Group #41009930 — subgroups

Subgroups — #2310×C₂₁₁ #41009930

Order 27008.

784 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 360 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	C₂³×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₂₁₁	C₂	1	2	yes	maximal
3	#391×C₂₁₁	C₂	1	8	yes	maximal, complemented
4	#381×C₂₁₁	C₂	1	4	yes	maximal
5	C₂²×Q₈×C₂₁₁	—	2	8	no	—
6	C₂²×Q₈×C₂₁₁	C₄	1	8	yes	complemented
7	C₂²×Q₈×C₂₁₁	C₂²	1	4	yes	—
8	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
9	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	#117×C₂₁₁	—	2	8	no	—
11	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
13	C₂×C₄²×C₂₁₁	C₂²	1	4	yes	—
14	#95×C₂₁₁	C₂²	1	16	yes	—
15	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
16	C₂×Q₈×C₂₁₁	—	2	32	no	—
17	C₂×Q₈×C₂₁₁	—	2	8	no	—
18	C₂×Q₈×C₂₁₁	—	2	8	no	—
19	C₂×Q₈×C₂₁₁	C₂×C₄	1	16	yes	—
20	C₂²×C₄×C₂₁₁	—	2	8	no	—
21	C₂²×C₄×C₂₁₁	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₂₁₁	D₄	1	8	yes	—
24	C₂²×C₄×C₂₁₁	C₂³	1	8	yes	—
25	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
27	C₄⋊C₄×C₂₁₁	—	2	8	no	—
28	C₂²⋊C₄×C₂₁₁	—	2	8	no	—
29	C₄²×C₂₁₁	—	2	8	no	—
30	C₂³×C₂₁₁	—	2	4	no	—
31	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
32	C₂³×C₂₁₁	C₂×D₄	1	4	yes	—
33	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
34	Q₈×C₂₁₁	—	2	32	no	—
35	C₂×C₄×C₂₁₁	—	2	16	no	—
36	C₂×C₄×C₂₁₁	—	2	8	no	—
37	C₂×C₄×C₂₁₁	—	2	8	no	—
38	C₂×C₄×C₂₁₁	—	2	16	no	—
39	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	16	yes	—
40	C₂×C₄×C₂₁₁	C₂²×C₄	1	8	yes	—
41	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
42	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
44	C₂²×C₂₁₁	—	2	4	no	—
45	C₂²×C₂₁₁	—	2	8	no	—
46	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	4	yes	—
47	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
48	C₂²×C₂₁₁	C₂²×D₄	1	2	yes	—
49	C₂²×C₂₁₁	2₋¹⁺⁴	1	4	yes	—
50	C₄×C₂₁₁	—	2	8	no	—
51	C₄×C₂₁₁	—	4	8	no	—
52	C₄×C₂₁₁	C₂×C₂²⋊C₄	1	8	yes	—
53	C₂×C₂₁₁	—	2	4	no	—
54	C₂×C₂₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₂₁₁	#519	1	2	yes	—
56	C₂×C₂₁₁	#536	1	4	yes	—
57	C₂₁₁	#2310	1	1	yes	characteristic, direct product factor, Sylow
58	#2310	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×Q₈	C₂×C₂₁₁	1	1	yes	characteristic
60	C₂×#117	C₂×C₂₁₁	1	2	yes	—
61	#391	C₂×C₂₁₁	1	8	yes	complemented
62	#381	C₂×C₂₁₁	1	4	yes	—
63	C₂²×Q₈	—	2	8	no	—
64	C₂²×Q₈	C₄×C₂₁₁	1	8	yes	complemented
65	C₂²×Q₈	C₂²×C₂₁₁	1	4	yes	—
66	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
67	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
68	#117	—	2	8	no	—
69	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	—
70	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
71	C₂×C₄²	C₂²×C₂₁₁	1	4	yes	—
72	#95	C₂²×C₂₁₁	1	16	yes	—
73	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
74	C₂×Q₈	—	2	32	no	—
75	C₂×Q₈	—	2	8	no	—
76	C₂×Q₈	—	2	8	no	—
77	C₂×Q₈	C₂×C₄×C₂₁₁	1	16	yes	—
78	C₂²×C₄	—	2	8	no	—
79	C₂²×C₄	C₂×C₄×C₂₁₁	1	8	yes	—
80	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
81	C₂²×C₄	D₄×C₂₁₁	1	8	yes	—
82	C₂²×C₄	C₂³×C₂₁₁	1	8	yes	—
83	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
84	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
85	C₄⋊C₄	—	2	8	no	—
86	C₂²⋊C₄	—	2	8	no	—
87	C₄²	—	2	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
90	C₂³	C₂×D₄×C₂₁₁	1	4	yes	—
91	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
92	Q₈	—	2	32	no	—
93	C₂×C₄	—	2	16	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	2	8	no	—
96	C₂×C₄	—	2	16	no	—
97	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	16	yes	—
98	C₂×C₄	C₂²×C₄×C₂₁₁	1	8	yes	—
99	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
100	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
101	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
102	C₂²	—	2	8	no	—
103	C₂²	—	2	4	no	—
104	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	4	yes	—
105	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
106	C₂²	C₂²×D₄×C₂₁₁	1	2	yes	—
107	C₂²	2₋¹⁺⁴×C₂₁₁	1	4	yes	—
108	C₄	—	2	8	no	—
109	C₄	—	4	8	no	—
110	C₄	C₂×C₂²⋊C₄×C₂₁₁	1	8	yes	—
111	C₂	—	2	4	no	—
112	C₂	C₂²×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
113	C₂	#519×C₂₁₁	1	2	yes	—
114	C₂	#536×C₂₁₁	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Q₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#117×C₂₁₁

