Group #20261304 — subgroups

Subgroups — #20261304

Order 6208.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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×#13095511	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690807	C₂	1	2	yes	maximal, complemented
4	#15690798	C₂	1	2	yes	maximal, complemented
5	#15690785	C₂	1	4	yes	maximal, complemented
6	#15690780	C₂	1	4	yes	maximal, complemented
7	#117×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₉₇	C₂²	1	2	yes	—
9	#13095511	—	2	4	no	—
10	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₃₈₈	—	2	4	no	—
12	C₂×D₃₈₈	C₂²	1	2	yes	—
13	C₂×D₃₈₈	C₂²	1	2	yes	—
14	C₂×D₃₈₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₃₈₈	C₂²	1	8	yes	complemented
16	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
17	C₂×C₄×D₉₇	C₂²	1	2	yes	complemented
18	#13095488	C₂²	1	8	yes	complemented
19	C₂²×C₄×C₉₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₉₇	C₂²	1	2	yes	complemented
21	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
22	C₄²×C₉₇	C₂²	1	2	yes	complemented
23	C₂³×C₉₇	C₂³	1	1	yes	characteristic
24	C₂²×D₉₇	—	2	4	no	—
25	C₂²×D₉₇	—	4	4	no	—
26	C₂²×D₉₇	C₂³	1	4	yes	—
27	C₂²×D₉₇	C₂³	1	2	yes	—
28	C₂×C₄×C₉₇	D₄	1	4	yes	complemented
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	C₂²⋊D₉₇	—	2	4	no	—
33	C₂×Dic₉₇	C₂³	1	2	yes	—
34	D₃₈₈	—	2	4	no	—
35	D₃₈₈	—	2	4	no	—
36	D₃₈₈	—	2	2	no	—
37	D₃₈₈	—	4	8	no	—
38	C₄×D₉₇	—	2	4	no	—
39	Dic₁₉₄	—	2	2	no	—
40	C₂²×C₉₇	—	2	2	no	—
41	C₂²×C₉₇	C₂×D₄	1	2	yes	—
42	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
43	C₂×D₉₇	—	2	2	no	—
44	C₂×D₉₇	—	2	4	no	—
45	C₂×D₉₇	—	2	4	no	—
46	C₂×D₉₇	—	2	8	no	—
47	C₂×D₉₇	—	4	8	no	—
48	C₄×C₉₇	—	2	4	no	—
49	C₄×C₉₇	C₂×D₄	1	2	yes	—
50	C₄×C₉₇	C₂×D₄	1	2	yes	—
51	Dic₉₇	—	2	2	no	—
52	C₂×C₉₇	—	2	2	no	—
53	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₉₇	H(5,𝔽₂)	1	2	yes	—
55	D₉₇	—	4	2	no	—
56	D₉₇	—	4	4	no	—
57	C₉₇	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	97	1	no	maximal, Sylow
59	C₂×#41	—	97	1	no	—
60	C₂²×D₄	—	97	1	no	—
61	#127	—	97	2	no	—
62	#124	—	97	2	no	—
63	#121	—	97	4	no	—
64	#120	—	97	4	no	—
65	#117	D₉₇	1	1	yes	characteristic, complemented
66	C₂⁴	—	97	2	no	—
67	#41	—	194	4	no	—
68	C₂×Q₈	—	97	1	no	—
69	C₂×D₄	—	97	2	no	—
70	C₂×D₄	—	97	8	no	—
71	C₂×D₄	—	97	2	no	—
72	C₂×D₄	—	97	2	no	—
73	C₂×D₄	—	97	1	no	—
74	C₂×D₄	—	194	4	no	—
75	C₂²×C₄	—	97	2	no	—
76	C₂²×C₄	C₂×D₉₇	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₉₇	1	2	yes	complemented
78	C₂²⋊C₄	—	97	8	no	—
79	C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
80	C₄²	C₂×D₉₇	1	2	yes	complemented
81	C₂³	—	97	4	no	—
82	C₂³	—	97	2	no	—
83	C₂³	—	388	4	no	—
84	C₂³	—	194	4	no	—
85	C₂³	C₂²×D₉₇	1	1	yes	characteristic
86	Q₈	—	194	2	no	—
87	D₄	—	388	8	no	—
88	D₄	—	194	4	no	—
89	D₄	—	194	4	no	—
90	D₄	—	194	4	no	—
91	D₄	—	194	2	no	—
92	C₂×C₄	—	97	2	no	—
93	C₂×C₄	—	194	4	no	—
94	C₂×C₄	D₃₈₈	1	4	yes	complemented
95	C₂×C₄	C₂²×D₉₇	1	4	yes	—
96	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	388	8	no	—
100	C₂²	—	194	2	no	—
101	C₂²	—	194	4	no	—
102	C₂²	—	194	4	no	—
103	C₂²	—	194	8	no	—
104	C₂²	C₂×D₃₈₈	1	2	yes	—
105	C₂²	C₂³×D₉₇	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	194	2	no	—
108	C₄	C₂×D₃₈₈	1	2	yes	—
109	C₄	C₂×D₃₈₈	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	388	4	no	—
112	C₂	—	388	2	no	—
113	C₂	C₂²×D₃₈₈	1	1	yes	characteristic
114	C₂	#15690856	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13095511

