Subgroups — #131×#10587739 #31797356

Order 17536.

572 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃₇⋊#131	C₂	1	1	yes	maximal, characteristic
2	#41×#10587739	C₂	1	1	yes	maximal, characteristic
3	#22578369	C₂	1	3	yes	maximal, complemented
4	#22578366	C₂	1	3	yes	maximal, complemented
5	M₁₆×#10587739	C₂	1	3	yes	maximal, complemented
6	C₂×C₈×#10587739	C₂	1	3	yes	maximal, complemented
7	#131×D₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#17984176	C₄	1	2	yes	complemented
9	#17984175	C₄	1	6	yes	complemented
10	D₁₃₇⋊M₁₆	C₂²	1	3	yes	—
11	C₂×#14555518	C₂²	1	3	yes	—
12	Q₈×#10587739	C₄	1	2	yes	—
13	D₄×#10587739	C₄	1	6	yes	—
14	#17984167	C₂²	1	3	yes	—
15	C₂×C₄×#10587739	C₂²	1	3	yes	—
16	#17984142	C₂²	1	6	yes	—
17	C₈×#10587739	C₂²	1	1	yes	characteristic
18	C₈×#10587739	C₂²	1	3	yes	—
19	C₈×#10587739	C₂²	1	3	yes	—
20	C₈×#10587739	C₂²	1	3	yes	—
21	#41×D₁₃₇	C₂²	1	1	yes	characteristic
22	#17984096	C₄	1	3	yes	complemented
23	#17984095	C₄	1	1	yes	characteristic, complemented
24	#17984094	C₄	1	3	yes	complemented
25	M₁₆×D₁₃₇	C₂²	1	3	yes	—
26	C₂×C₈×D₁₃₇	C₂²	1	3	yes	—
27	#131×C₁₃₇	C₄	1	1	yes	characteristic, complemented
28	C₂²×#10587739	C₂×C₄	1	6	yes	—
29	#14555520	C₂×C₄	1	6	yes	complemented
30	#14555519	C₂×C₄	1	6	yes	complemented
31	#14555518	C₂×C₄	1	6	yes	complemented
32	#14555518	C₂³	1	1	yes	characteristic
33	#14555518	C₂³	1	3	yes	—
34	C₂×#11764047	C₂×C₄	1	6	yes	complemented
35	C₄⋊#10587739	C₂×C₄	1	6	yes	—
36	C₂²⋊#10587739	C₂×C₄	1	6	yes	—
37	C₄×#10587739	C₂×C₄	1	6	yes	—
38	C₄×#10587739	C₂³	1	1	yes	characteristic
39	C₄×#10587739	C₂³	1	3	yes	—
40	#14555511	C₂×C₄	1	1	yes	characteristic
41	#14555510	C₂×C₄	1	3	yes	—
42	#14555509	C₂×C₄	1	3	yes	—
43	Q₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
44	D₄×D₁₃₇	C₂×C₄	1	3	yes	—
45	C₂×C₄×D₁₃₇	C₂³	1	3	yes	—
46	#14555494	C₂×C₄	1	3	yes	complemented
47	#14555493	C₂×C₄	1	6	yes	complemented
48	C₂×#11764040	C₂×C₄	1	3	yes	complemented
49	C₈×D₁₃₇	C₂×C₄	1	3	yes	complemented
50	C₈×D₁₃₇	C₂×C₄	1	3	yes	complemented
51	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
52	C₈×D₁₃₇	C₂³	1	3	yes	—
53	#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
54	M₁₆×C₁₃₇	C₂×C₄	1	3	yes	complemented
55	C₂×C₈×C₁₃₇	C₂×C₄	1	3	yes	complemented
56	C₂×#10587739	—	2	12	no	—
57	C₂×#10587739	C₂²×C₄	1	6	yes	—
58	C₂×#10587739	C₂²×C₄	1	2	yes	—
59	#11764047	C₂²×C₄	1	2	yes	—
