Subgroups — C₂²⋊(C₇⋊C₁₆) #89344

Order 448.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
4	C₂×C₇×C₈	—	2	2	no	—
5	C₂×C₇×C₈	C₄	1	1	yes	characteristic
6	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₇⋊C₁₆	—	2	2	no	—
8	C₂²⋊C₁₆	—	7	1	no	maximal, Sylow
9	C₂³×C₇	C₈	1	1	yes	characteristic
10	C₂×C₄×C₇	—	2	1	no	—
11	C₂×C₄×C₇	—	2	1	no	—
12	C₂×C₄×C₇	C₈	1	1	yes	characteristic
13	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₇×C₈	—	2	1	no	—
15	C₇×C₈	D₄	1	2	yes	—
16	C₂²×C₈	D₇	1	1	yes	characteristic
17	C₂×C₁₆	—	7	2	no	—
18	C₂²×C₇	—	2	1	no	—
19	C₂²×C₇	—	2	1	no	—
20	C₂²×C₇	C₁₆	1	2	yes	complemented
21	C₂²×C₇	C₂×C₈	1	1	yes	characteristic
22	C₄×C₇	—	2	1	no	—
23	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₄×C₇	M₁₆	1	1	yes	characteristic
25	C₂²×C₄	Dic₇	1	1	yes	characteristic
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	Dic₇	1	1	yes	characteristic
28	C₂×C₈	C₂×D₇	1	1	yes	characteristic
29	C₁₆	—	14	2	no	—
30	C₂×C₇	—	2	2	no	—
31	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
32	C₂×C₇	C₂×C₁₆	1	1	yes	characteristic
33	C₂×C₇	M₃₂	1	1	yes	characteristic
34	C₂³	C₇⋊C₈	1	1	yes	characteristic
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₇⋊C₈	1	1	yes	characteristic
38	C₂×C₄	C₂×Dic₇	1	1	yes	characteristic
39	C₈	—	2	1	no	—
40	C₈	C₂²⋊D₇	1	2	yes	—
41	C₇	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	2	1	no	—
43	C₂²	—	2	1	no	—
44	C₂²	C₇⋊C₁₆	1	2	yes	complemented
45	C₂²	C₂×C₇⋊C₈	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	#1136	1	1	yes	characteristic
48	C₄	C₂²⋊Dic₇	1	1	yes	characteristic
49	C₂	—	2	2	no	—
50	C₂	C₂×C₇⋊C₁₆	1	1	yes	characteristic
51	C₂	#6386	1	1	yes	characteristic
52	C₂	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₇⋊C₁₆	C₂	semidirect
20. C₂²×C₇	C₁₆	semidirect
41. C₇	C₂²⋊C₁₆	semidirect
44. C₂²	C₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₈	2
C₂×C₇⋊C₁₆	2
C₂²⋊C₁₆	7

Subgroups — C22⋊(C7⋊C16) #89344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊(C₇⋊C₁₆) #89344