Subgroups — #31797161

Order 17536.

128 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	#22578300	C₂	1	2	yes	maximal
3	#22578296	C₂	1	2	yes	maximal, complemented
4	C₄×#14555493	C₂	1	1	yes	maximal, characteristic
5	#17984162	C₂²	1	4	yes	—
6	C₂×#14555493	C₄	1	1	yes	characteristic
7	C₂×#14555493	C₂²	1	1	yes	characteristic
8	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
9	#17983998	C₄	1	1	yes	characteristic
10	#17983998	C₂²	1	1	yes	characteristic
11	C₄×#11764040	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
13	#14555513	—	2	4	no	—
14	C₂×C₄×D₁₃₇	—	2	1	no	—
15	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
16	#14555493	—	2	2	no	—
17	#14555493	—	2	2	no	—
18	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
19	C₂×#11764040	C₂³	1	1	yes	characteristic
20	C₄×Dic₁₃₇	—	2	1	no	—
21	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₁₃₇	M₁₆	1	1	yes	characteristic
26	C₂×Dic₁₃₇	—	2	1	no	—
27	C₂×Dic₁₃₇	M₁₆	1	1	yes	characteristic
28	C₄×D₁₃₇	—	2	1	no	—
29	C₄×D₁₃₇	—	2	1	no	—
30	C₄×D₁₃₇	—	4	2	no	—
31	#11764040	M₁₆	1	2	yes	—
32	#11764040	#41	1	2	yes	—
33	C₂×C₄×C₁₃₇	—	2	1	no	—
34	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₃₇	—	2	1	no	—
36	C₈×C₁₃₇	—	2	1	no	—
37	C₂²×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
38	C₂×D₁₃₇	—	2	1	no	—
39	C₂×D₁₃₇	—	4	1	no	—
40	C₄×C₁₃₇	—	2	2	no	—
41	C₄×C₁₃₇	#117	1	1	yes	characteristic
42	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
43	Dic₁₃₇	—	2	1	no	—
44	Dic₁₃₇	—	4	1	no	—
45	C₂×C₁₃₇	—	2	1	no	—
46	C₂×C₁₃₇	#432	1	1	yes	characteristic
47	D₁₃₇	—	4	1	no	—
48	C₁₃₇	#2165	1	1	yes	characteristic, complemented, Sylow
49	#2165	—	137	1	no	maximal, Sylow
50	C₄×M₁₆	—	137	1	no	—
51	#364	—	137	2	no	—
52	#349	—	137	2	no	—
53	#347	—	137	2	no	—
54	C₂×M₁₆	—	137	1	no	—
55	C₂×M₁₆	—	137	1	no	—
56	C₂×C₄²	—	137	1	no	—
57	M₃₂	—	137	4	no	—
58	#97	—	137	1	no	—
59	#97	—	137	1	no	—
60	C₄×C₈	—	137	1	no	—
61	C₄×C₈	#10587739	1	1	yes	characteristic
62	C₂²×C₄	—	274	1	no	—
63	C₂²×C₄	—	137	1	no	—
64	M₁₆	—	274	2	no	—
65	M₁₆	—	274	2	no	—
66	C₂×C₈	—	137	1	no	—
67	C₂×C₈	—	137	1	no	—
68	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
69	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
70	C₄²	—	274	1	no	—
71	C₄²	—	137	1	no	—
72	C₄²	C₂×#10587739	1	1	yes	characteristic
73	C₁₆	—	274	4	no	—
74	C₂³	—	137	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	274	1	no	—
77	C₂×C₄	—	274	1	no	—
78	C₂×C₄	—	274	1	no	—
79	C₂×C₄	—	548	2	no	—
80	C₂×C₄	—	137	1	no	—
81	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	2	1	no	—
84	C₈	—	137	2	no	—
85	C₈	—	137	2	no	—
86	C₂²	—	274	1	no	—
87	C₂²	—	548	1	no	—
88	C₂²	D₁₃₇⋊M₁₆	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	274	1	no	—
91	C₄	—	548	1	no	—
92	C₄	#17984167	1	1	yes	characteristic
93	C₄	D₁₃₇⋊M₁₆	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	548	1	no	—
96	C₂	#22578426	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984167
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄×C₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#432
prime radical	C₂²×D₁₃₇	M₁₆
cosocle	C₂×#11764040	C₂³
hypercenter	C₄×C₈	#10587739
hypocenter	C₁₃₇	#2165
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #22578326	C₂	semidirect
3. #22578296	C₂	semidirect
48. C₁₃₇	#2165	semidirect

Maximal subgroups

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

subgroup	index
#22578326	2
#22578300	2
#22578296	2
C₄×#14555493	2
#2165	137