Subgroups — C₁₃₇×#12917 #46862303

Order 35072.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2151×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#350×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1764×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1379×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1374×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₃₇	C₂²	1	1	yes	characteristic
9	#504×C₁₃₇	—	2	2	no	—
10	#504×C₁₃₇	—	2	2	no	—
11	C₂×M₃₂×C₁₃₇	—	2	1	no	—
12	C₂×M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₃₇	—	2	1	no	—
15	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#104×C₁₃₇	C₄	1	1	yes	characteristic
18	#416×C₁₃₇	C₄	1	1	yes	characteristic
19	#405×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	#364×C₁₃₇	—	2	2	no	—
22	C₄⋊C₁₆×C₁₃₇	—	2	1	no	—
23	C₄⋊C₁₆×C₁₃₇	—	2	1	no	—
24	#350×C₁₃₇	—	2	2	no	—
25	#341×C₁₃₇	C₄	1	1	yes	characteristic
26	#339×C₁₃₇	C₄	1	1	yes	characteristic
27	C₂×#41×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	#131×C₁₃₇	—	2	1	no	—
29	#131×C₁₃₇	—	2	1	no	—
30	#131×C₁₃₇	D₄	1	2	yes	—
31	#131×C₁₃₇	Q₈	1	2	yes	—
32	C₂×M₁₆×C₁₃₇	—	2	1	no	—
33	C₂×M₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₇	—	2	1	no	—
36	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
38	#117×C₁₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₁₃₇	—	2	4	no	—
41	M₃₂×C₁₃₇	—	2	1	no	—
42	M₃₂×C₁₃₇	—	2	1	no	—
43	M₃₂×C₁₃₇	—	2	1	no	—
44	M₃₂×C₁₃₇	D₄	1	2	yes	—
45	C₂×C₁₆×C₁₃₇	—	2	4	no	—
46	C₂×C₁₆×C₁₃₇	—	2	1	no	—
47	C₂×C₁₆×C₁₃₇	—	2	1	no	—
48	C₂×C₁₆×C₁₃₇	—	2	1	no	—
49	C₂×C₁₆×C₁₃₇	—	2	1	no	—
50	C₂×C₁₆×C₁₃₇	—	2	1	no	—
51	C₂×C₁₆×C₁₃₇	—	2	1	no	—
52	C₂×C₁₆×C₁₃₇	—	2	1	no	—
53	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
54	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
55	#104×C₁₃₇	—	2	2	no	—
56	#103×C₁₃₇	C₈	1	1	yes	characteristic
57	#103×C₁₃₇	C₈	1	1	yes	characteristic
58	#102×C₁₃₇	C₈	1	1	yes	characteristic
59	#102×C₁₃₇	C₈	1	1	yes	characteristic
60	#97×C₁₃₇	—	2	1	no	—
61	C₄×C₈×C₁₃₇	—	2	1	no	—
62	C₄×C₈×C₁₃₇	—	2	1	no	—
63	C₄×C₈×C₁₃₇	—	2	1	no	—
64	#41×C₁₃₇	—	2	1	no	—
65	#41×C₁₃₇	—	2	1	no	—
66	#41×C₁₃₇	C₄⋊C₄	1	2	yes	—
67	#41×C₁₃₇	M₁₆	1	2	yes	—
68	C₂×Q₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
69	C₂×D₄×C₁₃₇	—	2	1	no	—
70	C₂×D₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	—	2	1	no	—
72	C₂²×C₄×C₁₃₇	—	2	1	no	—
73	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
74	M₁₆×C₁₃₇	—	2	1	no	—
75	M₁₆×C₁₃₇	—	2	2	no	—
76	M₁₆×C₁₃₇	—	2	2	no	—
77	M₁₆×C₁₃₇	—	2	1	no	—
78	M₁₆×C₁₃₇	—	2	1	no	—
79	M₁₆×C₁₃₇	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₃₇	—	2	1	no	—
81	C₂×C₈×C₁₃₇	—	2	1	no	—
82	C₂×C₈×C₁₃₇	—	2	1	no	—
83	C₂×C₈×C₁₃₇	—	2	2	no	—
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	—	2	1	no	—
86	C₂×C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
87	C₂×C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
88	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
93	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
94	C₄⋊C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
95	C₄⋊C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
96	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
97	C₄²×C₁₃₇	—	2	1	no	—
98	C₄²×C₁₃₇	—	2	1	no	—
99	C₄²×C₁₃₇	—	2	1	no	—
100	C₁₆×C₁₃₇	—	2	1	no	—
101	C₁₆×C₁₃₇	—	2	2	no	—