C₂

yes

maximal

#391×C₂₁₁

C₂

yes

maximal, complemented

#381×C₂₁₁

C₂

yes

maximal

C₂²×Q₈×C₂₁₁

—

C₂²×Q₈×C₂₁₁

C₄

yes

complemented

C₂²×Q₈×C₂₁₁

C₂²

yes

—

C₂³×C₄×C₂₁₁

C₂²

yes

—

C₂³×C₄×C₂₁₁

C₂²

yes

characteristic

#117×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

—

#95×C₂₁₁

C₂²

yes

—

C₂⁴×C₂₁₁

C₂³

yes

characteristic

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₂²×C₄

yes

—

C₂³×C₂₁₁

C₂×D₄

yes

—

C₂³×C₂₁₁

C₂⁴

yes

characteristic

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₁₁

C₂³×C₄

yes

characteristic

C₂²×C₂₁₁

C₂²×D₄

yes

—

C₂²×C₂₁₁

2₋¹⁺⁴

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₂²⋊C₄

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₁₁

#519

yes

—

C₂×C₂₁₁

#536

yes

—

C₂₁₁

#2310

yes

characteristic, direct product factor, Sylow

#2310

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×Q₈

C₂×C₂₁₁

yes

characteristic

C₂×#117

C₂×C₂₁₁

yes

—

#391

C₂×C₂₁₁

yes

complemented

#381

C₂×C₂₁₁

yes

—

C₂²×Q₈

—

C₂²×Q₈

C₄×C₂₁₁

yes

complemented

C₂²×Q₈

C₂²×C₂₁₁

yes

—

C₂³×C₄

C₂²×C₂₁₁

yes

—

C₂³×C₄

C₂²×C₂₁₁

yes

characteristic

#117

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄²

C₂²×C₂₁₁

yes

—

#95

C₂²×C₂₁₁

yes

—

C₂⁴

C₂³×C₂₁₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₂²×C₄×C₂₁₁

yes

—

C₂³

C₂×D₄×C₂₁₁

yes

—

C₂³

C₂⁴×C₂₁₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

C₂×C₄

C₂²×C₄×C₂₁₁

yes

—

C₂×C₄

C₂²×C₄×C₂₁₁

yes

—

100

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

101

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₂²⋊C₄×C₂₁₁

yes

—

105

C₂²

C₂³×C₄×C₂₁₁

yes

characteristic

106

C₂²

C₂²×D₄×C₂₁₁

yes

—

107

C₂²

2₋¹⁺⁴×C₂₁₁

yes

—

108

C₄

—

109

C₄

—

110

C₄

C₂×C₂²⋊C₄×C₂₁₁

yes

—

111

C₂

—

112

C₂

C₂²×C₂²⋊C₄×C₂₁₁

yes

characteristic

113

C₂

#519×C₂₁₁

yes

—

114

C₂

#536×C₂₁₁

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
prime radical	C₂⁴×C₂₁₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₁₁

C₂⁴

commutator

C₂²

C₂³×C₄×C₂₁₁

Frattini, cosocle

C₂³

C₂⁴×C₂₁₁

prime radical

C₂⁴×C₂₁₁

C₂³

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₁₁ ◁ G

N	H ≅ G/N	type
3. #391×C₂₁₁	C₂	semidirect
6. C₂²×Q₈×C₂₁₁	C₄	semidirect
57. C₂₁₁	#2310	direct (N × H)
58. #2310	C₂₁₁	direct (N × H)
61. #391	C₂×C₂₁₁	semidirect
64. C₂²×Q₈	C₄×C₂₁₁	semidirect

H ≅ G/N

type

3. #391×C₂₁₁

C₂

semidirect

6. C₂²×Q₈×C₂₁₁

C₄

semidirect

57. C₂₁₁

#2310

direct (N × H)

58. #2310

C₂₁₁

direct (N × H)

61. #391

C₂×C₂₁₁

semidirect

64. C₂²×Q₈

C₄×C₂₁₁

semidirect

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

subgroup	index
C₂³×Q₈×C₂₁₁	2
C₂×#117×C₂₁₁	2
#391×C₂₁₁	2
#381×C₂₁₁	2
#2310	211

subgroup

index

C₂³×Q₈×C₂₁₁

C₂×#117×C₂₁₁

#391×C₂₁₁

#381×C₂₁₁

#2310

211

Subgroups — #2310×C₂₁₁ #41009930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2310×C211 #41009930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2310×C₂₁₁ #41009930