Subgroups — C₁₃₇×#23829 #46873215

Order 35072.

690 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2331×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2327×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2101×C₁₃₇	C₂	1	2	yes	maximal
4	#2074×C₁₃₇	C₂	1	2	yes	maximal
5	#2043×C₁₃₇	C₂	1	2	yes	maximal
6	#2024×C₁₃₇	C₂	1	2	yes	maximal
7	#1763×C₁₃₇	C₂	1	2	yes	maximal
8	C₄×#103×C₁₃₇	C₂	1	2	yes	maximal
9	#1746×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	#391×C₁₃₇	—	2	2	no	—
20	#391×C₁₃₇	C₂²	1	2	yes	—
21	#389×C₁₃₇	C₂²	1	2	yes	—
22	#387×C₁₃₇	—	2	2	no	—
23	#387×C₁₃₇	—	2	2	no	—
24	#387×C₁₃₇	—	2	2	no	—
25	#382×C₁₃₇	C₂²	1	1	yes	characteristic
26	#382×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
28	#376×C₁₃₇	C₂²	1	1	yes	characteristic
29	#340×C₁₃₇	C₂²	1	2	yes	—
30	#340×C₁₃₇	C₂²	1	2	yes	—
31	#340×C₁₃₇	C₂²	1	2	yes	—
32	#340×C₁₃₇	C₂²	1	2	yes	—
33	#336×C₁₃₇	C₂²	1	2	yes	—
34	#336×C₁₃₇	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃₇	—	2	1	no	—
36	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
40	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
41	C₄×Q₈×C₁₃₇	—	2	4	no	—
42	C₄×Q₈×C₁₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃₇	—	2	2	no	—
51	C₂×C₄²×C₁₃₇	—	2	2	no	—
52	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
55	#103×C₁₃₇	—	2	4	no	—
56	#103×C₁₃₇	—	2	4	no	—
57	#103×C₁₃₇	—	2	4	no	—
58	#103×C₁₃₇	—	2	4	no	—
59	#97×C₁₃₇	—	2	2	no	—
60	C₄×C₈×C₁₃₇	—	2	2	no	—
61	#95×C₁₃₇	—	2	2	no	—
62	#95×C₁₃₇	—	2	1	no	—
63	#95×C₁₃₇	—	2	2	no	—
64	#95×C₁₃₇	—	2	1	no	—
65	#95×C₁₃₇	—	2	2	no	—
66	#95×C₁₃₇	—	2	2	no	—
67	#95×C₁₃₇	—	2	2	no	—
68	#95×C₁₃₇	D₄	1	2	yes	—
69	#95×C₁₃₇	D₄	1	2	yes	—
70	C₂×Q₈×C₁₃₇	—	2	4	no	—
71	C₂×Q₈×C₁₃₇	—	2	2	no	—
72	C₂×Q₈×C₁₃₇	—	2	2	no	—
73	C₂×Q₈×C₁₃₇	—	2	2	no	—
74	C₂×Q₈×C₁₃₇	—	2	2	no	—
75	C₂×Q₈×C₁₃₇	—	2	2	no	—
76	C₂×Q₈×C₁₃₇	—	8	2	no	—
77	C₂×Q₈×C₁₃₇	#41	1	4	yes	—
78	C₂²×C₄×C₁₃₇	—	2	2	no	—
79	C₂²×C₄×C₁₃₇	—	2	2	no	—
80	C₂²×C₄×C₁₃₇	—	2	2	no	—
81	C₂²×C₄×C₁₃₇	—	2	2	no	—
82	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	2	no	—
89	C₂×C₈×C₁₃₇	—	2	2	no	—
90	C₂×C₈×C₁₃₇	—	2	2	no	—
91	C₂×C₈×C₁₃₇	—	2	2	no	—
92	C₂×C₈×C₁₃₇	—	2	2	no	—
93	C₄⋊C₄×C₁₃₇	—	2	8	no	—
94	C₄⋊C₄×C₁₃₇	—	2	4	no	—
95	C₄⋊C₄×C₁₃₇	—	2	2	no	—
96	C₄⋊C₄×C₁₃₇	—	2	2	no	—
97	C₄⋊C₄×C₁₃₇	—	2	2	no	—
98	C₄⋊C₄×C₁₃₇	—	2	2	no	—
99	C₄⋊C₄×C₁₃₇	—	2	2	no	—
100	C₄⋊C₄×C₁₃₇	—	4	4	no	—
101	C₄⋊C₄×C₁₃₇	—	8	2	no	—
102	C₄⋊C₄×C₁₃₇	#41	1	4	yes	—
103	C₄²×C₁₃₇	—	2	2	no	—
104	C₄²×C₁₃₇	—	2	2	no	—
105	C₄²×C₁₃₇	—	4	4	no	—
106	C₄²×C₁₃₇	—	8	2	no	—
107	C₄²×C₁₃₇	#41	1	2	yes	—
