Subgroups — #1534×C₂₁₁ #41009154

Order 27008.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#432×C₂₁₁	C₂	1	2	yes	maximal
3	#330×C₂₁₁	C₂	1	4	yes	maximal, complemented
4	#125×C₂₁₁	—	2	2	no	—
5	#125×C₂₁₁	C₄	1	2	yes	—
6	#125×C₂₁₁	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
8	C₂×C₄⋊C₄×C₂₁₁	C₄	1	2	yes	—
9	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₄⋊C₈×C₂₁₁	—	2	2	no	—
11	C₂²⋊C₈×C₂₁₁	—	2	2	no	—
12	#97×C₂₁₁	C₂²	1	4	yes	—
13	C₂²×C₄×C₂₁₁	—	2	2	no	—
14	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
16	C₂×C₈×C₂₁₁	—	2	4	no	—
17	C₄⋊C₄×C₂₁₁	—	2	4	no	—
18	C₄⋊C₄×C₂₁₁	—	2	2	no	—
19	C₄⋊C₄×C₂₁₁	—	2	4	no	—
20	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
21	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₁₁	D₄	1	2	yes	—
23	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₁₁	—	2	2	no	—
26	C₂×C₄×C₂₁₁	—	2	4	no	—
27	C₂×C₄×C₂₁₁	—	4	2	no	—
28	C₂×C₄×C₂₁₁	—	4	4	no	—
29	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
33	C₈×C₂₁₁	—	4	4	no	—
34	C₂²×C₂₁₁	—	2	2	no	—
35	C₂²×C₂₁₁	#101	1	2	yes	—
36	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₄×C₂₁₁	—	2	1	no	—
38	C₄×C₂₁₁	—	2	1	no	—
39	C₄×C₂₁₁	—	2	2	no	—
40	C₄×C₂₁₁	—	2	2	no	—
41	C₄×C₂₁₁	—	4	4	no	—
42	C₂×C₂₁₁	—	2	2	no	—
43	C₂×C₂₁₁	C₂×#101	1	1	yes	characteristic
44	C₂×C₂₁₁	#421	1	2	yes	—
45	C₂₁₁	#1534	1	1	yes	characteristic, direct product factor, Sylow
46	#1534	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂×#125	C₂×C₂₁₁	1	1	yes	characteristic
48	#432	C₂×C₂₁₁	1	2	yes	—
49	#330	C₂×C₂₁₁	1	4	yes	complemented
50	#125	—	2	2	no	—
51	#125	C₄×C₂₁₁	1	2	yes	—
52	#125	C₂²×C₂₁₁	1	2	yes	—
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄⋊C₄	C₄×C₂₁₁	1	2	yes	—
55	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
56	C₄⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	2	2	no	—
58	#97	C₂²×C₂₁₁	1	4	yes	—
59	C₂²×C₄	—	2	2	no	—
60	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
61	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
62	C₂×C₈	—	2	4	no	—
63	C₄⋊C₄	—	2	4	no	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
67	C₄²	C₂×C₄×C₂₁₁	1	1	yes	characteristic
68	C₄²	D₄×C₂₁₁	1	2	yes	—
69	C₄²	C₂³×C₂₁₁	1	1	yes	characteristic
70	C₂³	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	4	no	—
75	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
76	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
77	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
78	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
79	C₈	—	4	4	no	—
80	C₂²	—	2	2	no	—
81	C₂²	#101×C₂₁₁	1	2	yes	—
82	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	2	no	—
86	C₄	—	2	2	no	—
87	C₄	—	4	4	no	—
88	C₂	—	2	2	no	—
89	C₂	C₂×#101×C₂₁₁	1	1	yes	characteristic
90	C₂	#421×C₂₁₁	1	2	yes	—

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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #330×C₂₁₁	C₂	semidirect
45. C₂₁₁	#1534	direct (N × H)
46. #1534	C₂₁₁	direct (N × H)
49. #330	C₂×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#125×C₂₁₁	2
#432×C₂₁₁	2
#330×C₂₁₁	2
#1534	211

Subgroups — #1534×C211 #41009154

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1534×C₂₁₁ #41009154