Subgroups — C₇²×C₁₆² #72190895

Order 12544.

830 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₈×C₁₆	C₂	1	3	yes	maximal
2	C₇²×C₈²	C₂²	1	1	yes	characteristic
3	C₄×C₇²×C₁₆	C₄	1	6	yes	—
4	C₇×C₁₆²	C₇	1	8	yes	maximal, direct product factor
5	C₂×C₇²×C₁₆	C₈	1	12	yes	—
6	C₄×C₇²×C₈	C₂×C₄	1	3	yes	—
7	C₇×C₈×C₁₆	C₂×C₇	1	24	yes	—
8	C₂×C₇²×C₈	C₂×C₈	1	6	yes	—
9	C₄²×C₇²	C₄²	1	1	yes	characteristic
10	C₇²×C₁₆	C₁₆	1	24	yes	direct product factor
11	C₄×C₇×C₁₆	C₄×C₇	1	48	yes	—
12	C₇×C₈²	C₂²×C₇	1	8	yes	—
13	C₂×C₄×C₇²	C₄×C₈	1	3	yes	—
14	C₇²×C₈	C₂×C₁₆	1	12	yes	—
15	C₁₆²	C₇²	1	1	yes	characteristic, direct product factor, Sylow
16	C₂×C₇×C₁₆	C₇×C₈	1	96	yes	—
17	C₄×C₇×C₈	C₂×C₄×C₇	1	24	yes	—
18	C₂²×C₇²	C₈²	1	1	yes	characteristic
19	C₄×C₇²	C₄×C₁₆	1	6	yes	—
20	C₈×C₁₆	C₂×C₇²	1	3	yes	—
21	C₂×C₇×C₈	C₂×C₇×C₈	1	48	yes	—
22	C₄²×C₇	C₄²×C₇	1	8	yes	—
23	C₇×C₁₆	C₇×C₁₆	1	192	yes	direct product factor
24	C₂×C₇²	C₈×C₁₆	1	3	yes	—
25	C₄×C₁₆	C₄×C₇²	1	6	yes	—
26	C₈²	C₂²×C₇²	1	1	yes	characteristic
27	C₂×C₄×C₇	C₄×C₇×C₈	1	24	yes	—
28	C₇×C₈	C₂×C₇×C₁₆	1	96	yes	—
29	C₇²	C₁₆²	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₁₆	C₇²×C₈	1	12	yes	—
31	C₄×C₈	C₂×C₄×C₇²	1	3	yes	—
32	C₂²×C₇	C₇×C₈²	1	8	yes	—
33	C₄×C₇	C₄×C₇×C₁₆	1	48	yes	—
34	C₂×C₈	C₂×C₇²×C₈	1	6	yes	—
35	C₄²	C₄²×C₇²	1	1	yes	characteristic
36	C₁₆	C₇²×C₁₆	1	24	yes	direct product factor
37	C₂×C₇	C₇×C₈×C₁₆	1	24	yes	—
38	C₂×C₄	C₄×C₇²×C₈	1	3	yes	—
39	C₈	C₂×C₇²×C₁₆	1	12	yes	—
40	C₇	C₇×C₁₆²	1	8	yes	direct product factor
41	C₂²	C₇²×C₈²	1	1	yes	characteristic
42	C₄	C₄×C₇²×C₁₆	1	6	yes	—
43	C₂	C₇²×C₈×C₁₆	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈²	C₂²×C₇²
socle, prime radical	C₂²×C₇²	C₈²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×C₁₆²	C₇	direct (N × H)
10. C₇²×C₁₆	C₁₆	direct (N × H)
15. C₁₆²	C₇²	direct (N × H)
23. C₇×C₁₆	C₇×C₁₆	direct (N × H)
29. C₇²	C₁₆²	direct (N × H)
36. C₁₆	C₇²×C₁₆	direct (N × H)
40. C₇	C₇×C₁₆²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇²×C₈×C₁₆	2
C₇×C₁₆²	7

Subgroups — C72×C162 #72190895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇²×C₁₆² #72190895