Subgroups — #27213622

Order 13504.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₂₁₁	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×Dic₄₂₂	C₂	1	3	yes	maximal, complemented
4	#20269942	C₂	1	3	yes	maximal, complemented
5	#20269935	C₂	1	3	yes	maximal
6	#20269934	C₂	1	3	yes	maximal
7	C₄×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₄₂₂	C₂²	1	3	yes	—
9	Q₈×D₂₁₁	C₄	1	8	yes	complemented
10	C₂×C₄×D₂₁₁	C₂²	1	3	yes	—
11	C₄⋊Dic₂₁₁	C₂²	1	3	yes	—
12	Dic₂₁₁⋊C₄	C₂²	1	3	yes	—
13	Dic₂₁₁⋊C₄	C₂²	1	3	yes	—
14	(C₂×D₂₁₁)⋊C₄	C₂²	1	1	yes	characteristic
15	(C₂×D₂₁₁)⋊C₄	C₂²	1	3	yes	—
16	C₄×Dic₂₁₁	C₂²	1	3	yes	—
17	C₄×Dic₂₁₁	C₂²	1	6	yes	—
18	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₁₁	C₂²	1	3	yes	—
20	C₄²×C₂₁₁	C₂²	1	3	yes	—
21	C₂²×D₂₁₁	C₂³	1	1	yes	characteristic
22	Dic₄₂₂	C₂×C₄	1	12	yes	—
23	C₂×Dic₂₁₁	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₁₁	C₂³	1	3	yes	—
25	C₂×Dic₂₁₁	C₂³	1	3	yes	—
26	C₄×D₂₁₁	C₂×C₄	1	12	yes	—
27	Q₈×C₂₁₁	C₂×C₄	1	4	yes	—
28	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
30	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
31	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
32	C₂×D₂₁₁	—	2	1	no	—
33	C₂×D₂₁₁	—	2	1	no	—
34	C₂×D₂₁₁	C₂²×C₄	1	2	yes	—
35	C₄×C₂₁₁	—	2	3	no	—
36	C₄×C₂₁₁	—	2	1	no	—
37	C₄×C₂₁₁	C₂²×C₄	1	6	yes	—
38	Dic₂₁₁	—	2	1	no	—
39	Dic₂₁₁	—	2	3	no	—
40	Dic₂₁₁	C₂²×C₄	1	6	yes	—
41	C₂×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
42	C₂×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
43	C₂×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
44	D₂₁₁	—	2	2	no	—
45	C₂₁₁	#519	1	1	yes	characteristic, complemented, Sylow
46	#519	—	211	1	no	maximal, Sylow
47	C₂²×Q₈	—	211	1	no	—
48	C₄×Q₈	—	211	1	no	—
49	C₄×Q₈	—	211	3	no	—
50	C₄×Q₈	—	211	3	no	—
51	C₄×Q₈	D₂₁₁	1	1	yes	characteristic, complemented
52	#117	—	211	3	no	—
53	#117	—	211	3	no	—
54	C₂×Q₈	—	211	8	no	—
55	C₂×Q₈	—	211	3	no	—
56	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
57	C₂²×C₄	—	211	3	no	—
58	C₄⋊C₄	—	211	3	no	—
59	C₄⋊C₄	—	211	3	no	—
60	C₄⋊C₄	—	211	3	no	—
61	C₄⋊C₄	C₂×D₂₁₁	1	3	yes	—
62	C₂²⋊C₄	—	211	3	no	—
63	C₂²⋊C₄	—	211	1	no	—
64	C₄²	—	211	3	no	—
65	C₄²	—	211	6	no	—
66	C₄²	C₂×D₂₁₁	1	3	yes	—
67	C₂³	—	211	1	no	—
68	Q₈	—	211	12	no	—
69	Q₈	C₄×D₂₁₁	1	4	yes	—
70	C₂×C₄	—	211	1	no	—
71	C₂×C₄	—	211	3	no	—
72	C₂×C₄	—	211	3	no	—
73	C₂×C₄	—	211	12	no	—
74	C₂×C₄	C₂²×D₂₁₁	1	3	yes	—
75	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₂₁₁	1	3	yes	—
77	C₂²	—	422	1	no	—
78	C₂²	—	422	1	no	—
79	C₂²	—	211	2	no	—
80	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
81	C₄	—	2	3	no	—
82	C₄	—	2	1	no	—
83	C₄	—	422	1	no	—
84	C₄	—	422	3	no	—
85	C₄	—	211	6	no	—
86	C₄	C₂×C₄×D₂₁₁	1	6	yes	—
87	C₂	—	422	2	no	—
88	C₂	C₂²×C₄×D₂₁₁	1	1	yes	characteristic
89	C₂	#20270024	1	1	yes	characteristic
90	C₂	#20270025	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator	C₂×C₂₁₁	C₂³×C₄
Fitting	C₄×Q₈×C₂₁₁	C₂
socle, cosocle	C₂²×C₂₁₁	C₂⁴
prime radical	C₂²×D₂₁₁	C₂³
hypercenter	C₄×Q₈	D₂₁₁
hypocenter	C₂₁₁	#519
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₂₁₁	C₂	semidirect
3. C₄×Dic₄₂₂	C₂	semidirect
4. #20269942	C₂	semidirect
7. C₄×Q₈×C₂₁₁	C₂	semidirect
9. Q₈×D₂₁₁	C₄	semidirect
45. C₂₁₁	#519	semidirect
51. C₄×Q₈	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₂₁₁	2
Q₈×Dic₂₁₁	2
C₄×Dic₄₂₂	2
#20269942	2
#20269935	2
#20269934	2
C₄×Q₈×C₂₁₁	2
#519	211