Subgroups — C₂²×#22575399 #46823997

Order 34432.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₀₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#22575399	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#16864377	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₁₆×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₃₈	—	2	1	no	—
6	C₂×Dic₁₀₇₆	—	2	3	no	—
7	C₂×Dic₁₀₇₆	C₂²	1	6	yes	—
8	#22575399	C₂²	1	16	yes	direct product factor
9	C₂×#16864377	C₂²	1	6	yes	—
10	C₂²×Q₈×C₂₆₉	—	2	1	no	—
11	C₂×Q₁₆×C₂₆₉	—	2	3	no	—
12	C₂×Q₁₆×C₂₆₉	C₂²	1	6	yes	—
13	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₅₃₈	—	2	6	no	—
15	C₂×Dic₅₃₈	—	4	3	no	—
16	C₂²×Dic₂₆₉	—	4	1	no	—
17	Dic₁₀₇₆	—	2	6	no	—
18	Dic₁₀₇₆	C₂³	1	4	yes	—
19	#16864377	C₂³	1	4	yes	—
20	C₂×Q₈×C₂₆₉	—	2	6	no	—
21	C₂×Q₈×C₂₆₉	—	4	3	no	—
22	C₂²×C₄×C₂₆₉	—	4	1	no	—
23	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
24	Q₁₆×C₂₆₉	—	2	6	no	—
25	Q₁₆×C₂₆₉	C₂³	1	4	yes	—
26	C₂×C₈×C₂₆₉	D₄	1	3	yes	—
27	C₂×C₈×C₂₆₉	C₂³	1	3	yes	—
28	Dic₅₃₈	—	2	4	no	—
29	Dic₅₃₈	—	4	6	no	—
30	C₂×Dic₂₆₉	—	4	6	no	—
31	C₂³×C₂₆₉	D₈	1	1	yes	characteristic
32	Q₈×C₂₆₉	—	2	4	no	—
33	Q₈×C₂₆₉	—	4	6	no	—
34	C₂×C₄×C₂₆₉	—	4	6	no	—
35	C₂×C₄×C₂₆₉	D₈	1	3	yes	—
36	C₂×C₄×C₂₆₉	C₂×D₄	1	3	yes	—
37	C₈×C₂₆₉	C₂×D₄	1	3	yes	—
38	C₈×C₂₆₉	C₂⁴	1	1	yes	characteristic
39	C₂²×C₂₆₉	Q₃₂	1	4	yes	complemented
40	C₂²×C₂₆₉	C₂×D₈	1	3	yes	—
41	C₄×C₂₆₉	—	4	4	no	—
42	C₄×C₂₆₉	C₂×D₈	1	3	yes	—
43	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
44	Dic₂₆₉	—	4	4	no	—
45	C₂×C₂₆₉	C₂×Q₃₂	1	6	yes	complemented
46	C₂×C₂₆₉	C₂²×D₈	1	1	yes	characteristic
47	C₂₆₉	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
48	C₂²×Q₃₂	—	269	1	no	maximal, Sylow
49	C₂²×Q₁₆	—	269	1	no	—
50	C₂²×Q₁₆	D₂₆₉	1	1	yes	characteristic
51	C₂×Q₃₂	—	269	12	no	—
52	C₂²×C₁₆	—	269	1	no	—
53	C₂²×Q₈	—	2	1	no	—
54	C₂²×Q₈	—	538	1	no	—
55	C₂×Q₁₆	—	2	3	no	—
56	C₂×Q₁₆	—	269	6	no	—
57	C₂×Q₁₆	—	538	3	no	—
58	C₂×Q₁₆	C₂×D₂₆₉	1	6	yes	—
59	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
60	Q₃₂	—	269	16	no	—
61	C₂×C₁₆	—	269	6	no	—
62	C₂×Q₈	—	2	6	no	—
63	C₂×Q₈	—	4	3	no	—
64	C₂×Q₈	—	538	6	no	—
65	C₂×Q₈	—	1076	3	no	—
66	C₂²×C₄	—	4	1	no	—
67	C₂²×C₄	—	1076	1	no	—
68	C₂²×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
69	Q₁₆	—	2	6	no	—
70	Q₁₆	—	269	4	no	—
71	Q₁₆	—	538	6	no	—
72	Q₁₆	C₂²×D₂₆₉	1	4	yes	—
73	C₂×C₈	C₂²⋊D₂₆₉	1	3	yes	—
74	C₂×C₈	C₂²×D₂₆₉	1	3	yes	—
75	C₁₆	—	269	4	no	—
76	C₂³	#16864387	1	1	yes	characteristic
77	Q₈	—	2	4	no	—
78	Q₈	—	4	6	no	—
79	Q₈	—	538	4	no	—
80	Q₈	—	1076	6	no	—
81	C₂×C₄	—	4	6	no	—
82	C₂×C₄	—	1076	6	no	—
83	C₂×C₄	#16864387	1	3	yes	—
84	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	3	yes	—
85	C₈	C₂×C₂²⋊D₂₆₉	1	3	yes	—
86	C₈	C₂³×D₂₆₉	1	1	yes	characteristic
87	C₂²	#22575399	1	4	yes	direct product factor
88	C₂²	C₂×#16864387	1	3	yes	—
89	C₄	—	4	4	no	—
90	C₄	—	1076	4	no	—
91	C₄	C₂×#16864387	1	3	yes	—
92	C₄	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
93	C₂	C₂×#22575399	1	6	yes	direct product factor
94	C₂	C₂²×#16864387	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#16864387
commutator, cosocle	C₈×C₂₆₉	C₂⁴
Frattini	C₈	C₂³×D₂₆₉
Fitting	C₂²×Q₁₆×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	D₈
hypercenter	C₂²×Q₁₆	D₂₆₉
hypocenter	C₂₆₉	C₂²×Q₃₂
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#22575399	C₂	direct (N × H)
8. #22575399	C₂²	direct (N × H)
39. C₂²×C₂₆₉	Q₃₂	semidirect
45. C₂×C₂₆₉	C₂×Q₃₂	semidirect
47. C₂₆₉	C₂²×Q₃₂	semidirect
87. C₂²	#22575399	direct (N × H)
93. C₂	C₂×#22575399	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₀₇₆	2
C₂×#22575399	2
C₂²×#16864377	2
C₂²×Q₁₆×C₂₆₉	2
C₂²×Q₃₂	269

Subgroups — C22×#22575399 #46823997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#22575399 #46823997