Subgroups — #27213620

Order 13504.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈₄₄	C₂	1	1	yes	maximal, characteristic
2	C₄×D₈₄₄	C₂	1	4	yes	maximal, complemented
3	#20269939	C₂	1	4	yes	maximal, complemented
4	#20269936	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₄×D₂₁₁	C₂	1	4	yes	maximal, complemented
6	#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₁₁	C₂²	1	2	yes	—
8	C₂×D₈₄₄	C₄	1	8	yes	complemented
9	C₂×D₈₄₄	C₂²	1	2	yes	—
10	C₂×D₈₄₄	C₂²	1	2	yes	—
11	C₂×C₄×D₂₁₁	C₂²	1	8	yes	complemented
12	C₄⋊Dic₂₁₁	C₂²	1	2	yes	complemented
13	C₂²⋊Dic₂₁₁	C₂²	1	2	yes	—
14	(C₂×D₂₁₁)⋊C₄	C₂²	1	8	yes	complemented
15	C₄×Dic₂₁₁	C₂²	1	2	yes	complemented
16	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	complemented
18	C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₄²×C₂₁₁	C₂²	1	2	yes	complemented
20	C₂²×D₂₁₁	—	4	4	no	—
21	C₂²×D₂₁₁	C₂×C₄	1	8	yes	complemented
22	C₂²×D₂₁₁	C₂³	1	4	yes	—
23	D₈₄₄	C₂×C₄	1	8	yes	complemented
24	D₈₄₄	C₂×C₄	1	8	yes	complemented
25	C₂×Dic₂₁₁	C₂³	1	4	yes	—
26	C₄×D₂₁₁	—	4	8	no	—
27	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
29	C₂×C₄×C₂₁₁	C₂³	1	4	yes	—
30	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
32	C₂²×C₂₁₁	—	2	2	no	—
33	C₂²×C₂₁₁	C₂²×C₄	1	2	yes	—
34	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
35	C₂×D₂₁₁	—	2	8	no	—
36	C₂×D₂₁₁	—	4	8	no	—
37	C₂×D₂₁₁	C₂²×C₄	1	8	yes	—
38	C₄×C₂₁₁	—	2	4	no	—
39	C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
40	C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
41	Dic₂₁₁	—	2	4	no	—
42	C₂×C₂₁₁	—	2	2	no	—
43	C₂×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
44	C₂×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
45	D₂₁₁	—	2	8	no	—
46	C₂₁₁	#518	1	1	yes	characteristic, complemented, Sylow
47	#518	—	211	1	no	maximal, Sylow
48	C₂²×D₄	—	211	1	no	—
49	C₄×D₄	—	211	4	no	—
50	C₄×D₄	—	211	4	no	—
51	#117	—	211	1	no	—
52	#117	D₂₁₁	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊C₄	—	211	4	no	—
54	C₂⁴	—	211	2	no	—
55	C₂×D₄	—	211	8	no	—
56	C₂×D₄	—	211	2	no	—
57	C₂×D₄	—	211	2	no	—
58	C₂²×C₄	—	211	8	no	—
59	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
60	C₄⋊C₄	—	211	2	no	—
61	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	complemented
62	C₂²⋊C₄	—	211	2	no	—
63	C₂²⋊C₄	—	211	8	no	—
64	C₂²⋊C₄	C₂×D₂₁₁	1	2	yes	—
65	C₄²	—	211	2	no	—
66	C₄²	C₂×D₂₁₁	1	2	yes	complemented
67	C₂³	—	844	4	no	—
68	C₂³	—	211	8	no	—
69	C₂³	—	211	4	no	—
70	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
71	D₄	—	211	8	no	—
72	D₄	—	211	8	no	—
73	C₂×C₄	—	844	8	no	—
74	C₂×C₄	—	211	4	no	—
75	C₂×C₄	C₄×D₂₁₁	1	4	yes	complemented
76	C₂×C₄	C₂²×D₂₁₁	1	4	yes	—
77	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	—	844	8	no	—
81	C₂²	—	422	8	no	—
82	C₂²	—	211	8	no	—
83	C₂²	C₂×C₄×D₂₁₁	1	2	yes	—
84	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	422	4	no	—
87	C₄	C₂×C₄×D₂₁₁	1	2	yes	—
88	C₄	C₂×C₄×D₂₁₁	1	2	yes	—
89	C₂	—	2	2	no	—
90	C₂	—	422	8	no	—
91	C₂	C₂²×C₄×D₂₁₁	1	1	yes	characteristic
92	C₂	#20270023	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator	C₂×C₂₁₁	C₂³×C₄
Fitting	#117×C₂₁₁	C₂
socle, cosocle	C₂²×C₂₁₁	C₂⁴
prime radical	C₂²×D₈₄₄	C₂
hypercenter	#117	D₂₁₁
hypocenter	C₂₁₁	#518
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×D₈₄₄	C₂	semidirect
3. #20269939	C₂	semidirect
4. #20269936	C₂	semidirect
5. C₂²⋊C₄×D₂₁₁	C₂	semidirect
6. #117×C₂₁₁	C₂	semidirect
8. C₂×D₈₄₄	C₄	semidirect
11. C₂×C₄×D₂₁₁	C₂²	semidirect
12. C₄⋊Dic₂₁₁	C₂²	semidirect
14. (C₂×D₂₁₁)⋊C₄	C₂²	semidirect
15. C₄×Dic₂₁₁	C₂²	semidirect
17. C₄⋊C₄×C₂₁₁	C₂²	semidirect
19. C₄²×C₂₁₁	C₂²	semidirect
21. C₂²×D₂₁₁	C₂×C₄	semidirect
23. D₈₄₄	C₂×C₄	semidirect
24. D₈₄₄	C₂×C₄	semidirect
28. C₂×C₄×C₂₁₁	C₂×C₄	semidirect
46. C₂₁₁	#518	semidirect
52. #117	D₂₁₁	semidirect
61. C₄⋊C₄	C₂×D₂₁₁	semidirect
66. C₄²	C₂×D₂₁₁	semidirect
75. C₂×C₄	C₄×D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₈₄₄	2
C₄×D₈₄₄	2
#20269939	2
#20269936	2
C₂²⋊C₄×D₂₁₁	2
#117×C₂₁₁	2
#518	211