Subgroups — C₇×H(3,𝔽₄) #90582

Order 448.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#124	C₂	1	9	yes	maximal, complemented
2	C₇×#120	C₂	1	6	yes	maximal, complemented
3	C₂⁴×C₇	C₂²	1	2	yes	complemented
4	C₂×C₇×Q₈	C₂²	1	3	yes	complemented
5	C₂×C₇×D₄	C₂²	1	9	yes	—
6	C₇×C₂²⋊C₄	C₂²	1	18	yes	complemented
7	C₄²×C₇	C₂²	1	3	yes	complemented
8	H(3,𝔽₄)	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂³×C₇	—	4	6	no	—
10	C₂³×C₇	C₂³	1	6	yes	—
11	C₇×Q₈	—	4	3	no	—
12	C₇×D₄	—	4	9	no	—
13	C₂×C₄×C₇	C₂³	1	9	yes	—
14	#124	C₂×C₇	1	9	yes	complemented
15	#120	C₂×C₇	1	6	yes	complemented
16	C₂²×C₇	—	2	18	no	—
17	C₂²×C₇	—	4	8	no	—
18	C₂²×C₇	C₂⁴	1	1	yes	characteristic
19	C₄×C₇	—	2	9	no	—
20	C₂⁴	C₂²×C₇	1	2	yes	complemented
21	C₂×Q₈	C₂²×C₇	1	3	yes	complemented
22	C₂×D₄	C₂²×C₇	1	9	yes	—
23	C₂²⋊C₄	C₂²×C₇	1	18	yes	complemented
24	C₄²	C₂²×C₇	1	3	yes	complemented
25	C₂×C₇	—	4	6	no	—
26	C₂×C₇	H(5,𝔽₂)	1	3	yes	—
27	C₂³	—	4	6	no	—
28	C₂³	C₂³×C₇	1	6	yes	—
29	Q₈	—	4	3	no	—
30	D₄	—	4	9	no	—
31	C₂×C₄	C₂³×C₇	1	9	yes	—
32	C₇	H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
33	C₂²	—	2	18	no	—
34	C₂²	—	4	8	no	—
35	C₂²	C₂⁴×C₇	1	1	yes	characteristic
36	C₄	—	2	9	no	—
37	C₂	—	4	6	no	—
38	C₂	C₇×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

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

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