Subgroups — #46808853

Order 34432.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₂₁₅₂	C₂	1	1	yes	maximal, characteristic
2	#31770356	C₂	1	1	yes	maximal, characteristic, complemented
3	#346×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₁₅₂	C₄	1	1	yes	characteristic
5	C₄×D₁₀₇₆	—	2	1	no	—
6	#22575387	C₄	1	1	yes	characteristic
7	C₄⋊#14524130	—	2	1	no	—
8	#22575375	—	2	1	no	—
9	C₂×C₁₆×C₂₆₉	—	2	1	no	—
10	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
11	C₂×D₁₀₇₆	—	2	1	no	—
12	C₂×C₄×D₂₆₉	—	4	1	no	—
13	D₂₁₅₂	—	2	1	no	—
14	D₂₁₅₂	C₈	1	2	yes	complemented
15	C₂×#14524130	—	4	1	no	—
16	C₄⋊Dic₂₆₉	—	2	1	no	—
17	#16864379	—	4	1	no	—
18	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₆₉	D₄	1	1	yes	characteristic
20	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
21	C₁₆×C₂₆₉	—	2	2	no	—
22	C₂²×D₂₆₉	—	4	1	no	—
23	D₁₀₇₆	—	2	2	no	—
24	D₁₀₇₆	—	4	1	no	—
25	C₂×Dic₂₆₉	—	4	1	no	—
26	C₄×D₂₆₉	—	8	2	no	—
27	#14524130	—	8	1	no	—
28	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₆₉	D₈	1	1	yes	characteristic
30	C₂×C₄×C₂₆₉	SD₁₆	1	1	yes	characteristic
31	C₈×C₂₆₉	—	2	1	no	—
32	C₈×C₂₆₉	C₂×C₈	1	1	yes	characteristic
33	C₈×C₂₆₉	M₁₆	1	1	yes	characteristic
34	C₂²×C₂₆₉	#102	1	1	yes	characteristic
35	C₂×D₂₆₉	—	4	2	no	—
36	C₂×D₂₆₉	—	8	1	no	—
37	C₂×D₂₆₉	—	8	1	no	—
38	C₄×C₂₆₉	—	2	1	no	—
39	C₄×C₂₆₉	—	2	1	no	—
40	C₄×C₂₆₉	C₂²⋊C₈	1	1	yes	characteristic
41	C₄×C₂₆₉	#104	1	1	yes	characteristic
42	Dic₂₆₉	—	8	1	no	—
43	C₂×C₂₆₉	D₄⋊C₈	1	1	yes	characteristic
44	C₂×C₂₆₉	#360	1	1	yes	characteristic
45	C₂×C₂₆₉	#361	1	1	yes	characteristic
46	D₂₆₉	—	8	2	no	—
47	C₂₆₉	#1333	1	1	yes	characteristic, complemented, Sylow
48	#1333	—	269	1	no	maximal, Sylow
49	C₄×D₈	—	269	1	no	—
50	#346	D₂₆₉	1	1	yes	characteristic, complemented
51	#334	—	269	1	no	—
52	C₂×D₈	—	269	1	no	—
53	C₄×D₄	—	538	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	#107	—	269	1	no	—
56	C₄⋊C₈	—	538	1	no	—
57	#102	—	538	1	no	—
58	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
59	C₂×D₄	—	538	1	no	—
60	C₂²×C₄	—	1076	1	no	—
61	D₈	—	269	2	no	—
62	D₈	—	538	1	no	—
63	C₂×C₈	—	1076	1	no	—
64	C₂×C₈	C₄×D₂₆₉	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
66	C₄⋊C₄	—	538	1	no	—
67	C₂²⋊C₄	—	1076	1	no	—
68	C₄²	D₁₀₇₆	1	1	yes	characteristic
69	C₁₆	—	2	2	no	—
70	C₂³	—	1076	1	no	—
71	D₄	—	538	2	no	—
72	D₄	—	1076	1	no	—
73	C₂×C₄	—	1076	1	no	—
74	C₂×C₄	—	2152	2	no	—
75	C₂×C₄	#16864379	1	1	yes	characteristic
76	C₂×C₄	D₂₁₅₂	1	1	yes	characteristic
77	C₂×C₄	#16864391	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	2152	1	no	—
80	C₈	C₈×D₂₆₉	1	1	yes	characteristic
81	C₈	#16864385	1	1	yes	characteristic
82	C₂²	—	1076	2	no	—
83	C₂²	—	2152	1	no	—
84	C₂²	—	2152	1	no	—
85	C₂²	#22575375	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	2152	1	no	—
89	C₄	#22575365	1	1	yes	characteristic
90	C₄	#22575381	1	1	yes	characteristic
91	C₂	—	2152	2	no	—
92	C₂	#31770334	1	1	yes	characteristic
93	C₂	#31770416	1	1	yes	characteristic
94	C₂	#31770419	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22575375
commutator	C₈×C₂₆₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₆₉
Fitting	#346×C₂₆₉	C₂
socle	C₂²×C₂₆₉	#102
prime radical	C₂×D₂₁₅₂	C₄
cosocle	C₄×C₈×C₂₆₉	C₂²
hypercenter	#346	D₂₆₉
hypocenter	C₂₆₉	#1333
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #31770356	C₂	semidirect
3. #346×C₂₆₉	C₂	semidirect
14. D₂₁₅₂	C₈	semidirect
47. C₂₆₉	#1333	semidirect
50. #346	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₂₁₅₂	2
#31770356	2
#346×C₂₆₉	2
#1333	269