Group #41009161 — subgroups

Subgroups — C₂×#333×C₂₁₁ #41009161

Order 27008.

256 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 136 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⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₂₁₁	C₂	1	2	yes	maximal
3	#333×C₂₁₁	C₂	1	4	yes	maximal, direct product factor
4	C₂²×Q₈×C₂₁₁	C₄	1	2	yes	—
5	C₂²×C₈×C₂₁₁	—	2	2	no	—
6	C₄⋊Q₈×C₂₁₁	—	2	2	no	—
7	C₄⋊Q₈×C₂₁₁	C₄	1	2	yes	—
8	C₄⋊Q₈×C₂₁₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
10	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₂₁₁	—	2	2	no	—
12	C₄⋊C₈×C₂₁₁	C₂²	1	4	yes	—
13	C₂×Q₈×C₂₁₁	—	2	2	no	—
14	C₂×Q₈×C₂₁₁	—	4	4	no	—
15	C₂×Q₈×C₂₁₁	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₂₁₁	—	2	2	no	—
17	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
19	C₂×C₈×C₂₁₁	—	2	4	no	—
20	C₂×C₈×C₂₁₁	—	4	4	no	—
21	C₄⋊C₄×C₂₁₁	—	2	4	no	—
22	C₄⋊C₄×C₂₁₁	—	2	4	no	—
23	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₁₁	D₄	1	2	yes	—
25	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₂₁₁	—	4	4	no	—
28	Q₈×C₂₁₁	—	4	4	no	—
29	C₂×C₄×C₂₁₁	—	2	2	no	—
30	C₂×C₄×C₂₁₁	—	2	4	no	—
31	C₂×C₄×C₂₁₁	—	4	4	no	—
32	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₂₁₁	SD₁₆	1	4	yes	—
35	C₂×C₄×C₂₁₁	Q₁₆	1	4	yes	—
36	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
38	C₈×C₂₁₁	—	4	4	no	—
39	C₂²×C₂₁₁	#100	1	2	yes	—
40	C₂²×C₂₁₁	#103	1	4	yes	—
41	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₂₁₁	—	2	1	no	—
43	C₄×C₂₁₁	—	2	1	no	—
44	C₄×C₂₁₁	—	4	4	no	—
45	C₄×C₂₁₁	#103	1	4	yes	—
46	C₄×C₂₁₁	C₂×SD₁₆	1	2	yes	—
47	C₄×C₂₁₁	C₂×Q₁₆	1	2	yes	—
48	C₂×C₂₁₁	#333	1	4	yes	direct product factor
49	C₂×C₂₁₁	C₂×#100	1	1	yes	characteristic
50	C₂×C₂₁₁	C₂×#103	1	2	yes	—
51	C₂₁₁	C₂×#333	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×#333	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×C₄⋊Q₈	C₂×C₂₁₁	1	1	yes	characteristic
54	C₂×C₄⋊C₈	C₂×C₂₁₁	1	2	yes	—
55	#333	C₂×C₂₁₁	1	4	yes	direct product factor
56	C₂²×Q₈	C₄×C₂₁₁	1	2	yes	—
57	C₂²×C₈	—	2	2	no	—
58	C₄⋊Q₈	—	2	2	no	—
59	C₄⋊Q₈	C₄×C₂₁₁	1	2	yes	—
60	C₄⋊Q₈	C₂²×C₂₁₁	1	2	yes	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	C₂²×C₂₁₁	1	4	yes	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	4	4	no	—
67	C₂×Q₈	C₂×C₄×C₂₁₁	1	4	yes	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
70	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	4	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄²	C₂×C₄×C₂₁₁	1	1	yes	characteristic
76	C₄²	D₄×C₂₁₁	1	2	yes	—
77	C₄²	C₂³×C₂₁₁	1	1	yes	characteristic
78	C₂³	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
79	Q₈	—	4	4	no	—
80	Q₈	—	4	4	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
86	C₂×C₄	SD₁₆×C₂₁₁	1	4	yes	—
87	C₂×C₄	Q₁₆×C₂₁₁	1	4	yes	—
88	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
89	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
90	C₈	—	4	4	no	—
91	C₂²	#100×C₂₁₁	1	2	yes	—
92	C₂²	#103×C₂₁₁	1	4	yes	—
93	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	2	1	no	—
96	C₄	—	4	4	no	—
97	C₄	#103×C₂₁₁	1	4	yes	—
98	C₄	C₂×SD₁₆×C₂₁₁	1	2	yes	—
99	C₄	C₂×Q₁₆×C₂₁₁	1	2	yes	—
100	C₂	#333×C₂₁₁	1	4	yes	direct product factor
101	C₂	C₂×#100×C₂₁₁	1	1	yes	characteristic
102	C₂	C₂×#103×C₂₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₁₁

