Subgroups — C₂×C₈×#11764047 #31795980

Order 17536.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11764047	C₂	1	2	yes	maximal
2	C₈×#11764047	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11764047	C₄	1	4	yes	—
5	C₄×#11764047	C₄	1	4	yes	—
6	C₄×#11764047	C₂²	1	4	yes	—
7	C₂²×#11764040	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₃₇	C₄	1	2	yes	—
10	C₈×Dic₁₃₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
12	C₂×#11764047	C₈	1	16	yes	direct product factor
13	C₂×#11764047	C₂×C₄	1	8	yes	—
14	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
15	C₂×#11764040	C₈	1	4	yes	complemented
16	C₂×#11764040	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₃₇	C₈	1	4	yes	complemented
22	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
23	#11764047	C₂×C₈	1	16	yes	direct product factor
24	C₂×Dic₁₃₇	C₄²	1	2	yes	—
25	C₂×Dic₁₃₇	C₂×C₈	1	8	yes	—
26	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
27	#11764040	C₂×C₈	1	4	yes	complemented
28	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₃₇	C₂×C₈	1	4	yes	complemented
34	C₂²×C₁₃₇	C₄×C₈	1	4	yes	—
35	C₂²×C₁₃₇	C₄×C₈	1	2	yes	—
36	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
37	C₄×C₁₃₇	C₄×C₈	1	2	yes	—
38	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
40	Dic₁₃₇	C₄×C₈	1	4	yes	—
41	Dic₁₃₇	C₂²×C₈	1	4	yes	—
42	C₂×C₁₃₇	C₈²	1	4	yes	complemented
43	C₂×C₁₃₇	C₂×C₄×C₈	1	2	yes	—
44	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
45	C₁₃₇	C₂×C₈²	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₈²	—	137	1	no	maximal, Sylow
47	C₂×C₄×C₈	—	137	2	no	—
48	C₂×C₄×C₈	—	137	1	no	—
49	C₈²	—	137	4	no	—
50	C₂²×C₈	—	137	4	no	—
51	C₂²×C₈	—	137	1	no	—
52	C₂²×C₈	#10587739	1	1	yes	characteristic
53	C₂×C₄²	—	137	1	no	—
54	C₄×C₈	—	137	4	no	—
55	C₄×C₈	—	137	4	no	—
56	C₄×C₈	—	137	2	no	—
57	C₄×C₈	—	137	2	no	—
58	C₂²×C₄	—	137	2	no	—
59	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
60	C₂×C₈	—	137	8	no	—
61	C₂×C₈	—	137	16	no	—
62	C₂×C₈	—	137	4	no	—
63	C₂×C₈	—	137	2	no	—
64	C₂×C₈	#11764047	1	4	yes	direct product factor
65	C₂×C₈	C₂×#10587739	1	2	yes	—
66	C₄²	—	137	2	no	—
67	C₄²	—	137	1	no	—
68	C₄²	—	137	1	no	—
69	C₂³	C₄×#10587739	1	1	yes	characteristic
70	C₂×C₄	—	137	8	no	—
71	C₂×C₄	—	137	2	no	—
72	C₂×C₄	—	137	2	no	—
73	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
74	C₂×C₄	C₂×#11764047	1	2	yes	—
75	C₂×C₄	#14555518	1	2	yes	—
76	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
77	C₈	—	137	16	no	—
78	C₈	—	137	4	no	—
79	C₈	C₂×#11764047	1	4	yes	direct product factor
80	C₂²	C₈×#10587739	1	4	yes	—
81	C₂²	C₄×#11764047	1	2	yes	—
82	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
83	C₄	—	137	4	no	—
84	C₄	—	137	4	no	—
85	C₄	C₄×#11764047	1	2	yes	—
86	C₄	C₂²×#11764047	1	1	yes	characteristic
87	C₄	C₂×#14555518	1	1	yes	characteristic
88	C₂	C₈×#11764047	1	4	yes	direct product factor
89	C₂	C₂×C₈×#10587739	1	2	yes	—
90	C₂	C₂×C₄×#11764047	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	#10587739
commutator, hypocenter	C₁₃₇	C₂×C₈²
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle, prime radical	C₂³×C₁₃₇	C₄²
cosocle	C₄×Dic₁₃₇	C₂³
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₈×#11764047	C₂	direct (N × H)
12. C₂×#11764047	C₈	direct (N × H)
15. C₂×#11764040	C₈	semidirect
21. C₂×C₈×C₁₃₇	C₈	semidirect
23. #11764047	C₂×C₈	direct (N × H)
27. #11764040	C₂×C₈	semidirect
33. C₈×C₁₃₇	C₂×C₈	semidirect
42. C₂×C₁₃₇	C₈²	semidirect
45. C₁₃₇	C₂×C₈²	semidirect
64. C₂×C₈	#11764047	direct (N × H)
79. C₈	C₂×#11764047	direct (N × H)
88. C₂	C₈×#11764047	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#11764047	2
C₈×#11764047	2
C₂×C₈×Dic₁₃₇	2
C₂×C₈²	137

Subgroups — C2×C8×#11764047 #31795980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×#11764047 #31795980