Subgroups — #46825932

Order 34432.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22575527	C₂	1	2	yes	maximal
2	C₂×#22575364	C₂	1	1	yes	maximal, characteristic
3	#22575527	—	2	2	no	—
4	#22575527	C₄	1	4	yes	complemented
5	C₂×#16864405	—	2	2	no	—
6	C₂²×#14524130	C₄	1	1	yes	characteristic
7	C₂²×#14524130	C₂²	1	1	yes	characteristic
8	#22575364	—	2	2	no	—
9	C₂×C₄²×C₂₆₉	C₄	1	1	yes	characteristic
10	#16864405	—	2	4	no	—
11	C₂×#14524130	—	2	1	no	—
12	C₂×#14524130	—	2	1	no	—
13	C₂×#14524130	C₈	1	2	yes	—
14	C₂×#14524130	C₂×C₄	1	2	yes	—
15	C₂×#14524130	D₄	1	2	yes	—
16	C₂×#14524130	D₄	1	1	yes	characteristic
17	C₂×#14524130	Q₈	1	1	yes	characteristic
18	C₂²×C₄×C₂₆₉	C₈	1	2	yes	—
19	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₂₆₉	—	2	1	no	—
21	C₄²×C₂₆₉	—	2	1	no	—
22	#14524130	—	2	2	no	—
23	#14524130	C₂²⋊C₄	1	2	yes	—
24	#14524130	C₄⋊C₄	1	2	yes	—
25	C₂³×C₂₆₉	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₂₆₉	—	2	2	no	—
27	C₂×C₄×C₂₆₉	—	4	2	no	—
28	C₂×C₄×C₂₆₉	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₂₆₉	M₁₆	1	1	yes	characteristic
33	C₂×C₄×C₂₆₉	M₁₆	1	1	yes	characteristic
34	C₂²×C₂₆₉	—	2	2	no	—
35	C₂²×C₂₆₉	C₄×C₈	1	1	yes	characteristic
36	C₂²×C₂₆₉	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₂₆₉	C₄⋊C₈	1	1	yes	characteristic
38	C₄×C₂₆₉	—	4	2	no	—
39	C₄×C₂₆₉	#95	1	1	yes	characteristic
40	C₄×C₂₆₉	#97	1	1	yes	characteristic
41	C₄×C₂₆₉	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₂₆₉	C₄⋊C₈	1	1	yes	characteristic
43	C₂×C₂₆₉	—	2	1	no	—
44	C₂×C₂₆₉	—	2	1	no	—
45	C₂×C₂₆₉	#336	1	1	yes	characteristic
46	C₂×C₂₆₉	#349	1	2	yes	—
47	C₂₆₉	#1377	1	1	yes	characteristic, complemented, Sylow
48	#1377	—	269	1	no	maximal, Sylow
49	C₂×M₃₂	—	269	2	no	—
50	C₂×#97	—	269	1	no	—
51	C₂²×C₈	—	269	1	no	—
52	C₂²×C₈	—	269	1	no	—
53	C₂×C₄²	#11761640	1	1	yes	characteristic
54	M₃₂	—	269	4	no	—
55	M₃₂	—	538	2	no	—
56	C₂×C₁₆	—	538	2	no	—
57	#97	—	538	2	no	—
58	C₂²×C₄	#14524137	1	2	yes	—
59	C₂²×C₄	C₂×#11761640	1	1	yes	characteristic
60	C₂×C₈	—	269	2	no	—
61	C₂×C₈	—	269	2	no	—
62	C₂×C₈	—	269	1	no	—
63	C₂×C₈	—	269	1	no	—
64	C₂×C₈	—	269	2	no	—
65	C₂×C₈	—	538	1	no	—
66	C₂×C₈	—	538	1	no	—
67	C₄²	—	2	1	no	—
68	C₄²	—	2	1	no	—
69	C₁₆	—	538	4	no	—
70	C₂³	C₂×#14524137	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	C₄×#11761640	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊#11761640	1	1	yes	characteristic
75	C₂×C₄	C₄⋊#11761640	1	1	yes	characteristic
76	C₂×C₄	D₂₆₉⋊C₈	1	1	yes	characteristic
77	C₂×C₄	#16864411	1	1	yes	characteristic
78	C₂×C₄	#16864412	1	1	yes	characteristic
79	C₈	—	269	2	no	—
80	C₈	—	269	2	no	—
81	C₈	—	538	2	no	—
82	C₂²	—	2	2	no	—
83	C₂²	C₄×#14524137	1	1	yes	characteristic
84	C₂²	C₂²⋊#14524137	1	1	yes	characteristic
85	C₂²	C₄⋊#14524137	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	(C₂×#11761640)⋊C₄	1	1	yes	characteristic
88	C₄	#22575508	1	1	yes	characteristic
89	C₄	#22575510	1	1	yes	characteristic
90	C₄	#22575520	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#31771471	1	1	yes	characteristic
94	C₂	#31771501	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#11761640
commutator	C₂²×C₂₆₉	C₄×C₈
Frattini	C₂²×C₄	C₂×#11761640
Fitting	C₂×C₄²×C₂₆₉	C₄
socle	C₂²×C₂₆₉	C₂²⋊C₈
prime radical	C₂³×C₂₆₉	C₂×C₈
cosocle	C₂²×#14524130	C₂²
hypercenter	C₂×C₄²	#11761640
hypocenter	C₂₆₉	#1377
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #22575527	C₄	semidirect
47. C₂₆₉	#1377	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22575527	2
C₂×#22575364	2
#1377	269