Subgroups — C₁₃₇×#13059 #46862445

Order 35072.

578 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#466×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#357×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1808×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1380×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×#121×C₁₃₇	—	2	1	no	—
9	C₂²×C₁₆×C₁₃₇	—	2	2	no	—
10	C₂²×C₁₆×C₁₃₇	—	2	4	no	—
11	C₂²×C₁₆×C₁₃₇	C₂²	1	2	yes	—
12	#466×C₁₃₇	—	2	2	no	—
13	#466×C₁₃₇	C₄	1	4	yes	complemented
14	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#102×C₁₃₇	—	2	1	no	—
17	#380×C₁₃₇	—	2	1	no	—
18	#357×C₁₃₇	—	2	4	no	—
19	#340×C₁₃₇	—	2	1	no	—
20	C₂²×D₄×C₁₃₇	—	2	1	no	—
21	C₂²×D₄×C₁₃₇	—	4	1	no	—
22	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
23	C₂×D₈×C₁₃₇	—	2	2	no	—
24	C₂×D₈×C₁₃₇	—	2	1	no	—
25	C₂×D₈×C₁₃₇	—	2	2	no	—
26	C₂×D₈×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃₇	—	2	2	no	—
28	C₂²×C₈×C₁₃₇	—	2	2	no	—
29	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	#121×C₁₃₇	—	2	4	no	—
34	#121×C₁₃₇	—	4	2	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₁₃₇	—	4	1	no	—
38	C₂×C₂²⋊C₄×C₁₃₇	—	4	1	no	—
39	C₂×C₁₆×C₁₃₇	—	2	8	no	—
40	C₂×C₁₆×C₁₃₇	—	2	2	no	—
41	C₂×C₁₆×C₁₃₇	—	2	2	no	—
42	C₂×C₁₆×C₁₃₇	D₄	1	4	yes	complemented
43	#107×C₁₃₇	—	2	1	no	—
44	#107×C₁₃₇	—	2	2	no	—
45	#107×C₁₃₇	—	2	2	no	—
46	#107×C₁₃₇	C₂×C₄	1	2	yes	—
47	#102×C₁₃₇	—	2	4	no	—
48	#95×C₁₃₇	—	4	1	no	—
49	C₂⁴×C₁₃₇	—	4	1	no	—
50	C₂⁴×C₁₃₇	D₈	1	1	yes	characteristic
51	C₂×D₄×C₁₃₇	—	2	2	no	—
52	C₂×D₄×C₁₃₇	—	4	4	no	—
53	C₂×D₄×C₁₃₇	—	4	2	no	—
54	C₂×D₄×C₁₃₇	—	4	1	no	—
55	C₂×D₄×C₁₃₇	—	4	2	no	—
56	C₂×D₄×C₁₃₇	—	8	2	no	—
57	C₂×D₄×C₁₃₇	—	8	2	no	—
58	C₂²×C₄×C₁₃₇	—	2	2	no	—
59	C₂²×C₄×C₁₃₇	—	2	2	no	—
60	C₂²×C₄×C₁₃₇	—	4	1	no	—
61	C₂²×C₄×C₁₃₇	—	4	2	no	—
62	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	D₈×C₁₃₇	—	2	4	no	—
67	D₈×C₁₃₇	—	4	1	no	—
68	D₈×C₁₃₇	—	4	1	no	—
69	C₂×C₈×C₁₃₇	—	2	2	no	—
70	C₂×C₈×C₁₃₇	—	2	4	no	—
71	C₂×C₈×C₁₃₇	—	2	2	no	—
72	C₂×C₈×C₁₃₇	—	2	2	no	—
73	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃₇	#41	1	2	yes	—
78	C₄⋊C₄×C₁₃₇	—	2	2	no	—
79	C₄⋊C₄×C₁₃₇	—	4	1	no	—
80	C₄⋊C₄×C₁₃₇	—	4	2	no	—
81	C₄⋊C₄×C₁₃₇	—	4	2	no	—
82	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
83	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
84	C₁₆×C₁₃₇	—	2	4	no	—
85	C₂³×C₁₃₇	—	2	2	no	—
86	C₂³×C₁₃₇	—	2	2	no	—
87	C₂³×C₁₃₇	—	4	2	no	—
88	C₂³×C₁₃₇	—	8	2	no	—
89	C₂³×C₁₃₇	—	8	1	no	—
90	C₂³×C₁₃₇	—	8	1	no	—
91	C₂³×C₁₃₇	—	8	2	no	—
92	C₂³×C₁₃₇	#102	1	2	yes	—
93	C₂³×C₁₃₇	D₁₆	1	2	yes	—
94	C₂³×C₁₃₇	SD₃₂	1	2	yes	—
95	C₂³×C₁₃₇	C₂×D₈	1	1	yes	characteristic
96	D₄×C₁₃₇	—	4	2	no	—
97	D₄×C₁₃₇	—	4	2	no	—
98	D₄×C₁₃₇	—	8	8	no	—
99	D₄×C₁₃₇	—	8	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	4	2	no	—
105	C₂×C₄×C₁₃₇	—	8	1	no	—
106	C₂×C₄×C₁₃₇	—	8	1	no	—
107	C₂×C₄×C₁₃₇	—	8	2	no	—
