Subgroups — #2089

Order 128.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#428	C₂	1	2	yes	maximal
2	C₂×#107	C₂	1	1	yes	maximal, characteristic
3	#424	C₂	1	1	yes	maximal, characteristic
4	#343	C₂	1	2	yes	maximal
5	#341	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆	C₂²	1	1	yes	characteristic
7	C₂×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	#117	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄	C₂²	1	2	yes	—
11	#107	—	2	2	no	—
12	#107	—	2	2	no	—
13	#106	—	2	2	no	—
14	C₄⋊C₈	—	2	2	no	—
15	C₂²×C₄	—	2	2	no	—
16	C₂²×C₄	C₂³	1	1	yes	characteristic
17	M₁₆	—	2	2	no	—
18	M₁₆	—	2	2	no	—
19	M₁₆	Q₈	1	4	yes	—
20	C₂×C₈	—	2	2	no	—
21	C₂×C₈	—	2	2	no	—
22	C₂×C₈	—	2	1	no	—
23	C₂×C₈	D₄	1	2	yes	—
24	C₄⋊C₄	—	2	2	no	—
25	C₄⋊C₄	—	2	2	no	—
26	C₄⋊C₄	D₄	1	1	yes	characteristic
27	C₄⋊C₄	D₄	1	1	yes	characteristic
28	C₂²⋊C₄	—	2	1	no	—
29	C₄²	—	2	1	no	—
30	C₂³	#41	1	1	yes	characteristic
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	—	4	2	no	—
34	C₂×C₄	—	8	2	no	—
35	C₂×C₄	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄	C₂×Q₈	1	2	yes	—
38	C₂×C₄	#41	1	2	yes	—
39	C₈	—	2	4	no	—
40	C₈	—	2	2	no	—
41	C₈	—	4	1	no	—
42	C₂²	—	2	1	no	—
43	C₂²	—	2	1	no	—
44	C₂²	#121	1	1	yes	characteristic
45	C₂²	#125	1	2	yes	—
46	C₄	—	4	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	8	2	no	—
49	C₄	C₂²⋊Q₈	1	2	yes	—
50	C₄	#123	1	1	yes	characteristic
51	C₄	C₄⋊Q₈	1	1	yes	characteristic
52	C₂	—	2	2	no	—
53	C₂	#398	1	1	yes	characteristic
54	C₂	#472	1	1	yes	characteristic
55	C₂	#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³
prime radical	C₂³	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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