Subgroups — #89533

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Dic₇	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	#6391	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#6382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₂²×Dic₇	—	2	1	no	—
10	C₂×Dic₁₄	—	2	1	no	—
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	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
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	#476	—	7	1	no	maximal, Sylow
20	C₂³×C₇	—	2	1	no	—
21	C₇×D₄	—	2	1	no	—
22	C₇×D₄	Q₈	1	2	yes	complemented
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	—	4	2	no	—
28	C₂×Dic₇	D₄	1	1	yes	characteristic
29	C₂×Dic₇	D₄	1	1	yes	characteristic
30	Dic₁₄	—	4	2	no	—
31	C₇×C₈	—	2	1	no	—
32	C₇⋊C₈	—	2	1	no	—
33	C₄⋊Q₈	—	7	1	no	—
34	C₄×D₄	—	7	1	no	—
35	#106	—	7	1	no	—
36	#106	—	7	1	no	—
37	C₄⋊C₈	—	7	1	no	—
38	#102	—	7	1	no	—
39	#102	D₇	1	1	yes	characteristic
40	C₂²×C₇	—	2	2	no	—
41	C₂²×C₇	—	4	1	no	—
42	C₂²×C₇	—	4	1	no	—
43	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
44	C₄×C₇	—	4	1	no	—
45	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
46	C₄×C₇	#41	1	1	yes	characteristic
47	Dic₇	—	2	1	no	—
48	Dic₇	—	4	1	no	—
49	Dic₇	—	4	1	no	—
50	Dic₇	SD₁₆	1	2	yes	—
51	C₂×Q₈	—	14	1	no	—
52	C₂×D₄	C₂×D₇	1	1	yes	characteristic
53	C₂²×C₄	—	14	1	no	—
54	C₂×C₈	—	7	1	no	—
55	C₂×C₈	C₂×D₇	1	1	yes	characteristic
56	C₄⋊C₄	—	7	1	no	—
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	C₄²	—	7	1	no	—
62	C₂×C₇	—	4	2	no	—
63	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂³	—	2	1	no	—
67	Q₈	—	28	2	no	—
68	D₄	—	2	1	no	—
69	D₄	Dic₁₄	1	2	yes	complemented
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	7	1	no	—
72	C₂×C₄	—	7	1	no	—
73	C₂×C₄	—	14	1	no	—
74	C₂×C₄	—	14	1	no	—
75	C₂×C₄	—	28	2	no	—
76	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	14	1	no	—
79	C₇	#476	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	2	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	D₄×D₇	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	7	2	no	—
86	C₄	—	14	1	no	—
87	C₄	—	28	1	no	—
88	C₄	—	28	1	no	—
89	C₄	C₂×Dic₁₄	1	1	yes	characteristic
90	C₄	#1159	1	1	yes	characteristic
91	C₂	—	4	2	no	—
92	C₂	#6441	1	1	yes	characteristic
93	C₂	#6474	1	1	yes	characteristic
94	C₂	D₇×SD₁₆	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	#476
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. #6391	C₂	semidirect
6. Dic₇⋊C₈	C₂	semidirect
7. #6382	C₂	semidirect
22. C₇×D₄	Q₈	semidirect
69. D₄	Dic₁₄	semidirect
79. C₇	#476	semidirect

Maximal subgroups

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

subgroup	index
D₄×Dic₇	2
#6451	2
C₇×#102	2
#6406	2
#6391	2
Dic₇⋊C₈	2
#6382	2
#476	7