Group #9803 — subgroups

Subgroups — #9803

Order 256.

201 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 71 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×#487	C₂	1	1	yes	maximal, characteristic
2	#1997	C₂	1	1	yes	maximal, characteristic
3	#1997	C₂	1	1	yes	maximal, characteristic
4	#1566	C₂	1	1	yes	maximal, characteristic
5	#1549	C₂	1	1	yes	maximal, characteristic
6	#1548	C₂	1	1	yes	maximal, characteristic
7	#1451	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#487	—	2	2	no	—
11	#487	—	2	2	no	—
12	#431	C₂²	1	1	yes	characteristic
13	#423	C₂²	1	1	yes	characteristic
14	#423	C₂²	1	1	yes	characteristic
15	C₂×#103	C₄	1	4	yes	—
16	C₄×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	#343	—	2	2	no	—
19	#343	—	2	2	no	—
20	#334	—	2	2	no	—
21	#333	—	2	2	no	—
22	#332	—	2	2	no	—
23	#322	—	2	2	no	—
24	C₂²×Q₈	—	2	1	no	—
25	C₂²×Q₈	C₂×C₄	1	2	yes	—
26	C₂×M₁₆	—	2	1	no	—
27	C₂×M₁₆	—	2	1	no	—
28	C₂×M₁₆	D₄	1	2	yes	—
29	C₂²×C₈	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈	—	2	1	no	—
31	C₄⋊Q₈	—	2	1	no	—
32	C₄⋊Q₈	—	2	1	no	—
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₄⋊Q₈	—	2	1	no	—
37	C₄⋊Q₈	—	2	1	no	—
38	C₂×C₄⋊C₄	—	2	1	no	—
39	C₂×C₄⋊C₄	—	2	1	no	—
40	C₂×C₄⋊C₄	—	2	1	no	—
41	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	C₄⋊C₈	—	2	1	no	—
44	C₄⋊C₈	—	2	1	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	D₄	1	2	yes	—
48	#103	—	2	8	no	—
49	C₂²⋊C₈	—	2	1	no	—
50	#97	—	2	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	D₄	1	2	yes	—
54	C₄×C₈	D₄	1	2	yes	—
55	C₂×Q₈	—	2	2	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	4	4	no	—
58	C₂×Q₈	—	4	1	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	8	2	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄	#41	1	1	yes	characteristic
66	M₁₆	—	2	2	no	—
67	M₁₆	—	2	2	no	—
68	M₁₆	—	4	2	no	—
69	M₁₆	—	4	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	4	2	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	C₂²⋊C₄	1	4	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	4	1	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄⋊C₄	—	4	1	no	—
81	C₄⋊C₄	—	4	1	no	—
82	C₄⋊C₄	—	4	1	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₄²	C₂×D₄	1	1	yes	characteristic
85	C₄²	C₂×D₄	1	1	yes	characteristic
86	C₄²	C₂×D₄	1	1	yes	characteristic
87	C₄²	#41	1	1	yes	characteristic
88	C₂³	C₄×D₄	1	1	yes	characteristic
89	Q₈	—	4	4	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	8	2	no	—
92	Q₈	—	8	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄	#121	1	1	yes	characteristic
102	C₂×C₄	#121	1	1	yes	characteristic
103	C₂×C₄	#124	1	1	yes	characteristic
104	C₂×C₄	#127	1	1	yes	characteristic
105	C₂×C₄	#137	1	2	yes	—
106	C₂×C₄	#137	1	2	yes	—
107	C₈	—	2	4	no	—
108	C₈	—	4	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	8	1	no	—
111	C₈	—	8	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	#390	1	1	yes	characteristic
115	C₂²	#441	1	2	yes	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	8	2	no	—
119	C₄	—	8	2	no	—
120	C₄	#413	1	2	yes	—
121	C₄	#470	1	2	yes	—
122	C₄	#471	1	2	yes	—
123	C₄	#497	1	1	yes	characteristic
124	C₄	#497	1	1	yes	characteristic
125	C₂	—	2	2	no	—
126	C₂	#1967	1	1	yes	characteristic
127	C₂	#1971	1	1	yes	characteristic
128	C₂	#1977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#487

C₂

yes

maximal, characteristic

#1997

C₂

yes

maximal, characteristic

#1997

C₂

yes

maximal, characteristic

#1566

C₂

yes

maximal, characteristic

#1549

C₂

yes

maximal, characteristic

#1548

C₂

yes

maximal, characteristic

#1451

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

—

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×#103

C₄

yes

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

—

C₂²×Q₈

—

C₂²×Q₈

C₂×C₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

#103

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₄

C₄×D₄

yes

characteristic

101

C₂×C₄

#121

yes

characteristic

102

C₂×C₄

#121

yes

characteristic

103

C₂×C₄

#124

yes

characteristic

104

C₂×C₄

#127

yes

characteristic

105

C₂×C₄

#137

yes

—

106

C₂×C₄

#137

yes

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₂²

—

113

C₂²

—

114

C₂²

#390

yes

characteristic

115

C₂²

#441

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#413

yes

—

121

C₄

#470

yes

—

122

C₄

#471

yes

—

123

C₄

#497

yes

characteristic

124

C₄

#497

yes

characteristic

125

C₂

—

126

C₂

#1967

yes

characteristic

127

C₂

#1971

yes

characteristic

128

C₂

#1977

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#390
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

C₂²

C₂²×C₄

C₂×C₄²

C₂³

prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#487	2
#1997	2
#1566	2
#1549	2
#1548	2
#1451	2

subgroup

index

Subgroups — #9803

Conjugacy classes of subgroups

Maximal subgroups