Subgroups — #31771626

Order 17216.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14524137	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#14524137	C₂	1	2	yes	maximal
3	#22575508	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524137	C₂²	1	4	yes	—
6	C₂²×Dic₂₆₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
8	C₄×Dic₂₆₉	C₄	1	1	yes	characteristic
9	C₄×Dic₂₆₉	C₄	1	1	yes	characteristic
10	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₆₉	C₄	1	1	yes	characteristic
13	#14524137	—	2	4	no	—
14	C₂×Dic₂₆₉	—	2	2	no	—
15	C₂×Dic₂₆₉	—	2	2	no	—
16	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
20	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₆₉	—	2	2	no	—
22	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₂₆₉	—	2	1	no	—
25	C₂²×C₂₆₉	—	2	1	no	—
26	C₂²×C₂₆₉	M₁₆	1	2	yes	—
27	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₂₆₉	—	2	1	no	—
29	C₄×C₂₆₉	M₁₆	1	2	yes	—
30	Dic₂₆₉	—	2	1	no	—
31	Dic₂₆₉	—	2	1	no	—
32	Dic₂₆₉	—	2	1	no	—
33	Dic₂₆₉	#41	1	2	yes	—
34	C₂×C₂₆₉	—	2	2	no	—
35	C₂×C₂₆₉	#117	1	1	yes	characteristic
36	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
38	C₂₆₉	#432	1	1	yes	characteristic, complemented, Sylow
39	#432	—	269	1	no	maximal, Sylow
40	C₂×C₄²	—	269	1	no	—
41	C₄⋊C₈	—	269	2	no	—
42	C₂²⋊C₈	—	269	2	no	—
43	#97	—	269	2	no	—
44	C₂²×C₄	—	269	1	no	—
45	C₂²×C₄	—	269	1	no	—
46	C₂²×C₄	#11761640	1	1	yes	characteristic
47	C₂×C₈	—	269	4	no	—
48	C₄²	—	269	1	no	—
49	C₄²	—	269	1	no	—
50	C₄²	—	269	1	no	—
51	C₄²	—	269	1	no	—
52	C₂³	C₂×#11761640	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	269	1	no	—
55	C₂×C₄	—	269	1	no	—
56	C₂×C₄	—	269	1	no	—
57	C₂×C₄	—	269	1	no	—
58	C₂×C₄	—	538	2	no	—
59	C₂×C₄	—	538	2	no	—
60	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
61	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
62	C₈	—	538	4	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	#16864412	1	2	yes	—
66	C₂²	C₂²×#11761640	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	269	2	no	—
69	C₄	—	538	1	no	—
70	C₄	—	538	1	no	—
71	C₄	—	538	1	no	—
72	C₄	#16864411	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#22575532	1	1	yes	characteristic
75	C₂	C₂×#16864411	1	1	yes	characteristic
76	C₂	C₂×#16864412	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₂²×C₄×C₂₆₉	C₄
prime radical	C₂³×C₂₆₉	C₂×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₂²×C₄	#11761640
hypocenter	C₂₆₉	#432
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#14524137	C₂	semidirect
3. #22575508	C₂	semidirect
38. C₂₆₉	#432	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14524137	2
C₂²⋊#14524137	2
#22575508	2
C₂×C₄×Dic₂₆₉	2
#432	269