Subgroups — C₇×Q₈² #90579

Order 448.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	9	yes	maximal
2	C₄×C₇×Q₈	C₂	1	6	yes	maximal
3	C₂×C₇×Q₈	C₂²	1	6	yes	—
4	C₂×C₇×Q₈	C₂²	1	2	yes	—
5	C₇×C₄⋊C₄	C₂²	1	18	yes	—
6	C₄²×C₇	C₂²	1	9	yes	—
7	Q₈²	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₇×Q₈	—	4	6	no	—
9	C₇×Q₈	Q₈	1	8	yes	direct product factor
10	C₂×C₄×C₇	C₂³	1	9	yes	—
11	C₂×C₄×C₇	C₂³	1	6	yes	—
12	C₄⋊Q₈	C₂×C₇	1	9	yes	—
13	C₄×Q₈	C₂×C₇	1	6	yes	—
14	C₂²×C₇	C₂⁴	1	1	yes	characteristic
15	C₄×C₇	—	2	9	no	—
16	C₄×C₇	C₂×Q₈	1	12	yes	—
17	C₂×Q₈	C₂²×C₇	1	6	yes	—
18	C₂×Q₈	C₂²×C₇	1	2	yes	—
19	C₄⋊C₄	C₂²×C₇	1	18	yes	—
20	C₄²	C₂²×C₇	1	9	yes	—
21	C₂×C₇	C₂²×Q₈	1	2	yes	—
22	C₂×C₇	H(5,𝔽₂)	1	1	yes	characteristic
23	Q₈	—	4	6	no	—
24	Q₈	C₇×Q₈	1	8	yes	direct product factor
25	C₂×C₄	C₂³×C₇	1	9	yes	—
26	C₂×C₄	C₂³×C₇	1	6	yes	—
27	C₇	Q₈²	1	1	yes	characteristic, direct product factor, Sylow
28	C₂²	C₂⁴×C₇	1	1	yes	characteristic
29	C₄	—	2	9	no	—
30	C₄	C₂×C₇×Q₈	1	12	yes	—
31	C₂	C₂²×C₇×Q₈	1	2	yes	—
32	C₂	C₇×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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.