Subgroups — C₁₃₇×#8186 #46857572

Order 35072.

498 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#131×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1476×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1476×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1452×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1278×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1278×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₃₇	C₄	1	2	yes	—
9	C₄×#41×C₁₃₇	C₂²	1	1	yes	characteristic
10	#431×C₁₃₇	C₂²	1	1	yes	characteristic
11	#431×C₁₃₇	C₂²	1	1	yes	characteristic
12	#405×C₁₃₇	—	2	2	no	—
13	#405×C₁₃₇	C₄	1	2	yes	—
14	C₄×M₁₆×C₁₃₇	—	2	1	no	—
15	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₇	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	Q₈⋊C₈×C₁₃₇	C₄	1	2	yes	—
21	Q₈⋊C₈×C₁₃₇	C₄	1	2	yes	—
22	D₄⋊C₈×C₁₃₇	C₄	1	2	yes	—
23	D₄⋊C₈×C₁₃₇	C₄	1	2	yes	—
24	#322×C₁₃₇	—	2	2	no	—
25	#322×C₁₃₇	—	2	1	no	—
26	#322×C₁₃₇	—	2	1	no	—
27	C₂×#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	#131×C₁₃₇	—	2	4	no	—
29	#131×C₁₃₇	—	2	4	no	—
30	C₂×M₁₆×C₁₃₇	—	2	2	no	—
31	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃₇	—	2	1	no	—
33	C₂²×C₈×C₁₃₇	—	2	2	no	—
34	C₂²×C₈×C₁₃₇	—	2	1	no	—
35	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₄×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₁₃₇	—	2	1	no	—
39	C₄×D₄×C₁₃₇	—	2	1	no	—
40	C₄×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₄×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	#117×C₁₃₇	—	2	1	no	—
43	#117×C₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₇	—	2	1	no	—
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₁₃₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₈×C₁₃₇	C₂×C₄	1	2	yes	—
48	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
49	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
50	#97×C₁₃₇	—	2	2	no	—
51	#97×C₁₃₇	—	2	1	no	—
52	C₄×C₈×C₁₃₇	—	2	2	no	—
53	C₄×C₈×C₁₃₇	—	2	1	no	—
54	C₄×C₈×C₁₃₇	—	2	1	no	—
55	C₄×C₈×C₁₃₇	—	2	1	no	—
56	C₄×C₈×C₁₃₇	—	2	1	no	—
57	C₄×C₈×C₁₃₇	—	2	1	no	—
58	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
61	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
63	#41×C₁₃₇	—	2	2	no	—
64	#41×C₁₃₇	M₁₆	1	4	yes	—
65	C₂×Q₈×C₁₃₇	C₄²	1	1	yes	characteristic
66	C₂×D₄×C₁₃₇	—	2	1	no	—
67	C₂×D₄×C₁₃₇	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	—	2	1	no	—
69	C₂²×C₄×C₁₃₇	—	2	1	no	—
70	C₂²×C₄×C₁₃₇	—	2	1	no	—
71	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
74	M₁₆×C₁₃₇	—	2	8	no	—
75	M₁₆×C₁₃₇	—	2	2	no	—
76	M₁₆×C₁₃₇	M₁₆	1	4	yes	—
77	C₂×C₈×C₁₃₇	—	2	4	no	—
78	C₂×C₈×C₁₃₇	—	2	8	no	—
79	C₂×C₈×C₁₃₇	—	2	2	no	—
80	C₂×C₈×C₁₃₇	—	2	2	no	—
81	C₂×C₈×C₁₃₇	—	2	2	no	—
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	—	4	2	no	—
87	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₁₃₇	—	2	1	no	—
90	C₄⋊C₄×C₁₃₇	—	2	1	no	—
91	C₄⋊C₄×C₁₃₇	C₄²	1	1	yes	characteristic
92	C₄⋊C₄×C₁₃₇	C₄²	1	1	yes	characteristic
93	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
94	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
95	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
96	C₄²×C₁₃₇	—	2	1	no	—
97	C₄²×C₁₃₇	—	2	1	no	—
98	C₄²×C₁₃₇	—	2	1	no	—
99	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₁₃₇	—	2	1	no	—
104	C₂³×C₁₃₇	#117	1	1	yes	characteristic
105	Q₈×C₁₃₇	—	2	1	no	—
106	Q₈×C₁₃₇	#97	1	2	yes	—
107	D₄×C₁₃₇	—	2	4	no	—
108	D₄×C₁₃₇	—	2	1	no	—
109	D₄×C₁₃₇	#97	1	2	yes	—
110	C₂×C₄×C₁₃₇	—	2	1	no	—
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	2	2	no	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	2	1	no	—
115	C₂×C₄×C₁₃₇	—	2	1	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
126	C₈×C₁₃₇	—	2	4	no	—
127	C₈×C₁₃₇	—	2	2	no	—
128	C₈×C₁₃₇	—	2	4	no	—
129	C₈×C₁₃₇	—	2	2	no	—
130	C₈×C₁₃₇	—	4	2	no	—
131	C₈×C₁₃₇	—	4	2	no	—
132	C₈×C₁₃₇	—	4	2	no	—
133	C₂²×C₁₃₇	—	2	1	no	—
134	C₂²×C₁₃₇	—	2	1	no	—
135	C₂²×C₁₃₇	—	2	2	no	—
136	C₂²×C₁₃₇	—	4	1	no	—
137	C₂²×C₁₃₇	—	4	1	no	—
138	C₂²×C₁₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₁₃₇	#432	1	2	yes	—
140	C₄×C₁₃₇	—	2	2	no	—
141	C₄×C₁₃₇	—	2	1	no	—
142	C₄×C₁₃₇	—	2	1	no	—
143	C₄×C₁₃₇	—	4	1	no	—
144	C₄×C₁₃₇	—	4	1	no	—
145	C₄×C₁₃₇	C₂×#97	1	1	yes	characteristic
146	C₄×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
147	C₄×C₁₃₇	#407	1	2	yes	—
148	C₄×C₁₃₇	#443	1	2	yes	—
149	C₄×C₁₃₇	#444	1	2	yes	—
150	C₂×C₁₃₇	—	2	2	no	—
151	C₂×C₁₃₇	—	4	2	no	—
152	C₂×C₁₃₇	#1749	1	1	yes	characteristic
