Subgroups — #89909

Order 448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#6487	C₂	1	2	yes	maximal
3	C₂×Dic₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6391	C₂	1	1	yes	maximal, characteristic, complemented
5	#6390	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₄	C₂²	1	2	yes	—
8	C₇×M₁₆	—	2	2	no	—
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₇	—	2	2	no	—
12	C₄⋊Dic₇	C₂²	1	2	yes	—
13	Dic₇⋊C₄	—	2	2	no	—
14	Dic₂₈	—	2	2	no	—
15	#470	—	7	1	no	maximal, Sylow
16	C₂³×C₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₇	—	2	1	no	—
18	C₂×C₄×C₇	—	2	1	no	—
19	C₂×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₇	—	2	2	no	—
22	C₂×Dic₇	—	2	2	no	—
23	Dic₁₄	—	2	2	no	—
24	Dic₁₄	—	2	2	no	—
25	C₇×C₈	—	2	1	no	—
26	C₇×C₈	D₄	1	2	yes	—
27	C₂×Q₁₆	—	7	1	no	—
28	C₂×M₁₆	D₇	1	1	yes	characteristic
29	C₂²⋊Q₈	—	7	2	no	—
30	#106	—	7	1	no	—
31	#103	—	7	2	no	—
32	C₂²×C₇	—	2	1	no	—
33	C₂²×C₇	—	2	2	no	—
34	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
35	C₄×C₇	—	2	1	no	—
36	C₄×C₇	C₂×D₄	1	1	yes	characteristic
37	C₄×C₇	#41	1	1	yes	characteristic
38	Dic₇	—	4	2	no	—
39	Dic₇	—	4	2	no	—
40	C₂×Q₈	—	7	2	no	—
41	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
42	Q₁₆	—	14	2	no	—
43	M₁₆	—	2	2	no	—
44	C₂×C₈	C₂×D₇	1	1	yes	characteristic
45	C₂×C₈	C₂×D₇	1	1	yes	characteristic
46	C₄⋊C₄	—	7	2	no	—
47	C₄⋊C₄	—	14	2	no	—
48	C₂²⋊C₄	—	14	2	no	—
49	C₂×C₇	—	4	1	no	—
50	C₂×C₇	#121	1	1	yes	characteristic
51	C₂×C₇	#137	1	2	yes	—
52	C₂³	D₂₈	1	1	yes	characteristic
53	Q₈	—	14	2	no	—
54	Q₈	—	14	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	14	2	no	—
58	C₂×C₄	—	14	2	no	—
59	C₂×C₄	D₂₈	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	C₂²⋊D₇	1	2	yes	—
63	C₇	#470	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	2	2	no	—
65	C₂²	—	2	1	no	—
66	C₂²	C₂×D₂₈	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	28	2	no	—
69	C₄	—	28	2	no	—
70	C₄	#1157	1	1	yes	characteristic
71	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
72	C₂	—	4	1	no	—
73	C₂	#6472	1	2	yes	—
74	C₂	#6493	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×M₁₆	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂³×C₇	D₄
hypercenter	C₂×M₁₆	D₇
hypocenter	C₇	#470
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×Dic₂₈	C₂	semidirect
4. #6391	C₂	semidirect
5. #6390	C₂	semidirect
63. C₇	#470	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×M₁₆	2
#6487	2
C₂×Dic₂₈	2
#6391	2
#6390	2
#470	7