Subgroups — #2552

Order 128.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125	C₂	1	2	yes	maximal
2	#398	C₂	1	4	yes	maximal
3	#389	C₂	1	2	yes	maximal
4	#389	C₂	1	2	yes	maximal
5	#384	C₂	1	2	yes	maximal
6	#384	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	#125	—	2	8	no	—
10	C₂×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄²	C₂²	1	2	yes	—
21	C₂×C₄²	C₂²	1	2	yes	—
22	C₂×C₄²	C₂²	1	1	yes	characteristic
23	#95	C₂²	1	2	yes	—
24	#95	C₂²	1	2	yes	—
25	#95	C₂²	1	4	yes	—
26	C₂²×C₄	C₂³	1	2	yes	—
27	C₂²×C₄	C₂³	1	2	yes	—
28	C₂²×C₄	C₂³	1	4	yes	—
29	C₂²×C₄	C₂³	1	2	yes	—
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₄	C₂³	1	2	yes	—
32	C₂²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₄	C₂³	1	1	yes	characteristic
34	C₄⋊C₄	—	2	8	no	—
35	C₄⋊C₄	—	2	8	no	—
36	C₄⋊C₄	—	2	4	no	—
37	C₄⋊C₄	—	2	2	no	—
38	C₄⋊C₄	—	2	2	no	—
39	C₄⋊C₄	—	2	4	no	—
40	C₄⋊C₄	—	4	4	no	—
41	C₄⋊C₄	—	4	2	no	—
42	C₄⋊C₄	D₄	1	4	yes	—
43	C₄⋊C₄	Q₈	1	4	yes	—
44	C₄²	—	2	4	no	—
45	C₄²	—	2	4	no	—
46	C₄²	Q₈	1	4	yes	—
47	C₂³	C₂⁴	1	1	yes	characteristic
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	C₂×D₄	1	4	yes	—
64	C₂×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₄	C₂×Q₈	1	2	yes	—
66	C₂×C₄	C₂×Q₈	1	4	yes	—
67	C₂×C₄	C₂×Q₈	1	2	yes	—
68	C₂×C₄	C₂×Q₈	1	2	yes	—
69	C₂×C₄	C₂×Q₈	1	2	yes	—
70	C₂×C₄	#41	1	2	yes	—
71	C₂×C₄	#41	1	2	yes	—
72	C₂²	C₂²×D₄	1	1	yes	characteristic
73	C₂²	C₂²×Q₈	1	1	yes	characteristic
74	C₂²	C₂²×Q₈	1	1	yes	characteristic
75	C₂²	C₂×#41	1	1	yes	characteristic
76	C₂²	C₂×#41	1	1	yes	characteristic
77	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	4	no	—
81	C₄	—	2	4	no	—
82	C₄	—	4	2	no	—
83	C₄	—	4	2	no	—
84	C₄	—	4	4	no	—
85	C₄	#125	1	4	yes	complemented
86	C₄	C₄⋊Q₈	1	4	yes	—
87	C₂	C₂×#125	1	1	yes	characteristic
88	C₂	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂	#544	1	1	yes	characteristic
90	C₂	#547	1	1	yes	characteristic
91	C₂	#550	1	1	yes	characteristic
92	C₂	#554	1	1	yes	characteristic
93	C₂	#557	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
85. C₄	#125	semidirect

Maximal subgroups

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

subgroup	index
C₂×#125	2
#398	2
#389	2
#384	2
C₄×C₄⋊C₄	2