Subgroups — #41013535

Order 27008.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#27212910	C₂	1	4	yes	maximal
4	#423×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16809946	C₂²	1	2	yes	—
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
8	C₄×C₂₁₁⋊C₈	C₄	1	2	yes	—
9	C₄×C₂₁₁⋊C₈	C₄	1	2	yes	—
10	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₁₁	—	2	2	no	—
13	#16809946	—	2	4	no	—
14	C₂×C₂₁₁⋊C₈	—	2	2	no	—
15	C₂×C₂₁₁⋊C₈	—	2	2	no	—
16	C₂×C₂₁₁⋊C₈	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
19	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
20	M₁₆×C₂₁₁	—	2	4	no	—
21	C₂×C₈×C₂₁₁	—	2	2	no	—
22	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₁₁	D₄	1	2	yes	—
25	C₂₁₁⋊C₈	—	2	4	no	—
26	C₂₁₁⋊C₈	—	4	2	no	—
27	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₂₁₁	—	2	4	no	—
29	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	#41	1	2	yes	—
34	C₂×C₄×C₂₁₁	#41	1	2	yes	—
35	C₈×C₂₁₁	—	4	2	no	—
36	C₂²×C₂₁₁	—	2	2	no	—
37	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₂₁₁	#125	1	2	yes	—
39	C₄×C₂₁₁	—	2	2	no	—
40	C₄×C₂₁₁	#108	1	4	yes	—
41	C₄×C₂₁₁	#117	1	2	yes	—
42	C₄×C₂₁₁	#124	1	1	yes	characteristic
43	C₄×C₂₁₁	#124	1	1	yes	characteristic
44	C₂×C₂₁₁	—	2	2	no	—
45	C₂×C₂₁₁	#382	1	1	yes	characteristic
46	C₂×C₂₁₁	C₂×#108	1	2	yes	—
47	C₂₁₁	#1837	1	1	yes	characteristic, complemented, Sylow
48	#1837	—	211	1	no	maximal, Sylow
49	#423	—	211	1	no	—
50	#423	D₂₁₁	1	1	yes	characteristic
51	C₂×C₄×C₈	—	211	1	no	—
52	#341	—	211	4	no	—
53	C₂×M₁₆	—	211	2	no	—
54	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
55	C₂²×C₈	—	211	2	no	—
56	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	422	2	no	—
59	C₄×C₈	—	211	2	no	—
60	C₄×C₈	—	211	2	no	—
61	C₂²×C₄	Dic₄₂₂	1	2	yes	—
62	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	—	422	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	422	2	no	—
67	C₂×C₈	—	422	2	no	—
68	C₂×C₈	—	422	2	no	—
69	C₂×C₈	—	211	4	no	—
70	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
71	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
72	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
73	C₂³	C₂×Dic₄₂₂	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
76	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
79	C₂×C₄	#16809961	1	2	yes	—
80	C₂×C₄	#16809963	1	2	yes	—
81	C₈	—	4	2	no	—
82	C₈	—	844	2	no	—
83	C₈	—	422	4	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
86	C₂²	#20269968	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#20269912	1	4	yes	—
89	C₄	#20269935	1	2	yes	—
90	C₄	#20269964	1	1	yes	characteristic
91	C₄	#20269966	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#27213011	1	1	yes	characteristic
94	C₂	C₂×#20269912	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#423×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#423	D₂₁₁
hypocenter	C₂₁₁	#1837
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
47. C₂₁₁	#1837	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×C₂₁₁⋊C₈	2
#27212910	2
#423×C₂₁₁	2
#1837	211