Group #20261298 — subgroups

Subgroups — #20261298

Order 6208.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#13095510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690806	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690798	C₂	1	1	yes	maximal, characteristic, complemented
5	#15690797	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690782	C₂	1	4	yes	maximal, complemented
7	#15690778	C₂	1	4	yes	maximal, complemented
8	#15690767	C₂	1	1	yes	maximal, characteristic, complemented
9	#127×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₉₇	C₂²	1	2	yes	—
11	#13095510	—	2	4	no	—
12	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₃₈₈	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₉₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₉₇	C₂²	1	4	yes	complemented
17	D₄×D₉₇	—	2	4	no	—
18	C₂²×Dic₉₇	C₂²	1	2	yes	complemented
19	C₂×C₄×D₉₇	C₂²	1	1	yes	characteristic
20	#13095490	C₂²	1	2	yes	complemented
21	C₂²⋊Dic₉₇	C₂²	1	4	yes	complemented
22	#13095488	C₂²	1	2	yes	—
23	#13095488	C₂²	1	4	yes	complemented
24	C₄×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
25	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×C₉₇	C₂²	1	4	yes	complemented
28	C₄²×C₉₇	C₂²	1	1	yes	characteristic, complemented
29	C₂³×C₉₇	C₂³	1	2	yes	—
30	C₂³×C₉₇	C₂³	1	2	yes	—
31	C₂²×D₉₇	—	2	4	no	—
32	C₂²×D₉₇	—	4	4	no	—
33	C₂²×D₉₇	C₂³	1	2	yes	—
34	C₂²×D₉₇	C₂³	1	1	yes	characteristic
35	D₄×C₉₇	—	2	2	no	—
36	D₄×C₉₇	—	2	2	no	—
37	D₄×C₉₇	—	4	4	no	—
38	C₂×C₄×C₉₇	C₂³	1	2	yes	—
39	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
40	C₂²⋊D₉₇	—	2	4	no	—
41	C₂²⋊D₉₇	—	2	4	no	—
42	C₂²⋊D₉₇	—	4	4	no	—
43	C₂×Dic₉₇	—	2	4	no	—
44	C₂×Dic₉₇	C₂³	1	2	yes	—
45	C₂×Dic₉₇	C₂³	1	2	yes	—
46	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
47	D₃₈₈	—	2	2	no	—
48	C₄×D₉₇	D₄	1	4	yes	complemented
49	Dic₁₉₄	—	2	2	no	—
50	C₂²×C₉₇	—	2	2	no	—
51	C₂²×C₉₇	—	2	2	no	—
52	C₂²×C₉₇	—	2	4	no	—
53	C₂²×C₉₇	—	2	4	no	—
54	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
55	C₂×D₉₇	—	2	2	no	—
56	C₂×D₉₇	—	2	2	no	—
57	C₂×D₉₇	—	2	4	no	—
58	C₂×D₉₇	—	4	8	no	—