108	C₄²×C₁₃₇	#41	1	2	yes	—
109	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₁₃₇	—	2	4	no	—
111	Q₈×C₁₃₇	—	2	4	no	—
112	Q₈×C₁₃₇	—	4	2	no	—
113	Q₈×C₁₃₇	—	4	2	no	—
114	Q₈×C₁₃₇	—	8	4	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	2	2	no	—
118	C₂×C₄×C₁₃₇	—	2	1	no	—
119	C₂×C₄×C₁₃₇	—	2	1	no	—
120	C₂×C₄×C₁₃₇	—	2	4	no	—
121	C₂×C₄×C₁₃₇	—	2	4	no	—
122	C₂×C₄×C₁₃₇	—	2	2	no	—
123	C₂×C₄×C₁₃₇	—	2	2	no	—
124	C₂×C₄×C₁₃₇	—	2	2	no	—
125	C₂×C₄×C₁₃₇	—	2	2	no	—
126	C₂×C₄×C₁₃₇	—	4	2	no	—
127	C₂×C₄×C₁₃₇	—	4	2	no	—
128	C₂×C₄×C₁₃₇	—	4	2	no	—
129	C₂×C₄×C₁₃₇	—	4	2	no	—
130	C₂×C₄×C₁₃₇	—	4	2	no	—
131	C₂×C₄×C₁₃₇	—	4	2	no	—
132	C₂×C₄×C₁₃₇	—	4	2	no	—
133	C₂×C₄×C₁₃₇	—	4	2	no	—
134	C₂×C₄×C₁₃₇	—	4	2	no	—
135	C₂×C₄×C₁₃₇	—	4	2	no	—
136	C₂×C₄×C₁₃₇	#124	1	4	yes	—
137	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
141	C₈×C₁₃₇	—	4	2	no	—
142	C₈×C₁₃₇	—	4	2	no	—
143	C₂²×C₁₃₇	#525	1	1	yes	characteristic
144	C₂²×C₁₃₇	#525	1	1	yes	characteristic
145	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
146	C₂²×C₁₃₇	#577	1	2	yes	—
147	C₂²×C₁₃₇	#578	1	2	yes	—
148	C₄×C₁₃₇	—	2	4	no	—
149	C₄×C₁₃₇	—	4	2	no	—
150	C₄×C₁₃₇	—	4	1	no	—
151	C₄×C₁₃₇	—	4	4	no	—
152	C₄×C₁₃₇	—	4	1	no	—
153	C₄×C₁₃₇	—	4	4	no	—
154	C₄×C₁₃₇	—	8	2	no	—
155	C₄×C₁₃₇	—	8	2	no	—
156	C₄×C₁₃₇	#529	1	2	yes	—
157	C₄×C₁₃₇	#529	1	2	yes	—
158	C₂×C₁₃₇	#2398	1	1	yes	characteristic
159	C₂×C₁₃₇	#3121	1	2	yes	—
160	C₂×C₁₃₇	#3123	1	1	yes	characteristic
161	C₂×C₁₃₇	#3127	1	1	yes	characteristic
162	C₂×C₁₃₇	#3137	1	2	yes	—
163	#23829	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₁₃₇	#23829	1	1	yes	characteristic, direct product factor, Sylow
165	#2331	C₂×C₁₃₇	1	1	yes	characteristic
166	#2327	C₂×C₁₃₇	1	1	yes	characteristic
167	#2101	C₂×C₁₃₇	1	2	yes	—
168	#2074	C₂×C₁₃₇	1	2	yes	—
169	#2043	C₂×C₁₃₇	1	2	yes	—
170	#2024	C₂×C₁₃₇	1	2	yes	—
171	#1763	C₂×C₁₃₇	1	2	yes	—
172	C₄×#103	C₂×C₁₃₇	1	2	yes	—
173	#1746	C₂×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
175	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
176	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
177	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
178	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
179	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
180	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
181	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
183	#391	—	2	2	no	—
184	#391	C₂²×C₁₃₇	1	2	yes	—
185	#389	C₂²×C₁₃₇	1	2	yes	—
186	#387	—	2	2	no	—
187	#387	—	2	2	no	—
188	#387	—	2	2	no	—
189	#382	C₂²×C₁₃₇	1	1	yes	characteristic
