Subgroups — #1942

Order 128.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	#389	C₂	1	2	yes	maximal
5	#340	C₂	1	2	yes	maximal
6	C₂²×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄	—	2	2	no	—
9	C₂×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄²	C₂²	1	1	yes	characteristic
12	#107	—	2	2	no	—
13	#107	C₄	1	4	yes	complemented
14	C₄×C₈	—	2	2	no	—
15	#95	—	2	2	no	—
16	C₂²×C₄	—	2	2	no	—
17	C₂²×C₄	—	2	2	no	—
18	C₂²×C₄	D₄	1	1	yes	characteristic
19	C₂²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₄	C₂³	1	1	yes	characteristic
21	C₂×C₈	—	2	1	no	—
22	C₂×C₈	—	2	1	no	—
23	C₂×C₈	C₂×C₄	1	2	yes	—
24	C₂×C₈	D₄	1	2	yes	—
25	C₂×C₈	Q₈	1	2	yes	—
26	C₂×C₈	Q₈	1	2	yes	—
27	C₄⋊C₄	—	2	2	no	—
28	C₄⋊C₄	—	2	2	no	—
29	C₄⋊C₄	—	2	2	no	—
30	C₄⋊C₄	—	4	4	no	—
31	C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₄²	—	2	1	no	—
33	C₄²	—	2	1	no	—
34	C₂³	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	2	4	no	—
40	C₂×C₄	—	4	2	no	—
41	C₂×C₄	—	4	2	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	—	4	2	no	—
44	C₂×C₄	—	4	2	no	—
45	C₂×C₄	D₈	1	2	yes	—
46	C₂×C₄	Q₁₆	1	2	yes	—
47	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄	#41	1	1	yes	characteristic
52	C₂×C₄	#41	1	1	yes	characteristic
53	C₈	—	2	2	no	—
54	C₈	C₄⋊C₄	1	4	yes	complemented
55	C₂²	C₄×D₄	1	1	yes	characteristic
56	C₂²	#121	1	1	yes	characteristic
57	C₂²	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²	C₂×D₈	1	1	yes	characteristic
59	C₂²	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²	#135	1	1	yes	characteristic
61	C₂²	#135	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	2	2	no	—
64	C₄	—	4	4	no	—
65	C₄	—	8	2	no	—
66	C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄	C₄×Q₈	1	1	yes	characteristic
68	C₄	C₂²⋊Q₈	1	1	yes	characteristic
69	C₄	#125	1	1	yes	characteristic
70	C₂	#389	1	1	yes	characteristic
71	C₂	C₄×D₈	1	1	yes	characteristic
72	C₂	C₄×Q₁₆	1	1	yes	characteristic
73	C₂	#466	1	1	yes	characteristic
74	C₂	#467	1	1	yes	characteristic
75	C₂	#499	1	1	yes	characteristic
76	C₂	#500	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
13. #107	C₄	semidirect
54. C₈	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107	2
C₂×C₄×C₈	2
#389	2
#340	2