Subgroups — C₂×#22575514 #31771589

Order 17216.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864411	C₂	1	2	yes	maximal
2	#22575514	C₂	1	4	yes	maximal, direct product factor
3	C₂²×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
4	#16864411	—	2	2	no	—
5	#16864411	C₂²	1	4	yes	—
6	C₂×#14524137	—	2	2	no	—
7	C₂×Dic₅₃₈	—	2	2	no	—
8	C₂×Dic₅₃₈	—	2	2	no	—
9	C₂×Dic₅₃₈	C₄	1	2	yes	—
10	C₂×Dic₅₃₈	C₂²	1	2	yes	—
11	C₂²×Dic₂₆₉	C₄	1	1	yes	characteristic
12	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₆₉	C₄	1	1	yes	characteristic
14	#14524137	—	2	4	no	—
15	Dic₅₃₈	—	2	4	no	—
16	Dic₅₃₈	—	2	4	no	—
17	C₂×Dic₂₆₉	—	2	1	no	—
18	C₂×Dic₂₆₉	—	2	1	no	—
19	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
20	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₆₉	D₄	1	2	yes	—
22	C₂×Dic₂₆₉	D₄	1	2	yes	—
23	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
24	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₆₉	—	2	1	no	—
26	C₂×C₄×C₂₆₉	—	2	1	no	—
27	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
28	C₂²×C₂₆₉	—	2	2	no	—
29	C₂²×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₆₉	—	2	2	no	—
33	Dic₂₆₉	—	2	2	no	—
34	Dic₂₆₉	C₂²⋊C₄	1	2	yes	—
35	Dic₂₆₉	C₂×D₄	1	2	yes	—
36	C₂×C₂₆₉	—	2	1	no	—
37	C₂×C₂₆₉	—	2	1	no	—
38	C₂×C₂₆₉	#101	1	2	yes	complemented
39	C₂×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂₆₉	C₂×#101	1	1	yes	characteristic, complemented, Sylow
41	C₂×#101	—	269	1	no	maximal, Sylow
42	C₂²×Q₈	—	269	1	no	—
43	C₂×M₁₆	—	269	2	no	—
44	#101	—	269	4	no	—
45	C₂×Q₈	—	269	2	no	—
46	C₂×Q₈	—	269	2	no	—
47	C₂×Q₈	—	538	2	no	—
48	C₂×Q₈	—	538	2	no	—
49	C₂²×C₄	—	269	1	no	—
50	C₂²×C₄	—	269	1	no	—
51	C₂²×C₄	#11761640	1	1	yes	characteristic
52	M₁₆	—	269	4	no	—
53	M₁₆	—	538	2	no	—
54	C₂×C₈	—	538	2	no	—
55	C₂³	C₂×#11761640	1	1	yes	characteristic
56	Q₈	—	538	4	no	—
57	Q₈	—	538	4	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	269	2	no	—
61	C₂×C₄	—	269	2	no	—
62	C₂×C₄	—	269	1	no	—
63	C₂×C₄	—	269	1	no	—
64	C₂×C₄	—	269	2	no	—
65	C₂×C₄	—	538	1	no	—
66	C₂×C₄	—	538	1	no	—
67	C₂×C₄	C₂×#11761640	1	2	yes	—
68	C₈	—	538	4	no	—
69	C₂²	—	2	2	no	—
70	C₂²	C₂²⋊#11761640	1	1	yes	characteristic
71	C₂²	C₂²⋊#11761640	1	1	yes	characteristic
72	C₂²	C₂²×#11761640	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	269	2	no	—
75	C₄	—	269	2	no	—
76	C₄	—	538	2	no	—
77	C₂	—	2	1	no	—
78	C₂	—	2	1	no	—
79	C₂	#22575514	1	2	yes	direct product factor
80	C₂	C₂×C₂²⋊#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11761640
commutator	C₂²×C₂₆₉	C₂²×C₄
Frattini	C₂²	C₂²×#11761640
Fitting	C₂²×C₄×C₂₆₉	C₄
socle	C₂²×C₂₆₉	C₂²⋊C₄
prime radical	C₂³×C₂₆₉	C₂×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₂²×C₄	#11761640
hypocenter	C₂₆₉	C₂×#101
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #22575514	C₂	direct (N × H)
38. C₂×C₂₆₉	#101	semidirect
40. C₂₆₉	C₂×#101	semidirect
79. C₂	#22575514	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16864411	2
#22575514	2
C₂²×Dic₅₃₈	2
C₂×#101	269

Subgroups — C2×#22575514 #31771589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575514 #31771589