190	#382	C₂²×C₁₃₇	1	1	yes	characteristic
191	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
192	#376	C₂²×C₁₃₇	1	1	yes	characteristic
193	#340	C₂²×C₁₃₇	1	2	yes	—
194	#340	C₂²×C₁₃₇	1	2	yes	—
195	#340	C₂²×C₁₃₇	1	2	yes	—
196	#340	C₂²×C₁₃₇	1	2	yes	—
197	#336	C₂²×C₁₃₇	1	2	yes	—
198	#336	C₂²×C₁₃₇	1	2	yes	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
201	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
203	C₄⋊Q₈	—	2	4	no	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×Q₈	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	4	no	—
222	#103	—	2	4	no	—
223	#97	—	2	2	no	—
224	C₄×C₈	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	1	no	—
227	#95	—	2	2	no	—
228	#95	—	2	1	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	D₄×C₁₃₇	1	2	yes	—
233	#95	D₄×C₁₃₇	1	2	yes	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×Q₈	#41×C₁₃₇	1	4	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
250	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	8	2	no	—
266	C₄⋊C₄	#41×C₁₃₇	1	4	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	4	no	—
270	C₄²	—	8	2	no	—
271	C₄²	#41×C₁₃₇	1	2	yes	—
272	C₄²	#41×C₁₃₇	1	2	yes	—
273	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
274	Q₈	—	2	4	no	—
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	8	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	#124×C₁₃₇	1	4	yes	—
301	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
302	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
303	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₂²	#525×C₁₃₇	1	1	yes	characteristic
308	C₂²	#525×C₁₃₇	1	1	yes	characteristic
309	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
310	C₂²	#577×C₁₃₇	1	2	yes	—
311	C₂²	#578×C₁₃₇	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	#529×C₁₃₇	1	2	yes	—
321	C₄	#529×C₁₃₇	1	2	yes	—
322	C₂	#2398×C₁₃₇	1	1	yes	characteristic
323	C₂	#3121×C₁₃₇	1	2	yes	—
324	C₂	#3123×C₁₃₇	1	1	yes	characteristic
325	C₂	#3127×C₁₃₇	1	1	yes	characteristic
326	C₂	#3137×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ #376×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
163. #23829	C₁₃₇	direct (N × H)
164. C₁₃₇	#23829	direct (N × H)

Maximal subgroups

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

subgroup	index
#2331×C₁₃₇	2
#2327×C₁₃₇	2
#2101×C₁₃₇	2
#2074×C₁₃₇	2
#2043×C₁₃₇	2
#2024×C₁₃₇	2
#1763×C₁₃₇	2
C₄×#103×C₁₃₇	2
#1746×C₁₃₇	2
#23829	137

Subgroups — C137×#23829 #46873215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#23829 #46873215