Subgroups — #2079

Order 128.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#106	C₂	1	1	yes	maximal, characteristic, complemented
2	#417	C₂	1	2	yes	maximal
3	#408	C₂	1	1	yes	maximal, characteristic
4	#343	C₂	1	2	yes	maximal, complemented
5	#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41	C₂²	1	1	yes	characteristic
7	C₂×M₁₆	C₂²	1	2	yes	—
8	C₂×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄	C₂²	1	2	yes	—
11	#106	—	2	2	no	—
12	#103	—	2	2	no	—
13	#102	—	2	2	no	—
14	C₂²⋊C₈	—	2	2	no	—
15	#41	—	2	2	no	—
16	#41	—	2	2	no	—
17	C₂×Q₈	D₄	1	1	yes	characteristic
18	C₂×D₄	—	2	1	no	—
19	C₂×D₄	D₄	1	1	yes	characteristic
20	C₂²×C₄	—	2	1	no	—
21	C₂²×C₄	—	2	2	no	—
22	C₂²×C₄	C₂³	1	1	yes	characteristic
23	M₁₆	—	2	2	no	—
24	M₁₆	—	2	2	no	—
25	M₁₆	—	2	2	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	1	no	—
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	D₄	1	2	yes	—
30	C₄⋊C₄	—	2	2	no	—
31	C₄⋊C₄	—	2	2	no	—
32	C₂³	—	2	1	no	—
33	C₂³	#41	1	1	yes	characteristic
34	Q₈	—	2	2	no	—
35	D₄	—	2	2	no	—
36	D₄	—	4	2	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	4	1	no	—
40	C₂×C₄	—	4	2	no	—
41	C₂×C₄	—	8	2	no	—
42	C₂×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄	#41	1	2	yes	—
45	C₂×C₄	#41	1	2	yes	—
46	C₈	—	2	2	no	—
47	C₈	—	4	2	no	—
48	C₈	—	4	1	no	—
49	C₂²	—	2	2	no	—
50	C₂²	—	4	1	no	—
51	C₂²	—	4	2	no	—
52	C₂²	#121	1	1	yes	characteristic
53	C₂²	#126	1	2	yes	—
54	C₄	—	4	1	no	—
55	C₄	—	8	2	no	—
56	C₄	#123	1	1	yes	characteristic
57	C₄	#123	1	2	yes	—
58	C₄	#124	1	1	yes	characteristic
59	C₂	—	2	2	no	—
60	C₂	—	8	1	no	—
61	C₂	#397	1	1	yes	characteristic
62	C₂	#471	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#106	C₂	semidirect
4. #343	C₂	semidirect
5. #341	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#106	2
#417	2
#408	2
#343	2
#341	2