59	C₂×D₉₇	C₂×D₄	1	2	yes	—
60	C₄×C₉₇	—	2	2	no	—
61	C₄×C₉₇	C₂×D₄	1	2	yes	—
62	Dic₉₇	—	2	2	no	—
63	Dic₉₇	—	2	2	no	—
64	Dic₉₇	C₂×D₄	1	2	yes	—
65	C₂×C₉₇	—	4	2	no	—
66	C₂×C₉₇	—	4	2	no	—
67	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₉₇	H(5,𝔽₂)	1	2	yes	—
69	D₉₇	—	2	2	no	—
70	D₉₇	—	4	2	no	—
71	C₉₇	#535	1	1	yes	characteristic, complemented, Sylow
72	#535	—	97	1	no	maximal, Sylow
73	C₂×#41	—	97	1	no	—
74	C₂²×D₄	—	97	1	no	—
75	#127	—	97	1	no	—
76	#127	D₉₇	1	1	yes	characteristic, complemented
77	#124	—	97	1	no	—
78	#124	—	97	1	no	—
79	#121	—	97	4	no	—
80	#120	—	97	4	no	—
81	#117	—	97	1	no	—
82	C₂⁴	—	97	2	no	—
83	#41	—	194	4	no	—
84	C₂×Q₈	—	97	1	no	—
85	C₂×D₄	—	97	4	no	—
86	C₂×D₄	—	97	2	no	—
87	C₂×D₄	—	97	2	no	—
88	C₂×D₄	—	97	1	no	—
89	C₂×D₄	—	194	4	no	—
90	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
91	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic, complemented
92	C₂×D₄	C₂×D₉₇	1	4	yes	complemented
93	C₂²×C₄	—	97	1	no	—
94	C₂²×C₄	—	97	2	no	—
95	C₄⋊C₄	—	97	2	no	—
96	C₂²⋊C₄	—	97	4	no	—
97	C₂²⋊C₄	—	97	2	no	—
98	C₂²⋊C₄	—	97	4	no	—
99	C₄²	—	97	1	no	—
100	C₄²	C₂×D₉₇	1	1	yes	characteristic, complemented
101	C₂³	—	97	1	no	—
102	C₂³	—	97	2	no	—
103	C₂³	—	388	4	no	—
104	C₂³	—	194	4	no	—
105	C₂³	C₂²×D₉₇	1	2	yes	—
106	C₂³	C₂²×D₉₇	1	2	yes	—
107	Q₈	—	194	2	no	—
108	D₄	—	2	2	no	—
109	D₄	—	2	2	no	—
110	D₄	—	4	4	no	—
111	D₄	—	388	4	no	—
112	D₄	—	194	4	no	—
113	D₄	—	194	4	no	—
114	D₄	—	194	2	no	—
115	C₂×C₄	—	97	1	no	—
116	C₂×C₄	—	97	4	no	—
117	C₂×C₄	—	97	2	no	—
118	C₂×C₄	—	97	2	no	—
119	C₂×C₄	—	194	4	no	—
120	C₂×C₄	C₂²×D₉₇	1	2	yes	—
121	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	97	2	no	—
127	C₂²	—	388	8	no	—
128	C₂²	—	194	2	no	—
129	C₂²	—	194	4	no	—
130	C₂²	—	194	2	no	—
131	C₂²	C₂³×D₉₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	97	2	no	—
134	C₄	—	194	2	no	—
135	C₄	—	194	2	no	—
136	C₄	D₄×D₉₇	1	2	yes	—
137	C₂	—	4	2	no	—
138	C₂	—	4	2	no	—
139	C₂	—	388	2	no	—
140	C₂	—	194	2	no	—
141	C₂	C₂×D₄×D₉₇	1	1	yes	characteristic
142	C₂	#15690855	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13095510

