Group #20261310 — subgroups

Subgroups — #20261310

Order 6208.

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

normal subgroups only

This group has 270 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×#13095512	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690808	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690799	C₂	1	1	yes	maximal, characteristic, complemented
5	#15690798	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690795	C₂	1	4	yes	maximal, complemented
7	#15690788	C₂	1	4	yes	maximal, complemented
8	#15690767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#13095512	—	2	4	no	—
11	C₂×Dic₁₉₄	C₂²	1	2	yes	—
12	C₂×Dic₁₉₄	C₂²	1	1	yes	characteristic
13	Q₈×D₉₇	—	2	4	no	—
14	C₂×D₃₈₈	C₂²	1	2	yes	—
15	C₂×D₃₈₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₉₇	C₂²	1	2	yes	—
17	C₂×C₄×D₉₇	C₂²	1	2	yes	—
18	C₂×C₄×D₉₇	C₂²	1	1	yes	characteristic
19	#13095490	C₂²	1	2	yes	—
20	#13095490	C₂²	1	4	yes	—
21	#13095488	C₂²	1	2	yes	—
22	#13095488	C₂²	1	4	yes	—
23	#13095488	C₂²	1	4	yes	—
24	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
28	C₄²×C₉₇	C₂²	1	1	yes	characteristic
29	C₂²×D₉₇	C₂³	1	2	yes	—
30	C₂²×D₉₇	C₂³	1	1	yes	characteristic
31	Q₈×C₉₇	—	2	2	no	—
32	Q₈×C₉₇	—	2	2	no	—
33	C₂×C₄×C₉₇	C₂³	1	2	yes	—
34	C₂×C₄×C₉₇	C₂³	1	2	yes	—
35	C₂×C₄×C₉₇	C₂³	1	2	yes	—
36	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
37	C₂×Dic₉₇	C₂³	1	2	yes	—
38	C₂×Dic₉₇	C₂³	1	2	yes	—
39	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
40	D₃₈₈	—	2	4	no	—
41	D₃₈₈	—	2	2	no	—
42	C₄×D₉₇	—	2	4	no	—
43	C₄×D₉₇	—	2	4	no	—
44	C₄×D₉₇	D₄	1	4	yes	—
45	Dic₁₉₄	—	2	2	no	—
46	Dic₁₉₄	—	2	4	no	—
47	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
48	C₂×D₉₇	—	2	2	no	—
49	C₂×D₉₇	—	2	2	no	—
50	C₂×D₉₇	—	2	4	no	—
51	C₂×D₉₇	C₂×D₄	1	2	yes	—
52	C₄×C₉₇	—	2	2	no	—
53	C₄×C₉₇	—	2	2	no	—
54	C₄×C₉₇	—	2	2	no	—
55	C₄×C₉₇	C₂×D₄	1	2	yes	—
56	Dic₉₇	—	2	2	no	—
57	Dic₉₇	—	2	2	no	—
58	Dic₉₇	C₂×D₄	1	2	yes	—
59	C₂×C₉₇	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₉₇	2₋¹⁺⁴	1	2	yes	—
61	D₉₇	—	2	2	no	—
62	D₉₇	—	4	2	no	—
63	C₉₇	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	97	1	no	maximal, Sylow
65	C₂×#41	—	97	1	no	—
66	C₂²×Q₈	—	97	1	no	—
67	C₄⋊Q₈	—	97	1	no	—
68	C₄⋊Q₈	D₉₇	1	1	yes	characteristic, complemented
69	#124	—	97	1	no	—
70	#124	—	97	1	no	—
71	#123	—	97	4	no	—
72	C₂²⋊Q₈	—	97	4	no	—
73	#117	—	97	1	no	—
74	#41	—	194	4	no	—
75	C₂×Q₈	—	97	2	no	—
76	C₂×Q₈	—	97	1	no	—
77	C₂×Q₈	—	194	4	no	—
78	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic
80	C₂×D₄	—	97	2	no	—
81	C₂×D₄	—	97	1	no	—
82	C₂²×C₄	—	97	1	no	—
83	C₂²×C₄	—	97	2	no	—
84	C₂²×C₄	—	97	2	no	—
85	C₄⋊C₄	—	97	2	no	—
86	C₄⋊C₄	—	97	4	no	—
87	C₄⋊C₄	C₂×D₉₇	1	4	yes	—
88	C₂²⋊C₄	—	97	2	no	—
89	C₂²⋊C₄	—	97	4	no	—
90	C₂²⋊C₄	—	97	4	no	—
91	C₄²	—	97	1	no	—
92	C₄²	C₂×D₉₇	1	1	yes	characteristic
93	C₂³	—	97	1	no	—
94	C₂³	—	97	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	194	4	no	—
98	Q₈	—	194	2	no	—
99	D₄	—	194	4	no	—
100	D₄	—	194	2	no	—
101	C₂×C₄	—	97	1	no	—
102	C₂×C₄	—	97	4	no	—
103	C₂×C₄	—	97	2	no	—
104	C₂×C₄	—	97	2	no	—
105	C₂×C₄	—	194	4	no	—
106	C₂×C₄	—	194	4	no	—
107	C₂×C₄	C₂²×D₉₇	1	2	yes	—
108	C₂×C₄	C₂²×D₉₇	1	2	yes	—
109	C₂×C₄	C₂²×D₉₇	1	2	yes	—
110	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
111	C₂²	—	97	2	no	—
112	C₂²	—	194	2	no	—
113	C₂²	—	194	4	no	—
114	C₂²	—	194	2	no	—
115	C₂²	C₂³×D₉₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	97	2	no	—
120	C₄	—	194	2	no	—
121	C₄	—	194	2	no	—
122	C₄	D₄×D₉₇	1	2	yes	—
123	C₂	—	388	2	no	—
124	C₂	—	194	2	no	—
125	C₂	C₂×D₄×D₉₇	1	1	yes	characteristic
126	C₂	#15690858	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13095512

