Subgroups — #27213154

Order 13504.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16809946	C₂	1	1	yes	maximal, characteristic
3	#20269900	C₂	1	4	yes	maximal, complemented
4	#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₂₂	—	2	1	no	—
6	C₂×Dic₄₂₂	—	2	2	no	—
7	C₂×Dic₄₂₂	C₄	1	4	yes	complemented
8	C₂×Dic₄₂₂	C₂²	1	2	yes	—
9	C₂²×Dic₂₁₁	—	2	1	no	—
10	#16809946	—	2	2	no	—
11	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
15	C₄²×C₂₁₁	—	2	1	no	—
16	Dic₄₂₂	—	2	4	no	—
17	Dic₄₂₂	—	2	2	no	—
18	Dic₄₂₂	C₂×C₄	1	4	yes	—
19	C₂×Dic₂₁₁	—	2	4	no	—
20	C₂×Dic₂₁₁	—	2	2	no	—
21	C₂₁₁⋊C₈	—	2	2	no	—
22	C₂³×C₂₁₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	—	2	2	no	—
24	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
27	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₂²×C₂₁₁	—	2	2	no	—
29	C₂²×C₂₁₁	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₁₁	—	4	2	no	—
32	C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
33	C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	Dic₂₁₁	—	2	4	no	—
36	C₂×C₂₁₁	—	2	2	no	—
37	C₂×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₂₁₁	#137	1	2	yes	—
39	C₂₁₁	#419	1	1	yes	characteristic, complemented, Sylow
40	#419	—	211	1	no	maximal, Sylow
41	C₂²×Q₈	—	211	1	no	—
42	C₂×M₁₆	—	211	1	no	—
43	#117	D₂₁₁	1	1	yes	characteristic
44	#103	—	211	4	no	—
45	C₂×Q₈	—	422	1	no	—
46	C₂×Q₈	—	422	2	no	—
47	C₂×Q₈	—	211	4	no	—
48	C₂×Q₈	—	211	2	no	—
49	C₂²×C₄	—	422	1	no	—
50	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
51	M₁₆	—	422	2	no	—
52	C₂×C₈	—	211	2	no	—
53	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	—	2	1	no	—
56	C₂³	C₂²⋊D₂₁₁	1	1	yes	characteristic
57	Q₈	—	422	4	no	—
58	Q₈	—	422	2	no	—
59	Q₈	—	211	4	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	422	4	no	—
62	C₂×C₄	—	422	2	no	—
63	C₂×C₄	C₄×D₂₁₁	1	2	yes	—
64	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
65	C₂×C₄	D₈₄₄	1	2	yes	—
66	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
67	C₈	—	422	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
71	C₄	—	4	2	no	—
72	C₄	—	422	4	no	—
73	C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
74	C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
75	C₄	C₂×D₈₄₄	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
78	C₂	#20270007	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₁₁
commutator	C₄×C₂₁₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#117×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂³×C₂₁₁	D₄
cosocle	C₂×C₄×C₂₁₁	C₂³
hypercenter	#117	D₂₁₁
hypocenter	C₂₁₁	#419
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #20269900	C₂	semidirect
7. C₂×Dic₄₂₂	C₄	semidirect
39. C₂₁₁	#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₂₂	2
C₂×#16809946	2
#20269900	2
#117×C₂₁₁	2
#419	211