Subgroups — C₁₃₇×#28757 #46878143

Order 35072.

870 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#491×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#490×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2858×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2042×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1983×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1961×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1960×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1767×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
15	#491×C₁₃₇	—	2	4	no	—
16	#490×C₁₃₇	—	2	4	no	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
19	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
20	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
21	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
22	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
24	#390×C₁₃₇	—	2	2	no	—
25	#390×C₁₃₇	—	2	2	no	—
26	#389×C₁₃₇	C₂²	1	2	yes	—
27	#387×C₁₃₇	—	2	2	no	—
28	#382×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
30	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
31	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
32	#336×C₁₃₇	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×D₄×C₁₃₇	—	2	2	no	—
35	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
39	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
40	#127×C₁₃₇	—	2	2	no	—
41	#127×C₁₃₇	—	2	2	no	—
42	#125×C₁₃₇	—	2	2	no	—
43	#125×C₁₃₇	D₄	1	4	yes	—
44	#124×C₁₃₇	—	2	8	no	—
45	#124×C₁₃₇	—	4	4	no	—
46	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
52	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
54	C₂×C₄²×C₁₃₇	—	2	2	no	—
55	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
56	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
57	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₁₃₇	—	2	4	no	—
61	#103×C₁₃₇	—	2	8	no	—
62	#103×C₁₃₇	—	4	4	no	—
63	#102×C₁₃₇	—	2	8	no	—
64	#102×C₁₃₇	—	4	4	no	—
65	#97×C₁₃₇	—	2	2	no	—
66	#97×C₁₃₇	—	2	2	no	—
67	#95×C₁₃₇	—	2	2	no	—
68	#95×C₁₃₇	—	2	2	no	—
69	#95×C₁₃₇	—	2	2	no	—
70	C₂⁴×C₁₃₇	—	2	2	no	—
71	C₂×Q₈×C₁₃₇	—	2	2	no	—
72	C₂×Q₈×C₁₃₇	—	2	4	no	—
73	C₂×Q₈×C₁₃₇	—	2	2	no	—
74	C₂×Q₈×C₁₃₇	—	2	4	no	—
75	C₂×D₄×C₁₃₇	—	2	2	no	—
76	C₂×D₄×C₁₃₇	—	2	4	no	—
77	C₂×D₄×C₁₃₇	—	2	2	no	—
78	C₂×D₄×C₁₃₇	—	2	4	no	—
79	C₂×D₄×C₁₃₇	—	4	4	no	—
80	C₂×D₄×C₁₃₇	—	4	4	no	—
81	C₂×D₄×C₁₃₇	—	4	4	no	—
82	C₂²×C₄×C₁₃₇	—	2	2	no	—
83	C₂²×C₄×C₁₃₇	—	2	2	no	—
84	C₂²×C₄×C₁₃₇	—	2	2	no	—
85	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₃₇	—	2	2	no	—
91	C₂×C₈×C₁₃₇	—	2	2	no	—
92	C₂×C₈×C₁₃₇	—	2	2	no	—
93	C₂×C₈×C₁₃₇	—	2	2	no	—
94	C₂×C₈×C₁₃₇	—	2	2	no	—
95	C₂×C₈×C₁₃₇	—	2	2	no	—
96	C₄⋊C₄×C₁₃₇	—	2	8	no	—
97	C₄⋊C₄×C₁₃₇	—	2	2	no	—
98	C₄⋊C₄×C₁₃₇	—	2	4	no	—
99	C₄⋊C₄×C₁₃₇	—	4	4	no	—
100	C₄⋊C₄×C₁₃₇	—	4	4	no	—
101	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
102	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
103	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
104	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
105	C₄²×C₁₃₇	—	2	2	no	—
106	C₄²×C₁₃₇	—	4	2	no	—
107	C₄²×C₁₃₇	—	4	2	no	—
108	C₄²×C₁₃₇	—	4	4	no	—
109	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
110	C₄²×C₁₃₇	#41	1	2	yes	—
