Subgroups — #1950

Order 128.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108	C₂	1	2	yes	maximal
2	#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
5	#341	C₂	1	2	yes	maximal
6	C₂×M₁₆	C₂²	1	2	yes	—
7	C₂×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₈	C₂²	1	2	yes	—
10	C₂×C₄²	C₂²	1	1	yes	characteristic
11	#108	—	2	4	no	—
12	C₄⋊C₈	—	2	1	no	—
13	C₄⋊C₈	—	2	1	no	—
14	C₄⋊C₈	C₄	1	2	yes	—
15	C₄⋊C₈	C₄	1	2	yes	—
16	#97	—	2	1	no	—
17	C₄×C₈	—	2	1	no	—
18	C₂²×C₄	Q₈	1	1	yes	characteristic
19	C₂²×C₄	Q₈	1	1	yes	characteristic
20	C₂²×C₄	C₂³	1	1	yes	characteristic
21	M₁₆	—	2	1	no	—
22	M₁₆	—	2	1	no	—
23	M₁₆	—	2	2	no	—
24	M₁₆	—	2	2	no	—
25	M₁₆	D₄	1	2	yes	—
26	M₁₆	Q₈	1	2	yes	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	2	2	no	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	C₂×C₄	1	4	yes	—
33	C₄²	C₂×C₄	1	1	yes	characteristic
34	C₄²	C₂×C₄	1	1	yes	characteristic
35	C₄²	D₄	1	1	yes	characteristic
36	C₄²	D₄	1	1	yes	characteristic
37	C₂³	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	4	1	no	—
40	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄	#41	1	1	yes	characteristic
49	C₂×C₄	#41	1	1	yes	characteristic
50	C₈	—	2	4	no	—
51	C₈	—	2	2	no	—
52	C₈	—	2	2	no	—
53	C₈	—	4	2	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²	#125	1	1	yes	characteristic
58	C₂²	C₄⋊Q₈	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	2	1	no	—
62	C₄	#108	1	2	yes	complemented
63	C₄	C₄×D₄	1	1	yes	characteristic
64	C₄	C₄×Q₈	1	1	yes	characteristic
65	C₄	#121	1	1	yes	characteristic
66	C₄	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂	—	2	1	no	—
68	C₂	—	2	1	no	—
69	C₂	#389	1	1	yes	characteristic
70	C₂	C₂×#108	1	1	yes	characteristic
71	C₂	#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₂×C₄⋊C₄
prime radical	C₂³	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
62. C₄	#108	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
#423	2
C₂×C₄⋊C₈	2
C₄×M₁₆	2
#341	2