60	#11764047	C₂²×C₄	1	6	yes	—
61	C₂²×D₁₃₇	C₂²×C₄	1	3	yes	—
62	Dic₂₇₄	C₄²	1	3	yes	—
63	D₅₄₈	C₄²	1	3	yes	—
64	C₂²⋊D₁₃₇	C₄²	1	6	yes	—
65	C₂×Dic₁₃₇	C₂²×C₄	1	3	yes	—
66	C₄×D₁₃₇	C₂²×C₄	1	3	yes	—
67	C₄×D₁₃₇	C₂²×C₄	1	3	yes	—
68	C₄×D₁₃₇	C₂²×C₄	1	3	yes	—
69	C₄×D₁₃₇	C₂⁴	1	1	yes	characteristic
70	#11764040	C₂²×C₄	1	1	yes	characteristic
71	#11764040	C₂²×C₄	1	3	yes	—
72	Q₈×C₁₃₇	C₄²	1	1	yes	characteristic
73	D₄×C₁₃₇	C₄²	1	3	yes	—
74	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
75	C₈×C₁₃₇	C₂²×C₄	1	3	yes	—
76	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₁₃₇	C₂×C₄²	1	3	yes	—
78	C₂×D₁₃₇	—	2	6	no	—
79	C₂×D₁₃₇	C₂×C₄²	1	3	yes	—
80	C₂×D₁₃₇	C₂³×C₄	1	1	yes	characteristic
81	#10587739	—	2	6	no	—
82	#10587739	#131	1	4	yes	direct product factor
83	C₄×C₁₃₇	C₂×C₄²	1	3	yes	—
84	C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
85	Dic₁₃₇	C₂×C₄²	1	3	yes	—
86	Dic₁₃₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₁₃₇	—	2	3	no	—
88	C₂×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
89	D₁₃₇	—	2	3	no	—
90	D₁₃₇	C₂×#131	1	2	yes	—
91	C₁₃₇	C₄×#131	1	1	yes	characteristic, complemented, Sylow
92	C₄×#131	—	137	1	no	maximal, Sylow
93	C₂×#131	—	137	1	no	—
94	C₂×#131	—	137	1	no	—
95	C₄×#41	—	137	1	no	—
96	#405	—	137	3	no	—
97	#405	—	137	3	no	—
98	C₄×M₁₆	—	137	3	no	—
99	C₂×C₄×C₈	—	137	3	no	—
100	C₂×#41	—	137	1	no	—
101	#131	—	137	6	no	—
102	#131	—	137	2	no	—
103	#131	—	137	1	no	—
104	#131	—	137	3	no	—
105	#131	—	137	3	no	—
106	#131	#10587739	1	1	yes	characteristic, direct product factor
107	C₂×M₁₆	—	137	3	no	—
108	C₂×M₁₆	—	137	3	no	—
109	C₂²×C₈	—	137	3	no	—
110	C₂²×C₈	—	137	3	no	—
111	C₄×Q₈	—	137	2	no	—
112	C₄×D₄	—	137	6	no	—
113	#117	—	137	3	no	—
114	C₂×C₄²	—	137	3	no	—
115	#97	—	137	6	no	—
116	C₄×C₈	—	137	1	no	—
117	C₄×C₈	—	137	3	no	—
118	C₄×C₈	—	137	3	no	—
119	C₄×C₈	—	137	3	no	—
120	#41	—	137	1	no	—
121	#41	—	137	3	no	—
122	#41	—	137	3	no	—
123	#41	C₂×#10587739	1	1	yes	characteristic
124	C₂×Q₈	—	137	1	no	—
125	C₂×D₄	—	137	3	no	—
126	C₂²×C₄	—	137	6	no	—
127	C₂²×C₄	—	137	3	no	—
128	M₁₆	—	137	6	no	—
129	M₁₆	—	137	6	no	—
130	M₁₆	—	137	3	no	—
131	M₁₆	—	137	6	no	—
132	M₁₆	C₂×#10587739	1	3	yes	complemented
133	C₂×C₈	—	137	1	no	—
134	C₂×C₈	—	137	3	no	—
135	C₂×C₈	—	137	6	no	—
136	C₂×C₈	—	137	6	no	—
137	C₂×C₈	—	137	3	no	—
138	C₂×C₈	—	137	3	no	—
139	C₂×C₈	—	137	3	no	—
140	C₂×C₈	—	137	1	no	—
