Subgroups — C₂×#22578511 #31798019

Order 17536.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578511	C₂	1	4	yes	maximal, direct product factor
2	C₂×#17984178	C₂	1	2	yes	maximal, complemented
3	C₂²×#14555513	C₂	1	1	yes	maximal, characteristic
4	#17984178	C₂²	1	4	yes	complemented
5	C₂×#14555513	C₄	1	2	yes	—
6	C₂×#14555513	C₄	1	1	yes	characteristic
7	C₂×#14555513	C₂²	1	1	yes	characteristic
8	C₂×#14555513	C₂²	1	2	yes	—
9	C₂²×#11764040	C₄	1	1	yes	characteristic
10	#14555513	C₂×C₄	1	1	yes	characteristic
11	#14555513	C₂×C₄	1	2	yes	—
12	#14555513	C₂³	1	1	yes	characteristic
13	C₂×#11764040	C₈	1	2	yes	—
14	C₂×#11764040	C₈	1	1	yes	characteristic
15	C₂×#11764040	C₂×C₄	1	2	yes	—
16	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₃₇	C₈	1	1	yes	characteristic
18	#11764040	C₂×C₈	1	1	yes	characteristic
19	#11764040	C₂×C₈	1	2	yes	—
20	#11764040	C₂²×C₄	1	1	yes	characteristic
21	C₂³×C₁₃₇	C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₁₃₇	C₁₆	1	2	yes	—
23	C₂×C₄×C₁₃₇	C₁₆	1	1	yes	characteristic
24	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
25	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₁₃₇	—	2	2	no	—
27	C₂²×C₁₃₇	C₂×C₁₆	1	2	yes	—
28	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
29	C₄×C₁₃₇	C₂×C₁₆	1	2	yes	—
30	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
31	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₁₃₇	—	2	2	no	—
33	C₂×C₁₃₇	M₆₄	1	2	yes	complemented
34	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
35	C₁₃₇	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×M₆₄	—	137	1	no	maximal, Sylow
37	C₂²×C₁₆	—	137	1	no	—
38	M₆₄	—	137	4	no	—
39	C₂×C₃₂	—	137	2	no	—
40	C₂²×C₈	—	137	1	no	—
41	C₂×C₁₆	—	137	1	no	—
42	C₂×C₁₆	—	137	2	no	—
43	C₂×C₁₆	—	137	2	no	—
44	C₂×C₁₆	—	137	1	no	—
45	C₃₂	—	137	4	no	—
46	C₂²×C₄	#11764049	1	1	yes	characteristic
47	C₂×C₈	—	137	2	no	—
48	C₂×C₈	—	137	2	no	—
49	C₂×C₈	—	137	1	no	—
50	C₂×C₈	—	137	1	no	—
51	C₁₆	—	137	1	no	—
52	C₁₆	—	137	1	no	—
53	C₁₆	—	137	2	no	—
54	C₂³	#14555522	1	1	yes	characteristic
55	C₂×C₄	#14555522	1	1	yes	characteristic
56	C₂×C₄	#14555522	1	2	yes	—
57	C₂×C₄	C₂×#11764049	1	2	yes	—
58	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
59	C₈	—	137	1	no	—
60	C₈	—	137	1	no	—
61	C₈	—	137	2	no	—
62	C₂²	—	2	2	no	—
63	C₂²	C₂×#14555522	1	2	yes	—
64	C₂²	C₂×#14555522	1	1	yes	characteristic
65	C₄	C₂×#14555522	1	2	yes	—
66	C₄	C₂×#14555522	1	1	yes	characteristic
67	C₄	C₂²×#11764049	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	#22578511	1	2	yes	direct product factor
70	C₂	C₂²×#14555522	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11764049
commutator	C₂×C₁₃₇	C₂²×C₁₆
Frattini	C₄	C₂²×#11764049
Fitting	C₂²×C₄×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂×C₁₆
prime radical	C₂³×C₁₃₇	C₁₆
cosocle	#14555513	C₂³
hypercenter	C₂²×C₄	#11764049
hypocenter	C₁₃₇	C₂×M₆₄
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #22578511	C₂	direct (N × H)
2. C₂×#17984178	C₂	semidirect
4. #17984178	C₂²	semidirect
33. C₂×C₁₃₇	M₆₄	semidirect
35. C₁₃₇	C₂×M₆₄	semidirect
69. C₂	#22578511	direct (N × H)

Maximal subgroups

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

subgroup	index
#22578511	2
C₂×#17984178	2
C₂²×#14555513	2
C₂×M₆₄	137

Subgroups — C2×#22578511 #31798019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578511 #31798019