Subgroups — C₁₃₇×#28727 #46878113

Order 35072.

798 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#489×C₁₃₇	C₂	1	2	yes	maximal
2	#2862×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#124×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2074×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2073×C₁₃₇	C₂	1	2	yes	maximal
6	#1993×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1805×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1804×C₁₃₇	C₂	1	2	yes	maximal
9	#1767×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#124×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	#489×C₁₃₇	—	2	8	no	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
17	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
18	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
19	#391×C₁₃₇	—	2	2	no	—
20	#391×C₁₃₇	—	2	2	no	—
21	#391×C₁₃₇	C₂²	1	1	yes	characteristic
22	#390×C₁₃₇	C₂²	1	1	yes	characteristic
23	#388×C₁₃₇	—	2	2	no	—
24	#382×C₁₃₇	C₂²	1	1	yes	characteristic
25	#382×C₁₃₇	C₂²	1	2	yes	—
26	#382×C₁₃₇	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
28	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
29	#340×C₁₃₇	C₂²	1	4	yes	—
30	#340×C₁₃₇	C₂²	1	4	yes	—
31	#336×C₁₃₇	C₂²	1	2	yes	—
32	C₂²×Q₈×C₁₃₇	—	2	2	no	—
33	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₃₇	—	2	1	no	—
36	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
37	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
38	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
39	#124×C₁₃₇	—	2	2	no	—
40	#124×C₁₃₇	D₄	1	4	yes	—
41	C₄×Q₈×C₁₃₇	—	2	4	no	—
42	C₄×D₄×C₁₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
51	C₂×C₄²×C₁₃₇	—	2	1	no	—
52	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
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	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₁₃₇	—	2	4	no	—
58	#103×C₁₃₇	—	2	8	no	—
59	#102×C₁₃₇	—	2	8	no	—
60	#97×C₁₃₇	—	2	4	no	—
61	#95×C₁₃₇	—	2	2	no	—
62	#95×C₁₃₇	—	2	2	no	—
63	#95×C₁₃₇	—	2	4	no	—
64	#95×C₁₃₇	—	2	2	no	—
65	#95×C₁₃₇	—	2	2	no	—
66	C₂⁴×C₁₃₇	—	2	1	no	—
67	C₂×Q₈×C₁₃₇	—	2	1	no	—
68	C₂×Q₈×C₁₃₇	—	2	2	no	—
69	C₂×Q₈×C₁₃₇	—	2	2	no	—
70	C₂×Q₈×C₁₃₇	—	4	4	no	—
71	C₂×Q₈×C₁₃₇	—	4	4	no	—
72	C₂×Q₈×C₁₃₇	—	4	4	no	—
73	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂×D₄×C₁₃₇	—	2	2	no	—
75	C₂×D₄×C₁₃₇	—	2	1	no	—
76	C₂×D₄×C₁₃₇	—	2	2	no	—
77	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃₇	—	2	1	no	—
79	C₂²×C₄×C₁₃₇	—	2	1	no	—
80	C₂²×C₄×C₁₃₇	—	2	4	no	—
81	C₂²×C₄×C₁₃₇	—	2	1	no	—
82	C₂²×C₄×C₁₃₇	—	4	4	no	—
83	C₂²×C₄×C₁₃₇	—	4	2	no	—
84	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₁₃₇	—	2	4	no	—
90	C₂×C₈×C₁₃₇	—	2	4	no	—
91	C₂×C₈×C₁₃₇	—	2	4	no	—
92	C₄⋊C₄×C₁₃₇	—	2	4	no	—
93	C₄⋊C₄×C₁₃₇	—	2	2	no	—
94	C₄⋊C₄×C₁₃₇	—	2	2	no	—
95	C₄⋊C₄×C₁₃₇	—	2	4	no	—
96	C₄⋊C₄×C₁₃₇	—	4	4	no	—
97	C₄⋊C₄×C₁₃₇	—	4	2	no	—
98	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
99	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
100	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
101	C₄²×C₁₃₇	—	2	2	no	—
