Subgroups — #89466

Order 448.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6437	C₂	1	1	yes	maximal, characteristic
2	#6432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395	C₂	1	4	yes	maximal, complemented
5	C₂×D₂₈	—	2	1	no	—
6	C₂×D₂₈	—	2	1	no	—
7	C₂×D₂₈	C₂²	1	2	yes	—
8	C₂×Dic₁₄	—	2	1	no	—
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₇	—	2	1	no	—
12	C₄⋊Dic₇	C₂²	1	2	yes	—
13	#486	—	7	1	no	maximal, Sylow
14	C₂²×D₇	—	2	2	no	—
15	C₂×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₇	—	2	2	no	—
19	D₂₈	—	2	4	no	—
20	D₂₈	—	4	2	no	—
21	D₂₈	—	4	2	no	—
22	Dic₁₄	—	4	2	no	—
23	C₇×C₈	—	2	2	no	—
24	C₄⋊Q₈	—	7	1	no	—
25	#127	—	7	1	no	—
26	#102	—	7	4	no	—
27	C₄×C₈	D₇	1	1	yes	characteristic, complemented
28	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
29	C₂×D₇	—	4	2	no	—
30	C₂×D₇	—	4	2	no	—
31	C₂×D₇	—	4	2	no	—
32	C₄×C₇	D₈	1	2	yes	—
33	C₄×C₇	SD₁₆	1	2	yes	—
34	C₄×C₇	#41	1	2	yes	—
35	Dic₇	—	4	2	no	—
36	C₂×Q₈	—	14	1	no	—
37	C₂×D₄	—	7	2	no	—
38	C₂×D₄	—	14	1	no	—
39	C₂×D₄	—	14	1	no	—
40	C₂×C₈	C₂×D₇	1	2	yes	—
41	C₄⋊C₄	—	7	2	no	—
42	C₄⋊C₄	—	14	1	no	—
43	C₄²	C₂×D₇	1	1	yes	characteristic
44	C₂×C₇	#124	1	1	yes	characteristic
45	C₂×C₇	C₂×D₈	1	1	yes	characteristic
46	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
47	D₇	—	8	2	no	—
48	C₂³	—	14	2	no	—
49	Q₈	—	28	2	no	—
50	D₄	—	14	4	no	—
51	D₄	—	28	2	no	—
52	D₄	—	28	2	no	—
53	C₂×C₄	—	14	2	no	—
54	C₂×C₄	D₂₈	1	1	yes	characteristic
55	C₂×C₄	D₂₈	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₇	#486	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	28	2	no	—
60	C₂²	—	28	2	no	—
61	C₂²	—	28	2	no	—
62	C₂²	C₂×D₂₈	1	1	yes	characteristic
63	C₄	—	28	2	no	—
64	C₄	#1132	1	2	yes	—
65	C₄	D₅₆	1	2	yes	—
66	C₄	#1157	1	2	yes	—
67	C₂	—	56	2	no	—
68	C₂	#6438	1	1	yes	characteristic
69	C₂	C₂×#1132	1	1	yes	characteristic
70	C₂	C₂×D₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₄×C₇×C₈	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	#6437	C₂
hypercenter	C₄×C₈	D₇
hypocenter	C₇	#486
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #6432	C₂	semidirect
3. C₄×C₇×C₈	C₂	semidirect
4. #6395	C₂	semidirect
27. C₄×C₈	D₇	semidirect
58. C₇	#486	semidirect

Maximal subgroups

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

subgroup	index
#6437	2
#6432	2
C₄×C₇×C₈	2
#6395	2
#486	7