111	C₄²×C₁₃₇	#41	1	4	yes	—
112	C₂³×C₁₃₇	—	4	2	no	—
113	C₂³×C₁₃₇	—	4	2	no	—
114	C₂³×C₁₃₇	—	4	2	no	—
115	C₂³×C₁₃₇	—	4	4	no	—
116	C₂³×C₁₃₇	—	4	4	no	—
117	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₁₃₇	—	4	4	no	—
119	Q₈×C₁₃₇	—	4	4	no	—
120	D₄×C₁₃₇	—	4	4	no	—
121	D₄×C₁₃₇	—	4	4	no	—
122	D₄×C₁₃₇	—	8	8	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	—	2	4	no	—
128	C₂×C₄×C₁₃₇	—	2	2	no	—
129	C₂×C₄×C₁₃₇	—	2	2	no	—
130	C₂×C₄×C₁₃₇	—	2	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	—	4	2	no	—
137	C₂×C₄×C₁₃₇	—	4	2	no	—
138	C₂×C₄×C₁₃₇	—	4	2	no	—
139	C₂×C₄×C₁₃₇	#121	1	4	yes	—
140	C₂×C₄×C₁₃₇	#124	1	2	yes	—
141	C₂×C₄×C₁₃₇	#124	1	2	yes	—
142	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
144	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
146	C₈×C₁₃₇	—	4	2	no	—
147	C₈×C₁₃₇	—	4	2	no	—
148	C₂²×C₁₃₇	—	8	2	no	—
149	C₂²×C₁₃₇	—	8	2	no	—
150	C₂²×C₁₃₇	—	8	4	no	—
151	C₂²×C₁₃₇	—	8	4	no	—
152	C₂²×C₁₃₇	—	8	2	no	—
153	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
154	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
155	C₂²×C₁₃₇	#535	1	1	yes	characteristic
156	C₂²×C₁₃₇	#575	1	2	yes	—
157	C₂²×C₁₃₇	#575	1	2	yes	—
158	C₄×C₁₃₇	—	2	2	no	—
159	C₄×C₁₃₇	—	2	2	no	—
160	C₄×C₁₃₇	—	2	4	no	—
161	C₄×C₁₃₇	—	4	2	no	—
162	C₄×C₁₃₇	—	4	4	no	—
163	C₄×C₁₃₇	—	8	2	no	—
164	C₄×C₁₃₇	—	8	2	no	—
165	C₄×C₁₃₇	#550	1	2	yes	—
166	C₄×C₁₃₇	#559	1	2	yes	—
167	C₂×C₁₃₇	—	16	2	no	—
168	C₂×C₁₃₇	#2548	1	1	yes	characteristic
169	C₂×C₁₃₇	#3040	1	2	yes	—
170	C₂×C₁₃₇	#3135	1	1	yes	characteristic
171	C₂×C₁₃₇	#3142	1	1	yes	characteristic
172	C₂×C₁₃₇	#3219	1	2	yes	—
173	#28757	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₁₃₇	#28757	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×#491	C₂×C₁₃₇	1	1	yes	characteristic, complemented
176	C₂×#490	C₂×C₁₃₇	1	1	yes	characteristic
177	#2858	C₂×C₁₃₇	1	1	yes	characteristic
178	C₄×#125	C₂×C₁₃₇	1	1	yes	characteristic, complemented
179	#2042	C₂×C₁₃₇	1	4	yes	complemented
180	#1983	C₂×C₁₃₇	1	4	yes	complemented
181	#1961	C₂×C₁₃₇	1	1	yes	characteristic
182	#1960	C₂×C₁₃₇	1	1	yes	characteristic, complemented
183	#1767	C₂×C₁₃₇	1	1	yes	characteristic, complemented
184	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
185	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
186	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
187	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
188	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
189	#491	—	2	4	no	—
190	#490	—	2	4	no	—
191	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
192	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
193	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
194	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
195	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
196	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
197	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
198	#390	—	2	2	no	—
199	#390	—	2	2	no	—
200	#389	C₂²×C₁₃₇	1	2	yes	—
201	#387	—	2	2	no	—
202	#382	C₂²×C₁₃₇	1	1	yes	characteristic
203	C₄×C₄⋊C₄	—	2	2	no	—
204	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
205	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
206	#336	C₂²×C₁₃₇	1	2	yes	—
