Subgroups — #31796718

Order 17536.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578269	C₂	1	6	yes	maximal
2	#22577248	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764047	C₂²	1	4	yes	—
4	C₂×C₄×Dic₁₃₇	C₂²	1	3	yes	—
5	#17983995	C₄	1	3	yes	—
6	#95×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×#11764047	—	2	12	no	—
8	C₂²×Dic₁₃₇	C₂×C₄	1	3	yes	—
9	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
10	C₄×Dic₁₃₇	—	2	6	no	—
11	C₂²×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
12	#11764047	—	4	4	no	—
13	C₂×Dic₁₃₇	—	2	3	no	—
14	C₂×Dic₁₃₇	—	2	3	no	—
15	C₂×Dic₁₃₇	—	2	3	no	—
16	C₂×Dic₁₃₇	#41	1	6	yes	—
17	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₃₇	—	2	3	no	—
19	C₂×C₄×C₁₃₇	—	2	6	no	—
20	C₂²×C₁₃₇	#117	1	3	yes	—
21	C₂²×C₁₃₇	#131	1	3	yes	—
22	C₂²×C₁₃₇	#131	1	1	yes	characteristic
23	C₄×C₁₃₇	—	4	3	no	—
24	Dic₁₃₇	—	4	3	no	—
25	Dic₁₃₇	#126	1	4	yes	—
26	C₂×C₁₃₇	#383	1	1	yes	characteristic
27	C₂×C₁₃₇	#433	1	3	yes	—
28	C₂×C₁₃₇	#433	1	3	yes	—
29	C₁₃₇	#1840	1	1	yes	characteristic, complemented, Sylow
30	#1840	—	137	1	no	maximal, Sylow
31	#376	—	137	1	no	—
32	#336	—	137	6	no	—
33	C₂²×C₈	—	137	4	no	—
34	C₂×C₄²	—	137	3	no	—
35	#95	—	137	3	no	—
36	#95	#10587739	1	1	yes	characteristic
37	C₂²×C₄	—	137	3	no	—
38	C₂²×C₄	—	137	1	no	—
39	C₂²×C₄	C₂×#10587739	1	3	yes	—
40	C₂×C₈	—	274	12	no	—
41	C₄²	—	274	6	no	—
42	C₂³	C₂²×#10587739	1	1	yes	characteristic
43	C₂×C₄	—	2	3	no	—
44	C₂×C₄	—	2	6	no	—
45	C₂×C₄	—	274	3	no	—
46	C₂×C₄	—	274	3	no	—
47	C₂×C₄	—	274	3	no	—
48	C₂×C₄	—	137	6	no	—
49	C₈	—	548	4	no	—
50	C₂²	#17984167	1	3	yes	—
51	C₂²	#17984175	1	3	yes	—
52	C₂²	#17984176	1	1	yes	characteristic
53	C₄	—	4	3	no	—
54	C₄	—	548	3	no	—
55	C₄	—	137	4	no	—
56	C₂	#22578348	1	1	yes	characteristic
57	C₂	#22578427	1	3	yes	—
58	C₂	#22578428	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10587739
commutator, socle, prime radical	C₂³×C₁₃₇	C₂²×C₄
Fitting	#95×C₁₃₇	C₄
cosocle	C₂²×Dic₁₃₇	C₂³
hypercenter	#95	#10587739
hypocenter	C₁₃₇	#1840
LCS	G ▷ C₂³×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
29. C₁₃₇	#1840	semidirect

Maximal subgroups

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

subgroup	index
#22578269	2
#22577248	2
#1840	137