Group #21688 — subgroups

Subgroups — C₄×#494 #21688

Order 256.

543 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 231 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆	C₂	1	4	yes	maximal
3	C₄×C₄⋊Q₈	C₂	1	2	yes	maximal
4	#1969	C₂	1	4	yes	maximal
5	#1956	C₂	1	2	yes	maximal
6	#1845	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈	C₂²	1	4	yes	—
11	#494	C₄	1	8	yes	direct product factor
12	C₄×Q₁₆	—	2	16	no	—
13	C₂×#107	C₂²	1	4	yes	—
14	C₂×#103	C₂²	1	8	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	2	yes	—
19	#391	C₂²	1	4	yes	—
20	#389	—	2	4	no	—
21	#384	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄	—	2	4	no	—
23	C₄³	C₂²	1	1	yes	characteristic
24	C₂²×Q₈	C₂³	1	4	yes	—
25	C₂×Q₁₆	—	2	8	no	—
26	C₂×Q₁₆	—	2	8	no	—
27	C₂×Q₁₆	C₂×C₄	1	16	yes	—
28	C₂²×C₈	C₂³	1	2	yes	—
29	C₂²×C₈	C₂³	1	2	yes	—
30	C₄⋊Q₈	—	2	4	no	—
31	C₄⋊Q₈	C₂×C₄	1	8	yes	—
32	C₄×Q₈	—	2	16	no	—
33	C₂×C₄⋊C₄	—	2	4	no	—
34	C₂×C₄⋊C₄	—	2	4	no	—
35	C₂×C₄⋊C₄	—	2	4	no	—
36	C₂×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₄²	—	2	4	no	—
38	C₂×C₄²	D₄	1	2	yes	—
39	C₂×C₄²	D₄	1	2	yes	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	#107	—	2	8	no	—
44	#103	—	2	16	no	—
45	C₄×C₈	—	2	4	no	—
46	C₄×C₈	—	2	2	no	—
47	C₄×C₈	C₂×C₄	1	4	yes	—
48	C₄×C₈	D₄	1	8	yes	—
49	#95	—	2	8	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	—	2	8	no	—
52	C₂×Q₈	—	2	8	no	—
53	C₂×Q₈	C₂²×C₄	1	8	yes	—
54	C₂²×C₄	—	2	4	no	—
55	C₂²×C₄	—	2	4	no	—
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₄	1	2	yes	—
58	C₂²×C₄	C₂×D₄	1	2	yes	—
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	C₂⁴	1	1	yes	characteristic
61	Q₁₆	—	2	32	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	C₂²×C₄	1	4	yes	—
65	C₂×C₈	C₂×D₄	1	4	yes	—
66	C₂×C₈	C₂×D₄	1	4	yes	—
67	C₂×C₈	#41	1	4	yes	—
68	C₄⋊C₄	—	2	16	no	—
69	C₄⋊C₄	—	2	8	no	—
70	C₄⋊C₄	—	4	8	no	—
71	C₄⋊C₄	—	4	8	no	—
72	C₄²	—	2	4	no	—
73	C₄²	—	2	2	no	—
74	C₄²	—	4	8	no	—
75	C₄²	Q₁₆	1	8	yes	complemented
76	C₄²	C₂²×C₄	1	2	yes	—
77	C₄²	C₂×D₄	1	4	yes	—
78	C₄²	#41	1	2	yes	—
79	C₂³	C₂²×D₄	1	1	yes	characteristic
80	Q₈	—	2	16	no	—
81	Q₈	—	4	8	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	8	no	—
86	C₂×C₄	—	4	4	no	—
87	C₂×C₄	—	4	4	no	—
88	C₂×C₄	—	4	4	no	—
89	C₂×C₄	—	4	8	no	—
90	C₂×C₄	C₄×D₄	1	4	yes	—
91	C₂×C₄	#127	1	2	yes	—
92	C₂×C₄	#127	1	2	yes	—
93	C₂×C₄	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄	C₂×Q₁₆	1	4	yes	—
95	C₂×C₄	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄	#135	1	4	yes	—
97	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₄	1	2	yes	—
99	C₂×C₄	C₂×#41	1	2	yes	—
100	C₂×C₄	C₂×#41	1	1	yes	characteristic
101	C₈	—	2	4	no	—
102	C₈	C₄×D₄	1	8	yes	—
103	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
104	C₂²	C₂×#127	1	1	yes	characteristic
105	C₂²	#532	1	1	yes	characteristic
106	C₂²	C₂²×Q₁₆	1	2	yes	—
107	C₂²	C₂×#135	1	2	yes	—
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	—	4	8	no	—
111	C₄	—	8	4	no	—
112	C₄	C₄×Q₁₆	1	8	yes	complemented
113	C₄	#494	1	4	yes	direct product factor
114	C₄	C₂×C₄×D₄	1	2	yes	—
115	C₄	#532	1	2	yes	—
116	C₂	C₄×#127	1	1	yes	characteristic
117	C₂	C₂×C₄×Q₁₆	1	2	yes	—
118	C₂	C₂×#494	1	1	yes	characteristic
119	C₂	#3147	1	1	yes	characteristic
120	C₂	#3170	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

