Subgroups — #41011273

Order 27008.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213170	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16809946	—	2	2	no	—
4	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
5	C₄⋊(C₂₁₁⋊C₈)	C₄	1	4	yes	—
6	C₂²×C₈×C₂₁₁	C₄	1	2	yes	—
7	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₂₁₁	—	2	2	no	—
9	#16809946	—	4	2	no	—
10	#16809946	—	4	2	no	—
11	C₂×C₂₁₁⋊C₈	—	2	4	no	—
12	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₁₁	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₂₁₁	—	2	2	no	—
16	C₂×C₈×C₂₁₁	—	4	2	no	—
17	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
18	C₄²×C₂₁₁	D₄	1	2	yes	—
19	C₂₁₁⋊C₈	—	4	4	no	—
20	C₂³×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₁₁	—	2	1	no	—
22	C₂×C₄×C₂₁₁	—	2	1	no	—
23	C₂×C₄×C₂₁₁	—	4	1	no	—
24	C₂×C₄×C₂₁₁	C₄²	1	1	yes	characteristic
25	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
30	C₂×C₄×C₂₁₁	Q₁₆	1	1	yes	characteristic
31	C₈×C₂₁₁	—	4	2	no	—
32	C₂²×C₂₁₁	—	2	1	no	—
33	C₂²×C₂₁₁	—	2	1	no	—
34	C₂²×C₂₁₁	#95	1	1	yes	characteristic
35	C₂²×C₂₁₁	#106	1	1	yes	characteristic
36	C₂²×C₂₁₁	#107	1	1	yes	characteristic
37	C₄×C₂₁₁	—	2	2	no	—
38	C₄×C₂₁₁	—	2	1	no	—
39	C₄×C₂₁₁	#100	1	1	yes	characteristic
40	C₄×C₂₁₁	#101	1	1	yes	characteristic
41	C₄×C₂₁₁	#102	1	2	yes	—
42	C₄×C₂₁₁	#103	1	2	yes	—
43	C₂×C₂₁₁	—	2	1	no	—
44	C₂×C₂₁₁	—	2	1	no	—
45	C₂×C₂₁₁	#338	1	1	yes	characteristic
46	C₂×C₂₁₁	#340	1	1	yes	characteristic
47	C₂×C₂₁₁	#343	1	1	yes	characteristic
48	C₂₁₁	#1292	1	1	yes	characteristic, complemented, Sylow
49	#1292	—	211	1	no	maximal, Sylow
50	#423	—	211	2	no	—
51	C₂×C₄⋊C₈	D₂₁₁	1	1	yes	characteristic
52	C₂×M₁₆	—	422	2	no	—
53	C₂²×C₈	Dic₂₁₁	1	2	yes	—
54	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	422	2	no	—
57	C₄⋊C₈	—	211	4	no	—
58	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
59	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
60	C₂²×C₄	Dic₄₂₂	1	1	yes	characteristic
61	M₁₆	—	844	2	no	—
62	M₁₆	—	844	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	422	4	no	—
66	C₄²	C₄×D₂₁₁	1	2	yes	—
67	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
68	C₂³	C₄⋊Dic₂₁₁	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	C₄×Dic₂₁₁	1	1	yes	characteristic
73	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
74	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
75	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
76	C₂×C₄	D₁₆₈₈	1	1	yes	characteristic
77	C₂×C₄	#16809951	1	2	yes	—
78	C₂×C₄	Dic₈₄₄	1	1	yes	characteristic
79	C₈	—	4	2	no	—
80	C₈	—	844	4	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂₁₁⋊#95	1	1	yes	characteristic
84	C₂²	#20269909	1	1	yes	characteristic
85	C₂²	#20269911	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	#20269894	1	1	yes	characteristic
89	C₄	#20269896	1	1	yes	characteristic
90	C₄	#20269899	1	2	yes	—
91	C₄	#20269902	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	C₂₁₁⋊#338	1	1	yes	characteristic
95	C₂	#27212909	1	1	yes	characteristic
96	C₂	#27212915	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂₁₁⋊#95
commutator	C₂×C₄×C₂₁₁	C₄²
Frattini	C₂×C₄²	C₂×D₂₁₁
Fitting	C₂×C₄⋊C₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#95
prime radical	C₂³×C₂₁₁	C₄⋊C₄
cosocle	C₂×C₄²×C₂₁₁	C₂²
hypercenter	C₂×C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	#1292
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
48. C₂₁₁	#1292	semidirect

Maximal subgroups

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

subgroup	index
#27213170	2
C₂×C₄⋊C₈×C₂₁₁	2
#1292	211