Subgroups — #31771211

Order 17216.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575450	C₂	1	1	yes	maximal, characteristic, complemented
2	#22575438	C₂	1	4	yes	maximal, complemented
3	#22575428	C₂	1	2	yes	maximal, complemented
4	#22575428	C₂	1	4	yes	maximal, complemented
5	#22575415	C₂	1	2	yes	maximal, complemented
6	#22575410	C₂	1	1	yes	maximal, characteristic, complemented
7	#125×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₀₇₆	C₂²	1	2	yes	—
9	C₂×D₁₀₇₆	C₂²	1	4	yes	—
10	C₂×D₁₀₇₆	C₂²	1	4	yes	—
11	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₆₉	C₂²	1	4	yes	—
13	#16864381	C₂²	1	2	yes	—
14	#16864379	C₂²	1	2	yes	—
15	#16864379	C₂²	1	4	yes	—
16	#16864379	C₂²	1	4	yes	—
17	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
19	C₄⋊C₄×C₂₆₉	C₂²	1	4	yes	—
20	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
21	C₂²×D₂₆₉	C₂³	1	1	yes	characteristic
22	C₂²×D₂₆₉	C₂³	1	4	yes	—
23	D₁₀₇₆	—	2	4	no	—
24	D₁₀₇₆	—	4	4	no	—
25	D₁₀₇₆	—	4	4	no	—
26	C₂×Dic₂₆₉	C₂³	1	2	yes	—
27	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
28	C₄×D₂₆₉	—	2	2	no	—
29	C₄×D₂₆₉	—	4	4	no	—
30	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
31	C₂×C₄×C₂₆₉	C₂³	1	4	yes	—
32	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
33	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
34	C₂×D₂₆₉	—	2	1	no	—
35	C₂×D₂₆₉	—	2	4	no	—
36	C₂×D₂₆₉	—	2	4	no	—
37	C₂×D₂₆₉	—	2	2	no	—
38	C₂×D₂₆₉	—	2	4	no	—
39	C₄×C₂₆₉	—	2	4	no	—
40	C₄×C₂₆₉	—	2	2	no	—
41	C₄×C₂₆₉	#41	1	2	yes	—
42	Dic₂₆₉	—	2	1	no	—
43	Dic₂₆₉	—	2	2	no	—
44	C₂×C₂₆₉	C₂×#41	1	1	yes	characteristic
45	C₂×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
46	D₂₆₉	—	4	1	no	—
47	D₂₆₉	—	4	4	no	—
48	C₂₆₉	#540	1	1	yes	characteristic, complemented, Sylow
49	#540	—	269	1	no	maximal, Sylow
50	#127	—	269	1	no	—
51	#125	D₂₆₉	1	1	yes	characteristic, complemented
52	#123	—	269	4	no	—
53	#121	—	269	4	no	—
54	#121	—	269	2	no	—
55	C₄×D₄	—	269	2	no	—
56	#117	—	269	1	no	—
57	C₂×D₄	—	269	2	no	—
58	C₂×D₄	—	269	4	no	—
59	C₂×D₄	—	269	4	no	—
60	C₂²×C₄	—	269	1	no	—
61	C₂²×C₄	—	269	4	no	—
62	C₄⋊C₄	—	269	2	no	—
63	C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
64	C₄⋊C₄	C₂×D₂₆₉	1	4	yes	—
65	C₂²⋊C₄	—	269	2	no	—
66	C₂²⋊C₄	—	269	4	no	—
67	C₂²⋊C₄	—	269	4	no	—
68	C₄²	—	269	1	no	—
69	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
70	C₂³	—	269	1	no	—
71	C₂³	—	269	4	no	—
72	D₄	—	538	4	no	—
73	D₄	—	1076	4	no	—
74	D₄	—	1076	4	no	—
75	C₂×C₄	—	269	2	no	—
76	C₂×C₄	—	269	1	no	—
77	C₂×C₄	—	538	2	no	—
78	C₂×C₄	—	1076	4	no	—
79	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
80	C₂×C₄	C₂²×D₂₆₉	1	4	yes	—
81	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
82	C₂²	—	538	1	no	—
83	C₂²	—	538	4	no	—
84	C₂²	—	538	4	no	—
85	C₂²	—	538	2	no	—
86	C₂²	—	538	4	no	—
87	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	2	no	—
90	C₄	—	538	1	no	—
91	C₄	—	538	2	no	—
92	C₄	#16864403	1	2	yes	—
93	C₂	—	1076	1	no	—
94	C₂	—	1076	4	no	—
95	C₂	C₂×#16864403	1	1	yes	characteristic
96	C₂	#22575499	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁴
Fitting	#125×C₂₆₉	C₂
hypercenter	#125	D₂₆₉
hypocenter	C₂₆₉	#540
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #22575450	C₂	semidirect
2. #22575438	C₂	semidirect
3. #22575428	C₂	semidirect
4. #22575428	C₂	semidirect
5. #22575415	C₂	semidirect
6. #22575410	C₂	semidirect
7. #125×C₂₆₉	C₂	semidirect
48. C₂₆₉	#540	semidirect
51. #125	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575450	2
#22575438	2
#22575428	2
#22575415	2
#22575410	2
#125×C₂₆₉	2
#540	269