Group #20261406 — subgroups

Subgroups — #20261406

Order 6208.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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	2	yes	maximal, complemented
2	#15690807	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690806	C₂	1	2	yes	maximal, complemented
4	#15690784	C₂	1	2	yes	maximal, complemented
5	#15690783	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#15690772	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#13095511	—	2	8	no	—
10	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic
11	C₂×D₃₈₈	C₂²	1	2	yes	—
12	C₂×D₃₈₈	C₂²	1	2	yes	—
13	C₂×D₃₈₈	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₉₇	C₂²	1	4	yes	—
15	C₂×C₂²⋊D₉₇	C₂²	1	4	yes	—
16	C₂×C₄×D₉₇	C₂²	1	4	yes	—
17	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
18	#13095490	C₂²	1	2	yes	—
19	#13095488	C₂²	1	4	yes	—
20	C₄×Dic₉₇	C₂²	1	2	yes	—
21	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₉₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₉₇	C₂²	1	2	yes	—
25	C₄²×C₉₇	C₂²	1	1	yes	characteristic
26	C₂³×C₉₇	C₂³	1	2	yes	—
27	C₂²×D₉₇	C₂³	1	4	yes	—
28	D₄×C₉₇	—	2	2	no	—
29	C₂×C₄×C₉₇	—	2	4	no	—
30	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂³	1	2	yes	—
32	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
34	C₂²⋊D₉₇	—	2	8	no	—
35	C₂²⋊D₉₇	—	4	4	no	—
36	C₂×Dic₉₇	C₂³	1	2	yes	—
37	C₂×Dic₉₇	C₂³	1	2	yes	—
38	D₃₈₈	—	2	4	no	—
39	D₃₈₈	—	2	4	no	—
40	D₃₈₈	—	4	2	no	—
41	C₄×D₉₇	—	2	8	no	—
42	Dic₁₉₄	D₄	1	4	yes	complemented
43	C₂²×C₉₇	—	2	2	no	—
44	C₂²×C₉₇	—	2	2	no	—
45	C₂²×C₉₇	—	2	2	no	—
46	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
47	C₂×D₉₇	—	2	4	no	—
48	C₂×D₉₇	—	2	4	no	—
49	C₂×D₉₇	—	2	4	no	—
50	C₄×C₉₇	—	2	2	no	—
51	C₄×C₉₇	—	2	1	no	—
52	C₄×C₉₇	C₂×D₄	1	2	yes	—
53	C₄×C₉₇	#41	1	2	yes	complemented
54	Dic₉₇	—	2	2	no	—
55	Dic₉₇	C₂×D₄	1	4	yes	—
56	C₂×C₉₇	—	4	2	no	—
57	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
59	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
60	D₉₇	—	4	4	no	—
61	C₉₇	#550	1	1	yes	characteristic, complemented, Sylow
62	#550	—	97	1	no	maximal, Sylow
63	C₂×#41	—	97	2	no	—
64	#127	—	97	2	no	—
65	#127	—	97	1	no	—
66	#121	—	97	2	no	—
67	#121	—	97	4	no	—
68	C₄×Q₈	—	97	1	no	—
69	C₄×D₄	—	97	2	no	—
70	C₄×D₄	D₉₇	1	1	yes	characteristic, complemented
71	#41	—	194	8	no	—
72	C₂×Q₈	—	97	1	no	—
73	C₂×D₄	—	97	4	no	—
74	C₂×D₄	—	97	4	no	—
75	C₂×D₄	—	97	2	no	—
76	C₂×D₄	—	97	2	no	—
77	C₂×D₄	—	97	2	no	—
78	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
79	C₂²×C₄	—	97	4	no	—
80	C₂²×C₄	C₂×D₉₇	1	2	yes	—
81	C₄⋊C₄	—	97	2	no	—
82	C₄⋊C₄	—	97	1	no	—
83	C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
84	C₂²⋊C₄	—	97	4	no	—
85	C₂²⋊C₄	C₂×D₉₇	1	2	yes	—
86	C₄²	—	97	2	no	—
87	C₄²	C₂×D₉₇	1	1	yes	characteristic
88	C₂³	—	97	4	no	—
89	C₂³	C₂²×D₉₇	1	2	yes	—
90	Q₈	—	97	4	no	—
91	D₄	—	2	2	no	—
92	D₄	—	388	2	no	—
93	D₄	—	388	4	no	—
94	D₄	—	194	8	no	—
95	D₄	—	194	4	no	—
96	D₄	—	194	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	97	2	no	—
99	C₂×C₄	—	97	2	no	—
100	C₂×C₄	—	194	8	no	—
101	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₉₇	1	2	yes	—
103	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	194	4	no	—
109	C₂²	—	194	4	no	—
110	C₂²	—	194	4	no	—
111	C₂²	C₂³×D₉₇	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	97	4	no	—
115	C₄	—	194	2	no	—
116	C₄	D₄×D₉₇	1	2	yes	—
117	C₄	#13095511	1	2	yes	complemented
118	C₂	—	4	2	no	—
119	C₂	—	388	4	no	—
120	C₂	C₂×D₄×D₉₇	1	1	yes	characteristic
121	C₂	C₂×#13095511	1	1	yes	characteristic
122	C₂	#15690856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13095511