C₂

yes

maximal

#333×C₂₁₁

C₂

yes

maximal, direct product factor

C₂²×Q₈×C₂₁₁

C₄

yes

—

C₂²×C₈×C₂₁₁

—

C₄⋊Q₈×C₂₁₁

—

C₄⋊Q₈×C₂₁₁

C₄

yes

—

C₄⋊Q₈×C₂₁₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₄⋊C₈×C₂₁₁

C₂²

yes

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂³×C₂₁₁

C₂²⋊C₄

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

#100

yes

—

C₂²×C₂₁₁

#103

yes

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#103

yes

—

C₄×C₂₁₁

C₂×SD₁₆

yes

—

C₄×C₂₁₁

C₂×Q₁₆

yes

—

C₂×C₂₁₁

#333

yes

direct product factor

C₂×C₂₁₁

C₂×#100

yes

characteristic

C₂×C₂₁₁

C₂×#103

yes

—

C₂₁₁

C₂×#333

yes

characteristic, direct product factor, Sylow

C₂×#333

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₂₁₁

yes

characteristic

C₂×C₄⋊C₈

C₂×C₂₁₁

yes

—

#333

C₂×C₂₁₁

yes

direct product factor

C₂²×Q₈

C₄×C₂₁₁

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₂₁₁

yes

—

C₄⋊Q₈

C₂²×C₂₁₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₂₁₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

characteristic

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₂₁₁

yes

characteristic

C₄²

D₄×C₂₁₁

yes

—

C₄²

C₂³×C₂₁₁

yes

characteristic

C₂³

C₂²⋊C₄×C₂₁₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

C₂×C₄

SD₁₆×C₂₁₁

yes

—

C₂×C₄

Q₁₆×C₂₁₁

yes

—

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

C₈

—

C₂²

#100×C₂₁₁

yes

—

C₂²

#103×C₂₁₁

yes

—

C₂²

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#103×C₂₁₁

yes

—

C₄

C₂×SD₁₆×C₂₁₁

yes

—

C₄

C₂×Q₁₆×C₂₁₁

yes

—

100

C₂

#333×C₂₁₁

yes

direct product factor

101

C₂

C₂×#100×C₂₁₁

yes

characteristic

102

C₂

C₂×#103×C₂₁₁

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₁₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₄²	C₂³×C₂₁₁
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₁₁

C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₂₁₁

Frattini, cosocle

C₄²

C₂³×C₂₁₁

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

N	H ≅ G/N	type
3. #333×C₂₁₁	C₂	direct (N × H)
48. C₂×C₂₁₁	#333	direct (N × H)
51. C₂₁₁	C₂×#333	direct (N × H)
52. C₂×#333	C₂₁₁	direct (N × H)
55. #333	C₂×C₂₁₁	direct (N × H)
100. C₂	#333×C₂₁₁	direct (N × H)

H ≅ G/N

type

3. #333×C₂₁₁

C₂

direct (N × H)

48. C₂×C₂₁₁

#333

direct (N × H)

51. C₂₁₁

C₂×#333

direct (N × H)

52. C₂×#333

C₂₁₁

direct (N × H)

55. #333

C₂×C₂₁₁

direct (N × H)

100. C₂

#333×C₂₁₁

direct (N × H)

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

subgroup	index
C₂×C₄⋊Q₈×C₂₁₁	2
C₂×C₄⋊C₈×C₂₁₁	2
#333×C₂₁₁	2
C₂×#333	211

subgroup

index

C₂×C₄⋊Q₈×C₂₁₁

C₂×C₄⋊C₈×C₂₁₁

#333×C₂₁₁

C₂×#333

211

Subgroups — C₂×#333×C₂₁₁ #41009161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#333×C211 #41009161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#333×C₂₁₁ #41009161