Subgroups — C₂²×#17984184 #31798036

Order 17536.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984184	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#14555522	C₂	1	2	yes	maximal, complemented
3	C₂³×#11764047	C₂	1	1	yes	maximal, characteristic
4	#17984184	C₂²	1	16	yes	direct product factor
5	C₂×#14555522	C₂²	1	12	yes	complemented
6	C₂²×#11764047	C₄	1	6	yes	—
7	C₂²×#11764047	C₄	1	1	yes	characteristic
8	C₂²×#11764047	C₂²	1	1	yes	characteristic
9	C₂²×#11764047	C₂²	1	6	yes	—
10	C₂³×Dic₁₃₇	C₄	1	1	yes	characteristic
11	#14555522	C₂³	1	8	yes	complemented
12	C₂×#11764047	C₂×C₄	1	3	yes	—
13	C₂×#11764047	C₂×C₄	1	12	yes	—
14	C₂×#11764047	C₂×C₄	1	6	yes	—
15	C₂×#11764047	C₂³	1	3	yes	—
16	C₂×#11764047	C₂³	1	4	yes	—
17	C₂²×Dic₁₃₇	C₈	1	6	yes	—
18	C₂²×Dic₁₃₇	C₈	1	1	yes	characteristic
19	C₂²×Dic₁₃₇	C₂×C₄	1	6	yes	—
20	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂⁴×C₁₃₇	C₈	1	1	yes	characteristic
22	#11764047	C₂²×C₄	1	3	yes	—
23	#11764047	C₂²×C₄	1	4	yes	—
24	#11764047	C₂⁴	1	1	yes	characteristic
25	C₂×Dic₁₃₇	C₂×C₈	1	12	yes	—
26	C₂×Dic₁₃₇	C₂×C₈	1	3	yes	—
27	C₂×Dic₁₃₇	C₂×C₈	1	6	yes	—
28	C₂×Dic₁₃₇	C₂²×C₄	1	4	yes	—
29	C₂×Dic₁₃₇	C₂²×C₄	1	3	yes	—
30	C₂³×C₁₃₇	—	2	4	no	—
31	C₂³×C₁₃₇	C₂×C₈	1	6	yes	—
32	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₁₃₇	—	2	12	no	—
34	C₂²×C₁₃₇	M₃₂	1	4	yes	complemented
35	C₂²×C₁₃₇	C₂²×C₈	1	4	yes	—
36	C₂²×C₁₃₇	C₂²×C₈	1	3	yes	—
37	Dic₁₃₇	C₂²×C₈	1	3	yes	—
38	Dic₁₃₇	C₂²×C₈	1	4	yes	—
39	Dic₁₃₇	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₁₃₇	—	2	4	no	—
41	C₂×C₁₃₇	C₂×M₃₂	1	6	yes	complemented
42	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
43	C₁₃₇	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
44	C₂²×M₃₂	—	137	1	no	maximal, Sylow
45	C₂³×C₈	—	137	1	no	—
46	C₂×M₃₂	—	137	12	no	—
47	C₂²×C₁₆	—	137	2	no	—
48	C₂³×C₄	—	137	1	no	—
49	C₂²×C₈	—	137	6	no	—
50	C₂²×C₈	—	137	1	no	—
51	C₂²×C₈	—	137	1	no	—
52	C₂²×C₈	—	137	6	no	—
53	M₃₂	—	137	16	no	—
54	C₂×C₁₆	—	137	12	no	—
55	C₂⁴	#11764049	1	1	yes	characteristic
56	C₂²×C₄	—	137	6	no	—
57	C₂²×C₄	—	137	6	no	—
58	C₂²×C₄	—	137	1	no	—
59	C₂²×C₄	—	137	1	no	—
60	C₂×C₈	—	137	3	no	—
61	C₂×C₈	—	137	4	no	—
62	C₂×C₈	—	137	3	no	—
63	C₂×C₈	—	137	12	no	—
64	C₂×C₈	—	137	6	no	—
65	C₁₆	—	137	8	no	—
66	C₂³	—	2	4	no	—
67	C₂³	C₂×#11764049	1	6	yes	—
68	C₂³	C₂×#11764049	1	1	yes	characteristic
69	C₂×C₄	—	137	4	no	—
70	C₂×C₄	—	137	3	no	—
71	C₂×C₄	—	137	12	no	—
72	C₂×C₄	—	137	3	no	—
73	C₂×C₄	—	137	6	no	—
74	C₈	—	137	1	no	—
75	C₈	—	137	3	no	—
76	C₈	—	137	4	no	—
77	C₂²	—	2	12	no	—
78	C₂²	#17984184	1	4	yes	direct product factor
79	C₂²	C₂²×#11764049	1	4	yes	—
80	C₂²	C₂²×#11764049	1	3	yes	—
81	C₄	—	137	1	no	—
82	C₄	—	137	4	no	—
83	C₄	—	137	3	no	—
84	C₂	—	2	4	no	—
85	C₂	C₂×#17984184	1	6	yes	direct product factor
86	C₂	C₂³×#11764049	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11764049
commutator	C₂×C₁₃₇	C₂³×C₈
Frattini	C₂	C₂³×#11764049
Fitting, prime radical	C₂⁴×C₁₃₇	C₈
socle	C₂³×C₁₃₇	C₂×C₈
cosocle	#11764047	C₂⁴
hypercenter	C₂⁴	#11764049
hypocenter	C₁₃₇	C₂²×M₃₂
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#17984184	C₂	direct (N × H)
2. C₂²×#14555522	C₂	semidirect
4. #17984184	C₂²	direct (N × H)
5. C₂×#14555522	C₂²	semidirect
11. #14555522	C₂³	semidirect
34. C₂²×C₁₃₇	M₃₂	semidirect
41. C₂×C₁₃₇	C₂×M₃₂	semidirect
43. C₁₃₇	C₂²×M₃₂	semidirect
78. C₂²	#17984184	direct (N × H)
85. C₂	C₂×#17984184	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×#17984184	2
C₂²×#14555522	2
C₂³×#11764047	2
C₂²×M₃₂	137

Subgroups — C22×#17984184 #31798036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17984184 #31798036