Group #12601 — subgroups

Subgroups — C₄×#349 #12601

Order 256.

261 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 127 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×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#349	C₂	1	2	yes	maximal
3	C₄×M₃₂	C₂	1	2	yes	maximal, complemented
4	#1377	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄²	C₄	1	1	yes	characteristic
8	C₂×M₃₂	C₂²	1	4	yes	—
9	C₄×M₁₆	—	2	2	no	—
10	C₄×M₁₆	—	2	2	no	—
11	C₂×#97	C₄	1	1	yes	characteristic
12	C₂×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#349	C₄	1	8	yes	direct product factor
16	#346	—	2	2	no	—
17	C₄×C₁₆	—	2	2	no	—
18	C₂³×C₄	C₈	1	2	yes	—
19	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆	—	2	2	no	—
21	C₂×M₁₆	—	2	2	no	—
22	C₂×M₁₆	—	2	2	no	—
23	C₂×M₁₆	—	2	2	no	—
24	C₂×M₁₆	C₈	1	4	yes	—
25	C₂×M₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₄²	—	2	1	no	—
31	C₂×C₄²	—	2	1	no	—
32	C₂×C₄²	—	2	2	no	—
33	C₂×C₄²	C₈	1	2	yes	—
34	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	M₃₂	—	2	4	no	—
37	M₃₂	C₂×C₄	1	8	yes	complemented
38	C₂×C₁₆	—	2	4	no	—
39	#97	—	2	2	no	—
40	#97	—	2	2	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	D₄	1	2	yes	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₂⁴	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	—	2	1	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	—	2	2	no	—
51	C₂²×C₄	—	2	2	no	—
52	C₂²×C₄	—	4	4	no	—
53	C₂²×C₄	C₄²	1	2	yes	—
54	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄	C₂×C₈	1	2	yes	—
56	C₂²×C₄	C₂×C₈	1	2	yes	—
57	C₂²×C₄	C₂×C₈	1	2	yes	—
58	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
59	M₁₆	—	2	8	no	—
60	M₁₆	—	2	8	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₄²	1	2	yes	—
66	C₂×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈	C₂×C₈	1	2	yes	—
68	C₂×C₈	C₂×C₈	1	2	yes	—
69	C₂×C₈	M₁₆	1	2	yes	—
70	C₂×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₈	C₂×D₄	1	2	yes	—
72	C₂×C₈	#41	1	2	yes	—
73	C₄²	—	2	2	no	—
74	C₄²	—	2	2	no	—
75	C₄²	—	2	1	no	—
76	C₄²	—	2	1	no	—
77	C₄²	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²	M₁₆	1	2	yes	—
80	C₁₆	—	2	8	no	—
81	C₂³	—	2	1	no	—
82	C₂³	—	2	1	no	—
83	C₂³	—	4	2	no	—
84	C₂³	C₄×C₈	1	2	yes	—
85	C₂³	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	4	no	—
95	C₂×C₄	—	4	4	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	C₄×C₈	1	2	yes	—
98	C₂×C₄	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄	#117	1	1	yes	characteristic
103	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	2	2	no	—
108	C₈	C₂²⋊C₈	1	4	yes	—
109	C₈	C₄×D₄	1	4	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	2	no	—
114	C₂²	—	4	2	no	—
115	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
116	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²	#431	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	2	no	—
121	C₄	#349	1	4	yes	direct product factor
122	C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
123	C₄	C₄×M₁₆	1	1	yes	characteristic
124	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₄	#431	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	—	4	2	no	—
129	C₂	C₄×C₂²⋊C₈	1	1	yes	characteristic
130	C₂	C₂×#349	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×M₁₆

C₂

yes

maximal, characteristic

C₂×#349

C₂

yes

maximal

C₄×M₃₂

C₂

yes

maximal, complemented

#1377

C₂

yes

maximal, complemented

C₂²×M₁₆

C₄

yes

characteristic

C₂²×M₁₆

C₂²

yes

characteristic

C₂²×C₄²

C₄

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₄×M₁₆

—

C₄×M₁₆

—

C₂×#97

C₄

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#349

C₄

yes

direct product factor

#346

—

C₄×C₁₆

—

C₂³×C₄

C₈

yes

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₈

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

C₂×C₄

yes

complemented

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂⁴

C₂×C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄²

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

M₁₆

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄×C₈

yes

—

C₂³

C₂²×C₈

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×C₈

yes

—

C₂×C₄

C₂²⋊C₈

yes

—

C₂×C₄

C₂²⋊C₈

yes

—

100

C₂×C₄

C₂×C₄²

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

102

C₂×C₄

#117

yes

characteristic

103

C₂×C₄

C₂²×C₈

yes

characteristic

104

C₂×C₄

C₂×M₁₆

yes

characteristic

105

C₂×C₄

C₂×M₁₆

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

C₂²⋊C₈

yes

—

109

C₈

C₄×D₄

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄×C₈

yes

characteristic

116

C₂²

C₂×C₂²⋊C₈

yes

characteristic

117

C₂²

#431

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#349

yes

direct product factor

122

C₄

C₄×C₂²⋊C₄

yes

characteristic

123

C₄

C₄×M₁₆

yes

characteristic

124

C₄

C₂×C₂²⋊C₈

yes

characteristic

125

C₄

#431

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

C₄×C₂²⋊C₈

yes

characteristic

130

C₂

C₂×#349

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄²

C₂²⋊C₄

commutator

C₂²

C₂×C₄×C₈

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂²

C₂×C₂²⋊C₈

prime radical

C₂⁴

C₂×C₈

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. C₄×M₃₂	C₂	semidirect
4. #1377	C₂	semidirect
15. #349	C₄	direct (N × H)
37. M₃₂	C₂×C₄	semidirect
121. C₄	#349	direct (N × H)

H ≅ G/N

type

3. C₄×M₃₂

C₂

semidirect

4. #1377

C₂

semidirect

15. #349

C₄

direct (N × H)

37. M₃₂

C₂×C₄

semidirect

121. C₄

#349

direct (N × H)

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

subgroup	index
C₂×C₄×M₁₆	2
C₂×#349	2
C₄×M₃₂	2
#1377	2

subgroup

index

C₂×C₄×M₁₆

C₂×#349

C₄×M₃₂

#1377

Subgroups — C₄×#349 #12601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#349 #12601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#349 #12601