102	C₁₆×C₁₃₇	—	2	1	no	—
103	C₁₆×C₁₃₇	—	4	1	no	—
104	C₁₆×C₁₃₇	—	4	1	no	—
105	C₂³×C₁₃₇	—	2	1	no	—
106	C₂³×C₁₃₇	#131	1	1	yes	characteristic
107	Q₈×C₁₃₇	—	2	1	no	—
108	Q₈×C₁₃₇	C₄⋊C₈	1	2	yes	—
109	D₄×C₁₃₇	—	2	2	no	—
110	D₄×C₁₃₇	—	2	2	no	—
111	D₄×C₁₃₇	—	2	1	no	—
112	D₄×C₁₃₇	C₄⋊C₈	1	2	yes	—
113	C₂×C₄×C₁₃₇	—	2	1	no	—
114	C₂×C₄×C₁₃₇	—	2	2	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	2	1	no	—
118	C₂×C₄×C₁₃₇	—	2	1	no	—
119	C₂×C₄×C₁₃₇	—	4	1	no	—
120	C₂×C₄×C₁₃₇	—	4	1	no	—
121	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	#131	1	1	yes	characteristic
127	C₈×C₁₃₇	—	2	2	no	—
128	C₈×C₁₃₇	—	2	1	no	—
129	C₈×C₁₃₇	—	2	1	no	—
130	C₈×C₁₃₇	—	4	1	no	—
131	C₈×C₁₃₇	#120	1	1	yes	characteristic
132	C₈×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
133	C₈×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
134	C₈×C₁₃₇	#123	1	1	yes	characteristic
135	C₂²×C₁₃₇	—	2	1	no	—
136	C₂²×C₁₃₇	—	2	1	no	—
137	C₂²×C₁₃₇	—	2	2	no	—
138	C₂²×C₁₃₇	—	4	1	no	—
139	C₂²×C₁₃₇	—	4	1	no	—
140	C₂²×C₁₃₇	#424	1	1	yes	characteristic
141	C₂²×C₁₃₇	C₈×D₄	1	1	yes	characteristic
142	C₂²×C₁₃₇	#435	1	1	yes	characteristic
143	C₄×C₁₃₇	—	2	2	no	—
144	C₄×C₁₃₇	—	4	2	no	—
145	C₄×C₁₃₇	—	4	1	no	—
146	C₄×C₁₃₇	—	4	1	no	—
147	C₄×C₁₃₇	#385	1	1	yes	characteristic
148	C₄×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
149	C₄×C₁₃₇	C₈×D₄	1	1	yes	characteristic
150	C₄×C₁₃₇	#435	1	1	yes	characteristic
151	C₂×C₁₃₇	—	2	1	no	—
152	C₂×C₁₃₇	—	2	1	no	—
153	C₂×C₁₃₇	—	4	2	no	—
154	C₂×C₁₃₇	#1850	1	1	yes	characteristic
155	C₂×C₁₃₇	#2170	1	1	yes	characteristic
156	C₂×C₁₃₇	#2171	1	1	yes	characteristic
157	#12917	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₁₃₇	#12917	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×#504	C₂×C₁₃₇	1	1	yes	characteristic
160	C₂×#364	C₂×C₁₃₇	1	1	yes	characteristic, complemented
161	#2151	C₂×C₁₃₇	1	1	yes	characteristic, complemented
162	C₂×#350	C₂×C₁₃₇	1	1	yes	characteristic
163	#1764	C₂×C₁₃₇	1	1	yes	characteristic
164	#1379	C₂×C₁₃₇	1	1	yes	characteristic, complemented
165	#1374	C₂×C₁₃₇	1	1	yes	characteristic, complemented
166	C₂×#131	C₂²×C₁₃₇	1	1	yes	characteristic
167	#504	—	2	2	no	—
168	#504	—	2	2	no	—
169	C₂×M₃₂	—	2	1	no	—
170	C₂×M₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₂×M₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂²×C₁₆	—	2	1	no	—
173	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
174	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
175	C₂×#104	C₄×C₁₃₇	1	1	yes	characteristic
176	#416	C₄×C₁₃₇	1	1	yes	characteristic
177	#405	C₂²×C₁₃₇	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
179	#364	—	2	2	no	—
180	C₄⋊C₁₆	—	2	1	no	—
181	C₄⋊C₁₆	—	2	1	no	—
182	#350	—	2	2	no	—
183	#341	C₄×C₁₃₇	1	1	yes	characteristic
184	#339	C₄×C₁₃₇	1	1	yes	characteristic
185	C₂×#41	C₂×C₄×C₁₃₇	1	1	yes	characteristic
186	#131	—	2	1	no	—
187	#131	—	2	1	no	—
188	#131	D₄×C₁₃₇	1	2	yes	—
189	#131	Q₈×C₁₃₇	1	2	yes	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
192	C₂×M₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
196	#117	C₂×C₄×C₁₃₇	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
198	M₃₂	—	2	4	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	D₄×C₁₃₇	1	2	yes	—
203	C₂×C₁₆	—	2	4	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	2	1	no	—