108	C₂×C₄×C₁₃₇	—	8	2	no	—
109	C₂×C₄×C₁₃₇	—	8	2	no	—
110	C₂×C₄×C₁₃₇	#102	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	#121	1	2	yes	—
113	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	#135	1	2	yes	—
115	C₈×C₁₃₇	—	2	2	no	—
116	C₈×C₁₃₇	C₄×D₄	1	2	yes	—
117	C₈×C₁₃₇	#120	1	1	yes	characteristic
118	C₈×C₁₃₇	#123	1	1	yes	characteristic
119	C₂²×C₁₃₇	—	2	2	no	—
120	C₂²×C₁₃₇	—	2	2	no	—
121	C₂²×C₁₃₇	—	2	4	no	—
122	C₂²×C₁₃₇	—	2	4	no	—
123	C₂²×C₁₃₇	—	8	2	no	—
124	C₂²×C₁₃₇	—	8	2	no	—
125	C₂²×C₁₃₇	—	8	2	no	—
126	C₂²×C₁₃₇	—	16	2	no	—
127	C₂²×C₁₃₇	—	16	2	no	—
128	C₂²×C₁₃₇	#357	1	4	yes	complemented
129	C₂²×C₁₃₇	C₂×#102	1	1	yes	characteristic
130	C₂²×C₁₃₇	#466	1	2	yes	—
131	C₂²×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
132	C₂²×C₁₃₇	C₂×SD₃₂	1	1	yes	characteristic
133	C₂²×C₁₃₇	#508	1	2	yes	—
134	C₄×C₁₃₇	—	2	2	no	—
135	C₄×C₁₃₇	—	8	2	no	—
136	C₄×C₁₃₇	—	16	2	no	—
137	C₄×C₁₃₇	#386	1	1	yes	characteristic
138	C₄×C₁₃₇	#416	1	1	yes	characteristic
139	C₄×C₁₃₇	#465	1	2	yes	—
140	C₂×C₁₃₇	—	2	4	no	—
141	C₂×C₁₃₇	—	16	2	no	—
142	C₂×C₁₃₇	#1893	1	1	yes	characteristic
143	C₂×C₁₃₇	C₂×#357	1	1	yes	characteristic
144	C₂×C₁₃₇	#2138	1	1	yes	characteristic
145	C₂×C₁₃₇	#2215	1	2	yes	—
146	C₂×C₁₃₇	#2217	1	2	yes	—
147	#13059	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₁₃₇	#13059	1	1	yes	characteristic, direct product factor, Sylow
149	C₂³×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
150	C₂×#466	C₂×C₁₃₇	1	1	yes	characteristic
151	C₂×#357	C₂×C₁₃₇	1	2	yes	complemented
152	#1808	C₂×C₁₃₇	1	1	yes	characteristic, complemented
153	#1380	C₂×C₁₃₇	1	2	yes	complemented
154	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
156	C₂×#121	—	2	1	no	—
157	C₂²×C₁₆	—	2	2	no	—
158	C₂²×C₁₆	—	2	4	no	—
159	C₂²×C₁₆	C₂²×C₁₃₇	1	2	yes	—
160	#466	—	2	2	no	—
161	#466	C₄×C₁₃₇	1	4	yes	complemented
162	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
163	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
164	C₂×#102	—	2	1	no	—
165	#380	—	2	1	no	—
166	#357	—	2	4	no	—
167	#340	—	2	1	no	—
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	—	4	1	no	—
170	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
171	C₂×D₈	—	2	2	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	C₂×C₄×C₁₃₇	1	2	yes	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
178	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
180	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
181	#121	—	2	4	no	—
182	#121	—	4	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	4	1	no	—
186	C₂×C₂²⋊C₄	—	4	1	no	—
187	C₂×C₁₆	—	2	8	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	D₄×C₁₃₇	1	4	yes	complemented
191	#107	—	2	1	no	—
192	#107	—	2	2	no	—
193	#107	—	2	2	no	—
194	#107	C₂×C₄×C₁₃₇	1	2	yes	—
195	#102	—	2	4	no	—
196	#95	—	4	1	no	—
197	C₂⁴	—	4	1	no	—
198	C₂⁴	D₈×C₁₃₇	1	1	yes	characteristic