C₂

yes

maximal, complemented

#15690807

C₂

yes

maximal, characteristic, complemented

#15690806

C₂

yes

maximal, complemented

#15690784

C₂

yes

maximal, complemented

#15690783

C₂

yes

maximal, complemented

C₄×Dic₁₉₄

C₂

yes

maximal, characteristic, complemented

#15690772

C₂

yes

maximal, complemented

C₄×D₄×C₉₇

C₂

yes

maximal, characteristic, complemented

#13095511

—

C₂×Dic₁₉₄

C₂²

yes

characteristic

C₂×D₃₈₈

C₂²

yes

—

C₂×D₃₈₈

C₂²

yes

—

C₂×D₃₈₈

C₂²

yes

—

C₂×C₂²⋊D₉₇

C₂²

yes

—

C₂×C₂²⋊D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

—

C₄⋊Dic₉₇

C₂²

yes

characteristic

#13095490

C₂²

yes

—

#13095488

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

—

C₂²×D₉₇

C₂³

yes

—

D₄×C₉₇

—

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

C₂²⋊D₉₇

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

D₃₈₈

—

D₃₈₈

—

D₃₈₈

—

C₄×D₉₇

—

Dic₁₉₄

D₄

yes

complemented

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×D₄

yes

—

C₄×C₉₇

#41

yes

complemented

Dic₉₇

—

Dic₉₇

C₂×D₄

yes

—

C₂×C₉₇

—

C₂×C₉₇

C₂²×D₄

yes

characteristic

C₂×C₉₇

C₂×#41

yes

characteristic

C₂×C₉₇

H(5,𝔽₂)

yes

characteristic

D₉₇

—

C₉₇

#550

yes

characteristic, complemented, Sylow

#550

—

maximal, Sylow

C₂×#41

—

—

—

—

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₉₇

yes

characteristic, complemented

#41

—

194

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₉₇

yes

—

C₄²

—

C₄²

C₂×D₉₇

yes

characteristic

C₂³

—

C₂³

C₂²×D₉₇

yes

—

Q₈

—

D₄

—

D₄

—

388

—

D₄

—

388

—

D₄

—

194

—

D₄

—

194

—

D₄

—

194

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

194

—

101

C₂×C₄

C₂²×D₉₇

yes

characteristic

102

C₂×C₄

C₂²×D₉₇

yes

—

103

C₂×C₄

C₂²×D₉₇

yes

characteristic

104

C₂×C₄

C₂²×D₉₇

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

194

—

109

C₂²

—

194

—

110

C₂²

—

194

—

111

C₂²

C₂³×D₉₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

194

—

116

C₄

D₄×D₉₇

yes

—

117

C₄

#13095511

yes

complemented

118

C₂

—

119

C₂

—

388

—

120

C₂

C₂×D₄×D₉₇

yes

characteristic

121

C₂

C₂×#13095511

yes

characteristic

122

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	#550
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. C₂×#13095511	C₂	semidirect
2. #15690807	C₂	semidirect
3. #15690806	C₂	semidirect
4. #15690784	C₂	semidirect
5. #15690783	C₂	semidirect
6. C₄×Dic₁₉₄	C₂	semidirect
7. #15690772	C₂	semidirect
8. C₄×D₄×C₉₇	C₂	semidirect
42. Dic₁₉₄	D₄	semidirect
53. C₄×C₉₇	#41	semidirect
61. C₉₇	#550	semidirect
70. C₄×D₄	D₉₇	semidirect
117. C₄	#13095511	semidirect

H ≅ G/N

type

1. C₂×#13095511

C₂

semidirect

2. #15690807

C₂

semidirect

3. #15690806

C₂

semidirect

4. #15690784

C₂

semidirect

5. #15690783

C₂

semidirect

6. C₄×Dic₁₉₄

C₂

semidirect

7. #15690772

C₂

semidirect

8. C₄×D₄×C₉₇

C₂

semidirect

42. Dic₁₉₄

D₄

semidirect

53. C₄×C₉₇

#41

semidirect

61. C₉₇

#550

semidirect

70. C₄×D₄

D₉₇

semidirect

117. C₄

#13095511

semidirect

Subgroups — #20261406

Conjugacy classes of subgroups

Complements