Group #21673 — subgroups

Subgroups — #21673

Order 256.

347 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 191 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	C₄×#125	C₂	1	1	yes	maximal, characteristic
2	#1939	C₂	1	2	yes	maximal
3	#1916	C₂	1	4	yes	maximal
4	#1831	C₂	1	1	yes	maximal, characteristic
5	#1771	C₂	1	4	yes	maximal
6	#1770	C₂	1	2	yes	maximal
7	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#125	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×#97	C₂²	1	4	yes	—
12	C₂×C₄×C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#389	C₄	1	2	yes	—
17	#387	C₄	1	4	yes	—
18	#382	C₄	1	1	yes	characteristic
19	C₄×C₄⋊C₄	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄³	C₂²	1	1	yes	characteristic
22	#336	C₂²	1	4	yes	—
23	#336	C₂²	1	4	yes	—
24	#336	C₂²	1	4	yes	—
25	C₂²×C₈	C₂³	1	4	yes	—
26	C₂²×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₈	C₂³	1	2	yes	—
28	C₂²×C₈	C₂³	1	1	yes	characteristic
29	#125	—	2	4	no	—
30	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₄²	C₂×C₄	1	2	yes	—
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²	C₂³	1	2	yes	—
38	C₂×C₄²	C₂³	1	4	yes	—
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	C₄⋊C₈	—	2	8	no	—
41	C₄⋊C₈	—	2	4	no	—
42	#97	—	2	8	no	—
43	C₄×C₈	—	2	4	no	—
44	C₄×C₈	—	2	4	no	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	Q₈	1	4	yes	—
47	#95	C₂×C₄	1	4	yes	—
48	#95	C₂×C₄	1	4	yes	—
49	#95	C₂×C₄	1	4	yes	—
50	C₂²×C₄	C₂²×C₄	1	4	yes	—
51	C₂²×C₄	C₂²×C₄	1	4	yes	—
52	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄	C₂²×C₄	1	2	yes	—
54	C₂²×C₄	C₂²×C₄	1	2	yes	—
55	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄	C₂⁴	1	1	yes	characteristic
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	C₂×Q₈	1	2	yes	—
64	C₂×C₈	C₂×Q₈	1	2	yes	—
65	C₂×C₈	#41	1	4	yes	—
66	C₂×C₈	#41	1	4	yes	—
67	C₄⋊C₄	—	2	8	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	2	8	no	—
70	C₄⋊C₄	—	2	4	no	—
71	C₄²	—	2	4	no	—
72	C₄²	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	—	2	2	no	—
75	C₄²	—	2	4	no	—
76	C₄²	—	2	4	no	—
77	C₄²	M₁₆	1	4	yes	—
78	C₄²	C₂×Q₈	1	2	yes	—
79	C₄²	#41	1	2	yes	—
80	C₂³	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₄×Q₈	1	2	yes	—
93	C₂×C₄	C₄×Q₈	1	2	yes	—
94	C₂×C₄	C₂×M₁₆	1	4	yes	—
95	C₂×C₄	C₂×M₁₆	1	2	yes	—
96	C₂×C₄	#131	1	4	yes	—
97	C₂×C₄	#131	1	2	yes	—
98	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄	C₂×#41	1	2	yes	—
100	C₂×C₄	C₂×#41	1	1	yes	characteristic
101	C₂×C₄	C₂×#41	1	2	yes	—
102	C₈	—	2	4	no	—
103	C₈	—	2	2	no	—
104	C₈	—	4	4	no	—
105	C₈	#125	1	4	yes	complemented
106	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂²	C₄×#41	1	2	yes	—
108	C₂²	C₂²×M₁₆	1	1	yes	characteristic
109	C₂²	C₂×#131	1	1	yes	characteristic
110	C₂²	C₂×#131	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	2	4	no	—
113	C₄	—	2	4	no	—
114	C₄	—	4	4	no	—
115	C₄	—	4	4	no	—
116	C₄	#446	1	4	yes	—
117	C₄	C₂×#125	1	1	yes	characteristic
118	C₄	#529	1	2	yes	—
119	C₄	#533	1	1	yes	characteristic
120	C₂	C₄×#125	1	1	yes	characteristic
121	C₂	C₂×#446	1	1	yes	characteristic
122	C₂	#2960	1	1	yes	characteristic
123	C₂	#2965	1	2	yes	—
124	C₂	#2966	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#125

C₂

yes

maximal, characteristic

#1939

C₂

yes

maximal

#1916

C₂

yes

maximal

#1831

C₂

yes

maximal, characteristic

#1771

C₂

yes

maximal

#1770

C₂

yes

maximal

C₄²×C₈

C₂

yes

maximal, characteristic

C₂×#125

C₄

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#97

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#389

C₄

yes

—

#387

C₄

yes

—

#382

C₄

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

C₄×C₄⋊C₄

C₂²

yes

—

C₄³

C₂²

yes

characteristic

#336

C₂²

yes

—

#336

C₂²

yes

—

#336

C₂²

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

#125

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Q₈

yes

—

#95

C₂×C₄

yes

—

#95

C₂×C₄

yes

—

#95

C₂×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

M₁₆

yes

—

C₄²

C₂×Q₈

yes

—

C₄²

#41

yes

—

C₂³

C₂³×C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₄×Q₈

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

C₂²×Q₈

yes

characteristic

C₂×C₄

C₂×#41

yes

—

100

C₂×C₄

C₂×#41

yes

characteristic

101

C₂×C₄

C₂×#41

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

#125

yes

complemented

106

C₂²

C₂×C₄×Q₈

yes

characteristic

107

C₂²

C₄×#41

yes

—

108

C₂²

C₂²×M₁₆

yes

characteristic

109

C₂²

C₂×#131

yes

characteristic

110

C₂²

C₂×#131

yes

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#446

yes

—

117

C₄

C₂×#125

yes

characteristic

118

C₄

#529

yes

—

119

C₄

#533

yes

characteristic

120

C₂

C₄×#125

yes

characteristic

121

C₂

C₂×#446

yes

characteristic

122

C₂

#2960

yes

characteristic

123

C₂

#2965

yes

—

124

C₂

#2966

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂²×C₄

C₂⁴

commutator, socle, prime radical

C₂³

C₂³×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
105. C₈	#125	semidirect

H ≅ G/N

type

105. C₈

#125

semidirect

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

subgroup	index
C₄×#125	2
#1939	2
#1916	2
#1831	2
#1771	2
#1770	2
C₄²×C₈	2

subgroup

index

C₄²×C₈

Subgroups — #21673

Conjugacy classes of subgroups

Complements

Maximal subgroups