Subgroups — #31795732

Order 17536.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578248	C₂	1	2	yes	maximal
2	#22577272	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11764047	—	2	2	no	—
4	C₂×C₄⋊Dic₁₃₇	C₂²	1	1	yes	characteristic
5	C₂×#14555489	—	2	2	no	—
6	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×#11764047	—	4	2	no	—
9	C₂²×Dic₁₃₇	—	2	1	no	—
10	C₂²×Dic₁₃₇	D₄	1	2	yes	—
11	C₄⋊Dic₁₃₇	—	4	1	no	—
12	#14555489	—	4	4	no	—
13	C₄×Dic₁₃₇	—	2	1	no	—
14	C₄×Dic₁₃₇	C₈	1	2	yes	complemented
15	C₂²×C₄×C₁₃₇	—	2	1	no	—
16	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
17	C₄⋊C₄×C₁₃₇	—	4	1	no	—
18	#11764047	—	8	2	no	—
19	C₂×Dic₁₃₇	—	2	2	no	—
20	C₂×Dic₁₃₇	—	2	2	no	—
21	C₂×Dic₁₃₇	—	2	2	no	—
22	C₂×Dic₁₃₇	—	2	2	no	—
23	C₂×Dic₁₃₇	—	4	2	no	—
24	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₃₇	—	2	1	no	—
26	C₂×C₄×C₁₃₇	—	2	1	no	—
27	C₂×C₄×C₁₃₇	—	4	2	no	—
28	C₂×C₄×C₁₃₇	—	4	1	no	—
29	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
31	C₂²×C₁₃₇	—	2	1	no	—
32	C₂²×C₁₃₇	—	2	1	no	—
33	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₂²×C₁₃₇	#99	1	1	yes	characteristic
35	C₂²×C₁₃₇	#101	1	1	yes	characteristic
36	C₄×C₁₃₇	—	2	1	no	—
37	C₄×C₁₃₇	—	2	1	no	—
38	C₄×C₁₃₇	—	8	1	no	—
39	Dic₁₃₇	—	2	4	no	—
40	Dic₁₃₇	—	4	2	no	—
41	C₂×C₁₃₇	—	2	1	no	—
42	C₂×C₁₃₇	—	2	1	no	—
43	C₂×C₁₃₇	#324	1	1	yes	characteristic
44	C₂×C₁₃₇	#353	1	1	yes	characteristic
45	C₂×C₁₃₇	#356	1	1	yes	characteristic
46	C₁₃₇	#1325	1	1	yes	characteristic, complemented, Sylow
47	#1325	—	137	1	no	maximal, Sylow
48	#384	—	137	1	no	—
49	#324	—	137	2	no	—
50	C₂×C₄⋊C₄	—	274	2	no	—
51	C₂×C₄⋊C₄	—	137	1	no	—
52	C₂×C₄⋊C₄	#10587739	1	1	yes	characteristic
53	C₂×C₄²	—	137	1	no	—
54	C₂²⋊C₈	—	274	2	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	—	274	1	no	—
57	C₂²×C₄	—	137	2	no	—
58	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
59	C₂×C₈	—	548	2	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	548	1	no	—
62	C₄⋊C₄	—	548	4	no	—
63	C₄²	—	274	1	no	—
64	C₄²	—	137	2	no	—
65	C₂³	C₂²⋊#10587739	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	274	2	no	—
71	C₂×C₄	—	274	2	no	—
72	C₂×C₄	—	274	2	no	—
73	C₂×C₄	—	274	2	no	—
74	C₂×C₄	—	548	2	no	—
75	C₂×C₄	#14555518	1	1	yes	characteristic
76	C₂×C₄	#14555520	1	1	yes	characteristic
77	C₈	—	1096	2	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	#17984145	1	1	yes	characteristic
81	C₂²	#17984146	1	1	yes	characteristic
82	C₂²	#17984149	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	274	4	no	—
87	C₄	—	548	2	no	—
88	C₂	—	2	1	no	—
89	C₂	—	2	1	no	—
90	C₂	#22578249	1	1	yes	characteristic
91	C₂	#22578307	1	1	yes	characteristic
92	C₂	#22578313	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984146
commutator	C₂×C₄×C₁₃₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#10587739
Fitting	C₂×C₄⋊C₄×C₁₃₇	C₄
socle	C₂²×C₁₃₇	#99
prime radical	C₂³×C₁₃₇	C₂²⋊C₄
cosocle	C₂×C₄⋊Dic₁₃₇	C₂²
hypercenter	C₂×C₄⋊C₄	#10587739
hypocenter	C₁₃₇	#1325
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. C₄×Dic₁₃₇	C₈	semidirect
46. C₁₃₇	#1325	semidirect

Maximal subgroups

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

subgroup	index
#22578248	2
#22577272	2
#1325	137