Group #122890 — subgroups

Subgroups — #122890

Order 512.

288 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 86 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×#1713	C₂	1	1	yes	maximal, characteristic
2	#12013	C₂	1	2	yes	maximal
3	#7098	C₂	1	4	yes	maximal, complemented
4	C₂×#500	—	2	2	no	—
5	C₂×#494	C₄	1	2	yes	—
6	#2163	—	2	2	no	—
7	#1713	—	2	2	no	—
8	#1713	C₄	1	2	yes	—
9	#1713	C₂²	1	2	yes	—
10	C₂×C₈²	C₂²	1	1	yes	characteristic
11	#1370	C₂²	1	4	yes	—
12	#1367	—	2	2	no	—
13	C₂²×Q₁₆	—	4	2	no	—
14	C₂×C₄⋊Q₈	—	2	2	no	—
15	#500	—	2	4	no	—
16	#500	—	2	4	no	—
17	#500	—	4	2	no	—
18	#500	—	4	2	no	—
19	#494	—	2	2	no	—
20	#494	—	4	2	no	—
21	#494	C₂×C₄	1	4	yes	—
22	C₂×#107	—	4	2	no	—
23	C₂×#107	—	4	2	no	—
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	2	yes	—
26	C₄⋊C₁₆	—	2	4	no	—
27	C₂²⋊C₁₆	—	4	2	no	—
28	#346	—	4	2	no	—
29	C₈²	C₂×C₄	1	1	yes	characteristic
30	C₈²	D₄	1	2	yes	—
31	C₈²	C₂³	1	1	yes	characteristic
32	C₂²×Q₈	—	4	2	no	—
33	C₂×Q₁₆	—	4	4	no	—
34	C₂×Q₁₆	—	4	4	no	—
35	C₂×Q₁₆	—	8	8	no	—
36	C₂²×C₈	—	2	1	no	—
37	C₂²×C₈	D₈	1	2	yes	—
38	C₂²×C₈	SD₁₆	1	2	yes	—
39	C₄⋊Q₈	—	2	4	no	—
40	C₄⋊Q₈	—	4	2	no	—
41	C₄⋊Q₈	—	4	4	no	—
42	C₂×C₄⋊C₄	—	4	4	no	—
43	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₁₆	—	4	4	no	—
45	#107	—	4	4	no	—
46	#107	—	4	4	no	—
47	#107	—	4	4	no	—
48	#107	—	4	4	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₄×C₈	C₂²⋊C₄	1	2	yes	—
52	C₄×C₈	D₈	1	2	yes	—
53	C₄×C₈	SD₁₆	1	2	yes	—
54	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₄×C₈	C₂×D₄	1	2	yes	—
56	C₂×Q₈	—	4	4	no	—
57	C₂×Q₈	—	8	8	no	—
58	C₂×Q₈	—	8	2	no	—
59	C₂²×C₄	—	8	2	no	—
60	C₂²×C₄	#100	1	1	yes	characteristic
61	C₂²×C₄	#102	1	2	yes	—
62	Q₁₆	—	8	8	no	—
63	Q₁₆	—	8	8	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	#102	1	2	yes	—
70	C₂×C₈	#102	1	2	yes	—
71	C₂×C₈	C₂×D₈	1	2	yes	—
72	C₂×C₈	C₂×SD₁₆	1	2	yes	—
73	C₄⋊C₄	—	4	8	no	—
74	C₄⋊C₄	—	8	4	no	—
75	C₄²	#100	1	1	yes	characteristic
76	C₄²	#102	1	2	yes	—
77	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₁₆	—	8	4	no	—
79	C₂³	#331	1	1	yes	characteristic
80	Q₈	—	8	8	no	—
81	Q₈	—	16	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	8	4	no	—
86	C₂×C₄	—	16	4	no	—
87	C₂×C₄	#331	1	2	yes	—
88	C₂×C₄	#331	1	1	yes	characteristic
89	C₂×C₄	C₂×#100	1	1	yes	characteristic
90	C₂×C₄	C₂×#102	1	2	yes	—
91	C₈	—	2	2	no	—
92	C₈	—	2	2	no	—
93	C₈	—	2	2	no	—
94	C₈	—	2	2	no	—
95	C₈	#362	1	4	yes	—
96	C₈	#510	1	4	yes	—
97	C₂²	—	2	2	no	—
98	C₂²	#1357	1	2	yes	—
99	C₂²	C₂×#331	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	2	no	—
103	C₄	—	16	4	no	—
104	C₄	#2141	1	2	yes	—
105	C₄	C₂×#362	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	C₂×#1357	1	1	yes	characteristic
108	C₂	#11932	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1713

C₂

yes

maximal, characteristic

#12013

C₂

yes

maximal

#7098

C₂

yes

maximal, complemented

C₂×#500

—

C₂×#494

C₄

yes

—

#2163

—

#1713

—

#1713

C₄

yes

—

#1713

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1370

C₂²

yes

—

#1367

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#107

—

C₂×#107

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

—

C₈²

C₂³

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

SD₁₆

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#100

yes

characteristic

C₂²×C₄

#102

yes

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#100

yes

characteristic

C₄²

#102

yes

—

C₄²

C₂×C₂²⋊C₄

yes

characteristic

C₁₆

—

C₂³

#331

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#331

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#362

yes

—

C₈

#510

yes

—

C₂²

—

C₂²

#1357

yes

—

C₂²

C₂×#331

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#2141

yes

—

105

C₄

C₂×#362

yes

—

106

C₂

—

107

C₂

C₂×#1357

yes

characteristic

108

C₂

#11932

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
prime radical	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#331

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₈²

C₂³

prime radical

C₂³

#331

LCS

G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. #7098	C₂	semidirect

H ≅ G/N

type

3. #7098

C₂

semidirect

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

subgroup	index
C₂×#1713	2
#12013	2
#7098	2

subgroup

index

C₂×#1713

#12013

#7098

Subgroups — #122890

Conjugacy classes of subgroups

Complements

Maximal subgroups