Subgroups — C₂²×C₃×C₇⋊C₁₆ #13055529

Order 1344.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇⋊C₁₆	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₇⋊C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₄×C₇	C₄	1	1	yes	characteristic
5	C₂×C₃×C₇×C₈	C₄	1	3	yes	—
6	C₂×C₃×C₇×C₈	C₂²	1	3	yes	—
7	C₃×C₇⋊C₁₆	C₂²	1	4	yes	direct product factor
8	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
9	C₂×C₇⋊C₁₆	C₂×C₃	1	6	yes	direct product factor
10	C₂²×C₃×C₁₆	—	7	1	no	maximal, Hall
11	C₂³×C₃×C₇	C₈	1	1	yes	characteristic
12	C₂×C₃×C₄×C₇	C₈	1	3	yes	—
13	C₂×C₃×C₄×C₇	C₂×C₄	1	3	yes	—
14	C₃×C₇×C₈	C₂×C₄	1	3	yes	—
15	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
16	C₂²×C₄×C₇	C₃×C₄	1	1	yes	characteristic
17	C₂×C₇×C₈	C₃×C₄	1	3	yes	—
18	C₂×C₇×C₈	C₂²×C₃	1	3	yes	—
19	C₇⋊C₁₆	C₂²×C₃	1	4	yes	direct product factor
20	C₂²×C₃×C₈	D₇	1	1	yes	characteristic
21	C₂×C₃×C₁₆	—	7	6	no	—
22	C₂²×C₃×C₇	C₁₆	1	4	yes	complemented
23	C₂²×C₃×C₇	C₂×C₈	1	3	yes	—
24	C₃×C₄×C₇	C₂×C₈	1	3	yes	—
25	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	—	7	1	no	Sylow
27	C₂³×C₇	C₃×C₈	1	1	yes	characteristic
28	C₂×C₄×C₇	C₃×C₈	1	3	yes	—
29	C₂×C₄×C₇	C₂×C₃×C₄	1	3	yes	—
30	C₇×C₈	C₂×C₃×C₄	1	3	yes	—
31	C₇×C₈	C₂³×C₃	1	1	yes	characteristic
32	C₂²×C₃×C₄	Dic₇	1	1	yes	characteristic
33	C₂×C₃×C₈	Dic₇	1	3	yes	—
34	C₂×C₃×C₈	C₂×D₇	1	3	yes	—
35	C₃×C₁₆	—	7	4	no	—
36	C₂×C₃×C₇	C₂×C₁₆	1	6	yes	complemented
37	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
38	C₂²×C₈	C₃×D₇	1	1	yes	characteristic
39	C₂×C₁₆	—	7	6	no	—
40	C₂²×C₇	C₃×C₁₆	1	4	yes	complemented
41	C₂²×C₇	C₂×C₃×C₈	1	3	yes	—
42	C₄×C₇	C₂×C₃×C₈	1	3	yes	—
43	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
44	C₂³×C₃	C₇⋊C₈	1	1	yes	characteristic
45	C₂×C₃×C₄	C₇⋊C₈	1	3	yes	—
46	C₂×C₃×C₄	C₂×Dic₇	1	3	yes	—
47	C₃×C₈	C₂×Dic₇	1	3	yes	—
48	C₃×C₈	C₂²×D₇	1	1	yes	characteristic
49	C₃×C₇	C₂²×C₁₆	1	1	yes	characteristic, complemented, Hall
50	C₂²×C₄	C₃×Dic₇	1	1	yes	characteristic
51	C₂×C₈	C₃×Dic₇	1	3	yes	—
52	C₂×C₈	C₂×C₃×D₇	1	3	yes	—
53	C₁₆	—	7	4	no	—
54	C₂×C₇	C₂×C₃×C₁₆	1	6	yes	complemented
55	C₂×C₇	C₂²×C₃×C₈	1	1	yes	characteristic
56	C₂²×C₃	C₇⋊C₁₆	1	4	yes	direct product factor
57	C₂²×C₃	C₂×C₇⋊C₈	1	3	yes	—
58	C₃×C₄	C₂×C₇⋊C₈	1	3	yes	—
59	C₃×C₄	C₂²×Dic₇	1	1	yes	characteristic
60	C₂³	C₃×C₇⋊C₈	1	1	yes	characteristic
61	C₂×C₄	C₃×C₇⋊C₈	1	3	yes	—
62	C₂×C₄	C₂×C₃×Dic₇	1	3	yes	—
63	C₈	C₂×C₃×Dic₇	1	3	yes	—
64	C₈	C₂²×C₃×D₇	1	1	yes	characteristic
65	C₇	C₂²×C₃×C₁₆	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	C₂×C₇⋊C₁₆	1	6	yes	direct product factor
67	C₂×C₃	C₂²×C₇⋊C₈	1	1	yes	characteristic
68	C₂²	C₃×C₇⋊C₁₆	1	4	yes	direct product factor
69	C₂²	C₂×C₃×C₇⋊C₈	1	3	yes	—
70	C₄	C₂×C₃×C₇⋊C₈	1	3	yes	—
71	C₄	C₂²×C₃×Dic₇	1	1	yes	characteristic
72	C₃	C₂²×C₇⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
73	C₂	C₂×C₃×C₇⋊C₁₆	1	6	yes	direct product factor
74	C₂	C₂²×C₃×C₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₈	D₇
commutator, hypocenter	C₇	C₂²×C₃×C₁₆
Frattini	C₈	C₂²×C₃×D₇
Fitting	C₂²×C₃×C₇×C₈	C₂
socle, prime radical	C₂³×C₃×C₇	C₈
cosocle	C₇×C₈	C₂³×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇⋊C₁₆	C₂	direct (N × H)
3. C₂²×C₇⋊C₁₆	C₃	direct (N × H)
7. C₃×C₇⋊C₁₆	C₂²	direct (N × H)
9. C₂×C₇⋊C₁₆	C₂×C₃	direct (N × H)
19. C₇⋊C₁₆	C₂²×C₃	direct (N × H)
22. C₂²×C₃×C₇	C₁₆	semidirect
36. C₂×C₃×C₇	C₂×C₁₆	semidirect
40. C₂²×C₇	C₃×C₁₆	semidirect
49. C₃×C₇	C₂²×C₁₆	semidirect
54. C₂×C₇	C₂×C₃×C₁₆	semidirect
56. C₂²×C₃	C₇⋊C₁₆	direct (N × H)
65. C₇	C₂²×C₃×C₁₆	semidirect
66. C₂×C₃	C₂×C₇⋊C₁₆	direct (N × H)
68. C₂²	C₃×C₇⋊C₁₆	direct (N × H)
72. C₃	C₂²×C₇⋊C₁₆	direct (N × H)
73. C₂	C₂×C₃×C₇⋊C₁₆	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×C₃×C₇×C₈	2
C₂×C₃×C₇⋊C₁₆	2
C₂²×C₇⋊C₁₆	3
C₂²×C₃×C₁₆	7

Subgroups — C22×C3×C7⋊C16 #13055529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₇⋊C₁₆ #13055529