Group #122213 — subgroups

Subgroups — #122213

Order 512.

220 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 162 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	#11410	C₂	1	1	yes	maximal, characteristic
2	C₂×#1312	C₂	1	1	yes	maximal, characteristic
3	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#7053	C₂	1	4	yes	maximal, complemented
5	#2162	C₄	1	2	yes	—
6	C₄×M₃₂	C₄	1	1	yes	characteristic
7	C₄×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₄	1	2	yes	—
9	C₂×C₈²	C₂²	1	1	yes	characteristic
10	#1312	C₄	1	2	yes	—
11	#1312	C₂²	1	2	yes	—
12	#1312	C₂²	1	2	yes	—
13	#1311	C₂²	1	2	yes	—
14	C₈×C₁₆	C₄	1	2	yes	—
15	C₈×C₁₆	C₄	1	2	yes	—
16	C₂×M₃₂	C₈	1	2	yes	—
17	C₂×M₃₂	C₈	1	2	yes	—
18	C₂²×C₁₆	—	2	2	no	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₁₆	—	2	2	no	—
23	C₂²⋊C₁₆	—	2	2	no	—
24	#346	C₈	1	2	yes	—
25	#346	C₈	1	2	yes	—
26	C₄×C₁₆	—	2	2	no	—
27	C₄×C₁₆	C₂×C₄	1	4	yes	—
28	C₄×C₁₆	C₂×C₄	1	4	yes	—
29	C₄×C₁₆	C₂×C₄	1	2	yes	—
30	C₄×C₁₆	C₂×C₄	1	2	yes	—
31	C₈²	C₂×C₄	1	1	yes	characteristic
32	C₈²	C₂×C₄	1	1	yes	characteristic
33	C₈²	C₂×C₄	1	1	yes	characteristic
34	C₈²	C₂³	1	1	yes	characteristic
35	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₈	M₁₆	1	2	yes	—
38	C₂²×C₈	M₁₆	1	2	yes	—
39	C₂×C₄²	C₄²	1	1	yes	characteristic
40	M₃₂	—	2	4	no	—
41	M₃₂	C₁₆	1	8	yes	complemented
42	C₂×C₁₆	—	2	4	no	—
43	C₂×C₁₆	—	2	4	no	—
44	C₂×C₁₆	—	4	4	no	—
45	C₂×C₁₆	C₂×C₈	1	4	yes	—
46	C₂×C₁₆	C₂×C₈	1	4	yes	—
47	C₄×C₈	C₄²	1	1	yes	characteristic
48	C₄×C₈	C₄²	1	1	yes	characteristic
49	C₄×C₈	C₄²	1	1	yes	characteristic
50	C₄×C₈	C₂×C₈	1	1	yes	characteristic
51	C₄×C₈	C₂×C₈	1	1	yes	characteristic
52	C₄×C₈	M₁₆	1	2	yes	—
53	C₄×C₈	M₁₆	1	2	yes	—
54	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
58	C₂²×C₄	#97	1	1	yes	characteristic
59	C₂²×C₄	#97	1	1	yes	characteristic
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	C₄×C₈	1	1	yes	characteristic
63	C₂×C₈	C₄×C₈	1	1	yes	characteristic
64	C₂×C₈	#97	1	2	yes	—
65	C₂×C₈	#97	1	2	yes	—
66	C₂×C₈	C₂×C₁₆	1	4	yes	—
67	C₂×C₈	M₃₂	1	4	yes	—
68	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₈	C₂×M₁₆	1	2	yes	—
71	C₂×C₈	C₂×M₁₆	1	2	yes	—
72	C₄²	C₄×C₈	1	1	yes	characteristic
73	C₄²	#97	1	1	yes	characteristic
74	C₄²	#97	1	1	yes	characteristic
75	C₄²	C₂×C₄²	1	1	yes	characteristic
76	C₁₆	—	2	4	no	—
77	C₁₆	—	4	4	no	—
78	C₁₆	—	4	4	no	—
79	C₁₆	C₂×C₁₆	1	8	yes	complemented
80	C₂³	#322	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	#322	1	1	yes	characteristic
84	C₂×C₄	#322	1	1	yes	characteristic
85	C₂×C₄	#322	1	1	yes	characteristic
86	C₂×C₄	C₄×C₁₆	1	2	yes	—
87	C₂×C₄	#346	1	2	yes	—
88	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
89	C₂×C₄	C₂×#97	1	1	yes	characteristic
90	C₂×C₄	C₂×#97	1	1	yes	characteristic
91	C₈	—	2	2	no	—
92	C₈	—	2	2	no	—
93	C₈	—	2	2	no	—
94	C₈	C₄×C₁₆	1	2	yes	—
95	C₈	#346	1	2	yes	—
96	C₈	#347	1	4	yes	—
97	C₈	C₂²×C₁₆	1	2	yes	—
98	C₈	C₂×M₃₂	1	2	yes	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#1312	1	2	yes	—
102	C₂²	C₂×#322	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#1312	1	2	yes	—
106	C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
107	C₄	C₂×#346	1	1	yes	characteristic
108	C₄	C₂×#347	1	2	yes	—
109	C₄	#2110	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	C₂×#1312	1	1	yes	characteristic
112	C₂	#11404	1	1	yes	characteristic
113	C₂	#11406	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11410

