Subgroups — C₁₃₇×#25815 #46875201

Order 35072.

1046 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#499×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2851×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2059×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1991×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1853×C₁₃₇	C₂	1	2	yes	maximal
7	#1852×C₁₃₇	C₂	1	2	yes	maximal
8	C₄×C₂²⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₁₃₇	—	2	8	no	—
10	C₂×#125×C₁₃₇	—	2	2	no	—
11	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₃₇	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
17	#499×C₁₃₇	—	2	8	no	—
18	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
19	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
23	#389×C₁₃₇	C₂²	1	2	yes	—
24	#387×C₁₃₇	—	2	2	no	—
25	#385×C₁₃₇	—	2	2	no	—
26	#382×C₁₃₇	C₂²	1	1	yes	characteristic
27	#380×C₁₃₇	C₂²	1	2	yes	—
28	#340×C₁₃₇	C₂²	1	4	yes	—
29	#340×C₁₃₇	C₂²	1	4	yes	—
30	#336×C₁₃₇	C₂²	1	2	yes	—
31	C₂³×C₄×C₁₃₇	—	2	2	no	—
32	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
33	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
36	#125×C₁₃₇	—	2	16	no	—
37	#125×C₁₃₇	—	2	8	no	—
38	#125×C₁₃₇	—	2	4	no	—
39	#125×C₁₃₇	—	2	4	no	—
40	#125×C₁₃₇	D₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	16	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄²×C₁₃₇	—	2	4	no	—
53	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
54	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
56	#107×C₁₃₇	—	2	8	no	—
57	#106×C₁₃₇	—	2	8	no	—
58	C₂²⋊C₈×C₁₃₇	Q₈	1	4	yes	—
59	C₂²⋊C₈×C₁₃₇	Q₈	1	4	yes	—
60	C₄×C₈×C₁₃₇	—	2	4	no	—
61	#95×C₁₃₇	—	2	2	no	—
62	#95×C₁₃₇	—	2	2	no	—
63	#95×C₁₃₇	—	2	2	no	—
64	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	—	2	2	no	—
67	C₂²×C₄×C₁₃₇	—	2	8	no	—
68	C₂²×C₄×C₁₃₇	—	2	4	no	—
69	C₂²×C₄×C₁₃₇	—	2	4	no	—
70	C₂²×C₄×C₁₃₇	—	2	4	no	—
71	C₂²×C₄×C₁₃₇	—	2	2	no	—
72	C₂²×C₄×C₁₃₇	—	2	2	no	—
73	C₂²×C₄×C₁₃₇	—	4	2	no	—
74	C₂²×C₄×C₁₃₇	—	4	4	no	—
75	C₂²×C₄×C₁₃₇	—	4	4	no	—
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₁₃₇	—	2	2	no	—
83	C₂×C₈×C₁₃₇	—	2	2	no	—
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	—	2	2	no	—
86	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
87	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
88	C₄⋊C₄×C₁₃₇	—	2	8	no	—
89	C₄⋊C₄×C₁₃₇	—	2	16	no	—
90	C₄⋊C₄×C₁₃₇	—	2	16	no	—
91	C₄⋊C₄×C₁₃₇	—	2	4	no	—
92	C₄⋊C₄×C₁₃₇	—	2	4	no	—
93	C₄⋊C₄×C₁₃₇	—	2	4	no	—
94	C₄⋊C₄×C₁₃₇	—	4	4	no	—
95	C₄⋊C₄×C₁₃₇	—	4	4	no	—
96	C₄⋊C₄×C₁₃₇	C₂×D₄	1	8	yes	—
97	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
98	C₄²×C₁₃₇	—	2	4	no	—
99	C₄²×C₁₃₇	—	2	2	no	—
100	C₄²×C₁₃₇	C₂×D₄	1	4	yes	—
101	C₂³×C₁₃₇	—	2	1	no	—
102	C₂³×C₁₃₇	—	2	1	no	—
103	C₂³×C₁₃₇	—	2	2	no	—
104	C₂³×C₁₃₇	C₄⋊Q₈	1	2	yes	—
105	C₂³×C₁₃₇	#135	1	4	yes	—
106	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	—	2	1	no	—
108	C₂×C₄×C₁₃₇	—	2	1	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	2	4	no	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	2	8	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	4	8	no	—
118	C₂×C₄×C₁₃₇	—	4	16	no	—
119	C₂×C₄×C₁₃₇	—	4	4	no	—
120	C₂×C₄×C₁₃₇	—	4	4	no	—
121	C₂×C₄×C₁₃₇	—	4	2	no	—
122	C₂×C₄×C₁₃₇	—	4	2	no	—
123	C₂×C₄×C₁₃₇	—	4	2	no	—
124	C₂×C₄×C₁₃₇	—	4	2	no	—
125	C₂×C₄×C₁₃₇	—	4	2	no	—
126	C₂×C₄×C₁₃₇	—	4	2	no	—
127	C₂×C₄×C₁₃₇	#120	1	4	yes	—
128	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
133	C₈×C₁₃₇	—	2	4	no	—
134	C₈×C₁₃₇	—	2	4	no	—
135	C₂²×C₁₃₇	—	2	2	no	—
136	C₂²×C₁₃₇	—	2	2	no	—
137	C₂²×C₁₃₇	—	2	4	no	—
138	C₂²×C₁₃₇	—	2	4	no	—
139	C₂²×C₁₃₇	#499	1	4	yes	complemented
140	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
141	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₂²×C₁₃₇	#536	1	1	yes	characteristic
143	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
144	C₂²×C₁₃₇	#575	1	2	yes	—
145	C₄×C₁₃₇	—	2	2	no	—
146	C₄×C₁₃₇	—	2	4	no	—
147	C₄×C₁₃₇	—	4	2	no	—
148	C₄×C₁₃₇	—	4	8	no	—
149	C₄×C₁₃₇	—	8	2	no	—
150	C₄×C₁₃₇	—	8	2	no	—
151	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
152	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
153	C₂×C₁₃₇	—	2	4	no	—
