Subgroups — #31779087

Order 17536.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577454	C₂	1	2	yes	maximal
2	C₄×#14555493	C₂	1	1	yes	maximal, characteristic
3	#22577124	C₂	1	2	yes	maximal, complemented
4	C₈×#11764040	C₂	1	1	yes	maximal, characteristic, complemented
5	#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555493	C₄	1	2	yes	—
7	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
8	#17984017	C₄	1	4	yes	—
9	#17984000	C₄	1	4	yes	—
10	#17983998	C₄	1	2	yes	—
11	C₄×#11764040	C₂²	1	2	yes	—
12	C₄×#11764040	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
17	#14555493	—	2	4	no	—
18	C₂×#11764040	C₂×C₄	1	4	yes	—
19	C₂×#11764040	C₂×C₄	1	2	yes	—
20	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
21	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₂²×D₁₃₇	C₄²	1	1	yes	characteristic
26	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
27	C₂×Dic₁₃₇	C₄²	1	2	yes	—
28	C₄×D₁₃₇	—	2	2	no	—
29	C₄×D₁₃₇	—	2	4	no	—
30	#11764040	—	2	2	no	—
31	#11764040	—	2	4	no	—
32	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
34	C₈×C₁₃₇	—	2	4	no	—
35	C₈×C₁₃₇	M₁₆	1	4	yes	—
36	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
37	C₂×D₁₃₇	—	2	1	no	—
38	C₂×D₁₃₇	—	2	2	no	—
39	C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
40	C₄×C₁₃₇	#131	1	4	yes	—
41	Dic₁₃₇	—	2	1	no	—
42	Dic₁₃₇	—	2	2	no	—
43	C₂×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
44	C₂×C₁₃₇	#405	1	2	yes	—
45	D₁₃₇	—	4	1	no	—
46	C₁₃₇	#1453	1	1	yes	characteristic, complemented, Sylow
47	#1453	—	137	1	no	maximal, Sylow
48	#431	—	137	2	no	—
49	C₄×M₁₆	—	137	1	no	—
50	#322	—	137	2	no	—
51	#322	D₁₃₇	1	1	yes	characteristic, complemented
52	C₈²	—	137	1	no	—
53	C₂×M₁₆	—	137	2	no	—
54	C₂×C₄²	—	137	1	no	—
55	C₄⋊C₈	—	137	4	no	—
56	C₂²⋊C₈	—	137	4	no	—
57	#97	—	137	2	no	—
58	C₄×C₈	—	137	2	no	—
59	C₄×C₈	—	137	1	no	—
60	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
61	C₄×C₈	C₂×D₁₃₇	1	2	yes	—
62	C₂²×C₄	—	137	2	no	—
63	C₂²×C₄	—	137	1	no	—
64	M₁₆	—	274	4	no	—
65	C₂×C₈	—	137	2	no	—
66	C₂×C₈	—	137	4	no	—
67	C₂×C₈	C₄×D₁₃₇	1	4	yes	—
68	C₂×C₈	C₄×D₁₃₇	1	2	yes	—
69	C₄²	—	137	1	no	—
70	C₄²	—	137	2	no	—
71	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
72	C₂³	—	137	1	no	—
73	C₂×C₄	—	274	2	no	—
74	C₂×C₄	—	274	4	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	2	no	—
77	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
79	C₈	—	2	4	no	—
80	C₈	—	274	2	no	—
81	C₈	—	274	4	no	—
82	C₈	#14555493	1	4	yes	—
83	C₂²	—	274	1	no	—
84	C₂²	—	274	2	no	—
85	C₂²	C₄²×D₁₃₇	1	1	yes	characteristic
86	C₄	—	274	1	no	—
87	C₄	—	274	2	no	—
88	C₄	C₂×#14555493	1	2	yes	—
89	C₄	#17984096	1	4	yes	—
90	C₂	—	548	1	no	—
91	C₂	C₄×#14555493	1	1	yes	characteristic
92	C₂	#22577360	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂×C₄²
Fitting	#322×C₁₃₇	C₂
prime radical	C₂²×D₁₃₇	C₄²
cosocle	C₄²×C₁₃₇	C₂³
hypercenter	#322	D₁₃₇
hypocenter	C₁₃₇	#1453
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. #22577124	C₂	semidirect
4. C₈×#11764040	C₂	semidirect
5. #322×C₁₃₇	C₂	semidirect
46. C₁₃₇	#1453	semidirect
51. #322	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577454	2
C₄×#14555493	2
#22577124	2
C₈×#11764040	2
#322×C₁₃₇	2
#1453	137