Subgroups — #41013537

Order 27008.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213169	C₂	1	2	yes	maximal
2	C₂×#20269887	C₂	1	1	yes	maximal, characteristic
3	#27212911	C₂	1	1	yes	maximal, characteristic
4	#27212910	C₂	1	2	yes	maximal
5	#341×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
9	Dic₂₁₁⋊C₈	—	2	4	no	—
10	#20269887	C₄	1	4	yes	—
11	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
12	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₁₁	Q₈	1	2	yes	—
14	#16809946	—	2	2	no	—
15	#16809946	—	2	2	no	—
16	C₂×C₂₁₁⋊C₈	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₄×Dic₂₁₁	C₂×C₄	1	2	yes	—
20	C₄×Dic₂₁₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₂₁₁	—	2	2	no	—
23	M₁₆×C₂₁₁	—	2	2	no	—
24	C₂×C₈×C₂₁₁	—	2	2	no	—
25	C₂×C₈×C₂₁₁	—	2	2	no	—
26	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
27	C₂×Dic₂₁₁	—	4	2	no	—
28	C₂×Dic₂₁₁	C₄⋊C₄	1	4	yes	—
29	C₂₁₁⋊C₈	—	2	2	no	—
30	C₂₁₁⋊C₈	—	4	2	no	—
31	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	#41	1	2	yes	—
35	C₂×C₄×C₂₁₁	#41	1	2	yes	—
36	C₈×C₂₁₁	—	2	2	no	—
37	C₈×C₂₁₁	—	4	2	no	—
38	C₂²×C₂₁₁	—	2	2	no	—
39	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₁₁	#125	1	2	yes	—
41	C₄×C₂₁₁	#117	1	1	yes	characteristic
42	C₄×C₂₁₁	#117	1	1	yes	characteristic
43	C₄×C₂₁₁	#124	1	2	yes	—
44	Dic₂₁₁	—	2	4	no	—
45	C₂×C₂₁₁	—	2	2	no	—
46	C₂×C₂₁₁	#382	1	1	yes	characteristic
47	C₂×C₂₁₁	#430	1	2	yes	—
48	C₂₁₁	#1838	1	1	yes	characteristic, complemented, Sylow
49	#1838	—	211	1	no	maximal, Sylow
50	#423	—	211	2	no	—
51	C₂×#97	—	211	1	no	—
52	#341	—	211	2	no	—
53	#341	—	211	1	no	—
54	#341	D₂₁₁	1	1	yes	characteristic
55	C₂×M₁₆	—	211	2	no	—
56	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
57	C₂²×C₈	—	211	1	no	—
58	C₂²×C₈	C₂×D₂₁₁	1	1	yes	characteristic
59	C₂×C₄²	—	211	1	no	—
60	C₄⋊C₈	—	422	4	no	—
61	#97	—	211	4	no	—
62	C₂²×C₄	—	211	2	no	—
63	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
64	M₁₆	—	2	2	no	—
65	M₁₆	—	2	2	no	—
66	M₁₆	—	422	2	no	—
67	M₁₆	—	422	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	422	2	no	—
71	C₂×C₈	—	422	2	no	—
72	C₂×C₈	—	211	2	no	—
73	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
74	C₄²	—	211	2	no	—
75	C₄²	—	211	2	no	—
76	C₂³	Q₈×D₂₁₁	1	1	yes	characteristic
77	C₂×C₄	—	844	2	no	—
78	C₂×C₄	—	211	4	no	—
79	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
80	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
81	C₂×C₄	#16809961	1	2	yes	—
82	C₂×C₄	#16809962	1	2	yes	—
83	C₈	—	2	2	no	—
84	C₈	—	4	2	no	—
85	C₈	—	844	2	no	—
86	C₈	—	422	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
89	C₂²	#20269967	1	2	yes	—
90	C₄	—	422	4	no	—
91	C₄	#20269933	1	1	yes	characteristic
92	C₄	#20269934	1	1	yes	characteristic
93	C₄	#20269963	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	#27213010	1	1	yes	characteristic
96	C₂	#27213193	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#341×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#341	D₂₁₁
hypocenter	C₂₁₁	#1838
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

Complements

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

N	H ≅ G/N	type
48. C₂₁₁	#1838	semidirect

Maximal subgroups

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

subgroup	index
#27213169	2
C₂×#20269887	2
#27212911	2
#27212910	2
#341×C₂₁₁	2
#1838	211