Subgroups — #46825791

Order 34432.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771626	C₂	1	2	yes	maximal
2	#31770628	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14524137	—	2	2	no	—
4	C₂²⋊#14524137	C₄	1	4	yes	—
5	#22575508	—	2	2	no	—
6	C₂×#16864386	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
8	#22575382	—	2	2	no	—
9	C₂×M₁₆×C₂₆₉	C₄	1	1	yes	characteristic
10	C₂×#14524137	—	2	4	no	—
11	C₂²×Dic₂₆₉	C₂×C₄	1	2	yes	—
12	#16864386	—	4	1	no	—
13	C₂×#14524130	—	2	1	no	—
14	C₄×Dic₂₆₉	D₄	1	2	yes	—
15	C₄×Dic₂₆₉	D₄	1	1	yes	characteristic
16	C₄×Dic₂₆₉	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₂₆₉	—	4	1	no	—
19	C₂×C₈×C₂₆₉	—	2	1	no	—
20	#14524137	—	4	4	no	—
21	C₂×Dic₂₆₉	—	4	2	no	—
22	C₂×Dic₂₆₉	C₂²⋊C₄	1	2	yes	—
23	C₂×Dic₂₆₉	C₄⋊C₄	1	2	yes	—
24	#14524130	—	4	1	no	—
25	C₂³×C₂₆₉	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₂₆₉	—	2	1	no	—
27	C₂×C₄×C₂₆₉	—	2	1	no	—
28	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
30	C₈×C₂₆₉	—	4	1	no	—
31	C₂²×C₂₆₉	—	2	1	no	—
32	C₂²×C₂₆₉	—	2	2	no	—
33	C₂²×C₂₆₉	#95	1	1	yes	characteristic
34	C₄×C₂₆₉	—	2	1	no	—
35	C₄×C₂₆₉	#100	1	1	yes	characteristic
36	C₄×C₂₆₉	#101	1	1	yes	characteristic
37	Dic₂₆₉	—	2	2	no	—
38	Dic₂₆₉	—	2	2	no	—
39	C₂×C₂₆₉	—	4	1	no	—
40	C₂×C₂₆₉	#343	1	1	yes	characteristic
41	C₂×C₂₆₉	#344	1	2	yes	—
42	C₂₆₉	#1307	1	1	yes	characteristic, complemented, Sylow
43	#1307	—	269	1	no	maximal, Sylow
44	#432	—	269	2	no	—
45	#423	—	269	1	no	—
46	C₂×M₁₆	—	269	1	no	—
47	C₂×M₁₆	#11761640	1	1	yes	characteristic
48	C₂×C₄²	—	269	1	no	—
49	C₄⋊C₈	—	538	2	no	—
50	C₄⋊C₈	—	538	2	no	—
51	C₂²⋊C₈	—	269	4	no	—
52	#97	—	538	2	no	—
53	C₂²×C₄	—	269	2	no	—
54	C₂²×C₄	C₂×#11761640	1	1	yes	characteristic
55	M₁₆	—	4	1	no	—
56	M₁₆	—	1076	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	538	4	no	—
59	C₂×C₈	—	538	1	no	—
60	C₄²	—	269	2	no	—
61	C₄²	—	269	1	no	—
62	C₄²	—	269	1	no	—
63	C₂³	C₄×#11761640	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	269	2	no	—
67	C₂×C₄	—	269	2	no	—
68	C₂×C₄	—	1076	2	no	—
69	C₂×C₄	C₂²⋊#11761640	1	1	yes	characteristic
70	C₂×C₄	C₄⋊#11761640	1	1	yes	characteristic
71	C₈	—	4	1	no	—
72	C₈	—	1076	4	no	—
73	C₈	—	1076	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	2	no	—
76	C₂²	(C₂×#11761640)⋊C₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	538	2	no	—
79	C₄	—	538	2	no	—
80	C₄	#22575513	1	1	yes	characteristic
81	C₄	#22575514	1	1	yes	characteristic
82	C₂	—	4	1	no	—
83	C₂	#31771485	1	1	yes	characteristic
84	C₂	#31771488	1	2	yes	—

	subgroup	G/H
center	C₂²	(C₂×#11761640)⋊C₄
commutator, prime radical	C₂³×C₂₆₉	C₄²
Frattini	C₂²×C₄	C₂×#11761640
Fitting	C₂×M₁₆×C₂₆₉	C₄
socle	C₂²×C₂₆₉	#95
cosocle	C₂×C₄×Dic₂₆₉	C₂²
hypercenter	C₂×M₁₆	#11761640
hypocenter	C₂₆₉	#1307
LCS	G ▷ C₂³×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
42. C₂₆₉	#1307	semidirect

Maximal subgroups

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

subgroup	index
#31771626	2
#31770628	2
#1307	269