Subgroups — C₂²×#137 #3579

Order 128.

732 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41	C₂	1	1	yes	maximal, characteristic
2	C₂³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#137	C₂	1	24	yes	maximal, direct product factor
4	C₂²×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×SD₁₆	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41	—	2	6	no	—
8	C₂×#41	C₂²	1	12	yes	—
9	C₂²×Q₈	—	2	6	no	—
10	C₂²×Q₈	—	2	1	no	—
11	C₂²×Q₈	C₂²	1	2	yes	—
12	C₂²×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×Q₈	C₂²	1	12	yes	complemented
14	C₂²×D₄	—	2	1	no	—
15	C₂²×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₄	—	2	1	no	—
17	C₂³×C₄	—	2	1	no	—
18	C₂³×C₄	C₂²	1	1	yes	characteristic
19	#137	C₂²	1	64	yes	direct product factor
20	C₂×Q₁₆	C₂²	1	24	yes	complemented
21	C₂×SD₁₆	C₂²	1	24	yes	complemented
22	C₂×M₁₆	C₂²	1	12	yes	complemented
23	C₂²×C₈	C₂²	1	2	yes	—
24	C₂⁴	—	2	1	no	—
25	C₂⁴	D₄	1	1	yes	characteristic
26	#41	—	2	24	no	—
27	#41	C₂³	1	16	yes	—
28	C₂×Q₈	—	2	6	no	—
29	C₂×Q₈	—	2	12	no	—
30	C₂×Q₈	—	2	24	no	—
31	C₂×Q₈	—	2	3	no	—
32	C₂×Q₈	C₂³	1	12	yes	—
33	C₂×Q₈	C₂³	1	6	yes	—
34	C₂×Q₈	C₂³	1	16	yes	complemented
35	C₂×D₄	—	2	12	no	—
36	C₂×D₄	—	2	3	no	—
37	C₂×D₄	C₂³	1	6	yes	—
38	C₂²×C₄	—	2	12	no	—
39	C₂²×C₄	—	2	1	no	—
40	C₂²×C₄	—	2	2	no	—
41	C₂²×C₄	—	2	12	no	—
42	C₂²×C₄	D₄	1	6	yes	—
43	C₂²×C₄	D₄	1	1	yes	characteristic
44	C₂²×C₄	C₂³	1	6	yes	—
45	C₂²×C₄	C₂³	1	1	yes	characteristic
46	Q₁₆	C₂³	1	32	yes	complemented
47	SD₁₆	C₂³	1	32	yes	complemented
48	M₁₆	C₂³	1	16	yes	complemented
49	C₂×C₈	C₂³	1	12	yes	—
50	C₂³	—	2	4	no	—
51	C₂³	—	2	6	no	—
52	C₂³	—	4	4	no	—
53	C₂³	C₂×D₄	1	6	yes	—
54	C₂³	C₂×D₄	1	1	yes	characteristic
55	Q₈	—	2	12	no	—
56	Q₈	—	2	16	no	—
57	Q₈	—	2	6	no	—
58	Q₈	C₂⁴	1	8	yes	—
59	Q₈	C₂⁴	1	4	yes	—
60	D₄	—	2	16	no	—
61	D₄	—	2	6	no	—
62	D₄	C₂⁴	1	4	yes	—
63	C₂×C₄	—	2	6	no	—
64	C₂×C₄	—	2	12	no	—
65	C₂×C₄	—	2	16	no	—
66	C₂×C₄	—	2	16	no	—
67	C₂×C₄	C₂×D₄	1	6	yes	—
68	C₂×C₄	C₂×D₄	1	3	yes	—
69	C₂×C₄	C₂×D₄	1	12	yes	—
70	C₂×C₄	C₂⁴	1	3	yes	—
71	C₂×C₄	C₂⁴	1	4	yes	—
72	C₈	C₂⁴	1	8	yes	—
73	C₂²	—	2	12	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	4	12	no	—
76	C₂²	#137	1	4	yes	direct product factor
77	C₂²	C₂²×D₄	1	4	yes	—
78	C₂²	C₂²×D₄	1	3	yes	—
79	C₄	—	2	4	no	—
80	C₄	—	2	8	no	—
81	C₄	C₂²×D₄	1	4	yes	—
82	C₄	C₂²×D₄	1	3	yes	—
83	C₄	C₂⁵	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	—	4	4	no	—
86	C₂	C₂×#137	1	6	yes	direct product factor
87	C₂	C₂³×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵
prime radical	C₂²×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₈	C₂	semidirect
3. C₂×#137	C₂	direct (N × H)
4. C₂²×Q₁₆	C₂	semidirect
5. C₂²×SD₁₆	C₂	semidirect
6. C₂²×M₁₆	C₂	semidirect
13. C₂²×Q₈	C₂²	semidirect
19. #137	C₂²	direct (N × H)
20. C₂×Q₁₆	C₂²	semidirect
21. C₂×SD₁₆	C₂²	semidirect
22. C₂×M₁₆	C₂²	semidirect
34. C₂×Q₈	C₂³	semidirect
46. Q₁₆	C₂³	semidirect
47. SD₁₆	C₂³	semidirect
48. M₁₆	C₂³	semidirect
76. C₂²	#137	direct (N × H)
86. C₂	C₂×#137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41	2
C₂³×Q₈	2
C₂×#137	2
C₂²×Q₁₆	2
C₂²×SD₁₆	2
C₂²×M₁₆	2

Subgroups — C22×#137 #3579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#137 #3579