Subgroups — #1952

Order 128.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108	C₂	1	1	yes	maximal, characteristic
2	#428	C₂	1	2	yes	maximal
3	C₂×#107	C₂	1	1	yes	maximal, characteristic
4	#405	C₂	1	1	yes	maximal, characteristic
5	#343	C₂	1	2	yes	maximal
6	C₂×M₁₆	C₂²	1	2	yes	—
7	C₂×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	#117	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄	C₂²	1	2	yes	—
11	#108	C₄	1	4	yes	complemented
12	#107	—	2	1	no	—
13	#107	—	2	1	no	—
14	#107	—	2	2	no	—
15	#106	—	2	2	no	—
16	#97	—	2	1	no	—
17	C₄×C₈	—	2	1	no	—
18	C₂²×C₄	—	2	2	no	—
19	C₂²×C₄	C₂³	1	1	yes	characteristic
20	M₁₆	—	2	2	no	—
21	M₁₆	—	2	1	no	—
22	M₁₆	C₂×C₄	1	4	yes	—
23	M₁₆	Q₈	1	2	yes	—
24	C₂×C₈	—	2	1	no	—
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₈	D₄	1	1	yes	characteristic
28	C₂×C₈	D₄	1	1	yes	characteristic
29	C₂×C₈	Q₈	1	2	yes	—
30	C₄⋊C₄	—	2	2	no	—
31	C₄⋊C₄	—	2	2	no	—
32	C₄⋊C₄	D₄	1	1	yes	characteristic
33	C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₂²⋊C₄	—	2	1	no	—
35	C₄²	—	2	1	no	—
36	C₂³	#41	1	1	yes	characteristic
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	4	2	no	—
40	C₂×C₄	—	8	2	no	—
41	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄	#41	1	1	yes	characteristic
47	C₈	—	2	4	no	—
48	C₈	—	2	2	no	—
49	C₈	C₄⋊C₄	1	2	yes	—
50	C₈	C₄⋊C₄	1	2	yes	—
51	C₂²	—	2	1	no	—
52	C₂²	—	2	1	no	—
53	C₂²	C₄×Q₈	1	1	yes	characteristic
54	C₂²	#121	1	1	yes	characteristic
55	C₂²	#125	1	1	yes	characteristic
56	C₄	—	4	1	no	—
57	C₄	—	4	1	no	—
58	C₄	—	8	2	no	—
59	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄	C₄×D₄	1	1	yes	characteristic
61	C₄	C₂²⋊Q₈	1	1	yes	characteristic
62	C₄	C₄⋊Q₈	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	—	2	1	no	—
65	C₂	#389	1	1	yes	characteristic
66	C₂	#472	1	1	yes	characteristic
67	C₂	#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
prime radical	C₂³	#41
LCS	G ▷ C₂×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
11. #108	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108	2
#428	2
C₂×#107	2
#405	2
#343	2