211	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
212	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
213	#104	—	2	2	no	—
214	#103	C₈×C₁₃₇	1	1	yes	characteristic
215	#103	C₈×C₁₃₇	1	1	yes	characteristic
216	#102	C₈×C₁₃₇	1	1	yes	characteristic
217	#102	C₈×C₁₃₇	1	1	yes	characteristic
218	#97	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	#41	—	2	1	no	—
223	#41	—	2	1	no	—
224	#41	C₄⋊C₄×C₁₃₇	1	2	yes	—
225	#41	M₁₆×C₁₃₇	1	2	yes	—
226	C₂×Q₈	C₂×C₈×C₁₃₇	1	1	yes	characteristic
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
232	M₁₆	—	2	1	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	1	no	—
236	M₁₆	—	2	1	no	—
237	M₁₆	C₄⋊C₄×C₁₃₇	1	2	yes	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	C₂×C₈×C₁₃₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₈×C₁₃₇	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
247	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
249	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
250	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
251	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
252	C₄⋊C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
253	C₄⋊C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₁₆	—	2	1	no	—
259	C₁₆	—	2	2	no	—
260	C₁₆	—	2	1	no	—
261	C₁₆	—	4	1	no	—
262	C₁₆	—	4	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	#131×C₁₃₇	1	1	yes	characteristic
265	Q₈	—	2	1	no	—
266	Q₈	C₄⋊C₈×C₁₃₇	1	2	yes	—
267	D₄	—	2	2	no	—
268	D₄	—	2	2	no	—
269	D₄	—	2	1	no	—
270	D₄	C₄⋊C₈×C₁₃₇	1	2	yes	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
280	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
281	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
283	C₂×C₄	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
284	C₂×C₄	#131×C₁₃₇	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	4	1	no	—
289	C₈	#120×C₁₃₇	1	1	yes	characteristic
290	C₈	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
291	C₈	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
292	C₈	#123×C₁₃₇	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	#424×C₁₃₇	1	1	yes	characteristic
299	C₂²	C₈×D₄×C₁₃₇	1	1	yes	characteristic
300	C₂²	#435×C₁₃₇	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	#385×C₁₃₇	1	1	yes	characteristic
306	C₄	C₂×C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
307	C₄	C₈×D₄×C₁₃₇	1	1	yes	characteristic
308	C₄	#435×C₁₃₇	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	4	2	no	—
312	C₂	#1850×C₁₃₇	1	1	yes	characteristic
313	C₂	#2170×C₁₃₇	1	1	yes	characteristic
314	C₂	#2171×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₈×D₄
prime radical	C₂×#41×C₁₃₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₇ ◁ C₂²×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₁₃₇	C₂	semidirect
3. #2151×C₁₃₇	C₂	semidirect
6. #1379×C₁₃₇	C₂	semidirect
7. #1374×C₁₃₇	C₂	semidirect
157. #12917	C₁₃₇	direct (N × H)
158. C₁₃₇	#12917	direct (N × H)
160. C₂×#364	C₂×C₁₃₇	semidirect
161. #2151	C₂×C₁₃₇	semidirect
164. #1379	C₂×C₁₃₇	semidirect
165. #1374	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×#504×C₁₃₇	2
C₂×#364×C₁₃₇	2
#2151×C₁₃₇	2
C₂×#350×C₁₃₇	2
#1764×C₁₃₇	2
#1379×C₁₃₇	2
#1374×C₁₃₇	2
#12917	137

Subgroups — C137×#12917 #46862303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#12917 #46862303