Subgroups — #89956

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₇	C₂	1	2	yes	maximal
3	#6409	C₂	1	2	yes	maximal
4	#6391	C₂	1	1	yes	maximal, characteristic
5	#6389	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₇×Q₁₆	C₄	1	4	yes	complemented
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₇	—	2	2	no	—
10	C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₄×Dic₇	—	2	2	no	—
12	C₄×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	#441	—	7	1	no	maximal, Sylow
15	C₇×Q₈	—	2	2	no	—
16	C₇×Q₈	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₇	—	2	2	no	—
20	C₂×Dic₇	D₄	1	2	yes	—
21	C₇×C₈	C₂×C₄	1	2	yes	—
22	C₇⋊C₈	—	2	1	no	—
23	C₂×Q₁₆	D₇	1	1	yes	characteristic
24	C₄×Q₈	—	7	2	no	—
25	#106	—	7	1	no	—
26	#103	—	7	2	no	—
27	#97	—	7	1	no	—
28	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
29	C₄×C₇	—	2	4	no	—
30	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₇	#41	1	1	yes	characteristic
32	Dic₇	—	2	2	no	—
33	Dic₇	—	4	2	no	—
34	C₂×Q₈	C₂×D₇	1	2	yes	—
35	Q₁₆	Dic₇	1	4	yes	complemented
36	C₂×C₈	—	7	1	no	—
37	C₂×C₈	C₂×D₇	1	1	yes	characteristic
38	C₄⋊C₄	—	7	2	no	—
39	C₄⋊C₄	—	14	2	no	—
40	C₄²	—	7	1	no	—
41	C₄²	—	14	2	no	—
42	C₂×C₇	C₄×D₄	1	1	yes	characteristic
43	C₂×C₇	#137	1	2	yes	—
44	Q₈	—	2	2	no	—
45	Q₈	C₂×Dic₇	1	4	yes	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	7	2	no	—
48	C₂×C₄	—	14	2	no	—
49	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
50	C₈	—	14	1	no	—
51	C₈	C₂×Dic₇	1	2	yes	—
52	C₇	#441	1	1	yes	characteristic, complemented, Sylow
53	C₂²	D₄×D₇	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	—	14	2	no	—
56	C₄	—	28	2	no	—
57	C₄	#1159	1	1	yes	characteristic
58	C₄	C₂²×Dic₇	1	1	yes	characteristic
59	C₂	#6481	1	2	yes	—
60	C₂	D₄×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×Q₁₆	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×Q₁₆	D₇
hypocenter	C₇	#441
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
7. C₇×Q₁₆	C₄	semidirect
35. Q₁₆	Dic₇	semidirect
52. C₇	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₁₆	2
Q₈×Dic₇	2
#6409	2
#6391	2
#6389	2
#441	7