Subgroups — C₁₃₇×#12686 #46862072

Order 35072.

904 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	C₂×#568×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2117×C₁₃₇	C₂	1	4	yes	maximal, complemented
3	C₂×#349×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	C₂²×#41×C₁₃₇	C₄	1	1	yes	characteristic
5	#568×C₁₃₇	—	2	8	no	—
6	C₂×#131×C₁₃₇	—	2	2	no	—
7	C₂×#131×C₁₃₇	C₄	1	2	yes	—
8	C₂×#131×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₁₃₇	—	2	2	no	—
10	C₂²×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₁₃₇	C₂²	1	4	yes	complemented
13	#350×C₁₃₇	—	2	8	no	—
14	#349×C₁₃₇	—	2	2	no	—
15	#349×C₁₃₇	—	2	2	no	—
16	C₂×#41×C₁₃₇	—	2	4	no	—
17	C₂×#41×C₁₃₇	—	2	4	no	—
18	C₂×#41×C₁₃₇	—	2	1	no	—
19	C₂×#41×C₁₃₇	—	2	1	no	—
20	C₂×#41×C₁₃₇	C₈	1	2	yes	—
21	C₂×#41×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₂²×Q₈×C₁₃₇	C₈	1	1	yes	characteristic
23	C₂²×D₄×C₁₃₇	—	2	1	no	—
24	C₂²×D₄×C₁₃₇	C₈	1	1	yes	characteristic
25	C₂³×C₄×C₁₃₇	—	2	1	no	—
26	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	#131×C₁₃₇	—	2	16	no	—
28	#131×C₁₃₇	—	2	4	no	—
29	#131×C₁₃₇	—	2	4	no	—
30	#131×C₁₃₇	—	2	4	no	—
31	#131×C₁₃₇	D₄	1	8	yes	—
32	C₂×M₁₆×C₁₃₇	—	2	16	no	—
33	C₂×M₁₆×C₁₃₇	—	2	1	no	—
34	C₂×M₁₆×C₁₃₇	—	2	4	no	—
35	C₂×M₁₆×C₁₃₇	—	2	2	no	—
36	C₂×M₁₆×C₁₃₇	—	2	2	no	—
37	C₂×M₁₆×C₁₃₇	—	2	4	no	—
38	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
40	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
41	C₂²×C₈×C₁₃₇	—	2	1	no	—
42	C₂²×C₈×C₁₃₇	—	2	2	no	—
43	C₂²×C₈×C₁₃₇	—	2	2	no	—
44	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	M₃₂×C₁₃₇	—	2	4	no	—
47	M₃₂×C₁₃₇	—	2	4	no	—
48	C₂×C₁₆×C₁₃₇	—	2	4	no	—
49	C₂⁴×C₁₃₇	—	2	1	no	—
50	C₂⁴×C₁₃₇	M₁₆	1	1	yes	characteristic
51	#41×C₁₃₇	—	2	4	no	—
52	#41×C₁₃₇	—	2	8	no	—
53	#41×C₁₃₇	—	2	4	no	—
54	#41×C₁₃₇	—	2	4	no	—
55	#41×C₁₃₇	—	2	4	no	—
56	#41×C₁₃₇	—	2	2	no	—
57	#41×C₁₃₇	—	2	2	no	—
58	#41×C₁₃₇	C₂²⋊C₄	1	4	yes	—
59	#41×C₁₃₇	M₁₆	1	4	yes	—
60	C₂×Q₈×C₁₃₇	—	2	4	no	—
61	C₂×Q₈×C₁₃₇	—	2	1	no	—
62	C₂×Q₈×C₁₃₇	C₂×C₈	1	2	yes	—
63	C₂×D₄×C₁₃₇	—	2	8	no	—
64	C₂×D₄×C₁₃₇	—	2	2	no	—
65	C₂×D₄×C₁₃₇	—	2	2	no	—
66	C₂×D₄×C₁₃₇	—	2	4	no	—
67	C₂×D₄×C₁₃₇	—	2	1	no	—
68	C₂×D₄×C₁₃₇	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₁₃₇	—	2	4	no	—
70	C₂²×C₄×C₁₃₇	—	2	4	no	—
71	C₂²×C₄×C₁₃₇	—	2	1	no	—
72	C₂²×C₄×C₁₃₇	—	2	2	no	—
73	C₂²×C₄×C₁₃₇	—	2	2	no	—
74	C₂²×C₄×C₁₃₇	—	2	2	no	—
75	C₂²×C₄×C₁₃₇	—	2	2	no	—
76	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
82	M₁₆×C₁₃₇	—	2	4	no	—
83	M₁₆×C₁₃₇	—	2	8	no	—
84	M₁₆×C₁₃₇	—	2	2	no	—
85	M₁₆×C₁₃₇	—	2	4	no	—
86	M₁₆×C₁₃₇	—	2	8	no	—
87	M₁₆×C₁₃₇	—	2	4	no	—
88	M₁₆×C₁₃₇	C₂²⋊C₄	1	4	yes	—
89	C₂×C₈×C₁₃₇	—	2	4	no	—
90	C₂×C₈×C₁₃₇	—	2	8	no	—
91	C₂×C₈×C₁₃₇	—	2	2	no	—
92	C₂×C₈×C₁₃₇	—	2	2	no	—
93	C₂×C₈×C₁₃₇	—	2	8	no	—
94	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
97	C₁₆×C₁₃₇	—	4	4	no	—
98	C₂³×C₁₃₇	—	2	4	no	—
99	C₂³×C₁₃₇	—	2	2	no	—
100	C₂³×C₁₃₇	—	2	2	no	—
