Subgroups — C₁₃₇×#28731 #46878117

Order 35072.

698 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	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3141×C₁₃₇	C₂	1	2	yes	maximal
3	#2931×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2078×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1997×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1816×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₃₇	C₂²	1	2	yes	—
9	#532×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
11	#500×C₁₃₇	—	2	1	no	—
12	#500×C₁₃₇	—	2	1	no	—
13	#500×C₁₃₇	—	2	1	no	—
14	#500×C₁₃₇	—	2	1	no	—
15	#489×C₁₃₇	—	2	4	no	—
16	#487×C₁₃₇	—	2	2	no	—
17	#486×C₁₃₇	—	2	2	no	—
18	#436×C₁₃₇	—	2	1	no	—
19	#436×C₁₃₇	—	2	1	no	—
20	C₈×D₄×C₁₃₇	—	2	1	no	—
21	C₈×D₄×C₁₃₇	—	2	1	no	—
22	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
23	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
24	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
25	#423×C₁₃₇	C₂²	1	1	yes	characteristic
26	#417×C₁₃₇	C₂²	1	8	yes	—
27	#408×C₁₃₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	#343×C₁₃₇	C₂²	1	8	yes	—
31	#341×C₁₃₇	C₂²	1	2	yes	—
32	C₂×#41×C₁₃₇	C₂³	1	2	yes	—
33	C₂²×Q₈×C₁₃₇	—	2	2	no	—
34	#131×C₁₃₇	—	2	8	no	—
35	C₂×M₁₆×C₁₃₇	—	2	2	no	—
36	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
37	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₇	—	2	2	no	—
39	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
42	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
43	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
44	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
45	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
46	#127×C₁₃₇	D₄	1	1	yes	characteristic
47	#124×C₁₃₇	D₄	1	2	yes	—
48	#121×C₁₃₇	—	2	1	no	—
49	#121×C₁₃₇	—	2	1	no	—
50	C₄×D₄×C₁₃₇	—	2	1	no	—
51	C₄×D₄×C₁₃₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
55	#108×C₁₃₇	—	2	4	no	—
56	#107×C₁₃₇	—	2	2	no	—
57	#107×C₁₃₇	—	2	2	no	—
58	#107×C₁₃₇	—	2	2	no	—
59	#107×C₁₃₇	—	2	2	no	—
60	C₄⋊C₈×C₁₃₇	—	2	1	no	—
61	C₄⋊C₈×C₁₃₇	—	2	1	no	—
62	#103×C₁₃₇	—	2	8	no	—
63	#102×C₁₃₇	—	2	8	no	—
64	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
65	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
66	#97×C₁₃₇	—	2	2	no	—
67	C₄×C₈×C₁₃₇	—	2	2	no	—
68	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
72	#41×C₁₃₇	—	2	4	no	—
73	#41×C₁₃₇	—	2	4	no	—
74	C₂×Q₈×C₁₃₇	—	4	2	no	—
75	C₂×Q₈×C₁₃₇	—	4	2	no	—
76	C₂×Q₈×C₁₃₇	—	8	4	no	—
77	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
78	C₂×D₄×C₁₃₇	—	2	2	no	—
79	C₂×D₄×C₁₃₇	—	2	1	no	—
80	C₂×D₄×C₁₃₇	—	2	1	no	—
81	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₃₇	—	2	2	no	—
83	C₂²×C₄×C₁₃₇	—	2	4	no	—
84	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
86	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₁₃₇	—	2	4	no	—
88	M₁₆×C₁₃₇	—	2	4	no	—
89	M₁₆×C₁₃₇	—	2	4	no	—
90	M₁₆×C₁₃₇	—	4	4	no	—
91	C₂×C₈×C₁₃₇	—	2	2	no	—
92	C₂×C₈×C₁₃₇	—	2	4	no	—
93	C₂×C₈×C₁₃₇	—	2	2	no	—
94	C₂×C₈×C₁₃₇	—	4	4	no	—
95	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃₇	#41	1	4	yes	—
100	C₄⋊C₄×C₁₃₇	—	2	1	no	—
101	C₄⋊C₄×C₁₃₇	—	2	1	no	—
102	C₄⋊C₄×C₁₃₇	—	2	4	no	—
103	C₄⋊C₄×C₁₃₇	—	2	4	no	—
104	C₄⋊C₄×C₁₃₇	—	4	2	no	—
105	C₄⋊C₄×C₁₃₇	—	4	2	no	—
106	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
107	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
108	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₁₃₇	—	2	2	no	—
113	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
114	Q₈×C₁₃₇	—	2	4	no	—
115	Q₈×C₁₃₇	—	8	4	no	—
116	Q₈×C₁₃₇	—	8	4	no	—
117	D₄×C₁₃₇	—	2	4	no	—
118	D₄×C₁₃₇	—	4	4	no	—
119	D₄×C₁₃₇	—	4	2	no	—
120	D₄×C₁₃₇	—	4	2	no	—
121	C₂×C₄×C₁₃₇	—	2	2	no	—
122	C₂×C₄×C₁₃₇	—	2	2	no	—
123	C₂×C₄×C₁₃₇	—	2	2	no	—
124	C₂×C₄×C₁₃₇	—	4	2	no	—
125	C₂×C₄×C₁₃₇	—	4	2	no	—
126	C₂×C₄×C₁₃₇	—	4	4	no	—
127	C₂×C₄×C₁₃₇	—	8	4	no	—
128	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
135	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
136	C₈×C₁₃₇	—	2	1	no	—
137	C₈×C₁₃₇	—	2	1	no	—
138	C₈×C₁₃₇	—	4	2	no	—
139	C₈×C₁₃₇	—	4	4	no	—
140	C₈×C₁₃₇	#124	1	2	yes	—
141	C₈×C₁₃₇	#124	1	2	yes	—
142	C₂²×C₁₃₇	—	2	1	no	—
143	C₂²×C₁₃₇	—	2	1	no	—
144	C₂²×C₁₃₇	—	4	2	no	—
145	C₂²×C₁₃₇	—	4	2	no	—
146	C₂²×C₁₃₇	—	4	2	no	—
147	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
148	C₂²×C₁₃₇	#556	1	2	yes	—
149	C₄×C₁₃₇	—	2	1	no	—
150	C₄×C₁₃₇	—	2	1	no	—
151	C₄×C₁₃₇	—	4	2	no	—
152	C₄×C₁₃₇	—	8	4	no	—
153	C₄×C₁₃₇	#472	1	2	yes	—
154	C₄×C₁₃₇	#473	1	2	yes	—
155	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
156	C₄×C₁₃₇	#536	1	1	yes	characteristic
157	C₄×C₁₃₇	#547	1	2	yes	—
