Subgroups — #31771630

Order 17216.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575520	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#14524137	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575510	C₂	1	1	yes	maximal, characteristic
4	#22575510	C₂	1	1	yes	maximal, characteristic
5	#22575508	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14524137	C₂	1	1	yes	maximal, characteristic, complemented
7	#22575411	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524137	C₂²	1	1	yes	characteristic
9	C₂×#14524137	C₂²	1	1	yes	characteristic
10	C₂×#14524137	C₂²	1	1	yes	characteristic
11	C₂×#14524137	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₂₆₉	C₄	1	1	yes	characteristic
14	#16864379	C₄	1	1	yes	characteristic
15	#16864379	C₄	1	1	yes	characteristic
16	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₆₉	C₄	1	1	yes	characteristic
19	#14524137	—	2	1	no	—
20	#14524137	—	2	1	no	—
21	#14524137	—	2	1	no	—
22	#14524137	—	2	1	no	—
23	C₂²×D₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
27	C₄×D₂₆₉	—	2	1	no	—
28	C₄×D₂₆₉	—	2	1	no	—
29	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₂₆₉	—	2	1	no	—
34	C₂×D₂₆₉	—	2	1	no	—
35	C₂×D₂₆₉	—	2	1	no	—
36	C₄×C₂₆₉	—	2	1	no	—
37	C₄×C₂₆₉	—	2	1	no	—
38	C₄×C₂₆₉	—	2	1	no	—
39	Dic₂₆₉	—	2	1	no	—
40	Dic₂₆₉	—	2	1	no	—
41	Dic₂₆₉	#41	1	1	yes	characteristic
42	Dic₂₆₉	#41	1	1	yes	characteristic
43	C₂×C₂₆₉	#117	1	1	yes	characteristic
44	C₂×C₂₆₉	#131	1	1	yes	characteristic
45	C₂×C₂₆₉	#131	1	1	yes	characteristic
46	D₂₆₉	—	4	1	no	—
47	C₂₆₉	#433	1	1	yes	characteristic, complemented, Sylow
48	#433	—	269	1	no	maximal, Sylow
49	#117	—	269	1	no	—
50	C₄⋊C₈	—	269	1	no	—
51	C₄⋊C₈	—	269	1	no	—
52	C₂²⋊C₈	—	269	1	no	—
53	C₂²⋊C₈	—	269	1	no	—
54	#97	—	269	1	no	—
55	C₄×C₈	—	269	1	no	—
56	C₂²×C₄	—	269	1	no	—
57	C₂×C₈	—	269	1	no	—
58	C₂×C₈	—	269	1	no	—
59	C₂×C₈	—	269	1	no	—
60	C₂×C₈	—	269	1	no	—
61	C₄⋊C₄	—	269	1	no	—
62	C₄⋊C₄	#11761640	1	1	yes	characteristic
63	C₂²⋊C₄	—	269	1	no	—
64	C₂²⋊C₄	—	269	1	no	—
65	C₄²	—	269	1	no	—
66	C₄²	—	269	1	no	—
67	C₂³	—	269	1	no	—
68	C₂×C₄	—	269	1	no	—
69	C₂×C₄	—	269	1	no	—
70	C₂×C₄	—	269	1	no	—
71	C₂×C₄	—	538	1	no	—
72	C₂×C₄	—	538	1	no	—
73	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
74	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
75	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
76	C₈	—	538	1	no	—
77	C₈	—	538	1	no	—
78	C₈	—	538	1	no	—
79	C₈	—	538	1	no	—
80	C₂²	—	538	1	no	—
81	C₂²	—	538	1	no	—
82	C₂²	—	538	1	no	—
83	C₂²	C₂²×#11761640	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	269	1	no	—
88	C₄	—	269	1	no	—
89	C₄	—	538	1	no	—
90	C₄	—	538	1	no	—
91	C₂	—	1076	1	no	—
92	C₂	#22575532	1	1	yes	characteristic
93	C₂	#22575540	1	1	yes	characteristic
94	C₂	#22575541	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₄⋊C₄×C₂₆₉	C₄
prime radical	C₂²×D₂₆₉	C₂×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₄⋊C₄	#11761640
hypocenter	C₂₆₉	#433
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #22575520	C₂	semidirect
2. C₄⋊#14524137	C₂	semidirect
5. #22575508	C₂	semidirect
6. C₄×#14524137	C₂	semidirect
47. C₂₆₉	#433	semidirect

Maximal subgroups

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

subgroup	index
#22575520	2
C₄⋊#14524137	2
#22575510	2
#22575508	2
C₄×#14524137	2
#22575411	2
#433	269