Subgroups — C₁₃₇×#28834 #46878220

Order 35072.

594 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2570×C₁₃₇	C₂	1	2	yes	maximal
2	#2085×C₁₃₇	C₂	1	2	yes	maximal
3	#2084×C₁₃₇	C₂	1	2	yes	maximal
4	#1949×C₁₃₇	C₂	1	2	yes	maximal
5	#1943×C₁₃₇	C₂	1	2	yes	maximal
6	#1847×C₁₃₇	C₂	1	2	yes	maximal
7	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#1770×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
12	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	#400×C₁₃₇	—	2	2	no	—
21	#398×C₁₃₇	—	2	2	no	—
22	#389×C₁₃₇	C₂²	1	2	yes	—
23	#389×C₁₃₇	C₂²	1	2	yes	—
24	#387×C₁₃₇	—	2	2	no	—
25	#387×C₁₃₇	—	2	2	no	—
26	#384×C₁₃₇	C₂²	1	2	yes	—
27	#382×C₁₃₇	C₂²	1	2	yes	—
28	#382×C₁₃₇	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
31	#340×C₁₃₇	C₂²	1	2	yes	—
32	#340×C₁₃₇	C₂²	1	2	yes	—
33	#340×C₁₃₇	C₂²	1	2	yes	—
34	#336×C₁₃₇	C₂²	1	2	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃₇	—	2	2	no	—
48	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
50	#107×C₁₃₇	—	2	4	no	—
51	#107×C₁₃₇	—	4	2	no	—
52	#106×C₁₃₇	—	2	2	no	—
53	#106×C₁₃₇	—	2	2	no	—
54	#106×C₁₃₇	—	4	1	no	—
55	#106×C₁₃₇	—	4	2	no	—
56	#106×C₁₃₇	—	4	1	no	—
57	C₄⋊C₈×C₁₃₇	Q₈	1	4	yes	—
58	#97×C₁₃₇	—	2	4	no	—
59	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
60	#95×C₁₃₇	—	2	2	no	—
61	#95×C₁₃₇	—	2	2	no	—
62	#95×C₁₃₇	—	2	2	no	—
63	#95×C₁₃₇	—	2	2	no	—
64	#95×C₁₃₇	—	2	2	no	—
65	#95×C₁₃₇	—	2	2	no	—
66	#95×C₁₃₇	—	2	2	no	—
67	#95×C₁₃₇	D₄	1	1	yes	characteristic
68	#95×C₁₃₇	D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	—	2	2	no	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	2	no	—
72	C₂²×C₄×C₁₃₇	—	2	2	no	—
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	2	1	no	—
80	C₂×C₈×C₁₃₇	—	2	2	no	—
81	C₂×C₈×C₁₃₇	—	2	1	no	—
82	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₁₃₇	#41	1	2	yes	—
86	C₂×C₈×C₁₃₇	#41	1	2	yes	—
87	C₄⋊C₄×C₁₃₇	—	2	2	no	—
88	C₄⋊C₄×C₁₃₇	—	2	2	no	—
89	C₄⋊C₄×C₁₃₇	—	2	2	no	—
90	C₄⋊C₄×C₁₃₇	—	2	2	no	—
91	C₄⋊C₄×C₁₃₇	—	2	2	no	—
92	C₄⋊C₄×C₁₃₇	—	2	2	no	—
93	C₄⋊C₄×C₁₃₇	—	2	2	no	—
94	C₄⋊C₄×C₁₃₇	—	2	2	no	—
95	C₄⋊C₄×C₁₃₇	—	4	2	no	—
96	C₄⋊C₄×C₁₃₇	—	4	4	no	—
97	C₄⋊C₄×C₁₃₇	—	4	4	no	—
98	C₄⋊C₄×C₁₃₇	—	8	2	no	—
99	C₄⋊C₄×C₁₃₇	—	8	2	no	—
100	C₄²×C₁₃₇	—	2	2	no	—
101	C₄²×C₁₃₇	—	2	2	no	—
102	C₄²×C₁₃₇	—	8	2	no	—
103	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
104	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
105	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	2	no	—
111	C₂×C₄×C₁₃₇	—	2	2	no	—
112	C₂×C₄×C₁₃₇	—	2	2	no	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	4	2	no	—
115	C₂×C₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	—	4	2	no	—
119	C₂×C₄×C₁₃₇	—	4	2	no	—
120	C₂×C₄×C₁₃₇	—	4	2	no	—
121	C₂×C₄×C₁₃₇	—	4	2	no	—
122	C₂×C₄×C₁₃₇	—	4	2	no	—
123	C₂×C₄×C₁₃₇	—	4	2	no	—
124	C₂×C₄×C₁₃₇	—	4	2	no	—
125	C₂×C₄×C₁₃₇	—	4	2	no	—
126	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
134	C₈×C₁₃₇	—	2	2	no	—
135	C₈×C₁₃₇	—	2	4	no	—
136	C₈×C₁₃₇	#125	1	4	yes	—
137	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₂²×C₁₃₇	#547	1	1	yes	characteristic
139	C₂²×C₁₃₇	#550	1	1	yes	characteristic
140	C₂²×C₁₃₇	#575	1	2	yes	—
141	C₂²×C₁₃₇	#577	1	2	yes	—
142	C₄×C₁₃₇	—	2	2	no	—
143	C₄×C₁₃₇	—	2	2	no	—
144	C₄×C₁₃₇	—	4	2	no	—
145	C₄×C₁₃₇	—	4	2	no	—
146	C₄×C₁₃₇	—	8	2	no	—
147	C₄×C₁₃₇	—	8	2	no	—
148	C₄×C₁₃₇	—	8	2	no	—
149	C₄×C₁₃₇	—	8	2	no	—
150	C₄×C₁₃₇	C₂×#125	1	1	yes	characteristic
151	C₄×C₁₃₇	#544	1	1	yes	characteristic
152	C₄×C₁₃₇	#554	1	1	yes	characteristic
153	C₄×C₁₃₇	#557	1	1	yes	characteristic
154	C₂×C₁₃₇	#2552	1	1	yes	characteristic
155	C₂×C₁₃₇	#3162	1	1	yes	characteristic
156	C₂×C₁₃₇	#3163	1	1	yes	characteristic
157	C₂×C₁₃₇	#3351	1	2	yes	—
158	C₂×C₁₃₇	#3401	1	2	yes	—
159	#28834	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₁₃₇	#28834	1	1	yes	characteristic, direct product factor, Sylow
