Subgroups — #31770525

Order 17216.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16864381	C₂	1	2	yes	maximal
3	#22575360	C₂	1	1	yes	maximal, characteristic
4	#22575360	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	—	2	4	no	—
7	C₂×Dic₅₃₈	—	2	2	no	—
8	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
9	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
10	#16864381	—	2	4	no	—
11	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₂₆₉	—	2	2	no	—
15	Dic₅₃₈	—	2	8	no	—
16	C₂×Dic₂₆₉	—	2	2	no	—
17	C₂×Dic₂₆₉	—	2	2	no	—
18	C₂×Dic₂₆₉	—	2	2	no	—
19	C₂×Dic₂₆₉	—	2	2	no	—
20	C₂×Dic₂₆₉	D₄	1	4	yes	—
21	C₂×Dic₂₆₉	Q₈	1	4	yes	complemented
22	C₂³×C₂₆₉	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₂₆₉	—	2	1	no	—
24	C₂×C₄×C₂₆₉	—	2	2	no	—
25	C₂×C₄×C₂₆₉	—	2	1	no	—
26	C₂×C₄×C₂₆₉	—	2	1	no	—
27	C₂×C₄×C₂₆₉	—	2	2	no	—
28	C₂×C₄×C₂₆₉	D₄	1	2	yes	—
29	C₂×C₄×C₂₆₉	Q₈	1	2	yes	complemented
30	C₂²×C₂₆₉	C₂×D₄	1	2	yes	—
31	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₂₆₉	C₂×Q₈	1	2	yes	—
33	C₂²×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₂₆₉	#41	1	1	yes	characteristic
35	C₄×C₂₆₉	—	2	2	no	—
36	C₄×C₂₆₉	—	2	2	no	—
37	C₄×C₂₆₉	—	4	1	no	—
38	Dic₂₆₉	—	2	4	no	—
39	Dic₂₆₉	—	2	4	no	—
40	C₂×C₂₆₉	#120	1	1	yes	characteristic
41	C₂×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
42	C₂×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
44	C₂×C₂₆₉	C₄⋊Q₈	1	2	yes	—
45	C₂₆₉	#395	1	1	yes	characteristic, complemented, Sylow
46	#395	—	269	1	no	maximal, Sylow
47	C₂²×Q₈	—	269	1	no	—
48	C₂×C₄⋊C₄	—	269	2	no	—
49	C₂×C₄⋊C₄	D₂₆₉	1	1	yes	characteristic
50	#95	—	269	1	no	—
51	#95	—	269	2	no	—
52	C₂×Q₈	—	538	4	no	—
53	C₂×Q₈	—	538	2	no	—
54	C₂²×C₄	—	269	2	no	—
55	C₂²×C₄	—	269	2	no	—
56	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
58	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	—	538	4	no	—
61	C₂³	C₂²×D₂₆₉	1	1	yes	characteristic
62	Q₈	—	538	8	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	269	4	no	—
69	C₂×C₄	—	269	4	no	—
70	C₂×C₄	—	538	2	no	—
71	C₂×C₄	—	538	2	no	—
72	C₂×C₄	—	538	2	no	—
73	C₂×C₄	—	538	2	no	—
74	C₂×C₄	C₂²⋊D₂₆₉	1	2	yes	—
75	C₂×C₄	Dic₅₃₈	1	2	yes	complemented
76	C₂²	D₄×D₂₆₉	1	2	yes	—
77	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
78	C₂²	Q₈×D₂₆₉	1	2	yes	—
79	C₂²	C₂×Dic₅₃₈	1	1	yes	characteristic
80	C₂²	#16864402	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	—	538	4	no	—
85	C₄	—	538	4	no	—
86	C₂	#22575421	1	1	yes	characteristic
87	C₂	#22575431	1	2	yes	—
88	C₂	#22575432	1	1	yes	characteristic
89	C₂	#22575451	1	1	yes	characteristic
90	C₂	#22575452	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₆₉
commutator, socle, prime radical, cosocle	C₂³×C₂₆₉	C₂³
Fitting	C₂×C₄⋊C₄×C₂₆₉	C₂
hypercenter	C₂×C₄⋊C₄	D₂₆₉
hypocenter	C₂₆₉	#395
LCS	G ▷ C₂³×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
21. C₂×Dic₂₆₉	Q₈	semidirect
29. C₂×C₄×C₂₆₉	Q₈	semidirect
45. C₂₆₉	#395	semidirect
75. C₂×C₄	Dic₅₃₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₃₈	2
C₂×#16864381	2
#22575360	2
C₂×C₄⋊C₄×C₂₆₉	2
#395	269