C₂×C₄×Q₁₆

C₂

yes

maximal

C₄×C₄⋊Q₈

C₂

yes

maximal

#1969

C₂

yes

maximal

#1956

C₂

yes

maximal

#1845

C₂

yes

maximal, characteristic

C₄²×C₈

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

—

#494

C₄

yes

direct product factor

C₄×Q₁₆

—

C₂×#107

C₂²

yes

—

C₂×#103

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

#391

C₂²

yes

—

#389

—

#384

C₂²

yes

—

C₄×C₄⋊C₄

—

C₄³

C₂²

yes

characteristic

C₂²×Q₈

C₂³

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

#107

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂⁴

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Q₁₆

yes

complemented

C₄²

C₂²×C₄

yes

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

C₂²×D₄

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

#127

yes

—

C₂×C₄

#127

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

#135

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₂×C₄

C₂²×D₄

yes

—

C₂×C₄

C₂×#41

yes

—

100

C₂×C₄

C₂×#41

yes

characteristic

101

C₈

—

102

C₈

C₄×D₄

yes

—

103

C₂²

C₂×C₄×D₄

yes

characteristic

104

C₂²

C₂×#127

yes

characteristic

105

C₂²

#532

yes

characteristic

106

C₂²

C₂²×Q₁₆

yes

—

107

C₂²

C₂×#135

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

C₄×Q₁₆

yes

complemented

113

C₄

#494

yes

direct product factor

114

C₄

C₂×C₄×D₄

yes

—

115

C₄

#532

yes

—

116

C₂

C₄×#127

yes

characteristic

117

C₂

C₂×C₄×Q₁₆

yes

—

118

C₂

C₂×#494

yes

characteristic

119

C₂

#3147

yes

characteristic

120

C₂

#3170

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle, prime radical	C₂³	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

socle, prime radical

C₂³

C₂²×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
11. #494	C₄	direct (N × H)
75. C₄²	Q₁₆	semidirect
112. C₄	C₄×Q₁₆	semidirect
113. C₄	#494	direct (N × H)

H ≅ G/N

type

11. #494

C₄

direct (N × H)

75. C₄²

Q₁₆

semidirect

112. C₄

C₄×Q₁₆

semidirect

113. C₄

#494

direct (N × H)

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

subgroup	index
C₂×#494	2
C₂×C₄×Q₁₆	2
C₄×C₄⋊Q₈	2
#1969	2
#1956	2
#1845	2
C₄²×C₈	2

subgroup

index

C₂×#494

C₂×C₄×Q₁₆

C₄×C₄⋊Q₈

#1969

#1956

#1845

C₄²×C₈

Subgroups — C₄×#494 #21688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#494 #21688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#494 #21688