Subgroups — #90207

Order 448.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₇×#117	C₂	1	1	yes	maximal, characteristic
3	#6489	C₂	1	1	yes	maximal, characteristic
4	#6450	C₂	1	4	yes	maximal, complemented
5	#6440	C₂	1	4	yes	maximal
6	C₄×Dic₁₄	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₇	C₂²	1	2	yes	—
9	C₂×Dic₁₄	C₄	1	8	yes	complemented
10	C₂×Dic₁₄	C₂²	1	2	yes	—
11	C₂×Dic₁₄	C₂²	1	2	yes	—
12	C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₇×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₄²×C₇	C₂²	1	2	yes	—
15	C₂²⋊Dic₇	C₂²	1	2	yes	—
16	C₄⋊Dic₇	C₂²	1	2	yes	—
17	Dic₇⋊C₄	C₂²	1	8	yes	—
18	C₄×Dic₇	C₂²	1	8	yes	—
19	C₄×Dic₇	C₂²	1	2	yes	—
20	#519	—	7	1	no	maximal, Sylow
21	C₂³×C₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₇	C₂³	1	4	yes	—
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₇	C₂×C₄	1	8	yes	—
27	C₂×Dic₇	C₂³	1	4	yes	—
28	C₂×Dic₇	C₂³	1	4	yes	—
29	Dic₁₄	C₂×C₄	1	8	yes	—
30	Dic₁₄	C₂×C₄	1	8	yes	—
31	C₂²×Q₈	—	7	1	no	—
32	C₄×Q₈	—	7	4	no	—
33	C₄×Q₈	—	7	4	no	—
34	#117	—	7	1	no	—
35	#117	—	7	4	no	—
36	#117	D₇	1	1	yes	characteristic
37	C₂²×C₇	—	2	2	no	—
38	C₂²×C₇	C₂²×C₄	1	2	yes	—
39	C₂²×C₇	C₂⁴	1	1	yes	characteristic
40	C₄×C₇	—	2	4	no	—
41	C₄×C₇	C₂²×C₄	1	2	yes	—
42	C₄×C₇	C₂²×C₄	1	2	yes	—
43	Dic₇	—	2	4	no	—
44	Dic₇	C₂²×C₄	1	8	yes	—
45	C₂×Q₈	—	7	8	no	—
46	C₂×Q₈	—	7	2	no	—
47	C₂×Q₈	—	7	2	no	—
48	C₂²×C₄	—	7	2	no	—
49	C₂²×C₄	C₂×D₇	1	1	yes	characteristic
50	C₄⋊C₄	—	7	8	no	—
51	C₄⋊C₄	—	7	2	no	—
52	C₄⋊C₄	C₂×D₇	1	2	yes	—
53	C₂²⋊C₄	—	7	2	no	—
54	C₂²⋊C₄	C₂×D₇	1	2	yes	—
55	C₄²	—	7	8	no	—
56	C₄²	—	7	2	no	—
57	C₄²	C₂×D₇	1	2	yes	—
58	C₂×C₇	—	2	2	no	—
59	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
60	C₂×C₇	2₋¹⁺⁴	1	2	yes	—
61	C₂³	C₂²×D₇	1	1	yes	characteristic
62	Q₈	—	7	8	no	—
63	Q₈	—	7	8	no	—
64	C₂×C₄	—	7	8	no	—
65	C₂×C₄	—	7	4	no	—
66	C₂×C₄	—	7	4	no	—
67	C₂×C₄	C₄×D₇	1	4	yes	—
68	C₂×C₄	C₂²×D₇	1	4	yes	—
69	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
71	C₇	#519	1	1	yes	characteristic, complemented, Sylow
72	C₂²	—	2	2	no	—
73	C₂²	C₂×C₄×D₇	1	2	yes	—
74	C₂²	C₂³×D₇	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	—	7	8	no	—
77	C₄	—	14	4	no	—
78	C₄	C₂×C₄×D₇	1	2	yes	—
79	C₄	C₂×C₄×D₇	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	C₂²×C₄×D₇	1	1	yes	characteristic
82	C₂	#6554	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator	C₂×C₇	C₂³×C₄
Fitting	C₇×#117	C₂
socle, cosocle	C₂²×C₇	C₂⁴
prime radical	C₂³×C₇	C₂³
hypercenter	#117	D₇
hypocenter	C₇	#519
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
4. #6450	C₂	semidirect
6. C₄×Dic₁₄	C₂	semidirect
9. C₂×Dic₁₄	C₄	semidirect
71. C₇	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₄	2
C₇×#117	2
#6489	2
#6450	2
#6440	2
C₄×Dic₁₄	2
#519	7