C₂

yes

maximal, characteristic

C₂×#1312

C₂

yes

maximal, characteristic

C₈×M₃₂

C₂

yes

maximal, characteristic

#7053

C₂

yes

maximal, complemented

#2162

C₄

yes

—

C₄×M₃₂

C₄

yes

characteristic

C₄×M₃₂

C₄

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1312

C₄

yes

—

#1312

C₂²

yes

—

#1312

C₂²

yes

—

#1311

C₂²

yes

—

C₈×C₁₆

C₄

yes

—

C₈×C₁₆

C₄

yes

—

C₂×M₃₂

C₈

yes

—

C₂×M₃₂

C₈

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

C₈

yes

—

#346

C₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂³

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

M₁₆

yes

—

C₂×C₄²

C₄²

yes

characteristic

M₃₂

—

M₃₂

C₁₆

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₂²×C₄

C₄×C₈

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

#97

yes

—

C₂×C₈

#97

yes

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₄²

C₄×C₈

yes

C₄²

yes

C₄²

yes

C₄²

C₂×C₄²

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₂×C₁₆

yes

complemented

C₂³

#322

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×C₁₆

yes

—

C₂×C₄

#346

yes

—

C₂×C₄

C₂×C₄×C₈

yes

characteristic

C₂×C₄

C₂×#97

yes

characteristic

C₂×C₄

C₂×#97

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

C₄×C₁₆

yes

—

C₈

#346

yes

—

C₈

#347

yes

—

C₈

C₂²×C₁₆

yes

—

C₈

C₂×M₃₂

yes

—

C₂²

—

100

C₂²

—

101

C₂²

#1312

yes

—

102

C₂²

C₂×#322

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

#1312

yes

—

106

C₄

C₂×C₄×C₁₆

yes

characteristic

107

C₄

C₂×#346

yes

characteristic

108

C₄

C₂×#347

yes

—

109

C₄

#2110

yes

—

110

C₂

—

111

C₂

C₂×#1312

yes

characteristic

112

C₂

#11404

yes

characteristic

113

C₂

#11406

yes

characteristic

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×#322
prime radical	C₂³	#322
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₄×C₈

C₄²

commutator

C₄

C₂×C₄×C₁₆

Frattini, cosocle

C₈²

C₂³

C₂²

C₂³

G ▷ C₄ ▷ C₁

C₁ ◁ C₄×C₈ ◁ G

G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
4. #7053	C₂	semidirect
41. M₃₂	C₁₆	semidirect
79. C₁₆	C₂×C₁₆	semidirect

H ≅ G/N

type

4. #7053

C₂

semidirect

41. M₃₂

C₁₆

semidirect

79. C₁₆

C₂×C₁₆

semidirect

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

subgroup	index
#11410	2
C₂×#1312	2
C₈×M₃₂	2
#7053	2

subgroup

index

#11410

C₂×#1312

C₈×M₃₂

#7053

Subgroups — #122213

Conjugacy classes of subgroups

Complements

Maximal subgroups