161	#2570	C₂×C₁₃₇	1	2	yes	—
162	#2085	C₂×C₁₃₇	1	2	yes	—
163	#2084	C₂×C₁₃₇	1	2	yes	—
164	#1949	C₂×C₁₃₇	1	2	yes	—
165	#1943	C₂×C₁₃₇	1	2	yes	—
166	#1847	C₂×C₁₃₇	1	2	yes	—
167	#1846	C₂×C₁₃₇	1	1	yes	characteristic
168	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic
169	#1770	C₂×C₁₃₇	1	1	yes	characteristic
170	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
171	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
172	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
173	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
175	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
176	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
178	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
180	#400	—	2	2	no	—
181	#398	—	2	2	no	—
182	#389	C₂²×C₁₃₇	1	2	yes	—
183	#389	C₂²×C₁₃₇	1	2	yes	—
184	#387	—	2	2	no	—
185	#387	—	2	2	no	—
186	#384	C₂²×C₁₃₇	1	2	yes	—
187	#382	C₂²×C₁₃₇	1	2	yes	—
188	#382	C₂²×C₁₃₇	1	2	yes	—
189	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
190	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
191	#340	C₂²×C₁₃₇	1	2	yes	—
192	#340	C₂²×C₁₃₇	1	2	yes	—
193	#340	C₂²×C₁₃₇	1	2	yes	—
194	#336	C₂²×C₁₃₇	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
197	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
209	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
210	#107	—	2	4	no	—
211	#107	—	4	2	no	—
212	#106	—	2	2	no	—
213	#106	—	2	2	no	—
214	#106	—	4	1	no	—
215	#106	—	4	2	no	—
216	#106	—	4	1	no	—
217	C₄⋊C₈	Q₈×C₁₃₇	1	4	yes	—
218	#97	—	2	4	no	—
219	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	D₄×C₁₃₇	1	1	yes	characteristic
228	#95	D₄×C₁₃₇	1	1	yes	characteristic
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
237	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
243	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
244	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
245	C₂×C₈	#41×C₁₃₇	1	2	yes	—
246	C₂×C₈	#41×C₁₃₇	1	2	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	8	2	no	—
259	C₄⋊C₄	—	8	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	8	2	no	—
263	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
264	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
265	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
287	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
288	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
289	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
290	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
292	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
293	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	2	4	no	—
296	C₈	#125×C₁₃₇	1	4	yes	—
297	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
298	C₂²	#547×C₁₃₇	1	1	yes	characteristic
299	C₂²	#550×C₁₃₇	1	1	yes	characteristic
300	C₂²	#575×C₁₃₇	1	2	yes	—
301	C₂²	#577×C₁₃₇	1	2	yes	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₂×#125×C₁₃₇	1	1	yes	characteristic
311	C₄	#544×C₁₃₇	1	1	yes	characteristic
312	C₄	#554×C₁₃₇	1	1	yes	characteristic
313	C₄	#557×C₁₃₇	1	1	yes	characteristic
314	C₂	#2552×C₁₃₇	1	1	yes	characteristic
315	C₂	#3162×C₁₃₇	1	1	yes	characteristic
316	C₂	#3163×C₁₃₇	1	1	yes	characteristic
317	C₂	#3351×C₁₃₇	1	2	yes	—
318	C₂	#3401×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 ◁ C₄×C₄⋊C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
159. #28834	C₁₃₇	direct (N × H)
160. C₁₃₇	#28834	direct (N × H)

Maximal subgroups

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

subgroup	index
#2570×C₁₃₇	2
#2085×C₁₃₇	2
#2084×C₁₃₇	2
#1949×C₁₃₇	2
#1943×C₁₃₇	2
#1847×C₁₃₇	2
#1846×C₁₃₇	2
C₄×#106×C₁₃₇	2
#1770×C₁₃₇	2
#28834	137

Subgroups — C137×#28834 #46878220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28834 #46878220