C₂

yes

maximal, characteristic

C₂×Q₈×D₉₇

C₂

yes

maximal, characteristic, complemented

#15690808

C₂

yes

maximal, characteristic, complemented

#15690799

C₂

yes

maximal, characteristic, complemented

#15690798

C₂

yes

maximal, characteristic, complemented

#15690795

C₂

yes

maximal, complemented

#15690788

C₂

yes

maximal, complemented

#15690767

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₉₇

C₂

yes

maximal, characteristic, complemented

#13095512

—

C₂×Dic₁₉₄

C₂²

yes

—

C₂×Dic₁₉₄

C₂²

yes

characteristic

Q₈×D₉₇

—

C₂×D₃₈₈

C₂²

yes

—

C₂×D₃₈₈

C₂²

yes

characteristic

C₂×C₄×D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

—

C₂×C₄×D₉₇

C₂²

yes

characteristic

#13095490

C₂²

yes

—

#13095490

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

#13095488

C₂²

yes

—

C₄×Dic₉₇

C₂²

yes

characteristic

C₂×Q₈×C₉₇

C₂²

yes

characteristic

C₂×Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊C₄×C₉₇

C₂²

yes

—

C₄²×C₉₇

C₂²

yes

characteristic

C₂²×D₉₇

C₂³

yes

—

C₂²×D₉₇

C₂³

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

—

C₂×Dic₉₇

C₂³

yes

characteristic

D₃₈₈

—

D₃₈₈

—

C₄×D₉₇

—

C₄×D₉₇

—

C₄×D₉₇

D₄

yes

—

Dic₁₉₄

—

Dic₁₉₄

—

C₂²×C₉₇

C₂⁴

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

C₂×D₄

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×D₄

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

C₂×D₄

yes

—

C₂×C₉₇

C₂²×D₄

yes

characteristic

C₂×C₉₇

2₋¹⁺⁴

yes

—

D₉₇

—

D₉₇

—

C₉₇

#536

yes

characteristic, complemented, Sylow

#536

—

maximal, Sylow

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₉₇

yes

characteristic, complemented

#124

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

#41

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

194

—

C₂×Q₈

C₂×D₉₇

yes

characteristic

C₂×Q₈

C₂×D₉₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₉₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₉₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

194

—

Q₈

—

194

—

D₄

—

194

—

100

D₄

—

194

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

194

—

106

C₂×C₄

—

194

—

107

C₂×C₄

C₂²×D₉₇

yes

—

108

C₂×C₄

C₂²×D₉₇

yes

—

109

C₂×C₄

C₂²×D₉₇

yes

—

110

C₂×C₄

C₂²×D₉₇

yes

characteristic

111

C₂²

—

112

C₂²

—

194

—

113

C₂²

—

194

—

114

C₂²

—

194

—

115

C₂²

C₂³×D₉₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

194

—

121

C₄

—

194

—

122

C₄

D₄×D₉₇

yes

—

123

C₂

—

388

—

124

C₂

—

194

—

125

C₂

C₂×D₄×D₉₇

yes

characteristic

126

C₂

#15690858

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₄⋊Q₈×C₉₇	C₂
hypercenter	C₄⋊Q₈	D₉₇
hypocenter	C₉₇	#536
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
2. C₂×Q₈×D₉₇	C₂	semidirect
3. #15690808	C₂	semidirect
4. #15690799	C₂	semidirect
5. #15690798	C₂	semidirect
6. #15690795	C₂	semidirect
7. #15690788	C₂	semidirect
8. #15690767	C₂	semidirect
9. C₄⋊Q₈×C₉₇	C₂	semidirect
63. C₉₇	#536	semidirect
68. C₄⋊Q₈	D₉₇	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₉₇

C₂

semidirect

3. #15690808

C₂

semidirect

4. #15690799

C₂

semidirect

5. #15690798

C₂

semidirect

6. #15690795

C₂

semidirect

7. #15690788

C₂

semidirect

8. #15690767

C₂

semidirect

9. C₄⋊Q₈×C₉₇

C₂

semidirect

63. C₉₇

#536

semidirect

68. C₄⋊Q₈

D₉₇

semidirect

Subgroups — #20261310

Conjugacy classes of subgroups

Complements