Subgroups — C₂×C₇×#108 #90075

Order 448.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	2	yes	maximal
2	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#108	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₇×M₁₆	—	2	2	no	—
6	C₇×M₁₆	C₂²	1	4	yes	—
7	C₂×C₇×C₈	—	2	2	no	—
8	C₂×C₇×C₈	C₄	1	2	yes	—
9	C₂×C₇×C₈	C₄	1	2	yes	—
10	C₂×C₇×C₈	C₂²	1	2	yes	—
11	C₂×#108	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂³×C₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×C₄×C₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₇	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₇×C₈	—	2	4	no	—
20	C₇×C₈	C₂×C₄	1	4	yes	—
21	C₂×M₁₆	C₂×C₇	1	2	yes	—
22	C₂²×C₈	C₂×C₇	1	1	yes	characteristic
23	#108	C₂×C₇	1	4	yes	direct product factor
24	C₂²×C₇	—	2	2	no	—
25	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₇	C₂×Q₈	1	1	yes	characteristic
28	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
33	M₁₆	—	2	2	no	—
34	M₁₆	C₂²×C₇	1	4	yes	—
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	C₄×C₇	1	2	yes	—
37	C₂×C₈	C₄×C₇	1	2	yes	—
38	C₂×C₈	C₂²×C₇	1	2	yes	—
39	C₂×C₇	—	2	1	no	—
40	C₂×C₇	—	2	1	no	—
41	C₂×C₇	#108	1	2	yes	direct product factor
42	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂³	C₇×Q₈	1	1	yes	characteristic
44	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
45	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
46	C₂×C₄	C₇×D₄	1	1	yes	characteristic
47	C₂×C₄	C₇×D₄	1	1	yes	characteristic
48	C₂×C₄	C₇×Q₈	1	1	yes	characteristic
49	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
50	C₈	—	2	4	no	—
51	C₈	C₂×C₄×C₇	1	4	yes	—
52	C₇	C₂×#108	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	2	2	no	—
54	C₂²	C₇×C₄⋊C₄	1	1	yes	characteristic
55	C₂²	C₇×C₄⋊C₄	1	1	yes	characteristic
56	C₂²	C₂×C₇×Q₈	1	1	yes	characteristic
57	C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
58	C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
59	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
60	C₄	C₂×C₇×D₄	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	—	2	1	no	—
63	C₂	C₇×#108	1	2	yes	direct product factor
64	C₂	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	D₄
commutator	C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₇
socle	C₂²×C₇	C₄⋊C₄
prime radical	C₂³×C₇	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#108	C₂	direct (N × H)
11. C₂×#108	C₇	direct (N × H)
23. #108	C₂×C₇	direct (N × H)
41. C₂×C₇	#108	direct (N × H)
52. C₇	C₂×#108	direct (N × H)
63. C₂	C₇×#108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×M₁₆	2
C₂²×C₇×C₈	2
C₇×#108	2
C₂×#108	7

Subgroups — C2×C7×#108 #90075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#108 #90075