101	C₂³×C₁₃₇	—	8	1	no	—
102	C₂³×C₁₃₇	—	8	2	no	—
103	C₂³×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
104	C₂³×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
105	C₂³×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
106	Q₈×C₁₃₇	—	2	4	no	—
107	Q₈×C₁₃₇	—	2	2	no	—
108	Q₈×C₁₃₇	C₂²⋊C₈	1	4	yes	—
109	D₄×C₁₃₇	—	2	8	no	—
110	D₄×C₁₃₇	—	2	4	no	—
111	D₄×C₁₃₇	—	2	4	no	—
112	D₄×C₁₃₇	—	2	4	no	—
113	D₄×C₁₃₇	—	2	2	no	—
114	D₄×C₁₃₇	C₂²⋊C₈	1	4	yes	—
115	C₂×C₄×C₁₃₇	—	2	4	no	—
116	C₂×C₄×C₁₃₇	—	2	8	no	—
117	C₂×C₄×C₁₃₇	—	2	2	no	—
118	C₂×C₄×C₁₃₇	—	2	2	no	—
119	C₂×C₄×C₁₃₇	—	2	4	no	—
120	C₂×C₄×C₁₃₇	—	2	1	no	—
121	C₂×C₄×C₁₃₇	—	2	1	no	—
122	C₂×C₄×C₁₃₇	—	2	2	no	—
123	C₂×C₄×C₁₃₇	—	2	2	no	—
124	C₂×C₄×C₁₃₇	—	2	1	no	—
125	C₂×C₄×C₁₃₇	—	2	4	no	—
126	C₂×C₄×C₁₃₇	—	2	1	no	—
127	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
133	C₈×C₁₃₇	—	2	4	no	—
134	C₈×C₁₃₇	—	2	2	no	—
135	C₈×C₁₃₇	#120	1	4	yes	—
136	C₂²×C₁₃₇	—	2	1	no	—
137	C₂²×C₁₃₇	—	2	1	no	—
138	C₂²×C₁₃₇	—	2	4	no	—
139	C₂²×C₁₃₇	—	4	1	no	—
140	C₂²×C₁₃₇	—	4	1	no	—
141	C₂²×C₁₃₇	—	4	1	no	—
142	C₂²×C₁₃₇	—	4	4	no	—
143	C₂²×C₁₃₇	—	8	2	no	—
144	C₂²×C₁₃₇	—	8	4	no	—
145	C₂²×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×C₁₃₇	#407	1	2	yes	—
147	C₄×C₁₃₇	—	2	1	no	—
148	C₄×C₁₃₇	—	2	4	no	—
149	C₄×C₁₃₇	—	2	1	no	—
150	C₄×C₁₃₇	#379	1	1	yes	characteristic
151	C₄×C₁₃₇	C₂×C₂²⋊C₈	1	2	yes	—
152	C₄×C₁₃₇	#407	1	1	yes	characteristic
153	C₂×C₁₃₇	—	2	1	no	—
154	C₂×C₁₃₇	—	2	2	no	—
155	C₂×C₁₃₇	—	4	4	no	—
156	C₂×C₁₃₇	—	4	1	no	—
157	C₂×C₁₃₇	—	4	1	no	—
158	C₂×C₁₃₇	#1779	1	1	yes	characteristic
159	#12686	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₁₃₇	#12686	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×#568	C₂×C₁₃₇	1	1	yes	characteristic
162	#2117	C₂×C₁₃₇	1	4	yes	complemented
163	C₂×#349	C₂×C₁₃₇	1	2	yes	complemented
164	C₂²×#41	C₄×C₁₃₇	1	1	yes	characteristic
165	#568	—	2	8	no	—
166	C₂×#131	—	2	2	no	—
167	C₂×#131	C₄×C₁₃₇	1	2	yes	—
168	C₂×#131	C₂²×C₁₃₇	1	2	yes	—
169	C₂²×M₁₆	—	2	2	no	—
170	C₂²×M₁₆	C₄×C₁₃₇	1	1	yes	characteristic
171	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂×M₃₂	C₂²×C₁₃₇	1	4	yes	complemented
173	#350	—	2	8	no	—
174	#349	—	2	2	no	—
175	#349	—	2	2	no	—
176	C₂×#41	—	2	4	no	—
177	C₂×#41	—	2	4	no	—
178	C₂×#41	—	2	1	no	—
179	C₂×#41	—	2	1	no	—
180	C₂×#41	C₈×C₁₃₇	1	2	yes	—
181	C₂×#41	C₂×C₄×C₁₃₇	1	2	yes	—
182	C₂²×Q₈	C₈×C₁₃₇	1	1	yes	characteristic
183	C₂²×D₄	—	2	1	no	—
184	C₂²×D₄	C₈×C₁₃₇	1	1	yes	characteristic
185	C₂³×C₄	—	2	1	no	—
186	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
187	#131	—	2	16	no	—
188	#131	—	2	4	no	—
189	#131	—	2	4	no	—
190	#131	—	2	4	no	—
191	#131	D₄×C₁₃₇	1	8	yes	—
192	C₂×M₁₆	—	2	16	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	2	4	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
199	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
200	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