154	C₂×C₁₃₇	#2434	1	1	yes	characteristic
155	C₂×C₁₃₇	#3002	1	2	yes	—
156	C₂×C₁₃₇	C₂×#499	1	1	yes	characteristic
157	C₂×C₁₃₇	#3166	1	1	yes	characteristic
158	C₂×C₁₃₇	#3232	1	2	yes	—
159	#25815	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₁₃₇	#25815	1	1	yes	characteristic, direct product factor, Sylow
161	C₂²×#125	C₂×C₁₃₇	1	1	yes	characteristic
162	C₂×#499	C₂×C₁₃₇	1	2	yes	complemented
163	#2851	C₂×C₁₃₇	1	1	yes	characteristic
164	#2059	C₂×C₁₃₇	1	4	yes	complemented
165	#1991	C₂×C₁₃₇	1	2	yes	complemented
166	#1853	C₂×C₁₃₇	1	2	yes	—
167	#1852	C₂×C₁₃₇	1	2	yes	—
168	C₄×C₂²⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic
169	C₂×#125	—	2	8	no	—
170	C₂×#125	—	2	2	no	—
171	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
172	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
173	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
174	C₂²×C₄⋊C₄	—	2	2	no	—
175	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
176	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
177	#499	—	2	8	no	—
178	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
179	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
180	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
181	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
183	#389	C₂²×C₁₃₇	1	2	yes	—
184	#387	—	2	2	no	—
185	#385	—	2	2	no	—
186	#382	C₂²×C₁₃₇	1	1	yes	characteristic
187	#380	C₂²×C₁₃₇	1	2	yes	—
188	#340	C₂²×C₁₃₇	1	4	yes	—
189	#340	C₂²×C₁₃₇	1	4	yes	—
190	#336	C₂²×C₁₃₇	1	2	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
193	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
196	#125	—	2	16	no	—
197	#125	—	2	8	no	—
198	#125	—	2	4	no	—
199	#125	—	2	4	no	—
200	#125	D₄×C₁₃₇	1	8	yes	—
201	C₂×C₄⋊C₄	—	2	16	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	8	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
214	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
216	#107	—	2	8	no	—
217	#106	—	2	8	no	—
218	C₂²⋊C₈	Q₈×C₁₃₇	1	4	yes	—
219	C₂²⋊C₈	Q₈×C₁₃₇	1	4	yes	—
220	C₄×C₈	—	2	4	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
240	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
247	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	16	no	—
250	C₄⋊C₄	—	2	16	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	C₂×D₄×C₁₃₇	1	8	yes	—
257	C₂²⋊C₄	—	4	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	2	no	—
260	C₄²	C₂×D₄×C₁₃₇	1	4	yes	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₄⋊Q₈×C₁₃₇	1	2	yes	—
265	C₂³	#135×C₁₃₇	1	4	yes	—
266	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	16	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	#120×C₁₃₇	1	4	yes	—
288	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
289	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
290	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
292	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	#499×C₁₃₇	1	4	yes	complemented
300	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
301	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
302	C₂²	#536×C₁₃₇	1	1	yes	characteristic
303	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
304	C₂²	#575×C₁₃₇	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	8	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
312	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
313	C₂	—	2	4	no	—
314	C₂	#2434×C₁₃₇	1	1	yes	characteristic
315	C₂	#3002×C₁₃₇	1	2	yes	—
316	C₂	C₂×#499×C₁₃₇	1	1	yes	characteristic
317	C₂	#3166×C₁₃₇	1	1	yes	characteristic
318	C₂	#3232×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#499×C₁₃₇	C₂	semidirect
4. #2059×C₁₃₇	C₂	semidirect
5. #1991×C₁₃₇	C₂	semidirect
139. C₂²×C₁₃₇	#499	semidirect
159. #25815	C₁₃₇	direct (N × H)
160. C₁₃₇	#25815	direct (N × H)
162. C₂×#499	C₂×C₁₃₇	semidirect
164. #2059	C₂×C₁₃₇	semidirect
165. #1991	C₂×C₁₃₇	semidirect
299. C₂²	#499×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#125×C₁₃₇	2
C₂×#499×C₁₃₇	2
#2851×C₁₃₇	2
#2059×C₁₃₇	2
#1991×C₁₃₇	2
#1853×C₁₃₇	2
#1852×C₁₃₇	2
C₄×C₂²⋊C₈×C₁₃₇	2
#25815	137

Subgroups — C137×#25815 #46875201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25815 #46875201