158	C₂×C₁₃₇	—	2	2	no	—
159	C₂×C₁₃₇	—	8	2	no	—
160	C₂×C₁₃₇	#2547	1	1	yes	characteristic
161	C₂×C₁₃₇	C₂×#472	1	1	yes	characteristic
162	C₂×C₁₃₇	C₂×#473	1	1	yes	characteristic
163	#28731	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₁₃₇	#28731	1	1	yes	characteristic, direct product factor, Sylow
165	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic, complemented
166	#3141	C₂×C₁₃₇	1	2	yes	—
167	#2931	C₂×C₁₃₇	1	1	yes	characteristic
168	#2078	C₂×C₁₃₇	1	4	yes	complemented
169	#1997	C₂×C₁₃₇	1	2	yes	complemented
170	#1816	C₂×C₁₃₇	1	4	yes	complemented
171	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic, complemented
172	C₂×#131	C₂²×C₁₃₇	1	2	yes	—
173	#532	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
175	#500	—	2	1	no	—
176	#500	—	2	1	no	—
177	#500	—	2	1	no	—
178	#500	—	2	1	no	—
179	#489	—	2	4	no	—
180	#487	—	2	2	no	—
181	#486	—	2	2	no	—
182	#436	—	2	1	no	—
183	#436	—	2	1	no	—
184	C₈×D₄	—	2	1	no	—
185	C₈×D₄	—	2	1	no	—
186	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
187	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
188	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
189	#423	C₂²×C₁₃₇	1	1	yes	characteristic
190	#417	C₂²×C₁₃₇	1	8	yes	—
191	#408	C₂²×C₁₃₇	1	2	yes	—
192	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
193	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
194	#343	C₂²×C₁₃₇	1	8	yes	—
195	#341	C₂²×C₁₃₇	1	2	yes	—
196	C₂×#41	C₂³×C₁₃₇	1	2	yes	—
197	C₂²×Q₈	—	2	2	no	—
198	#131	—	2	8	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
201	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	—	2	2	no	—
209	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
210	#127	D₄×C₁₃₇	1	1	yes	characteristic
211	#124	D₄×C₁₃₇	1	2	yes	—
212	#121	—	2	1	no	—
213	#121	—	2	1	no	—
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
218	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
219	#108	—	2	4	no	—
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#107	—	2	2	no	—
223	#107	—	2	2	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	#103	—	2	8	no	—
227	#102	—	2	8	no	—
228	C₂²⋊C₈	—	2	2	no	—
229	C₂²⋊C₈	—	2	2	no	—
230	#97	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
233	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
234	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
235	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
236	#41	—	2	4	no	—
237	#41	—	2	4	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	8	4	no	—
241	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
249	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
251	M₁₆	—	2	4	no	—
252	M₁₆	—	2	4	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	—	4	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	4	4	no	—
259	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂×C₈	#41×C₁₃₇	1	4	yes	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
278	Q₈	—	2	4	no	—
279	Q₈	—	8	4	no	—
280	Q₈	—	8	4	no	—
281	D₄	—	2	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
293	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
294	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
295	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
298	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
299	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	4	4	no	—
304	C₈	#124×C₁₃₇	1	2	yes	—
305	C₈	#124×C₁₃₇	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
312	C₂²	#556×C₁₃₇	1	2	yes	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	4	no	—
317	C₄	#472×C₁₃₇	1	2	yes	—
318	C₄	#473×C₁₃₇	1	2	yes	—
319	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
320	C₄	#536×C₁₃₇	1	1	yes	characteristic
321	C₄	#547×C₁₃₇	1	2	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	8	2	no	—
324	C₂	#2547×C₁₃₇	1	1	yes	characteristic
325	C₂	C₂×#472×C₁₃₇	1	1	yes	characteristic
326	C₂	C₂×#473×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	#532×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₃₇	C₂	semidirect
4. #2078×C₁₃₇	C₂	semidirect
5. #1997×C₁₃₇	C₂	semidirect
6. #1816×C₁₃₇	C₂	semidirect
7. C₄×#108×C₁₃₇	C₂	semidirect
163. #28731	C₁₃₇	direct (N × H)
164. C₁₃₇	#28731	direct (N × H)
165. C₂×#500	C₂×C₁₃₇	semidirect
168. #2078	C₂×C₁₃₇	semidirect
169. #1997	C₂×C₁₃₇	semidirect
170. #1816	C₂×C₁₃₇	semidirect
171. C₄×#108	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₃₇	2
#3141×C₁₃₇	2
#2931×C₁₃₇	2
#2078×C₁₃₇	2
#1997×C₁₃₇	2
#1816×C₁₃₇	2
C₄×#108×C₁₃₇	2
#28731	137

Subgroups — C137×#28731 #46878117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28731 #46878117