Subgroups — C₂²×#17984163 #31797752

Order 17536.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984163	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#14555513	C₂	1	2	yes	maximal, complemented
3	C₂²×C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#17984163	C₂²	1	16	yes	direct product factor
5	C₂×#14555513	C₂²	1	12	yes	complemented
6	C₂²×C₄×D₁₃₇	C₄	1	1	yes	characteristic
7	C₂²×#11764040	C₂²	1	1	yes	characteristic
8	C₂×C₈×D₁₃₇	C₄	1	6	yes	—
9	C₂×C₈×D₁₃₇	C₂²	1	6	yes	—
10	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555513	C₂³	1	8	yes	complemented
12	C₂³×D₁₃₇	C₈	1	1	yes	characteristic
13	C₂²×Dic₁₃₇	C₈	1	1	yes	characteristic
14	C₂×C₄×D₁₃₇	C₈	1	6	yes	—
15	C₂×C₄×D₁₃₇	C₂×C₄	1	6	yes	—
16	C₂×#11764040	C₂×C₄	1	3	yes	—
17	C₂×#11764040	C₂³	1	3	yes	—
18	C₈×D₁₃₇	C₂×C₄	1	12	yes	—
19	C₈×D₁₃₇	C₂³	1	4	yes	—
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₃₇	C₂×C₄	1	6	yes	—
22	C₂²×D₁₃₇	—	2	4	no	—
23	C₂²×D₁₃₇	C₂×C₈	1	6	yes	—
24	C₂×Dic₁₃₇	C₂×C₈	1	6	yes	—
25	C₄×D₁₃₇	C₂×C₈	1	12	yes	—
26	C₄×D₁₃₇	C₂²×C₄	1	4	yes	—
27	#11764040	C₂²×C₄	1	3	yes	—
28	#11764040	C₂⁴	1	1	yes	characteristic
29	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
31	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
32	C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
33	C₂²×C₁₃₇	M₃₂	1	4	yes	complemented
34	C₂²×C₁₃₇	C₂²×C₈	1	3	yes	—
35	C₂×D₁₃₇	—	2	12	no	—
36	C₂×D₁₃₇	C₂²×C₈	1	4	yes	—
37	C₄×C₁₃₇	C₂²×C₈	1	3	yes	—
38	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
39	Dic₁₃₇	C₂²×C₈	1	4	yes	—
40	C₂×C₁₃₇	C₂×M₃₂	1	6	yes	complemented
41	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
42	D₁₃₇	—	2	4	no	—
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	1	no	—
50	C₂²×C₈	—	137	6	no	—
51	C₂²×C₈	—	137	6	no	—
52	C₂²×C₈	#10587739	1	1	yes	characteristic
53	M₃₂	—	137	16	no	—
54	C₂×C₁₆	—	137	12	no	—
55	C₂⁴	—	137	1	no	—
56	C₂²×C₄	—	137	1	no	—
57	C₂²×C₄	—	137	6	no	—
58	C₂²×C₄	—	137	6	no	—
59	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
60	C₂×C₈	—	137	3	no	—
61	C₂×C₈	—	137	3	no	—
62	C₂×C₈	—	137	12	no	—
63	C₂×C₈	—	137	4	no	—
64	C₂×C₈	C₂×#10587739	1	6	yes	—
65	C₁₆	—	137	8	no	—
66	C₂³	—	274	4	no	—
67	C₂³	—	137	6	no	—
68	C₂³	#14555518	1	1	yes	characteristic
69	C₂×C₄	—	137	6	no	—
70	C₂×C₄	—	137	12	no	—
71	C₂×C₄	—	137	4	no	—
72	C₂×C₄	#14555518	1	3	yes	—
73	C₂×C₄	C₂²×#10587739	1	3	yes	—
74	C₈	—	137	1	no	—
75	C₈	—	137	3	no	—
76	C₈	C₂²×#10587739	1	4	yes	—
77	C₂²	—	274	12	no	—
78	C₂²	—	137	4	no	—
79	C₂²	#17984163	1	4	yes	direct product factor
80	C₂²	C₂×#14555518	1	3	yes	—
81	C₄	—	137	4	no	—
82	C₄	C₂×#14555518	1	3	yes	—
83	C₄	C₂³×#10587739	1	1	yes	characteristic
84	C₂	—	274	4	no	—
85	C₂	C₂×#17984163	1	6	yes	direct product factor
86	C₂	C₂²×#14555518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#10587739
commutator	C₂×C₁₃₇	C₂³×C₈
Frattini	C₄	C₂³×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle	C₂³×C₁₃₇	C₂×C₈
prime radical	C₂³×D₁₃₇	C₈
cosocle	#11764040	C₂⁴
hypercenter	C₂²×C₈	#10587739
hypocenter	C₁₃₇	C₂²×M₃₂
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#17984163	C₂	direct (N × H)
2. C₂²×#14555513	C₂	semidirect
4. #17984163	C₂²	direct (N × H)
5. C₂×#14555513	C₂²	semidirect
11. #14555513	C₂³	semidirect
33. C₂²×C₁₃₇	M₃₂	semidirect
40. C₂×C₁₃₇	C₂×M₃₂	semidirect
43. C₁₃₇	C₂²×M₃₂	semidirect
79. C₂²	#17984163	direct (N × H)
85. C₂	C₂×#17984163	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×#17984163	2
C₂²×#14555513	2
C₂²×C₈×D₁₃₇	2
C₂²×M₃₂	137

Subgroups — C22×#17984163 #31797752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17984163 #31797752