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₉₇

C₂

yes

maximal, characteristic, complemented

#15690806

C₂

yes

maximal, characteristic, complemented

#15690798

C₂

yes

maximal, characteristic, complemented

#15690797

C₂

yes

maximal, characteristic, complemented

#15690782

C₂

yes

maximal, complemented

#15690778

C₂

yes

maximal, complemented

#15690767

C₂

yes

maximal, characteristic, complemented

#127×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₉₇

C₂²

yes

—

#13095510

—

C₂×Dic₁₉₄

C₂²

yes

characteristic, complemented

C₂×D₃₈₈

C₂²

yes

characteristic

C₂×C₂²⋊D₉₇

C₂²

yes

—

C₂×C₂²⋊D₉₇

C₂²

yes

—

C₂×C₂²⋊D₉₇

C₂²

yes

complemented

D₄×D₉₇

—

C₂²×Dic₉₇

C₂²

yes

complemented

C₂×C₄×D₉₇

C₂²

yes

characteristic

#13095490

C₂²

yes

complemented

C₂²⋊Dic₉₇

C₂²

yes

complemented

#13095488

C₂²

yes

—

#13095488

C₂²

yes

complemented

C₄×Dic₉₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₉₇

C₂²

yes

characteristic

C₂×D₄×C₉₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₉₇

C₂²

yes

complemented

C₄²×C₉₇

C₂²

yes

characteristic, complemented

C₂³×C₉₇

C₂³

yes

—

C₂³×C₉₇

C₂³

yes

—

C₂²×D₉₇

—

C₂²×D₉₇

—

C₂²×D₉₇

C₂³

yes

—

C₂²×D₉₇

C₂³

yes

characteristic

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂²⋊D₉₇

—

C₂²⋊D₉₇

—

C₂²⋊D₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

characteristic

D₃₈₈

—

C₄×D₉₇

D₄

yes

complemented

Dic₁₉₄

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

C₂×D₄

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₂×D₄

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

C₂×D₄

yes

—

C₂×C₉₇

—

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

—

#127

—

#127

D₉₇

yes

characteristic, complemented

—

—

—

—

—

C₂⁴

—

#41

—

194

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

194

—

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂×D₄

C₂×D₉₇

yes

characteristic, complemented

C₂×D₄

C₂×D₉₇

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

C₂×D₉₇

yes

characteristic, complemented

101

C₂³

—

102

C₂³

—

103

C₂³

—

388

—

104

C₂³

—

194

—

105

C₂³

C₂²×D₉₇

yes

—

106

C₂³

C₂²×D₉₇

yes

—

107

Q₈

—

194

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

388

—

112

D₄

—

194

—

113

D₄

—

194

—

114

D₄

—

194

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

194

—

120

C₂×C₄

C₂²×D₉₇

yes

—

121

C₂×C₄

C₂²×D₉₇

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

388

—

128

C₂²

—

194

—

129

C₂²

—

194

—

130

C₂²

—

194

—

131

C₂²

C₂³×D₉₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

194

—

135

C₄

—

194

—

136

C₄

D₄×D₉₇

yes

—

137

C₂

—

138

C₂

—

139

C₂

—

388

—

140

C₂

—

194

—

141

C₂

C₂×D₄×D₉₇

yes

characteristic

142

C₂

#15690855

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	#127×C₉₇	C₂
hypercenter	#127	D₉₇
hypocenter	C₉₇	#535
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×D₉₇

commutator, socle

C₂²×C₉₇

C₂⁴

Fitting

#127×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₉₇

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. C₂×#13095510	C₂	semidirect
2. C₂×D₄×D₉₇	C₂	semidirect
3. #15690806	C₂	semidirect
4. #15690798	C₂	semidirect
5. #15690797	C₂	semidirect
6. #15690782	C₂	semidirect
7. #15690778	C₂	semidirect
8. #15690767	C₂	semidirect
9. #127×C₉₇	C₂	semidirect
12. C₂×Dic₁₉₄	C₂²	semidirect
16. C₂×C₂²⋊D₉₇	C₂²	semidirect
18. C₂²×Dic₉₇	C₂²	semidirect
20. #13095490	C₂²	semidirect
21. C₂²⋊Dic₉₇	C₂²	semidirect
23. #13095488	C₂²	semidirect
24. C₄×Dic₉₇	C₂²	semidirect
26. C₂×D₄×C₉₇	C₂²	semidirect
27. C₂×D₄×C₉₇	C₂²	semidirect
28. C₄²×C₉₇	C₂²	semidirect
48. C₄×D₉₇	D₄	semidirect
71. C₉₇	#535	semidirect
76. #127	D₉₇	semidirect
91. C₂×D₄	C₂×D₉₇	semidirect
92. C₂×D₄	C₂×D₉₇	semidirect
100. C₄²	C₂×D₉₇	semidirect

H ≅ G/N

type

1. C₂×#13095510

C₂

semidirect

2. C₂×D₄×D₉₇

C₂

semidirect

3. #15690806

C₂

semidirect

4. #15690798

C₂

semidirect

5. #15690797

C₂

semidirect

6. #15690782

C₂

semidirect

7. #15690778

C₂

semidirect

8. #15690767

C₂

semidirect

9. #127×C₉₇

C₂

semidirect

12. C₂×Dic₁₉₄

C₂²

semidirect

16. C₂×C₂²⋊D₉₇

C₂²

semidirect

18. C₂²×Dic₉₇

C₂²

semidirect

20. #13095490

C₂²

semidirect

21. C₂²⋊Dic₉₇

C₂²

semidirect

23. #13095488

C₂²

semidirect

24. C₄×Dic₉₇

C₂²

semidirect

26. C₂×D₄×C₉₇

C₂²

semidirect

27. C₂×D₄×C₉₇

C₂²

semidirect

28. C₄²×C₉₇

C₂²

semidirect

48. C₄×D₉₇

D₄

semidirect

71. C₉₇

#535

semidirect

76. #127

D₉₇

semidirect

91. C₂×D₄

C₂×D₉₇

semidirect

92. C₂×D₄

C₂×D₉₇

semidirect

100. C₄²

C₂×D₉₇

semidirect

Subgroups — #20261298

Conjugacy classes of subgroups

Complements