Subgroups — #89537

Order 448.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6499	C₂	1	1	yes	maximal, characteristic
2	#6451	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#102	C₂	1	1	yes	maximal, characteristic
4	#6406	C₂	1	1	yes	maximal, characteristic
5	#6390	C₂	1	1	yes	maximal, characteristic, complemented
6	#6389	C₂	1	1	yes	maximal, characteristic, complemented
7	#6384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₄	—	2	1	no	—
10	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₇	—	2	1	no	—
14	C₂²⋊Dic₇	—	2	1	no	—
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	Dic₇⋊C₄	—	2	1	no	—
17	C₄×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#489	—	7	1	no	maximal, Sylow
20	C₂³×C₇	—	2	1	no	—
21	C₇×D₄	—	2	1	no	—
22	C₇×D₄	—	2	1	no	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₇	—	2	1	no	—
26	C₂×Dic₇	—	2	1	no	—
27	C₂×Dic₇	D₄	1	1	yes	characteristic
28	C₂×Dic₇	D₄	1	1	yes	characteristic
29	Dic₁₄	—	2	1	no	—
30	Dic₁₄	—	2	1	no	—
31	Dic₁₄	—	4	2	no	—
32	C₇×C₈	—	2	1	no	—
33	C₇⋊C₈	—	2	1	no	—
34	C₄⋊Q₈	—	7	1	no	—
35	#124	—	7	1	no	—
36	#103	—	7	1	no	—
37	#103	—	7	1	no	—
38	#102	—	7	1	no	—
39	#102	D₇	1	1	yes	characteristic
40	#97	—	7	1	no	—
41	C₂²×C₇	—	4	1	no	—
42	C₂²×C₇	—	4	1	no	—
43	C₂²×C₇	—	4	1	no	—
44	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
45	C₄×C₇	—	4	1	no	—
46	C₄×C₇	#41	1	1	yes	characteristic
47	C₄×C₇	#41	1	1	yes	characteristic
48	Dic₇	—	2	1	no	—
49	Dic₇	—	2	1	no	—
50	Dic₇	—	4	1	no	—
51	Dic₇	—	4	1	no	—
52	C₂×Q₈	—	7	1	no	—
53	C₂×Q₈	—	14	1	no	—
54	C₂×D₄	C₂×D₇	1	1	yes	characteristic
55	C₂×C₈	—	7	1	no	—
56	C₂×C₈	C₂×D₇	1	1	yes	characteristic
57	C₄⋊C₄	—	7	1	no	—
58	C₄⋊C₄	—	14	1	no	—
59	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
60	C₂²⋊C₄	—	14	1	no	—
61	C₂²⋊C₄	—	14	1	no	—
62	C₄²	—	7	1	no	—
63	C₂×C₇	—	8	1	no	—
64	C₂×C₇	#124	1	1	yes	characteristic
65	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×C₇	#137	1	1	yes	characteristic
67	C₂³	—	2	1	no	—
68	Q₈	—	14	1	no	—
69	Q₈	—	14	1	no	—
70	Q₈	—	28	2	no	—
71	D₄	—	2	1	no	—
72	D₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	7	1	no	—
75	C₂×C₄	—	7	1	no	—
76	C₂×C₄	—	14	1	no	—
77	C₂×C₄	—	14	1	no	—
78	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	—	14	1	no	—
81	C₇	#489	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	4	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	D₄×D₇	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	14	1	no	—
88	C₄	—	14	1	no	—
89	C₄	—	28	1	no	—
90	C₄	—	28	1	no	—
91	C₄	#1157	1	1	yes	characteristic
92	C₄	#1159	1	1	yes	characteristic
93	C₂	—	8	1	no	—
94	C₂	#6448	1	1	yes	characteristic
95	C₂	#6474	1	1	yes	characteristic
96	C₂	#6478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#102	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×C₇×D₄	C₂²
hypercenter	#102	D₇
hypocenter	C₇	#489
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
2. #6451	C₂	semidirect
5. #6390	C₂	semidirect
6. #6389	C₂	semidirect
7. #6384	C₂	semidirect
81. C₇	#489	semidirect

Maximal subgroups

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

subgroup	index
#6499	2
#6451	2
C₇×#102	2
#6406	2
#6390	2
#6389	2
#6384	2
#489	7