207	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
210	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
211	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	2	2	no	—
214	#127	—	2	2	no	—
215	#127	—	2	2	no	—
216	#125	—	2	2	no	—
217	#125	D₄×C₁₃₇	1	4	yes	—
218	#124	—	2	8	no	—
219	#124	—	4	4	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
225	C₂×C₂²⋊C₄	—	2	4	no	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
230	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
231	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
234	C₄⋊C₈	—	2	4	no	—
235	#103	—	2	8	no	—
236	#103	—	4	4	no	—
237	#102	—	2	8	no	—
238	#102	—	4	4	no	—
239	#97	—	2	2	no	—
240	#97	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	C₂⁴	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	4	no	—
255	C₂×D₄	—	4	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
276	C₂²⋊C₄	—	4	8	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	4	2	no	—
282	C₄²	—	4	4	no	—
283	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
284	C₄²	#41×C₁₃₇	1	2	yes	—
285	C₄²	#41×C₁₃₇	1	4	yes	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	4	4	no	—
290	C₂³	—	4	4	no	—
291	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
292	Q₈	—	4	4	no	—
293	Q₈	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	4	4	no	—
296	D₄	—	8	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	#121×C₁₃₇	1	4	yes	—
314	C₂×C₄	#124×C₁₃₇	1	2	yes	—
315	C₂×C₄	#124×C₁₃₇	1	2	yes	—
316	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
317	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
318	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
319	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
320	C₈	—	4	2	no	—
321	C₈	—	4	2	no	—
322	C₂²	—	8	4	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	4	no	—
327	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
328	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
329	C₂²	#535×C₁₃₇	1	1	yes	characteristic
330	C₂²	#575×C₁₃₇	1	2	yes	—
331	C₂²	#575×C₁₃₇	1	2	yes	—
332	C₄	—	2	4	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	4	no	—
336	C₄	—	4	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	2	no	—
339	C₄	#550×C₁₃₇	1	2	yes	—
340	C₄	#559×C₁₃₇	1	2	yes	—
341	C₂	—	16	2	no	—
342	C₂	#2548×C₁₃₇	1	1	yes	characteristic
343	C₂	#3040×C₁₃₇	1	2	yes	—
344	C₂	#3135×C₁₃₇	1	1	yes	characteristic
345	C₂	#3142×C₁₃₇	1	1	yes	characteristic
346	C₂	#3219×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
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
1. C₂×#491×C₁₃₇	C₂	semidirect
4. C₄×#125×C₁₃₇	C₂	semidirect
5. #2042×C₁₃₇	C₂	semidirect
6. #1983×C₁₃₇	C₂	semidirect
8. #1960×C₁₃₇	C₂	semidirect
9. #1767×C₁₃₇	C₂	semidirect
173. #28757	C₁₃₇	direct (N × H)
174. C₁₃₇	#28757	direct (N × H)
175. C₂×#491	C₂×C₁₃₇	semidirect
178. C₄×#125	C₂×C₁₃₇	semidirect
179. #2042	C₂×C₁₃₇	semidirect
180. #1983	C₂×C₁₃₇	semidirect
182. #1960	C₂×C₁₃₇	semidirect
183. #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×#491×C₁₃₇	2
C₂×#490×C₁₃₇	2
#2858×C₁₃₇	2
C₄×#125×C₁₃₇	2
#2042×C₁₃₇	2
#1983×C₁₃₇	2
#1961×C₁₃₇	2
#1960×C₁₃₇	2
#1767×C₁₃₇	2
#28757	137

Subgroups — C137×#28757 #46878143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28757 #46878143