Subgroups — #31795654

Order 17536.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578424	C₂	1	2	yes	maximal
2	C₂×#17984017	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11764047	—	2	2	no	—
4	C₂²⋊#11764047	—	2	2	no	—
5	#17984143	C₄	1	4	yes	—
6	C₂²×#11764040	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
8	#17984017	—	2	2	no	—
9	C₂²×C₈×C₁₃₇	C₄	1	1	yes	characteristic
10	C₂×#11764047	—	2	4	no	—
11	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
12	C₂²×Dic₁₃₇	Q₈	1	1	yes	characteristic
13	C₂×#11764040	—	2	1	no	—
14	C₂×#11764040	—	4	1	no	—
15	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
16	C₄×Dic₁₃₇	D₄	1	2	yes	—
17	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₃₇	—	2	1	no	—
19	C₂×C₈×C₁₃₇	—	4	1	no	—
20	#11764047	—	4	4	no	—
21	C₂×Dic₁₃₇	—	2	1	no	—
22	C₂×Dic₁₃₇	—	2	1	no	—
23	C₂×Dic₁₃₇	—	4	1	no	—
24	C₂×Dic₁₃₇	C₂²⋊C₄	1	2	yes	—
25	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
26	#11764040	—	4	1	no	—
27	C₂³×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	—	2	1	no	—
29	C₂×C₄×C₁₃₇	—	2	1	no	—
30	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₈×C₁₃₇	—	4	1	no	—
33	C₂²×C₁₃₇	—	2	1	no	—
34	C₂²×C₁₃₇	—	2	1	no	—
35	C₂²×C₁₃₇	#95	1	1	yes	characteristic
36	C₂²×C₁₃₇	#108	1	1	yes	characteristic
37	C₂²×C₁₃₇	#108	1	1	yes	characteristic
38	C₄×C₁₃₇	—	2	1	no	—
39	C₄×C₁₃₇	#100	1	1	yes	characteristic
40	C₄×C₁₃₇	#101	1	1	yes	characteristic
41	Dic₁₃₇	—	2	2	no	—
42	Dic₁₃₇	—	2	2	no	—
43	C₂×C₁₃₇	—	2	1	no	—
44	C₂×C₁₃₇	—	2	1	no	—
45	C₂×C₁₃₇	H(3,ℤ/4)	1	1	yes	characteristic
46	C₂×C₁₃₇	#341	1	1	yes	characteristic
47	C₂×C₁₃₇	#343	1	1	yes	characteristic
48	C₁₃₇	#1290	1	1	yes	characteristic, complemented, Sylow
49	#1290	—	137	1	no	maximal, Sylow
50	#432	—	137	2	no	—
51	C₂×C₄⋊C₈	—	137	1	no	—
52	C₂²×C₈	—	137	1	no	—
53	C₂²×C₈	#10587739	1	1	yes	characteristic
54	C₂×C₄²	—	137	1	no	—
55	C₄⋊C₈	—	274	2	no	—
56	C₄⋊C₈	—	274	2	no	—
57	C₂²⋊C₈	—	274	2	no	—
58	#97	—	137	4	no	—
59	C₂²×C₄	—	137	1	no	—
60	C₂²×C₄	—	137	1	no	—
61	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	274	4	no	—
65	C₂×C₈	—	274	1	no	—
66	C₂×C₈	—	548	1	no	—
67	C₄²	—	137	2	no	—
68	C₄²	—	137	2	no	—
69	C₂³	C₄⋊#10587739	1	1	yes	characteristic
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	274	1	no	—
73	C₂×C₄	—	274	1	no	—
74	C₂×C₄	—	548	1	no	—
75	C₂×C₄	—	137	2	no	—
76	C₂×C₄	—	137	2	no	—
77	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
79	C₈	—	4	1	no	—
80	C₈	—	548	4	no	—
81	C₈	—	548	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
85	C₂²	#17984158	1	1	yes	characteristic
86	C₂²	#17984159	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	274	2	no	—
89	C₄	—	274	2	no	—
90	C₄	#17984148	1	1	yes	characteristic
91	C₄	#17984149	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	#22578271	1	1	yes	characteristic
95	C₂	#22578279	1	1	yes	characteristic
96	C₂	#22578283	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×#10587739)⋊C₄
commutator	C₂×C₄×C₁₃₇	C₄²
Frattini	C₂²×C₄	C₂×#10587739
Fitting	C₂²×C₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	#95
prime radical	C₂³×C₁₃₇	C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃₇	C₂²
hypercenter	C₂²×C₈	#10587739
hypocenter	C₁₃₇	#1290
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
48. C₁₃₇	#1290	semidirect

Maximal subgroups

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

subgroup	index
#22578424	2
C₂×#17984017	2
#1290	137