153	C₂×C₁₃₇	#1764	1	1	yes	characteristic
154	C₂×C₁₃₇	#1765	1	1	yes	characteristic
155	#8186	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₁₃₇	#8186	1	1	yes	characteristic, direct product factor, Sylow
157	C₄×#131	C₂×C₁₃₇	1	1	yes	characteristic
158	#1476	C₂×C₁₃₇	1	1	yes	characteristic
159	#1476	C₂×C₁₃₇	1	1	yes	characteristic
160	#1452	C₂×C₁₃₇	1	1	yes	characteristic, complemented
161	C₂×#322	C₂×C₁₃₇	1	1	yes	characteristic, complemented
162	#1278	C₂×C₁₃₇	1	1	yes	characteristic, complemented
163	#1278	C₂×C₁₃₇	1	1	yes	characteristic, complemented
164	C₂×#131	C₄×C₁₃₇	1	2	yes	—
165	C₄×#41	C₂²×C₁₃₇	1	1	yes	characteristic
166	#431	C₂²×C₁₃₇	1	1	yes	characteristic
167	#431	C₂²×C₁₃₇	1	1	yes	characteristic
168	#405	—	2	2	no	—
169	#405	C₄×C₁₃₇	1	2	yes	—
170	C₄×M₁₆	—	2	1	no	—
171	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂×C₄×C₈	—	2	1	no	—
173	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
176	Q₈⋊C₈	C₄×C₁₃₇	1	2	yes	—
177	Q₈⋊C₈	C₄×C₁₃₇	1	2	yes	—
178	D₄⋊C₈	C₄×C₁₃₇	1	2	yes	—
179	D₄⋊C₈	C₄×C₁₃₇	1	2	yes	—
180	#322	—	2	2	no	—
181	#322	—	2	1	no	—
182	#322	—	2	1	no	—
183	C₂×#41	C₂×C₄×C₁₃₇	1	1	yes	characteristic
184	#131	—	2	4	no	—
185	#131	—	2	4	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
192	C₄×Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
193	C₄×Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
197	C₄×D₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
198	#117	—	2	1	no	—
199	#117	C₂×C₄×C₁₃₇	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
202	C₄⋊C₈	C₂×C₄×C₁₃₇	1	2	yes	—
203	C₄⋊C₈	C₂×C₄×C₁₃₇	1	2	yes	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	#97	—	2	2	no	—
207	#97	—	2	1	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
215	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
216	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
217	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
218	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
219	#41	—	2	2	no	—
220	#41	M₁₆×C₁₃₇	1	4	yes	—
221	C₂×Q₈	C₄²×C₁₃₇	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₄²×C₁₃₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
228	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
229	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
230	M₁₆	—	2	8	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	M₁₆×C₁₃₇	1	4	yes	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
244	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	C₄²×C₁₃₇	1	1	yes	characteristic
248	C₄⋊C₄	C₄²×C₁₃₇	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
256	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	C₂³	#117×C₁₃₇	1	1	yes	characteristic
261	Q₈	—	2	1	no	—
262	Q₈	#97×C₁₃₇	1	2	yes	—
263	D₄	—	2	4	no	—
264	D₄	—	2	1	no	—
265	D₄	#97×C₁₃₇	1	2	yes	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
276	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
277	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
278	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
279	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
280	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
281	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	2	2	no	—
284	C₈	—	2	4	no	—
285	C₈	—	2	2	no	—
286	C₈	—	4	2	no	—
287	C₈	—	4	2	no	—
288	C₈	—	4	2	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	C₄×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
295	C₂²	#432×C₁₃₇	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₂×#97×C₁₃₇	1	1	yes	characteristic
302	C₄	C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
303	C₄	#407×C₁₃₇	1	2	yes	—
304	C₄	#443×C₁₃₇	1	2	yes	—
305	C₄	#444×C₁₃₇	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#1749×C₁₃₇	1	1	yes	characteristic
309	C₂	#1764×C₁₃₇	1	1	yes	characteristic
310	C₂	#1765×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₁₃₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1452×C₁₃₇	C₂	semidirect
5. C₂×#322×C₁₃₇	C₂	semidirect
6. #1278×C₁₃₇	C₂	semidirect
7. #1278×C₁₃₇	C₂	semidirect
155. #8186	C₁₃₇	direct (N × H)
156. C₁₃₇	#8186	direct (N × H)
160. #1452	C₂×C₁₃₇	semidirect
161. C₂×#322	C₂×C₁₃₇	semidirect
162. #1278	C₂×C₁₃₇	semidirect
163. #1278	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#131×C₁₃₇	2
#1476×C₁₃₇	2
#1452×C₁₃₇	2
C₂×#322×C₁₃₇	2
#1278×C₁₃₇	2
#8186	137

Subgroups — C137×#8186 #46857572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8186 #46857572