102	C₄²×C₁₃₇	—	4	2	no	—
103	C₄²×C₁₃₇	—	4	2	no	—
104	C₄²×C₁₃₇	—	4	2	no	—
105	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
106	C₄²×C₁₃₇	#41	1	2	yes	—
107	C₄²×C₁₃₇	#41	1	4	yes	—
108	C₂³×C₁₃₇	—	2	2	no	—
109	C₂³×C₁₃₇	—	4	1	no	—
110	C₂³×C₁₃₇	—	4	1	no	—
111	C₂³×C₁₃₇	—	4	2	no	—
112	C₂³×C₁₃₇	—	4	2	no	—
113	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₁₃₇	—	2	4	no	—
115	Q₈×C₁₃₇	—	4	2	no	—
116	Q₈×C₁₃₇	—	8	8	no	—
117	D₄×C₁₃₇	—	2	4	no	—
118	D₄×C₁₃₇	—	4	2	no	—
119	C₂×C₄×C₁₃₇	—	2	2	no	—
120	C₂×C₄×C₁₃₇	—	2	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	—	2	2	no	—
125	C₂×C₄×C₁₃₇	—	2	2	no	—
126	C₂×C₄×C₁₃₇	—	2	2	no	—
127	C₂×C₄×C₁₃₇	—	4	1	no	—
128	C₂×C₄×C₁₃₇	—	4	1	no	—
129	C₂×C₄×C₁₃₇	—	4	4	no	—
130	C₂×C₄×C₁₃₇	—	4	1	no	—
131	C₂×C₄×C₁₃₇	—	4	1	no	—
132	C₂×C₄×C₁₃₇	—	4	2	no	—
133	C₂×C₄×C₁₃₇	—	4	2	no	—
134	C₂×C₄×C₁₃₇	—	4	4	no	—
135	C₂×C₄×C₁₃₇	—	4	4	no	—
136	C₂×C₄×C₁₃₇	—	8	4	no	—
137	C₂×C₄×C₁₃₇	#121	1	4	yes	—
138	C₂×C₄×C₁₃₇	#124	1	2	yes	—
139	C₂×C₄×C₁₃₇	#124	1	2	yes	—
140	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
142	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
144	C₈×C₁₃₇	—	4	4	no	—
145	C₂²×C₁₃₇	—	4	4	no	—
146	C₂²×C₁₃₇	—	4	2	no	—
147	C₂²×C₁₃₇	—	8	1	no	—
148	C₂²×C₁₃₇	—	8	1	no	—
149	C₂²×C₁₃₇	—	8	2	no	—
150	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
151	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
152	C₂²×C₁₃₇	#536	1	1	yes	characteristic
153	C₂²×C₁₃₇	#575	1	2	yes	—
154	C₂²×C₁₃₇	#575	1	2	yes	—
155	C₄×C₁₃₇	—	2	2	no	—
156	C₄×C₁₃₇	—	2	2	no	—
157	C₄×C₁₃₇	—	2	4	no	—
158	C₄×C₁₃₇	—	4	2	no	—
159	C₄×C₁₃₇	—	4	2	no	—
160	C₄×C₁₃₇	—	8	1	no	—
161	C₄×C₁₃₇	—	8	1	no	—
162	C₄×C₁₃₇	—	8	4	no	—
163	C₄×C₁₃₇	#547	1	2	yes	—
164	C₄×C₁₃₇	#556	1	2	yes	—
165	C₂×C₁₃₇	—	8	2	no	—
166	C₂×C₁₃₇	#2547	1	1	yes	characteristic
167	C₂×C₁₃₇	#3040	1	2	yes	—
168	C₂×C₁₃₇	#3135	1	1	yes	characteristic
169	C₂×C₁₃₇	#3140	1	1	yes	characteristic
170	C₂×C₁₃₇	#3240	1	2	yes	—
171	#28727	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₁₃₇	#28727	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×#489	C₂×C₁₃₇	1	2	yes	—
174	#2862	C₂×C₁₃₇	1	1	yes	characteristic, complemented
175	C₄×#124	C₂×C₁₃₇	1	1	yes	characteristic
176	#2074	C₂×C₁₃₇	1	2	yes	complemented
177	#2073	C₂×C₁₃₇	1	2	yes	—
178	#1993	C₂×C₁₃₇	1	2	yes	complemented
179	#1805	C₂×C₁₃₇	1	2	yes	complemented
180	#1804	C₂×C₁₃₇	1	2	yes	—
181	#1767	C₂×C₁₃₇	1	1	yes	characteristic, complemented
182	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
183	C₂×#124	C₂²×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
186	#489	—	2	8	no	—
187	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
188	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
189	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
190	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
191	#391	—	2	2	no	—
192	#391	—	2	2	no	—
193	#391	C₂²×C₁₃₇	1	1	yes	characteristic
194	#390	C₂²×C₁₃₇	1	1	yes	characteristic
195	#388	—	2	2	no	—
196	#382	C₂²×C₁₃₇	1	2	yes	—
197	#382	C₂²×C₁₃₇	1	2	yes	—
