Subgroups — #31795704

Order 17536.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14555513	C₂	1	2	yes	maximal
2	C₈×#11764040	C₂	1	1	yes	maximal, characteristic
3	C₂×#14555513	C₄	1	4	yes	—
4	C₄×#11764040	C₄	1	1	yes	characteristic
5	C₄×#11764040	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
7	#14555513	—	2	4	no	—
8	C₂×#11764040	C₈	1	2	yes	—
9	C₂×#11764040	C₂×C₄	1	2	yes	—
10	C₂×C₈×C₁₃₇	C₈	1	2	yes	—
11	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
12	#11764040	C₁₆	1	4	yes	complemented
13	#11764040	M₁₆	1	4	yes	—
14	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
15	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
17	C₈×C₁₃₇	—	2	2	no	—
18	C₂²×C₁₃₇	C₄×C₈	1	1	yes	characteristic
19	C₄×C₁₃₇	#97	1	2	yes	—
20	C₄×C₁₃₇	C₂×C₁₆	1	2	yes	—
21	C₄×C₁₃₇	M₃₂	1	2	yes	—
22	C₂×C₁₃₇	#322	1	1	yes	characteristic
23	C₂×C₁₃₇	C₄×C₁₆	1	1	yes	characteristic
24	C₂×C₁₃₇	#346	1	1	yes	characteristic
25	C₁₃₇	#1312	1	1	yes	characteristic, complemented, Sylow
26	#1312	—	137	1	no	maximal, Sylow
27	C₄×C₁₆	—	137	2	no	—
28	C₈²	—	137	1	no	—
29	C₂×C₁₆	—	137	4	no	—
30	C₄×C₈	—	137	1	no	—
31	C₄×C₈	—	137	1	no	—
32	C₄×C₈	#10587739	1	1	yes	characteristic
33	C₂×C₈	—	137	2	no	—
34	C₂×C₈	—	137	2	no	—
35	C₂×C₈	#11764047	1	2	yes	—
36	C₄²	C₂×#10587739	1	1	yes	characteristic
37	C₁₆	—	274	4	no	—
38	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
39	C₂×C₄	C₂×#11764047	1	1	yes	characteristic
40	C₂×C₄	#14555518	1	1	yes	characteristic
41	C₈	—	2	2	no	—
42	C₈	—	137	4	no	—
43	C₈	—	137	4	no	—
44	C₂²	C₄×#11764047	1	1	yes	characteristic
45	C₄	#17984142	1	2	yes	—
46	C₄	#17984161	1	2	yes	—
47	C₄	#17984162	1	2	yes	—
48	C₂	#22578244	1	1	yes	characteristic
49	C₂	#22578289	1	1	yes	characteristic
50	C₂	#22578291	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×#10587739
commutator	C₂×C₁₃₇	C₄×C₁₆
Fitting	C₄×C₈×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	C₄×C₈
cosocle	C₄×#11764040	C₂²
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#1312
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. #11764040	C₁₆	semidirect
25. C₁₃₇	#1312	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14555513	2
C₈×#11764040	2
#1312	137