Subgroups — #2401

Order 128.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#400	C₂	1	3	yes	maximal, complemented
3	#397	C₂	1	3	yes	maximal
4	#396	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₄×C₂²⋊C₄	C₂	1	3	yes	maximal
7	#376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄	C₂²	1	3	yes	—
10	C₂×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₂²⋊C₄	C₂²	1	3	yes	—
12	C₂×C₂²⋊C₄	C₂²	1	3	yes	—
13	C₂×C₄²	C₂²	1	3	yes	—
14	C₂×C₄²	C₂²	1	3	yes	—
15	#95	C₂²	1	1	yes	characteristic
16	#95	C₂²	1	3	yes	—
17	#95	C₂²	1	3	yes	—
18	#95	C₂²	1	3	yes	—
19	#95	C₂²	1	3	yes	—
20	#95	C₂²	1	3	yes	—
21	C₂⁴	C₂³	1	1	yes	characteristic
22	C₂²×C₄	—	2	6	no	—
23	C₂²×C₄	C₂³	1	1	yes	characteristic
24	C₂²×C₄	C₂³	1	3	yes	—
25	C₂²×C₄	C₂³	1	3	yes	—
26	C₂²×C₄	C₂³	1	3	yes	—
27	C₂²×C₄	C₂³	1	3	yes	—
28	C₂²×C₄	C₂³	1	1	yes	characteristic
29	C₄⋊C₄	—	2	6	no	—
30	C₄⋊C₄	—	2	6	no	—
31	C₂²⋊C₄	—	2	6	no	—
32	C₂²⋊C₄	—	2	6	no	—
33	C₄²	—	2	6	no	—
34	C₄²	—	2	6	no	—
35	C₂³	—	2	3	no	—
36	C₂³	—	2	3	no	—
37	C₂³	#41	1	2	yes	—
38	C₂³	C₂⁴	1	1	yes	characteristic
39	C₂×C₄	—	2	3	no	—
40	C₂×C₄	—	2	3	no	—
41	C₂×C₄	—	2	3	no	—
42	C₂×C₄	—	2	3	no	—
43	C₂×C₄	—	2	3	no	—
44	C₂×C₄	—	2	3	no	—
45	C₂×C₄	—	2	3	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	2	3	no	—
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	4	4	no	—
52	C₂×C₄	#41	1	6	yes	—
53	C₂×C₄	#41	1	6	yes	—
54	C₂²	—	2	6	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	4	3	no	—
57	C₂²	C₂×#41	1	3	yes	—
58	C₂²	C₂×#41	1	1	yes	characteristic
59	C₂²	C₂×#41	1	3	yes	—
60	C₄	—	2	6	no	—
61	C₄	—	4	1	no	—
62	C₄	—	4	3	no	—
63	C₄	—	4	3	no	—
64	C₄	—	4	3	no	—
65	C₄	#126	1	4	yes	—
66	C₂	—	4	2	no	—
67	C₂	C₂×#126	1	1	yes	characteristic
68	C₂	#529	1	3	yes	—
69	C₂	#529	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³
commutator, Frattini, socle, cosocle	C₂³	C₂⁴
prime radical	C₂⁴	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #401	C₂	semidirect
2. #400	C₂	semidirect
5. C₄×C₄⋊C₄	C₂	semidirect
7. #376	C₂	semidirect

Maximal subgroups

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

subgroup	index
#401	2
#400	2
#397	2
#396	2
C₄×C₄⋊C₄	2
C₄×C₂²⋊C₄	2
#376	2