Subgroups — #89581

Order 448.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6509	C₂	1	1	yes	maximal, characteristic
2	D₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
4	#6409	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₇)⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#6392	C₂	1	1	yes	maximal, characteristic, complemented
7	#6381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₇	—	2	1	no	—
10	C₂×C₄×D₇	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	(C₂×D₇)⋊C₄	—	2	1	no	—
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
16	Dic₇⋊C₄	—	2	1	no	—
17	Dic₇⋊C₄	—	2	1	no	—
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#484	—	7	1	no	maximal, Sylow
20	C₂²×D₇	D₄	1	1	yes	characteristic
21	C₇×Q₈	—	2	1	no	—
22	C₇×Q₈	—	2	1	no	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	—	2	1	no	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₇	—	2	1	no	—
27	C₂×Dic₇	—	2	1	no	—
28	C₂×Dic₇	D₄	1	1	yes	characteristic
29	C₄×D₇	—	2	1	no	—
30	C₄×D₇	—	2	1	no	—
31	C₄×D₇	—	4	2	no	—
32	C₇×C₈	—	2	1	no	—
33	C₇⋊C₈	—	2	1	no	—
34	C₂²⋊Q₈	—	7	1	no	—
35	C₂×C₄⋊C₄	—	7	1	no	—
36	#107	—	7	1	no	—
37	#107	—	7	1	no	—
38	#103	—	7	1	no	—
39	#103	D₇	1	1	yes	characteristic, complemented
40	C₂²⋊C₈	—	7	1	no	—
41	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
42	C₂×D₇	—	2	1	no	—
43	C₂×D₇	—	2	1	no	—
44	C₂×D₇	Q₁₆	1	2	yes	—
45	C₄×C₇	—	4	1	no	—
46	C₄×C₇	—	4	1	no	—
47	C₄×C₇	#41	1	1	yes	characteristic
48	C₄×C₇	#41	1	1	yes	characteristic
49	Dic₇	—	2	1	no	—
50	Dic₇	—	4	1	no	—
51	Dic₇	—	4	1	no	—
52	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
53	C₂²×C₄	—	7	1	no	—
54	C₂²×C₄	—	14	1	no	—
55	C₂×C₈	—	7	1	no	—
56	C₂×C₈	C₂×D₇	1	1	yes	characteristic
57	C₄⋊C₄	—	7	1	no	—
58	C₄⋊C₄	—	7	1	no	—
59	C₄⋊C₄	—	14	1	no	—
60	C₄⋊C₄	—	14	1	no	—
61	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
62	C₂²⋊C₄	—	14	1	no	—
63	C₂×C₇	#123	1	1	yes	characteristic
64	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	D₇	—	2	2	no	—
67	C₂³	—	7	1	no	—
68	Q₈	—	2	1	no	—
69	Q₈	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	7	1	no	—
73	C₂×C₄	—	14	1	no	—
74	C₂×C₄	—	14	1	no	—
75	C₂×C₄	—	14	1	no	—
76	C₂×C₄	—	14	1	no	—
77	C₂×C₄	—	28	2	no	—
78	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	—	14	1	no	—
81	C₇	#484	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	7	2	no	—
83	C₂²	—	14	1	no	—
84	C₂²	—	14	1	no	—
85	C₂²	D₄×D₇	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	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	—	14	2	no	—
94	C₂	#6446	1	1	yes	characteristic
95	C₂	#6477	1	1	yes	characteristic
96	C₂	D₇×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#103	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂²×D₇	D₄
hypercenter	#103	D₇
hypocenter	C₇	#484
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
3. C₇×#103	C₂	semidirect
4. #6409	C₂	semidirect
6. #6392	C₂	semidirect
7. #6381	C₂	semidirect
39. #103	D₇	semidirect
81. C₇	#484	semidirect

Maximal subgroups

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

subgroup	index
#6509	2
D₇×C₄⋊C₄	2
C₇×#103	2
#6409	2
(C₂×D₇)⋊C₈	2
#6392	2
#6381	2
#484	7