Subgroups — #27213751

Order 13504.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269974	C₂	1	1	yes	maximal, characteristic, complemented
2	#20269969	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269962	C₂	1	4	yes	maximal, complemented
4	#20269951	C₂	1	2	yes	maximal, complemented
5	#20269950	C₂	1	4	yes	maximal, complemented
6	C₄×D₈₄₄	C₂	1	2	yes	maximal, complemented
7	#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈₄₄	C₂²	1	2	yes	—
9	C₂×D₈₄₄	C₂²	1	4	yes	—
10	C₂×C₂²⋊D₂₁₁	C₂²	1	4	yes	—
11	C₂×C₄×D₂₁₁	C₂²	1	4	yes	—
12	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
13	Dic₂₁₁⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₂₁₁)⋊C₄	C₂²	1	4	yes	—
15	(C₂×D₂₁₁)⋊C₄	C₂²	1	4	yes	—
16	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
21	C₂²×D₂₁₁	C₂³	1	4	yes	—
22	D₈₄₄	—	2	4	no	—
23	D₈₄₄	—	4	4	no	—
24	C₂²⋊D₂₁₁	—	4	4	no	—
25	C₂×Dic₂₁₁	C₂³	1	4	yes	—
26	C₄×D₂₁₁	—	4	4	no	—
27	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₂₁₁	—	2	2	no	—
29	C₂×C₄×C₂₁₁	C₂³	1	2	yes	—
30	C₂×C₄×C₂₁₁	C₂³	1	2	yes	—
31	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
33	C₂²×C₂₁₁	—	2	1	no	—
34	C₂²×C₂₁₁	—	2	2	no	—
35	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
36	C₂×D₂₁₁	—	2	4	no	—
37	C₂×D₂₁₁	—	2	4	no	—
38	C₂×D₂₁₁	—	2	4	no	—
39	C₄×C₂₁₁	—	2	1	no	—
40	C₄×C₂₁₁	—	2	2	no	—
41	C₄×C₂₁₁	—	2	2	no	—
42	C₄×C₂₁₁	#41	1	2	yes	—
43	Dic₂₁₁	—	2	4	no	—
44	C₂×C₂₁₁	—	4	1	no	—
45	C₂×C₂₁₁	C₂×#41	1	1	yes	characteristic
46	C₂×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
47	D₂₁₁	—	4	4	no	—
48	C₂₁₁	#540	1	1	yes	characteristic, complemented, Sylow
49	#540	—	211	1	no	maximal, Sylow
50	#127	—	211	1	no	—
51	#125	—	211	1	no	—
52	#123	—	211	4	no	—
53	#121	—	211	2	no	—
54	#121	—	211	4	no	—
55	C₄×D₄	—	211	2	no	—
56	#117	D₂₁₁	1	1	yes	characteristic, complemented
57	C₂×D₄	—	211	4	no	—
58	C₂×D₄	—	211	4	no	—
59	C₂×D₄	—	211	2	no	—
60	C₂²×C₄	—	211	4	no	—
61	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
62	C₄⋊C₄	—	211	2	no	—
63	C₄⋊C₄	—	211	4	no	—
64	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
65	C₂²⋊C₄	—	211	4	no	—
66	C₂²⋊C₄	—	211	4	no	—
67	C₂²⋊C₄	C₂×D₂₁₁	1	2	yes	—
68	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
69	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
70	C₂³	—	211	4	no	—
71	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
72	D₄	—	844	4	no	—
73	D₄	—	844	4	no	—
74	D₄	—	422	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	844	4	no	—
77	C₂×C₄	—	211	4	no	—
78	C₂×C₄	C₂²×D₂₁₁	1	2	yes	—
79	C₂×C₄	C₂²×D₂₁₁	1	2	yes	—
80	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	422	4	no	—
85	C₂²	—	422	4	no	—
86	C₂²	—	422	4	no	—
87	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	—	422	4	no	—
92	C₄	#16809962	1	2	yes	—
93	C₂	—	4	1	no	—
94	C₂	—	844	4	no	—
95	C₂	C₂×#16809962	1	1	yes	characteristic
96	C₂	#20270023	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator, socle, cosocle	C₂²×C₂₁₁	C₂⁴
Fitting	#117×C₂₁₁	C₂
hypercenter	#117	D₂₁₁
hypocenter	C₂₁₁	#540
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #20269974	C₂	semidirect
2. #20269969	C₂	semidirect
3. #20269962	C₂	semidirect
4. #20269951	C₂	semidirect
5. #20269950	C₂	semidirect
6. C₄×D₈₄₄	C₂	semidirect
7. #117×C₂₁₁	C₂	semidirect
48. C₂₁₁	#540	semidirect
56. #117	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269974	2
#20269969	2
#20269962	2
#20269951	2
#20269950	2
C₄×D₈₄₄	2
#117×C₂₁₁	2
#540	211