198	#382	C₂²×C₁₃₇	1	1	yes	characteristic
199	C₄×C₂²⋊C₄	—	2	2	no	—
200	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
201	#340	C₂²×C₁₃₇	1	4	yes	—
202	#340	C₂²×C₁₃₇	1	4	yes	—
203	#336	C₂²×C₁₃₇	1	2	yes	—
204	C₂²×Q₈	—	2	2	no	—
205	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
206	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
207	C₂³×C₄	—	2	1	no	—
208	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	#124	—	2	2	no	—
212	#124	D₄×C₁₃₇	1	4	yes	—
213	C₄×Q₈	—	2	4	no	—
214	C₄×D₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
225	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
226	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
228	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
229	C₄⋊C₈	—	2	4	no	—
230	#103	—	2	8	no	—
231	#102	—	2	8	no	—
232	#97	—	2	4	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	4	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	C₂⁴	—	2	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	—	4	4	no	—
245	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	4	4	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
260	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	2	8	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
278	C₄²	#41×C₁₃₇	1	2	yes	—
279	C₄²	#41×C₁₃₇	1	4	yes	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	8	no	—
289	D₄	—	2	4	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	#121×C₁₃₇	1	4	yes	—
310	C₂×C₄	#124×C₁₃₇	1	2	yes	—
311	C₂×C₄	#124×C₁₃₇	1	2	yes	—
312	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
313	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
314	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
315	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
316	C₈	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	2	no	—
322	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
323	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
324	C₂²	#536×C₁₃₇	1	1	yes	characteristic
325	C₂²	#575×C₁₃₇	1	2	yes	—
326	C₂²	#575×C₁₃₇	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	8	4	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	#547×C₁₃₇	1	2	yes	—
336	C₄	#556×C₁₃₇	1	2	yes	—
337	C₂	—	8	2	no	—
338	C₂	#2547×C₁₃₇	1	1	yes	characteristic
339	C₂	#3040×C₁₃₇	1	2	yes	—
340	C₂	#3135×C₁₃₇	1	1	yes	characteristic
341	C₂	#3140×C₁₃₇	1	1	yes	characteristic
342	C₂	#3240×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2862×C₁₃₇	C₂	semidirect
4. #2074×C₁₃₇	C₂	semidirect
6. #1993×C₁₃₇	C₂	semidirect
7. #1805×C₁₃₇	C₂	semidirect
9. #1767×C₁₃₇	C₂	semidirect
171. #28727	C₁₃₇	direct (N × H)
172. C₁₃₇	#28727	direct (N × H)
174. #2862	C₂×C₁₃₇	semidirect
176. #2074	C₂×C₁₃₇	semidirect
178. #1993	C₂×C₁₃₇	semidirect
179. #1805	C₂×C₁₃₇	semidirect
181. #1767	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×#489×C₁₃₇	2
#2862×C₁₃₇	2
C₄×#124×C₁₃₇	2
#2074×C₁₃₇	2
#2073×C₁₃₇	2
#1993×C₁₃₇	2
#1805×C₁₃₇	2
#1804×C₁₃₇	2
#1767×C₁₃₇	2
#28727	137

Subgroups — C137×#28727 #46878113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28727 #46878113