Subgroups — C₂×#22578525 #31798041

Order 17536.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578525	C₂	1	8	yes	maximal, direct product factor
2	C₂×#17984185	C₂	1	3	yes	maximal, complemented
3	C₂×#17984183	C₂	1	3	yes	maximal, complemented
4	C₂×#17984176	C₂	1	1	yes	maximal, characteristic
5	#17984185	C₂²	1	12	yes	complemented
6	#17984183	C₂²	1	12	yes	complemented
7	C₂×#14555522	C₂²	1	1	yes	characteristic
8	C₂×#14555522	C₂²	1	3	yes	—
9	#17984176	C₄	1	4	yes	—
10	#17984176	C₂²	1	4	yes	—
11	C₂×#14555520	C₄	1	3	yes	—
12	C₂×#14555518	C₂²	1	3	yes	—
13	C₂×#14555511	C₄	1	1	yes	characteristic
14	#14555522	C₂³	1	2	yes	—
15	#14555522	C₂³	1	6	yes	—
16	#14555520	C₂×C₄	1	12	yes	—
17	#14555518	C₂×C₄	1	6	yes	—
18	#14555518	C₂³	1	6	yes	—
19	C₂×#11764047	C₂×C₄	1	3	yes	—
20	C₂×#11764047	C₂³	1	1	yes	characteristic
21	#14555511	C₈	1	4	yes	—
22	#14555511	C₂×C₄	1	4	yes	—
23	C₂×D₅₄₈	C₈	1	3	yes	—
24	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
25	C₂×Q₈×C₁₃₇	C₈	1	1	yes	characteristic
26	#11764047	C₂²×C₄	1	6	yes	—
27	#11764047	C₂²×C₄	1	1	yes	characteristic
28	#11764047	C₂⁴	1	1	yes	characteristic
29	C₂²×D₁₃₇	C₂×C₈	1	3	yes	—
30	D₅₄₈	C₂×C₈	1	12	yes	—
31	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×D₁₃₇	C₂×C₈	1	6	yes	—
33	C₄×D₁₃₇	C₂²×C₄	1	6	yes	—
34	Q₈×C₁₃₇	C₂×C₈	1	4	yes	—
35	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
36	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
37	C₂×D₁₃₇	—	2	6	no	—
38	C₂×D₁₃₇	C₂²×C₈	1	6	yes	—
39	C₄×C₁₃₇	C₂²×C₈	1	6	yes	—
40	Dic₁₃₇	C₂²×C₈	1	1	yes	characteristic
41	Dic₁₃₇	C₂³×C₄	1	1	yes	characteristic
42	C₂×C₁₃₇	#504	1	2	yes	complemented
43	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
44	D₁₃₇	—	2	6	no	—
45	C₁₃₇	C₂×#504	1	1	yes	characteristic, complemented, Sylow
46	C₂×#504	—	137	1	no	maximal, Sylow
47	C₂×#131	—	137	1	no	—
48	#504	—	137	8	no	—
49	C₂×M₃₂	—	137	3	no	—
50	C₂²×C₁₆	—	137	3	no	—
51	C₂×#41	—	137	1	no	—
52	#131	—	137	4	no	—
53	#131	—	137	4	no	—
54	C₂×M₁₆	—	137	3	no	—
55	C₂²×C₈	—	137	3	no	—
56	M₃₂	—	137	12	no	—
57	C₂×C₁₆	—	137	1	no	—
58	C₂×C₁₆	—	137	3	no	—
59	C₂×C₁₆	—	137	12	no	—
60	#41	—	137	4	no	—
61	#41	—	137	4	no	—
62	C₂×Q₈	#11764049	1	1	yes	characteristic
63	C₂×D₄	—	137	3	no	—
64	C₂²×C₄	—	137	3	no	—
65	M₁₆	—	137	12	no	—
66	C₂×C₈	—	137	1	no	—
67	C₂×C₈	—	137	6	no	—
68	C₂×C₈	—	137	3	no	—
69	C₂×C₈	—	137	6	no	—
70	C₁₆	—	137	2	no	—
71	C₁₆	—	137	6	no	—
72	C₂³	—	137	3	no	—
73	Q₈	C₂×#11764049	1	4	yes	—
74	D₄	—	137	12	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	6	no	—
77	C₂×C₄	—	137	6	no	—
78	C₂×C₄	C₂×#11764049	1	3	yes	—
79	C₈	—	137	1	no	—
80	C₈	—	137	1	no	—
81	C₈	—	137	6	no	—
82	C₂²	—	274	6	no	—
83	C₂²	—	137	6	no	—
84	C₂²	C₂²×#11764049	1	1	yes	characteristic
85	C₄	—	137	1	no	—
86	C₄	—	137	1	no	—
87	C₄	C₂²×#11764049	1	6	yes	—
88	C₂	—	274	6	no	—
89	C₂	#22578525	1	2	yes	direct product factor
90	C₂	C₂³×#11764049	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11764049
commutator	C₂×C₁₃₇	C₂³×C₈
Frattini	C₂	C₂³×#11764049
Fitting	C₂×Q₈×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂²×C₈
prime radical	C₂×#14555511	C₄
cosocle	#11764047	C₂⁴
hypercenter	C₂×Q₈	#11764049
hypocenter	C₁₃₇	C₂×#504
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #22578525	C₂	direct (N × H)
2. C₂×#17984185	C₂	semidirect
3. C₂×#17984183	C₂	semidirect
5. #17984185	C₂²	semidirect
6. #17984183	C₂²	semidirect
42. C₂×C₁₃₇	#504	semidirect
45. C₁₃₇	C₂×#504	semidirect
89. C₂	#22578525	direct (N × H)

Maximal subgroups

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

subgroup	index
#22578525	2
C₂×#17984185	2
C₂×#17984183	2
C₂×#17984176	2
C₂×#504	137

Subgroups — C2×#22578525 #31798041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578525 #31798041