141	C₂×C₈	—	137	3	no	—
142	C₂×C₈	C₂×#10587739	1	3	yes	complemented
143	C₄⋊C₄	—	137	6	no	—
144	C₂²⋊C₄	—	137	6	no	—
145	C₄²	—	137	1	no	—
146	C₄²	—	137	3	no	—
147	C₄²	—	137	6	no	—
148	C₂³	—	137	3	no	—
149	Q₈	—	137	3	no	—
150	Q₈	C₄×#10587739	1	1	yes	characteristic
151	D₄	—	137	6	no	—
152	D₄	—	137	3	no	—
153	D₄	C₄×#10587739	1	3	yes	—
154	C₂×C₄	—	274	12	no	—
155	C₂×C₄	—	137	6	no	—
156	C₂×C₄	—	137	2	no	—
157	C₂×C₄	—	137	3	no	—
158	C₂×C₄	—	137	3	no	—
159	C₂×C₄	—	137	3	no	—
160	C₂×C₄	—	137	1	no	—
161	C₂×C₄	—	137	3	no	—
162	C₂×C₄	C₂²×#10587739	1	3	yes	—
163	C₈	—	137	2	no	—
164	C₈	—	137	6	no	—
165	C₈	—	137	1	no	—
166	C₈	—	137	3	no	—
167	C₈	C₂²×#10587739	1	3	yes	—
168	C₈	C₂²×#10587739	1	1	yes	characteristic
169	C₂²	—	274	6	no	—
170	C₂²	—	137	3	no	—
171	C₂²	—	137	1	no	—
172	C₂²	C₂×C₄×#10587739	1	3	yes	—
173	C₄	—	274	6	no	—
174	C₄	—	137	4	no	—
175	C₄	—	137	1	no	—
176	C₄	—	137	3	no	—
177	C₄	C₂×C₄×#10587739	1	3	yes	—
178	C₄	C₂³×#10587739	1	1	yes	characteristic
179	C₂	—	2	3	no	—
180	C₂	—	274	3	no	—
181	C₂	—	137	2	no	—
182	C₂	C₂²×C₄×#10587739	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×#10587739
commutator, socle	C₂×C₁₃₇	C₂²×C₄²
Frattini	C₄	C₂³×#10587739
Fitting	#131×C₁₃₇	C₄
prime radical	#41×D₁₃₇	C₂²
cosocle	C₄×D₁₃₇	C₂⁴
hypercenter	#131	#10587739
hypocenter	C₁₃₇	C₄×#131
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
3. #22578369	C₂	semidirect
4. #22578366	C₂	semidirect
5. M₁₆×#10587739	C₂	semidirect
6. C₂×C₈×#10587739	C₂	semidirect
8. #17984176	C₄	semidirect
9. #17984175	C₄	semidirect
22. #17984096	C₄	semidirect
23. #17984095	C₄	semidirect
24. #17984094	C₄	semidirect
27. #131×C₁₃₇	C₄	semidirect
29. #14555520	C₂×C₄	semidirect
30. #14555519	C₂×C₄	semidirect
31. #14555518	C₂×C₄	semidirect
34. C₂×#11764047	C₂×C₄	semidirect
46. #14555494	C₂×C₄	semidirect
47. #14555493	C₂×C₄	semidirect
48. C₂×#11764040	C₂×C₄	semidirect
49. C₈×D₁₃₇	C₂×C₄	semidirect
50. C₈×D₁₃₇	C₂×C₄	semidirect
54. M₁₆×C₁₃₇	C₂×C₄	semidirect
55. C₂×C₈×C₁₃₇	C₂×C₄	semidirect
82. #10587739	#131	direct (N × H)
91. C₁₃₇	C₄×#131	semidirect
106. #131	#10587739	direct (N × H)
132. M₁₆	C₂×#10587739	semidirect
142. C₂×C₈	C₂×#10587739	semidirect

Maximal subgroups

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

subgroup	index
D₁₃₇⋊#131	2
#41×#10587739	2
#22578369	2
#22578366	2
M₁₆×#10587739	2
C₂×C₈×#10587739	2
#131×D₁₃₇	2
C₄×#131	137