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	4	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	8	2	no	—
205	C₂×D₄	—	8	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
211	C₂²×C₄	D₈×C₁₃₇	1	1	yes	characteristic
212	C₂²×C₄	SD₁₆×C₁₃₇	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
214	D₈	—	2	4	no	—
215	D₈	—	4	1	no	—
216	D₈	—	4	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
222	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
224	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
225	C₂×C₈	#41×C₁₃₇	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	8	2	no	—
232	C₁₆	—	2	4	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	1	no	—
239	C₂³	—	8	2	no	—
240	C₂³	#102×C₁₃₇	1	2	yes	—
241	C₂³	D₁₆×C₁₃₇	1	2	yes	—
242	C₂³	SD₃₂×C₁₃₇	1	2	yes	—
243	C₂³	C₂×D₈×C₁₃₇	1	1	yes	characteristic
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	8	8	no	—
247	D₄	—	8	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	#102×C₁₃₇	1	2	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
260	C₂×C₄	#121×C₁₃₇	1	2	yes	—
261	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
262	C₂×C₄	#135×C₁₃₇	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	C₄×D₄×C₁₃₇	1	2	yes	—
265	C₈	#120×C₁₃₇	1	1	yes	characteristic
266	C₈	#123×C₁₃₇	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	#357×C₁₃₇	1	4	yes	complemented
277	C₂²	C₂×#102×C₁₃₇	1	1	yes	characteristic
278	C₂²	#466×C₁₃₇	1	2	yes	—
279	C₂²	C₂×D₁₆×C₁₃₇	1	1	yes	characteristic
280	C₂²	C₂×SD₃₂×C₁₃₇	1	1	yes	characteristic
281	C₂²	#508×C₁₃₇	1	2	yes	—
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	16	2	no	—
285	C₄	#386×C₁₃₇	1	1	yes	characteristic
286	C₄	#416×C₁₃₇	1	1	yes	characteristic
287	C₄	#465×C₁₃₇	1	2	yes	—
288	C₂	—	2	4	no	—
289	C₂	—	16	2	no	—
290	C₂	#1893×C₁₃₇	1	1	yes	characteristic
291	C₂	C₂×#357×C₁₃₇	1	1	yes	characteristic
292	C₂	#2138×C₁₃₇	1	1	yes	characteristic
293	C₂	#2215×C₁₃₇	1	2	yes	—
294	C₂	#2217×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
prime radical	C₂×#466×C₁₃₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ C₂³×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₆×C₁₃₇	C₂	semidirect
3. C₂×#357×C₁₃₇	C₂	semidirect
4. #1808×C₁₃₇	C₂	semidirect
5. #1380×C₁₃₇	C₂	semidirect
13. #466×C₁₃₇	C₄	semidirect
42. C₂×C₁₆×C₁₃₇	D₄	semidirect
128. C₂²×C₁₃₇	#357	semidirect
147. #13059	C₁₃₇	direct (N × H)
148. C₁₃₇	#13059	direct (N × H)
149. C₂³×C₁₆	C₂×C₁₃₇	semidirect
151. C₂×#357	C₂×C₁₃₇	semidirect
152. #1808	C₂×C₁₃₇	semidirect
153. #1380	C₂×C₁₃₇	semidirect
161. #466	C₄×C₁₃₇	semidirect
190. C₂×C₁₆	D₄×C₁₃₇	semidirect
276. C₂²	#357×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₁₃₇	2
C₂×#466×C₁₃₇	2
C₂×#357×C₁₃₇	2
#1808×C₁₃₇	2
#1380×C₁₃₇	2
#13059	137

Subgroups — C137×#13059 #46862445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13059 #46862445