Subgroups — C₂×#22578331 #31797142

Order 17536.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984159	C₂	1	2	yes	maximal
2	#22578331	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555520	—	2	2	no	—
5	#17984159	—	2	2	no	—
6	#17984159	C₂²	1	4	yes	—
7	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×#14555485	C₄	1	1	yes	characteristic
9	C₁₆×D₁₃₇	C₄	1	2	yes	—
10	C₁₆×D₁₃₇	C₂²	1	2	yes	—
11	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
12	#14555520	—	2	4	no	—
13	#14555520	—	4	2	no	—
14	C₂×#11764047	—	4	2	no	—
15	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
16	C₂×#11764040	Q₈	1	1	yes	characteristic
17	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₈×D₁₃₇	D₄	1	1	yes	characteristic
19	C₈×D₁₃₇	Q₈	1	1	yes	characteristic
20	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
21	#14555485	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
24	#11764047	—	4	4	no	—
25	C₂²×D₁₃₇	Q₁₆	1	1	yes	characteristic
26	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
27	C₄×D₁₃₇	C₄⋊C₄	1	1	yes	characteristic
28	C₄×D₁₃₇	D₈	1	1	yes	characteristic
29	C₄×D₁₃₇	Q₁₆	1	1	yes	characteristic
30	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
31	#11764040	C₄⋊C₄	1	1	yes	characteristic
32	#11764040	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
34	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
35	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₁₃₇	#107	1	1	yes	characteristic
37	C₂×D₁₃₇	—	2	2	no	—
38	C₂×D₁₃₇	#107	1	1	yes	characteristic
39	C₂×D₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
40	C₄×C₁₃₇	#107	1	1	yes	characteristic
41	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	Dic₁₃₇	#107	1	1	yes	characteristic
43	Dic₁₃₇	C₂×D₈	1	1	yes	characteristic
44	C₂×C₁₃₇	#368	1	2	yes	complemented
45	C₂×C₁₃₇	C₂×#107	1	1	yes	characteristic
46	D₁₃₇	—	2	1	no	—
47	D₁₃₇	—	2	1	no	—
48	C₁₃₇	C₂×#368	1	1	yes	characteristic, complemented, Sylow
49	C₂×#368	—	137	1	no	maximal, Sylow
50	C₂²×C₁₆	—	137	1	no	—
51	C₂×#108	—	137	2	no	—
52	#368	—	137	4	no	—
53	C₂×M₁₆	—	274	2	no	—
54	C₂²×C₈	—	137	1	no	—
55	C₂×C₁₆	—	137	1	no	—
56	C₂×C₁₆	—	137	2	no	—
57	C₂×C₁₆	—	137	2	no	—
58	C₂×C₁₆	#10587739	1	1	yes	characteristic
59	#108	—	274	2	no	—
60	#108	—	137	4	no	—
61	C₂²×C₄	—	137	1	no	—
62	M₁₆	—	274	4	no	—
63	M₁₆	—	548	2	no	—
64	C₂×C₈	—	548	2	no	—
65	C₂×C₈	—	137	1	no	—
66	C₂×C₈	—	137	1	no	—
67	C₂×C₈	—	137	1	no	—
68	C₂×C₈	—	137	1	no	—
69	C₂×C₈	—	137	1	no	—
70	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
71	C₁₆	—	137	2	no	—
72	C₁₆	C₂×#10587739	1	2	yes	—
73	C₂³	—	137	1	no	—
74	C₂×C₄	—	137	1	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	1	no	—
77	C₂×C₄	—	137	1	no	—
78	C₂×C₄	—	137	1	no	—
79	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
80	C₈	—	548	4	no	—
81	C₈	—	137	1	no	—
82	C₈	—	137	1	no	—
83	C₈	C₄⋊#10587739	1	1	yes	characteristic
84	C₈	C₂²×#10587739	1	1	yes	characteristic
85	C₂²	—	274	2	no	—
86	C₂²	—	137	1	no	—
87	C₂²	—	137	1	no	—
88	C₂²	#17984157	1	1	yes	characteristic
89	C₄	—	137	1	no	—
90	C₄	—	137	1	no	—
91	C₄	#17984157	1	1	yes	characteristic
92	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
93	C₂	—	274	1	no	—
94	C₂	—	274	1	no	—
95	C₂	#22578331	1	2	yes	direct product factor
96	C₂	C₂×#17984157	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984157
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₈	C₂²×#10587739
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle	C₂²×C₁₃₇	#107
prime radical	C₂²×D₁₃₇	Q₁₆
cosocle	C₈×D₁₃₇	C₂³
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	C₂×#368
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22578331	C₂	direct (N × H)
44. C₂×C₁₃₇	#368	semidirect
48. C₁₃₇	C₂×#368	semidirect
95. C₂	#22578331	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×#17984159	2
#22578331	2
C₂×C₁₆×D₁₃₇	2
C₂×#368	137

Subgroups — C2×#22578331 #31797142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578331 #31797142