Subgroups — #41011480

Order 27008.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269917	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#20269917	—	2	2	no	—
4	#20269917	C₄	1	4	yes	complemented
5	C₂×C₂₁₁⋊C₁₆	—	2	2	no	—
6	C₂²×C₈×C₂₁₁	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₂₁₁	C₄	1	1	yes	characteristic
9	C₄×C₈×C₂₁₁	—	2	2	no	—
10	C₂₁₁⋊C₁₆	—	2	4	no	—
11	C₂²×C₄×C₂₁₁	—	2	1	no	—
12	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₂₁₁	—	2	1	no	—
14	C₂×C₈×C₂₁₁	—	2	1	no	—
15	C₂×C₈×C₂₁₁	C₈	1	2	yes	—
16	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
18	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₂₁₁	Q₈	1	1	yes	characteristic
20	C₄²×C₂₁₁	—	2	1	no	—
21	C₄²×C₂₁₁	C₈	1	2	yes	—
22	C₂³×C₂₁₁	M₁₆	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	—	2	4	no	—
24	C₂×C₄×C₂₁₁	—	2	2	no	—
25	C₂×C₄×C₂₁₁	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
31	C₈×C₂₁₁	—	2	2	no	—
32	C₈×C₂₁₁	C₂²⋊C₄	1	2	yes	—
33	C₈×C₂₁₁	C₄⋊C₄	1	2	yes	—
34	C₂²×C₂₁₁	—	2	2	no	—
35	C₂²×C₂₁₁	#97	1	1	yes	characteristic
36	C₂²×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₂₁₁	C₄⋊C₈	1	1	yes	characteristic
38	C₄×C₂₁₁	—	2	4	no	—
39	C₄×C₂₁₁	#95	1	1	yes	characteristic
40	C₄×C₂₁₁	C₄×C₈	1	1	yes	characteristic
41	C₄×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₂₁₁	C₄⋊C₈	1	1	yes	characteristic
43	C₂×C₂₁₁	—	2	1	no	—
44	C₂×C₂₁₁	—	2	1	no	—
45	C₂×C₂₁₁	#336	1	1	yes	characteristic
46	C₂×C₂₁₁	#364	1	2	yes	—
47	C₂₁₁	#1376	1	1	yes	characteristic, complemented, Sylow
48	#1376	—	211	1	no	maximal, Sylow
49	C₂×M₃₂	—	211	2	no	—
50	C₂×C₄×C₈	D₂₁₁	1	1	yes	characteristic
51	C₂²×C₈	Dic₂₁₁	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
53	C₂×C₄²	Dic₂₁₁	1	1	yes	characteristic
54	M₃₂	—	422	2	no	—
55	M₃₂	—	211	4	no	—
56	C₂×C₁₆	—	422	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₂²×C₄	—	2	1	no	—
59	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	C₂₁₁⋊C₈	1	2	yes	—
63	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
64	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
65	C₂×C₈	D₈₄₄	1	1	yes	characteristic
66	C₂×C₈	Dic₄₂₂	1	1	yes	characteristic
67	C₄²	—	2	1	no	—
68	C₄²	C₂₁₁⋊C₈	1	2	yes	—
69	C₁₆	—	422	4	no	—
70	C₂³	#16809946	1	1	yes	characteristic
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	C₄×Dic₂₁₁	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
75	C₂×C₄	C₄⋊Dic₂₁₁	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂₁₁⋊C₈	1	1	yes	characteristic
77	C₂×C₄	C₂×C₂₁₁⋊C₈	1	1	yes	characteristic
78	C₂×C₄	#16809946	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
81	C₈	Dic₂₁₁⋊C₄	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	#20269888	1	1	yes	characteristic
84	C₂²	C₂²⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
85	C₂²	C₄⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₂₁₁⋊#95	1	1	yes	characteristic
88	C₄	C₄×C₂₁₁⋊C₈	1	1	yes	characteristic
89	C₄	C₂²⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
90	C₄	C₄⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#27212901	1	1	yes	characteristic
94	C₂	#27212967	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₈₄₄
commutator	C₄×C₂₁₁	C₄×C₈
Frattini	C₂²×C₈	C₂×D₂₁₁
Fitting	C₂×C₄×C₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₄⋊C₈
prime radical	C₂³×C₂₁₁	M₁₆
cosocle	C₂²×C₈×C₂₁₁	C₂²
hypercenter	C₂×C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	#1376
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #20269917	C₄	semidirect
47. C₂₁₁	#1376	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269917	2
C₂×C₄×C₈×C₂₁₁	2
#1376	211