Subgroups — #90174

Order 448.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₇	C₂	1	2	yes	maximal
3	#6487	C₂	1	4	yes	maximal, complemented
4	#6441	C₂	1	8	yes	maximal, complemented
5	C₂⁴×C₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₇	C₂²	1	2	yes	—
7	C₂²×Dic₇	C₂²	1	4	yes	—
8	C₂×Dic₁₄	C₂²	1	4	yes	complemented
9	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
10	C₂²⋊Dic₇	C₂²	1	4	yes	—
11	C₂²⋊Dic₇	C₂²	1	4	yes	—
12	C₄⋊Dic₇	C₂²	1	4	yes	complemented
13	Dic₇⋊C₄	C₂²	1	8	yes	complemented
14	#543	—	7	1	no	maximal, Sylow
15	C₂³×C₇	—	4	2	no	—
16	C₂³×C₇	Q₈	1	4	yes	complemented
17	C₂³×C₇	C₂³	1	1	yes	characteristic
18	C₂³×C₇	C₂³	1	2	yes	—
19	C₂×C₄×C₇	—	4	2	no	—
20	C₂×C₄×C₇	C₂³	1	4	yes	—
21	C₂×Dic₇	—	4	4	no	—
22	C₂×Dic₇	C₂³	1	8	yes	—
23	Dic₁₄	—	4	4	no	—
24	C₂²⋊Q₈	—	7	4	no	—
25	C₂²⋊Q₈	—	7	8	no	—
26	C₂×C₂²⋊C₄	—	7	2	no	—
27	C₂×C₂²⋊C₄	D₇	1	1	yes	characteristic
28	C₂²×C₇	—	2	4	no	—
29	C₂²×C₇	—	2	2	no	—
30	C₂²×C₇	—	4	4	no	—
31	C₂²×C₇	C₂×Q₈	1	2	yes	—
32	C₂²×C₇	C₂×Q₈	1	4	yes	—
33	C₂²×C₇	C₂⁴	1	1	yes	characteristic
34	C₄×C₇	—	2	4	no	—
35	Dic₇	—	2	8	no	—
36	C₂⁴	C₂×D₇	1	1	yes	characteristic
37	C₂×Q₈	—	7	4	no	—
38	C₂²×C₄	—	7	4	no	—
39	C₂²×C₄	C₂×D₇	1	2	yes	—
40	C₄⋊C₄	—	7	8	no	—
41	C₄⋊C₄	—	7	4	no	—
42	C₂²⋊C₄	—	7	4	no	—
43	C₂²⋊C₄	—	7	4	no	—
44	C₂²⋊C₄	C₂×D₇	1	4	yes	—
45	C₂×C₇	—	2	4	no	—
46	C₂×C₇	—	2	2	no	—
47	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
48	C₂×C₇	H(5,𝔽₂)	1	2	yes	—
49	C₂³	—	4	2	no	—
50	C₂³	Dic₁₄	1	4	yes	complemented
51	C₂³	C₂²×D₇	1	1	yes	characteristic
52	C₂³	C₂²×D₇	1	2	yes	—
53	Q₈	—	28	4	no	—
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	—	7	8	no	—
56	C₂×C₄	—	28	4	no	—
57	C₂×C₄	C₂²×D₇	1	4	yes	—
58	C₇	#543	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	2	2	no	—
60	C₂²	—	2	4	no	—
61	C₂²	—	4	4	no	—
62	C₂²	C₂×Dic₁₄	1	2	yes	—
63	C₂²	C₂×Dic₁₄	1	4	yes	—
64	C₂²	C₂³×D₇	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	14	8	no	—
67	C₂	—	2	2	no	—
68	C₂	—	2	4	no	—
69	C₂	C₂²×Dic₁₄	1	1	yes	characteristic
70	C₂	#6548	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁴
Fitting	C₂×C₇×C₂²⋊C₄	C₂
prime radical	C₂⁴×C₇	C₂²
hypercenter	C₂×C₂²⋊C₄	D₇
hypocenter	C₇	#543
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. #6487	C₂	semidirect
4. #6441	C₂	semidirect
8. C₂×Dic₁₄	C₂²	semidirect
12. C₄⋊Dic₇	C₂²	semidirect
13. Dic₇⋊C₄	C₂²	semidirect
16. C₂³×C₇	Q₈	semidirect
50. C₂³	Dic₁₄	semidirect
58. C₇	#543	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₂²⋊C₄	2
C₂×C₂²⋊Dic₇	2
#6487	2
#6441	2
#543	7