Subgroups — C₂×C₇×#102 #90060

Order 448.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#102	C₂	1	4	yes	maximal, direct product factor
5	C₂⁴×C₇	—	2	1	no	—
6	C₂×C₇×D₄	—	2	1	no	—
7	C₂×C₇×D₄	—	2	2	no	—
8	C₂×C₇×D₄	C₄	1	4	yes	complemented
9	C₂×C₇×D₄	C₂²	1	2	yes	—
10	C₂²×C₄×C₇	—	2	1	no	—
11	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₈	—	2	2	no	—
13	C₂×C₇×C₈	C₂²	1	2	yes	complemented
14	C₇×C₄⋊C₄	—	2	1	no	—
15	C₇×C₄⋊C₄	C₂²	1	2	yes	complemented
16	C₂×#102	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	—	2	4	no	—
18	C₂³×C₇	—	2	2	no	—
19	C₂³×C₇	—	4	2	no	—
20	C₂³×C₇	—	4	2	no	—
21	C₂³×C₇	D₄	1	1	yes	characteristic
22	C₇×D₄	—	2	4	no	—
23	C₇×D₄	—	2	2	no	—
24	C₇×D₄	C₂×C₄	1	4	yes	complemented
25	C₂×C₄×C₇	—	2	2	no	—
26	C₂×C₄×C₇	—	4	2	no	—
27	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₇	D₄	1	1	yes	characteristic
29	C₂×C₄×C₇	D₄	1	2	yes	—
30	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₇×C₈	—	2	2	no	—
32	C₂²×D₄	C₂×C₇	1	1	yes	characteristic
33	C₂²×C₈	C₂×C₇	1	1	yes	characteristic, complemented
34	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
35	#102	C₂×C₇	1	4	yes	direct product factor
36	C₂²×C₇	—	2	4	no	—
37	C₂²×C₇	—	4	8	no	—
38	C₂²×C₇	—	4	2	no	—
39	C₂²×C₇	—	4	2	no	—
40	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₇	D₈	1	2	yes	—
42	C₂²×C₇	SD₁₆	1	2	yes	—
43	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
44	C₄×C₇	—	4	2	no	—
45	C₄×C₇	C₂²⋊C₄	1	2	yes	—
46	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂⁴	—	2	1	no	—
49	C₂×D₄	—	2	1	no	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	C₄×C₇	1	4	yes	complemented
52	C₂×D₄	C₂²×C₇	1	2	yes	—
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	C₂²×C₇	1	2	yes	complemented
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	C₂²×C₇	1	2	yes	complemented
59	C₂×C₇	—	4	4	no	—
60	C₂×C₇	#102	1	4	yes	direct product factor
61	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂³	—	2	4	no	—
65	C₂³	—	2	2	no	—
66	C₂³	—	4	2	no	—
67	C₂³	—	4	2	no	—
68	C₂³	C₇×D₄	1	1	yes	characteristic
69	D₄	—	2	4	no	—
70	D₄	—	2	2	no	—
71	D₄	C₂×C₄×C₇	1	4	yes	complemented
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
75	C₂×C₄	C₇×D₄	1	1	yes	characteristic
76	C₂×C₄	C₇×D₄	1	2	yes	—
77	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₇	C₂×#102	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²	—	2	4	no	—
81	C₂²	—	4	8	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₇×C₂²⋊C₄	1	2	yes	—
85	C₂²	C₇×D₈	1	2	yes	—
86	C₂²	C₇×SD₁₆	1	2	yes	—
87	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
88	C₄	—	4	2	no	—
89	C₄	C₇×C₂²⋊C₄	1	2	yes	—
90	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
91	C₄	C₂×C₇×D₄	1	1	yes	characteristic
92	C₂	—	4	4	no	—
93	C₂	C₇×#102	1	4	yes	direct product factor
94	C₂	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
95	C₂	C₂×C₇×D₈	1	1	yes	characteristic
96	C₂	C₂×C₇×SD₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	D₄
commutator	C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₂²×C₇×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₈	C₂	semidirect
3. C₂×C₇×C₄⋊C₄	C₂	semidirect
4. C₇×#102	C₂	direct (N × H)
8. C₂×C₇×D₄	C₄	semidirect
13. C₂×C₇×C₈	C₂²	semidirect
15. C₇×C₄⋊C₄	C₂²	semidirect
16. C₂×#102	C₇	direct (N × H)
24. C₇×D₄	C₂×C₄	semidirect
33. C₂²×C₈	C₂×C₇	semidirect
34. C₂×C₄⋊C₄	C₂×C₇	semidirect
35. #102	C₂×C₇	direct (N × H)
51. C₂×D₄	C₄×C₇	semidirect
56. C₂×C₈	C₂²×C₇	semidirect
58. C₄⋊C₄	C₂²×C₇	semidirect
60. C₂×C₇	#102	direct (N × H)
71. D₄	C₂×C₄×C₇	semidirect
79. C₇	C₂×#102	direct (N × H)
93. C₂	C₇×#102	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×D₄	2
C₂²×C₇×C₈	2
C₂×C₇×C₄⋊C₄	2
C₇×#102	2
C₂×#102	7

Subgroups — C2×C7×#102 #90060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#102 #90060