206	M₃₂	—	2	4	no	—
207	M₃₂	—	2	4	no	—
208	C₂×C₁₆	—	2	4	no	—
209	C₂⁴	—	2	1	no	—
210	C₂⁴	M₁₆×C₁₃₇	1	1	yes	characteristic
211	#41	—	2	4	no	—
212	#41	—	2	8	no	—
213	#41	—	2	4	no	—
214	#41	—	2	4	no	—
215	#41	—	2	4	no	—
216	#41	—	2	2	no	—
217	#41	—	2	2	no	—
218	#41	C₂²⋊C₄×C₁₃₇	1	4	yes	—
219	#41	M₁₆×C₁₃₇	1	4	yes	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	C₂×C₈×C₁₃₇	1	2	yes	—
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	C₂×C₈×C₁₃₇	1	2	yes	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
240	C₂²×C₄	M₁₆×C₁₃₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	8	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	2	4	no	—
246	M₁₆	—	2	8	no	—
247	M₁₆	—	2	4	no	—
248	M₁₆	C₂²⋊C₄×C₁₃₇	1	4	yes	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	8	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	8	no	—
254	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
255	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
256	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
257	C₁₆	—	4	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	8	1	no	—
262	C₂³	—	8	2	no	—
263	C₂³	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
264	C₂³	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
265	C₂³	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
266	Q₈	—	2	4	no	—
267	Q₈	—	2	2	no	—
268	Q₈	C₂²⋊C₈×C₁₃₇	1	4	yes	—
269	D₄	—	2	8	no	—
270	D₄	—	2	4	no	—
271	D₄	—	2	4	no	—
272	D₄	—	2	4	no	—
273	D₄	—	2	2	no	—
274	D₄	C₂²⋊C₈×C₁₃₇	1	4	yes	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
288	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
290	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
292	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	2	2	no	—
295	C₈	#120×C₁₃₇	1	4	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	4	no	—
305	C₂²	C₂×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
306	C₂²	#407×C₁₃₇	1	2	yes	—
307	C₄	—	2	1	no	—
308	C₄	—	2	4	no	—
309	C₄	—	2	1	no	—
310	C₄	#379×C₁₃₇	1	1	yes	characteristic
311	C₄	C₂×C₂²⋊C₈×C₁₃₇	1	2	yes	—
312	C₄	#407×C₁₃₇	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	4	4	no	—
316	C₂	—	4	1	no	—
317	C₂	—	4	1	no	—
318	C₂	#1779×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	#379
commutator	C₂×C₄	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂×C₁₃₇	#1779
prime radical	C₂²×#41×C₁₃₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂²×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2117×C₁₃₇	C₂	semidirect
3. C₂×#349×C₁₃₇	C₂	semidirect
12. C₂×M₃₂×C₁₃₇	C₂²	semidirect
159. #12686	C₁₃₇	direct (N × H)
160. C₁₃₇	#12686	direct (N × H)
162. #2117	C₂×C₁₃₇	semidirect
163. C₂×#349	C₂×C₁₃₇	semidirect
172. C₂×M₃₂	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568×C₁₃₇	2
#2117×C₁₃₇	2
C₂×#349×C₁₃₇	2
#12686	137

Subgroups — C137×#12686 #46862072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#12686 #46862072