Subgroups — #90263

Order 448.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#6455	C₂	1	3	yes	maximal
5	#6450	C₂	1	3	yes	maximal, complemented
6	#6435	C₂	1	3	yes	maximal
7	C₄×Dic₁₄	C₂	1	3	yes	maximal, complemented
8	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
9	Q₈×D₇	C₄	1	8	yes	complemented
10	C₂×C₄×D₇	C₂²	1	3	yes	—
11	C₂×Dic₁₄	C₂²	1	3	yes	—
12	C₇×C₄⋊C₄	C₂²	1	3	yes	—
13	C₄²×C₇	C₂²	1	3	yes	—
14	(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
15	(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
16	C₄⋊Dic₇	C₂²	1	3	yes	—
17	Dic₇⋊C₄	C₂²	1	3	yes	—
18	Dic₇⋊C₄	C₂²	1	3	yes	—
19	C₄×Dic₇	C₂²	1	6	yes	—
20	C₄×Dic₇	C₂²	1	3	yes	—
21	#519	—	7	1	no	maximal, Sylow
22	C₂²×D₇	C₂³	1	1	yes	characteristic
23	C₇×Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₄×C₇	C₂³	1	3	yes	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₇	C₂³	1	3	yes	—
27	C₂×Dic₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₇	C₂³	1	3	yes	—
29	C₂×Dic₇	C₂³	1	3	yes	—
30	C₄×D₇	C₂×C₄	1	12	yes	—
31	Dic₁₄	C₂×C₄	1	12	yes	—
32	C₂²×Q₈	—	7	1	no	—
33	C₄×Q₈	—	7	1	no	—
34	C₄×Q₈	—	7	3	no	—
35	C₄×Q₈	—	7	3	no	—
36	C₄×Q₈	D₇	1	1	yes	characteristic, complemented
37	#117	—	7	3	no	—
38	#117	—	7	3	no	—
39	C₂²×C₇	C₂⁴	1	1	yes	characteristic
40	C₂×D₇	—	2	1	no	—
41	C₂×D₇	—	2	1	no	—
42	C₂×D₇	C₂²×C₄	1	2	yes	—
43	C₄×C₇	—	2	1	no	—
44	C₄×C₇	—	2	3	no	—
45	C₄×C₇	C₂²×C₄	1	6	yes	—
46	Dic₇	—	2	3	no	—
47	Dic₇	—	2	1	no	—
48	Dic₇	C₂²×C₄	1	6	yes	—
49	C₂×Q₈	—	7	8	no	—
50	C₂×Q₈	—	7	3	no	—
51	C₂×Q₈	C₂×D₇	1	1	yes	characteristic
52	C₂²×C₄	—	7	3	no	—
53	C₄⋊C₄	—	7	3	no	—
54	C₄⋊C₄	—	7	3	no	—
55	C₄⋊C₄	—	7	3	no	—
56	C₄⋊C₄	C₂×D₇	1	3	yes	—
57	C₂²⋊C₄	—	7	3	no	—
58	C₂²⋊C₄	—	7	1	no	—
59	C₄²	—	7	6	no	—
60	C₄²	—	7	3	no	—
61	C₄²	C₂×D₇	1	3	yes	—
62	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
64	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
65	D₇	—	2	2	no	—
66	C₂³	—	7	1	no	—
67	Q₈	—	7	12	no	—
68	Q₈	C₄×D₇	1	4	yes	—
69	C₂×C₄	—	7	3	no	—
70	C₂×C₄	—	7	12	no	—
71	C₂×C₄	—	7	3	no	—
72	C₂×C₄	—	7	1	no	—
73	C₂×C₄	C₂²×D₇	1	3	yes	—
74	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₇	1	3	yes	—
76	C₇	#519	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	7	2	no	—
78	C₂²	—	14	1	no	—
79	C₂²	—	14	1	no	—
80	C₂²	C₂³×D₇	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	3	no	—
83	C₄	—	7	6	no	—
84	C₄	—	14	3	no	—
85	C₄	—	14	1	no	—
86	C₄	C₂×C₄×D₇	1	6	yes	—
87	C₂	—	14	2	no	—
88	C₂	C₂²×C₄×D₇	1	1	yes	characteristic
89	C₂	#6551	1	1	yes	characteristic
90	C₂	#6554	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇
commutator	C₂×C₇	C₂³×C₄
Fitting	C₄×C₇×Q₈	C₂
socle, cosocle	C₂²×C₇	C₂⁴
prime radical	C₂²×D₇	C₂³
hypercenter	C₄×Q₈	D₇
hypocenter	C₇	#519
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₇×Q₈	C₂	semidirect
3. Q₈×Dic₇	C₂	semidirect
5. #6450	C₂	semidirect
7. C₄×Dic₁₄	C₂	semidirect
9. Q₈×D₇	C₄	semidirect
36. C₄×Q₈	D₇	semidirect
76. C₇	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₇	2
C₄×C₇×Q₈	2
Q₈×Dic₇	2
#6455	2
#6450	2
#6435	2
C₄×Dic₁₄	2
#519	7