Subgroups — #31797162

Order 17536.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578326	C₂	1	2	yes	maximal, complemented
2	#22578299	C₂	1	2	yes	maximal
3	#22578295	C₂	1	2	yes	maximal, complemented
4	#22577359	C₂	1	1	yes	maximal, characteristic
5	#17984162	C₂²	1	4	yes	—
6	C₂×#14555494	C₂²	1	1	yes	characteristic
7	C₂²×#11764040	C₄	1	1	yes	characteristic
8	#17983999	C₄	1	1	yes	characteristic
9	#17983999	C₂²	1	1	yes	characteristic
10	C₄×#11764040	C₄	1	1	yes	characteristic
11	C₄×#11764040	C₂²	1	1	yes	characteristic
12	#117×C₁₃₇	C₄	1	1	yes	characteristic
13	#14555513	—	2	4	no	—
14	#14555494	—	2	2	no	—
15	C₂×#11764040	—	2	2	no	—
16	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
17	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
18	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
19	C₂×#11764040	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₄×C₁₃₇	—	2	1	no	—
22	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
23	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	#11764040	—	2	1	no	—
26	#11764040	—	2	1	no	—
27	#11764040	M₁₆	1	2	yes	—
28	#11764040	#41	1	2	yes	—
29	C₂³×C₁₃₇	M₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	—	2	1	no	—
31	C₂×C₄×C₁₃₇	—	2	1	no	—
32	C₂×C₄×C₁₃₇	—	2	1	no	—
33	C₂×C₄×C₁₃₇	—	2	1	no	—
34	C₂×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₁₃₇	—	2	1	no	—
37	C₂²×C₁₃₇	—	4	1	no	—
38	C₂²×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
39	C₄×C₁₃₇	—	2	1	no	—
40	C₄×C₁₃₇	—	4	1	no	—
41	C₄×C₁₃₇	—	4	1	no	—
42	C₄×C₁₃₇	#117	1	1	yes	characteristic
43	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₁₃₇	—	2	1	no	—
45	C₂×C₁₃₇	—	4	1	no	—
46	C₂×C₁₃₇	#432	1	1	yes	characteristic
47	C₁₃₇	#2166	1	1	yes	characteristic, complemented, Sylow
48	#2166	—	137	1	no	maximal, Sylow
49	#405	—	137	1	no	—
50	#364	—	137	2	no	—
51	#349	—	137	2	no	—
52	#347	—	137	2	no	—
53	C₂×M₁₆	—	137	1	no	—
54	C₂²×C₈	—	137	1	no	—
55	#117	#10587739	1	1	yes	characteristic
56	M₃₂	—	137	4	no	—
57	#97	—	137	1	no	—
58	#97	—	137	1	no	—
59	C₄×C₈	—	137	1	no	—
60	C₄×C₈	—	137	1	no	—
61	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
62	M₁₆	—	274	2	no	—
63	C₂×C₈	—	274	2	no	—
64	C₂×C₈	—	137	1	no	—
65	C₂×C₈	—	137	1	no	—
66	C₂×C₈	—	137	1	no	—
67	C₂×C₈	—	137	1	no	—
68	C₄⋊C₄	—	2	1	no	—
69	C₂²⋊C₄	—	2	1	no	—
70	C₄²	C₂×#10587739	1	1	yes	characteristic
71	C₄²	C₂×#10587739	1	1	yes	characteristic
72	C₁₆	—	274	4	no	—
73	C₂³	#14555519	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	#14555519	1	1	yes	characteristic
79	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
80	C₈	—	274	1	no	—
81	C₈	—	274	1	no	—
82	C₈	—	137	2	no	—
83	C₈	—	137	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	C₂×#14555519	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	—	4	1	no	—
90	C₄	#17984167	1	1	yes	characteristic
91	C₄	D₁₃₇⋊M₁₆	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	#22578423	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984167
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	#117×C₁₃₇	C₄
socle	C₂×C₁₃₇	#432
prime radical	C₂³×C₁₃₇	M₁₆
cosocle	C₂×#11764040	C₂³
hypercenter	#117	#10587739
hypocenter	C₁₃₇	#2166
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₄² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #22578326	C₂	semidirect
3. #22578295	C₂	semidirect
47. C₁₃₇	#2166	semidirect

Maximal subgroups

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

subgroup	index
#22578326	2
#22578299	2
#22578295	2
#22577359	2
#2166	137