Subgroups — C₂×#6440 #90171

Order 448.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×Dic₇	C₂	1	2	yes	maximal, complemented
6	#6440	C₂	1	8	yes	maximal, direct product factor
7	C₂⁴×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₇	C₂²	1	2	yes	—
9	C₂²×Dic₇	C₄	1	8	yes	complemented
10	C₂²×Dic₇	C₂²	1	2	yes	—
11	C₂²×Dic₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₇	C₂²	1	4	yes	—
14	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₂²⋊Dic₇	C₂²	1	4	yes	—
16	Dic₇⋊C₄	C₂²	1	8	yes	complemented
17	C₄×Dic₇	C₂²	1	8	yes	complemented
18	C₂×#117	—	7	1	no	maximal, Sylow
19	C₂³×C₇	—	2	4	no	—
20	C₂³×C₇	C₂×C₄	1	4	yes	—
21	C₂³×C₇	C₂³	1	2	yes	—
22	C₂³×C₇	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₇	—	2	4	no	—
24	C₂×C₄×C₇	C₂³	1	4	yes	—
25	C₂×Dic₇	—	2	4	no	—
26	C₂×Dic₇	C₂×C₄	1	4	yes	—
27	C₂×Dic₇	C₂×C₄	1	4	yes	—
28	C₂×Dic₇	C₂×C₄	1	16	yes	complemented
29	C₂×Dic₇	C₂³	1	4	yes	—
30	C₂×Dic₇	C₂³	1	2	yes	—
31	C₂×Dic₇	C₂³	1	2	yes	—
32	C₂³×C₄	—	7	1	no	—
33	#117	—	7	8	no	—
34	C₂×C₄⋊C₄	—	7	2	no	—
35	C₂×C₂²⋊C₄	—	7	1	no	—
36	C₂×C₂²⋊C₄	D₇	1	1	yes	characteristic
37	C₂×C₄²	—	7	2	no	—
38	C₂²×C₇	—	2	8	no	—
39	C₂²×C₇	—	2	4	no	—
40	C₂²×C₇	C₂²×C₄	1	4	yes	—
41	C₂²×C₇	C₂²×C₄	1	2	yes	—
42	C₂²×C₇	#41	1	4	yes	—
43	C₂²×C₇	C₂⁴	1	1	yes	characteristic
44	C₄×C₇	—	2	4	no	—
45	Dic₇	—	2	4	no	—
46	Dic₇	C₂²×C₄	1	4	yes	—
47	Dic₇	C₂²×C₄	1	4	yes	—
48	C₂⁴	C₂×D₇	1	1	yes	characteristic
49	C₂²×C₄	—	7	8	no	—
50	C₂²×C₄	—	7	4	no	—
51	C₂²×C₄	—	7	1	no	—
52	C₂²×C₄	—	7	1	no	—
53	C₂²×C₄	—	7	2	no	—
54	C₂²×C₄	C₂×D₇	1	2	yes	—
55	C₄⋊C₄	—	7	8	no	—
56	C₂²⋊C₄	—	7	4	no	—
57	C₂²⋊C₄	C₂×D₇	1	4	yes	—
58	C₄²	—	7	8	no	—
59	C₂×C₇	—	2	4	no	—
60	C₂×C₇	#117	1	4	yes	complemented
61	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
62	C₂×C₇	C₂×#41	1	2	yes	—
63	C₂³	—	2	4	no	—
64	C₂³	C₄×D₇	1	4	yes	—
65	C₂³	C₂²×D₇	1	2	yes	—
66	C₂³	C₂²×D₇	1	1	yes	characteristic
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	—	7	4	no	—
69	C₂×C₄	—	7	4	no	—
70	C₂×C₄	—	7	16	no	—
71	C₂×C₄	—	7	2	no	—
72	C₂×C₄	—	7	2	no	—
73	C₂×C₄	—	7	4	no	—
74	C₂×C₄	—	14	4	no	—
75	C₂×C₄	C₂²×D₇	1	4	yes	—
76	C₇	C₂×#117	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	2	8	no	—
78	C₂²	—	2	4	no	—
79	C₂²	C₂×C₄×D₇	1	4	yes	—
80	C₂²	C₂×C₄×D₇	1	2	yes	—
81	C₂²	#1159	1	4	yes	—
82	C₂²	C₂³×D₇	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	7	4	no	—
85	C₄	—	7	4	no	—
86	C₄	—	14	4	no	—
87	C₂	—	2	4	no	—
88	C₂	#6440	1	4	yes	direct product factor
89	C₂	C₂²×C₄×D₇	1	1	yes	characteristic
90	C₂	C₂×#1159	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₇⋊C₄	C₂	semidirect
5. C₂×C₄×Dic₇	C₂	semidirect
6. #6440	C₂	direct (N × H)
9. C₂²×Dic₇	C₄	semidirect
16. Dic₇⋊C₄	C₂²	semidirect
17. C₄×Dic₇	C₂²	semidirect
28. C₂×Dic₇	C₂×C₄	semidirect
60. C₂×C₇	#117	semidirect
76. C₇	C₂×#117	semidirect
88. C₂	#6440	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₇	2
C₂×C₇×C₂²⋊C₄	2
C₂×C₂²⋊Dic₇	2
C₂×Dic₇⋊C₄	2
C₂×C₄×Dic₇	2
#6440	2
C₂×#117	7

Subgroups — C2×#6440 #90171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6440 #90171