dic₉₇	C₂²	1	2	yes	—
22	C₂×Q₈×C₉₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
24	C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
25	C₄²×C₉₇	C₂²	1	1	yes	characteristic
26	C₂²×D₉₇	C₂³	1	2	yes	—
27	Q₈×C₉₇	—	2	4	no	—
28	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂³	1	4	yes	—
30	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₉₇	C₂³	1	2	yes	—
33	C₂×Dic₉₇	C₂³	1	2	yes	—
34	C₂×Dic₉₇	C₂³	1	2	yes	—
35	D₃₈₈	Q₈	1	4	yes	complemented
36	C₄×D₉₇	—	2	4	no	—
37	C₄×D₉₇	—	2	8	no	—
38	Dic₁₉₄	—	2	8	no	—
39	Dic₁₉₄	D₄	1	4	yes	complemented
40	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₉₇	—	2	2	no	—
42	C₂×D₉₇	—	2	2	no	—
43	C₂×D₉₇	C₂×Q₈	1	4	yes	complemented
44	C₄×C₉₇	—	2	4	no	—
45	C₄×C₉₇	—	2	1	no	—
46	C₄×C₉₇	C₂×D₄	1	2	yes	—
47	C₄×C₉₇	C₂×Q₈	1	2	yes	complemented
48	Dic₉₇	—	2	2	no	—
49	Dic₉₇	—	2	2	no	—
50	Dic₉₇	C₂×D₄	1	4	yes	—
51	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₉₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
54	D₉₇	—	2	4	no	—
55	C₉₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
56	D₄×Q₈	—	97	1	no	maximal, Sylow
57	C₂²×Q₈	—	97	2	no	—
58	C₄⋊Q₈	—	97	2	no	—
59	C₄⋊Q₈	D₉₇	1	1	yes	characteristic, complemented
60	C₂²⋊Q₈	—	97	2	no	—
61	C₂²⋊Q₈	—	97	4	no	—
62	C₄×Q₈	—	97	1	no	—
63	C₄×D₄	—	97	2	no	—
64	C₄×D₄	—	97	1	no	—
65	C₂×Q₈	—	97	1	no	—
66	C₂×Q₈	—	97	4	no	—
67	C₂×Q₈	—	194	8	no	—
68	C₂×Q₈	C₂×D₉₇	1	2	yes	—
69	C₂×D₄	—	97	1	no	—
70	C₂²×C₄	—	97	4	no	—
71	C₂²×C₄	—	97	2	no	—
72	C₄⋊C₄	—	97	2	no	—
73	C₄⋊C₄	—	97	4	no	—
74	C₄⋊C₄	—	97	1	no	—
75	C₄⋊C₄	—	97	1	no	—
76	C₄⋊C₄	C₂×D₉₇	1	2	yes	—
77	C₄⋊C₄	C₂×D₉₇	1	2	yes	—
78	C₂²⋊C₄	—	97	4	no	—
79	C₂²⋊C₄	—	97	2	no	—
80	C₄²	—	97	2	no	—
81	C₄²	C₂×D₉₇	1	1	yes	characteristic
82	C₂³	—	97	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	97	4	no	—
85	Q₈	—	194	8	no	—
86	D₄	—	97	4	no	—
87	C₂×C₄	—	97	2	no	—
88	C₂×C₄	—	97	2	no	—
89	C₂×C₄	—	97	2	no	—
90	C₂×C₄	—	194	8	no	—
91	C₂×C₄	—	194	4	no	—
92	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₉₇	1	4	yes	—
94	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
96	C₂²	—	97	4	no	—
97	C₂²	—	194	2	no	—
98	C₂²	—	194	2	no	—
99	C₂²	C₂³×D₉₇	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	1	no	—
102	C₄	—	97	4	no	—
103	C₄	—	194	2	no	—
104	C₄	—	194	2	no	—
105	C₄	D₄×D₉₇	1	2	yes	—
106	C₄	Q₈×D₉₇	1	2	yes	complemented
107	C₂	—	194	4	no	—
108	C₂	C₂×D₄×D₉₇	1	1	yes	characteristic
109	C₂	C₂×Q₈×D₉₇	1	1	yes	characteristic
110	C₂	#15690858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₉₇

C₂

yes

maximal, complemented

#15690809

C₂

yes

maximal, complemented

#15690788

C₂

yes

maximal, complemented

#15690786

C₂

yes

maximal, complemented

C₄×Dic₁₉₄

C₂

yes

maximal, characteristic, complemented

C₄×D₃₈₈

C₂

yes

maximal, characteristic

#15690772

C₂

yes

maximal

C₄⋊Q₈×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₉₄

C₂²

yes

—

C₂×Dic₁₉₄

C₂²

yes

characteristic

Q₈×D₉₇

—

C₂×D₃₈₈

C₂²

yes

characteristic

C₂×C₄×D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

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

Q₈

yes

complemented

C₄×D₉₇

—

C₄×D₉₇

—

Dic₁₉₄

—

Dic₁₉₄

D₄

yes

complemented

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

C₂×Q₈

yes

complemented

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×D₄

yes

—

C₄×C₉₇

C₂×Q₈

yes

complemented

Dic₉₇

—

Dic₉₇

—

Dic₉₇

C₂×D₄

yes

—

C₂×C₉₇

C₂²×D₄

yes

characteristic

C₂×C₉₇

C₂²×Q₈

yes

characteristic

C₂×C₉₇

2₋¹⁺⁴

yes

characteristic

D₉₇

—

C₉₇

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

maximal, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₉₇

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

C₂×D₉₇

yes

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

—

C₄⋊C₄

C₂×D₉₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₉₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

194

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

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

—

194

—

C₂²

—

194

—

C₂²

C₂³×D₉₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

194

—

104

C₄

—

194

—

105

C₄

D₄×D₉₇

yes

—

106

C₄

Q₈×D₉₇

yes

complemented

107

C₂

—

194

—

108

C₂

C₂×D₄×D₉₇

yes

characteristic

109

C₂

C₂×Q₈×D₉₇

yes

characteristic

110

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	D₄×Q₈
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

hypocenter

C₉₇

D₄×Q₈

LCS

G ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₉₇	C₂	semidirect
2. #15690809	C₂	semidirect
3. #15690788	C₂	semidirect
4. #15690786	C₂	semidirect
5. C₄×Dic₁₉₄	C₂	semidirect
8. C₄⋊Q₈×C₉₇	C₂	semidirect
35. D₃₈₈	Q₈	semidirect
39. Dic₁₉₄	D₄	semidirect
43. C₂×D₉₇	C₂×Q₈	semidirect
47. C₄×C₉₇	C₂×Q₈	semidirect
55. C₉₇	D₄×Q₈	semidirect
59. C₄⋊Q₈	D₉₇	semidirect
106. C₄	Q₈×D₉₇	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₉₇

C₂

semidirect

2. #15690809

C₂

semidirect

3. #15690788

C₂

semidirect

4. #15690786

C₂

semidirect

5. C₄×Dic₁₉₄

C₂

semidirect

8. C₄⋊Q₈×C₉₇

C₂

semidirect

35. D₃₈₈

Q₈

semidirect

39. Dic₁₉₄

D₄

semidirect

43. C₂×D₉₇

C₂×Q₈

semidirect

47. C₄×C₉₇

C₂×Q₈

semidirect

55. C₉₇

D₄×Q₈

semidirect

59. C₄⋊Q₈

D₉₇

semidirect

106. C₄

Q₈×D₉₇

semidirect

Subgroups — #20261400

Conjugacy classes of subgroups

Complements