Subgroups — C₂²×#17984162 #31797750

Order 17536.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984162	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#14555513	C₂	1	2	yes	maximal, complemented
3	C₂³×#11764040	C₂	1	1	yes	maximal, characteristic
4	#17984162	C₂²	1	16	yes	direct product factor
5	C₂×#14555513	C₂²	1	12	yes	complemented
6	C₂²×#11764040	C₄	1	6	yes	—
7	C₂²×#11764040	C₄	1	1	yes	characteristic
8	C₂²×#11764040	C₂²	1	1	yes	characteristic
9	C₂²×#11764040	C₂²	1	6	yes	—
10	C₂³×C₄×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555513	C₂³	1	8	yes	complemented
12	C₂×#11764040	C₂×C₄	1	3	yes	—
13	C₂×#11764040	C₂×C₄	1	12	yes	—
14	C₂×#11764040	C₂×C₄	1	6	yes	—
15	C₂×#11764040	C₂³	1	3	yes	—
16	C₂×#11764040	C₂³	1	4	yes	—
17	C₂⁴×C₁₃₇	C₈	1	1	yes	characteristic
18	C₂²×C₄×C₁₃₇	C₈	1	6	yes	—
19	C₂²×C₄×C₁₃₇	C₈	1	1	yes	characteristic
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	6	yes	—
21	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	#11764040	C₂²×C₄	1	3	yes	—
23	#11764040	C₂²×C₄	1	4	yes	—
24	#11764040	C₂⁴	1	1	yes	characteristic
25	C₂³×C₁₃₇	—	2	4	no	—
26	C₂³×C₁₃₇	C₂×C₈	1	6	yes	—
27	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	C₂×C₈	1	12	yes	—
29	C₂×C₄×C₁₃₇	C₂×C₈	1	6	yes	—
30	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
31	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
32	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
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	C₄×C₁₃₇	C₂²×C₈	1	4	yes	—
38	C₄×C₁₃₇	C₂²×C₈	1	3	yes	—
39	C₄×C₁₃₇	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	#10587739	1	1	yes	characteristic
49	C₂²×C₈	—	137	6	no	—
50	C₂²×C₈	—	137	6	no	—
51	C₂²×C₈	—	137	1	no	—
52	C₂²×C₈	—	137	1	no	—
53	M₃₂	—	137	16	no	—
54	C₂×C₁₆	—	137	12	no	—
55	C₂⁴	#11764047	1	1	yes	characteristic
56	C₂²×C₄	#11764047	1	1	yes	characteristic
57	C₂²×C₄	#11764047	1	6	yes	—
58	C₂²×C₄	C₂×#10587739	1	6	yes	—
59	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
60	C₂×C₈	—	137	12	no	—
61	C₂×C₈	—	137	3	no	—
62	C₂×C₈	—	137	6	no	—
63	C₂×C₈	—	137	3	no	—
64	C₂×C₈	—	137	4	no	—
65	C₁₆	—	137	8	no	—
66	C₂³	—	2	4	no	—
67	C₂³	C₂×#11764047	1	6	yes	—
68	C₂³	C₂×#11764047	1	1	yes	characteristic
69	C₂×C₄	C₂×#11764047	1	12	yes	—
70	C₂×C₄	C₂×#11764047	1	6	yes	—
71	C₂×C₄	C₂×#11764047	1	3	yes	—
72	C₂×C₄	C₂²×#10587739	1	4	yes	—
73	C₂×C₄	C₂²×#10587739	1	3	yes	—
74	C₈	—	137	1	no	—
75	C₈	—	137	3	no	—
76	C₈	—	137	4	no	—
77	C₂²	—	2	12	no	—
78	C₂²	#17984162	1	4	yes	direct product factor
79	C₂²	C₂²×#11764047	1	4	yes	—
80	C₂²	C₂²×#11764047	1	3	yes	—
81	C₄	C₂²×#11764047	1	4	yes	—
82	C₄	C₂²×#11764047	1	3	yes	—
83	C₄	C₂³×#10587739	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×#17984162	1	6	yes	direct product factor
86	C₂	C₂³×#11764047	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#10587739
commutator	C₂×C₁₃₇	C₂³×C₈
Frattini	C₄	C₂³×#10587739
Fitting	C₂³×C₄×C₁₃₇	C₄
socle	C₂³×C₁₃₇	C₂×C₈
prime radical	C₂⁴×C₁₃₇	C₈
cosocle	#11764040	C₂⁴
hypercenter	C₂³×C₄	#10587739
hypocenter	C₁₃₇	C₂²×M₃₂
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#17984162	C₂	direct (N × H)
2. C₂²×#14555513	C₂	semidirect
4. #17984162	C₂²	direct (N × H)
5. C₂×#14555513	C₂²	semidirect
11. #14555513	C₂³	semidirect
34. C₂²×C₁₃₇	M₃₂	semidirect
41. C₂×C₁₃₇	C₂×M₃₂	semidirect
43. C₁₃₇	C₂²×M₃₂	semidirect
78. C₂²	#17984162	direct (N × H)
85. C₂	C₂×#17984162	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×#17984162	2
C₂²×#14555513	2
C₂³×#11764040	2
C₂²×M₃₂	137

Subgroups — C22×#17984162 #31797750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17984162 #31797750