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₈₈

C₂

yes

maximal, characteristic, complemented

#15690807

C₂

yes

maximal, complemented

#15690798

C₂

yes

maximal, complemented

#15690785

C₂

yes

maximal, complemented

#15690780

C₂

yes

maximal, complemented

#117×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₉₇

C₂²

yes

—

#13095511

—

C₂×Dic₁₉₄

C₂²

yes

characteristic, complemented

C₂×D₃₈₈

—

C₂×D₃₈₈

C₂²

yes

—

C₂×D₃₈₈

C₂²

yes

—

C₂×D₃₈₈

C₂²

yes

characteristic, complemented

C₂×D₃₈₈

C₂²

yes

complemented

C₂×C₂²⋊D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

complemented

#13095488

C₂²

yes

complemented

C₂²×C₄×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

C₂²

yes

complemented

C₂²⋊C₄×C₉₇

C₂²

yes

—

C₄²×C₉₇

C₂²

yes

complemented

C₂³×C₉₇

C₂³

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

C₂³

yes

—

C₂²×D₉₇

C₂³

yes

—

C₂×C₄×C₉₇

D₄

yes

complemented

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂²⋊D₉₇

—

C₂×Dic₉₇

C₂³

yes

—

D₃₈₈

—

D₃₈₈

—

D₃₈₈

—

D₃₈₈

—

C₄×D₉₇

—

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×D₄

yes

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×D₄

yes

—

C₄×C₉₇

C₂×D₄

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂²×D₄

yes

characteristic

C₂×C₉₇

H(5,𝔽₂)

yes

—

D₉₇

—

D₉₇

—

C₉₇

#535

yes

characteristic, complemented, Sylow

#535

—

maximal, Sylow

C₂×#41

—

C₂²×D₄

—

—

—

—

—

D₉₇

yes

characteristic, complemented

C₂⁴

—

#41

—

194

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

characteristic

C₄⋊C₄

C₂×D₉₇

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₉₇

yes

—

C₄²

C₂×D₉₇

yes

complemented

C₂³

—

C₂³

—

C₂³

—

388

—

C₂³

—

194

—

C₂³

C₂²×D₉₇

yes

characteristic

Q₈

—

194

—

D₄

—

388

—

D₄

—

194

—

D₄

—

194

—

D₄

—

194

—

D₄

—

194

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

D₃₈₈

yes

complemented

C₂×C₄

C₂²×D₉₇

yes

—

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₂²

—

C₂²

—

388

—

100

C₂²

—

194

—

101

C₂²

—

194

—

102

C₂²

—

194

—

103

C₂²

—

194

—

104

C₂²

C₂×D₃₈₈

yes

—

105

C₂²

C₂³×D₉₇

yes

characteristic

106

C₄

—

107

C₄

—

194

—

108

C₄

C₂×D₃₈₈

yes

—

109

C₄

C₂×D₃₈₈

yes

—

110

C₂

—

111

C₂

—

388

—

112

C₂

—

388

—

113

C₂

C₂²×D₃₈₈

yes

characteristic

114

C₂

#15690856

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	#117×C₉₇	C₂
hypercenter	#117	D₉₇
hypocenter	C₉₇	#535
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

#117×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#13095511	C₂	semidirect
2. C₂²×D₃₈₈	C₂	semidirect
3. #15690807	C₂	semidirect
4. #15690798	C₂	semidirect
5. #15690785	C₂	semidirect
6. #15690780	C₂	semidirect
7. #117×C₉₇	C₂	semidirect
10. C₂×Dic₁₉₄	C₂²	semidirect
14. C₂×D₃₈₈	C₂²	semidirect
15. C₂×D₃₈₈	C₂²	semidirect
17. C₂×C₄×D₉₇	C₂²	semidirect
18. #13095488	C₂²	semidirect
20. C₄⋊C₄×C₉₇	C₂²	semidirect
22. C₄²×C₉₇	C₂²	semidirect
28. C₂×C₄×C₉₇	D₄	semidirect
57. C₉₇	#535	semidirect
65. #117	D₉₇	semidirect
77. C₄⋊C₄	C₂×D₉₇	semidirect
80. C₄²	C₂×D₉₇	semidirect
94. C₂×C₄	D₃₈₈	semidirect

H ≅ G/N

type

1. C₂×#13095511

C₂

semidirect

2. C₂²×D₃₈₈

C₂

semidirect

3. #15690807

C₂

semidirect

4. #15690798

C₂

semidirect

5. #15690785

C₂

semidirect

6. #15690780

C₂

semidirect

7. #117×C₉₇

C₂

semidirect

10. C₂×Dic₁₉₄

C₂²

semidirect

14. C₂×D₃₈₈

C₂²

semidirect

15. C₂×D₃₈₈

C₂²

semidirect

17. C₂×C₄×D₉₇

C₂²

semidirect

18. #13095488

C₂²

semidirect

20. C₄⋊C₄×C₉₇

C₂²

semidirect

22. C₄²×C₉₇

C₂²

semidirect

28. C₂×C₄×C₉₇

D₄

semidirect

57. C₉₇

#535

semidirect

65. #117

D₉₇

semidirect

77. C₄⋊C₄

C₂×D₉₇

semidirect

80. C₄²

C₂×D₉₇

semidirect

94. C₂×C₄

D₃₈₈

semidirect

Subgroups — #20261304

Conjugacy classes of subgroups

Complements