Subgroups — C₂×#22578289 #31796987

Order 17536.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555513	C₂	1	2	yes	maximal
2	#22578289	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	C₄	1	8	yes	complemented
5	C₂×#14555513	C₂²	1	4	yes	—
6	C₂²×#11764040	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
8	C₈×Dic₁₃₇	C₄	1	2	yes	—
9	C₈×Dic₁₃₇	C₂²	1	2	yes	—
10	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555513	C₂×C₄	1	8	yes	complemented
12	C₂²×Dic₁₃₇	C₈	1	2	yes	—
13	C₂×#11764040	C₂×C₄	1	4	yes	—
14	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
15	C₂×#11764040	C₂³	1	1	yes	characteristic
16	C₄×Dic₁₃₇	C₈	1	2	yes	—
17	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₃₇	C₈	1	4	yes	—
20	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₃₇	C₁₆	1	8	yes	complemented
22	C₂×Dic₁₃₇	C₂×C₈	1	4	yes	—
23	#11764040	C₄²	1	2	yes	—
24	#11764040	C₂²×C₄	1	2	yes	—
25	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
27	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₁₃₇	C₂×C₈	1	4	yes	—
31	C₂²×C₁₃₇	C₄×C₈	1	2	yes	—
32	C₂²×C₁₃₇	C₂×C₁₆	1	4	yes	—
33	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
34	C₄×C₁₃₇	C₄×C₈	1	2	yes	—
35	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
36	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
37	Dic₁₃₇	C₂×C₁₆	1	8	yes	complemented
38	C₂×C₁₃₇	C₄×C₁₆	1	4	yes	complemented
39	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
40	C₂×C₁₃₇	C₂²×C₁₆	1	2	yes	—
41	C₁₃₇	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄×C₁₆	—	137	1	no	maximal, Sylow
43	C₂²×C₁₆	—	137	2	no	—
44	C₂×C₄×C₈	—	137	1	no	—
45	C₄×C₁₆	—	137	4	no	—
46	C₂²×C₈	—	137	1	no	—
47	C₂²×C₈	#10587739	1	1	yes	characteristic
48	C₂×C₄²	—	137	1	no	—
49	C₂×C₁₆	—	137	4	no	—
50	C₂×C₁₆	—	137	8	no	—
51	C₄×C₈	—	137	2	no	—
52	C₄×C₈	—	137	2	no	—
53	C₂²×C₄	—	137	2	no	—
54	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
55	C₂×C₈	—	137	4	no	—
56	C₂×C₈	—	137	1	no	—
57	C₂×C₈	—	137	1	no	—
58	C₂×C₈	#11764047	1	4	yes	—
59	C₂×C₈	C₂×#10587739	1	2	yes	—
60	C₄²	—	137	2	no	—
61	C₄²	—	137	2	no	—
62	C₁₆	—	137	8	no	—
63	C₂³	#14555518	1	1	yes	characteristic
64	C₂×C₄	—	137	8	no	—
65	C₂×C₄	—	137	4	no	—
66	C₂×C₄	C₄×#10587739	1	2	yes	—
67	C₂×C₄	C₂×#11764047	1	2	yes	—
68	C₂×C₄	#14555518	1	1	yes	characteristic
69	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
70	C₈	—	137	2	no	—
71	C₈	—	137	2	no	—
72	C₈	C₂×#11764047	1	4	yes	—
73	C₂²	C₄×#11764047	1	2	yes	—
74	C₂²	#17984161	1	4	yes	—
75	C₂²	C₂×#14555518	1	1	yes	characteristic
76	C₄	—	137	8	no	—
77	C₄	C₄×#11764047	1	2	yes	—
78	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
79	C₄	C₂²×#11764047	1	1	yes	characteristic
80	C₂	#22578289	1	4	yes	direct product factor
81	C₂	C₂×C₄×#11764047	1	1	yes	characteristic
82	C₂	C₂×#17984161	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₈	#10587739
commutator, hypocenter	C₁₃₇	C₂×C₄×C₁₆
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle, prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂×#11764040	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. #22578289	C₂	direct (N × H)
4. C₂×#14555513	C₄	semidirect
11. #14555513	C₂×C₄	semidirect
21. C₂×Dic₁₃₇	C₁₆	semidirect
37. Dic₁₃₇	C₂×C₁₆	semidirect
38. C₂×C₁₃₇	C₄×C₁₆	semidirect
41. C₁₃₇	C₂×C₄×C₁₆	semidirect
80. C₂	#22578289	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×#14555513	2
#22578289	2
C₂×C₈×Dic₁₃₇	2
C₂×C₄×C₁₆	137

Subgroups — C2×#22578289 #31796987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578289 #31796987