Subgroups — C₁₃₇×#33936 #46883322

Order 35072.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3393×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2243×C₁₃₇	C₂	1	2	yes	maximal
4	#2241×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2226×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#2225×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2211×C₁₃₇	C₂	1	2	yes	maximal
8	#2209×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#2183×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#2176×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₁₃₇	—	2	1	no	—
13	#556×C₁₃₇	—	2	1	no	—
14	C₂×Q₃₂×C₁₃₇	C₂²	1	2	yes	—
15	C₂×SD₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
16	#500×C₁₃₇	C₂²	1	2	yes	—
17	#500×C₁₃₇	C₂²	1	1	yes	characteristic
18	#495×C₁₃₇	C₂²	1	2	yes	—
19	#494×C₁₃₇	C₂²	1	1	yes	characteristic
20	#477×C₁₃₇	—	2	1	no	—
21	#477×C₁₃₇	—	2	2	no	—
22	#464×C₁₃₇	—	2	2	no	—
23	#461×C₁₃₇	—	2	1	no	—
24	C₈×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₄×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
28	C₄×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
29	#367×C₁₃₇	C₂²	1	1	yes	characteristic
30	#367×C₁₃₇	C₂²	1	2	yes	—
31	C₄⋊C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆×C₁₃₇	C₂²	1	2	yes	—
33	#358×C₁₃₇	C₂²	1	1	yes	characteristic
34	#358×C₁₃₇	C₂²	1	2	yes	—
35	#358×C₁₃₇	C₂²	1	2	yes	—
36	#358×C₁₃₇	C₂²	1	2	yes	—
37	#357×C₁₃₇	C₂²	1	1	yes	characteristic
38	#357×C₁₃₇	C₂²	1	2	yes	—
39	#346×C₁₃₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₁₃₇	—	2	1	no	—
42	C₂×Q₁₆×C₁₃₇	C₂³	1	2	yes	—
43	C₂×Q₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
45	C₂×D₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
47	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
48	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
49	C₄⋊Q₈×C₁₃₇	—	4	1	no	—
50	C₄⋊Q₈×C₁₃₇	—	4	2	no	—
51	#126×C₁₃₇	—	4	2	no	—
52	#124×C₁₃₇	—	2	2	no	—
53	C₂²⋊Q₈×C₁₃₇	—	4	1	no	—
54	C₄×Q₈×C₁₃₇	—	2	2	no	—
55	C₄×Q₈×C₁₃₇	—	2	1	no	—
56	C₄×Q₈×C₁₃₇	—	2	2	no	—
57	C₄×Q₈×C₁₃₇	—	4	1	no	—
58	C₄×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
59	C₄×D₄×C₁₃₇	—	2	1	no	—
60	#117×C₁₃₇	—	4	1	no	—
61	Q₃₂×C₁₃₇	—	2	4	no	—
62	SD₃₂×C₁₃₇	—	2	2	no	—
63	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₁₃₇	C₂³	1	2	yes	—
65	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
66	#107×C₁₃₇	—	2	1	no	—
67	#107×C₁₃₇	—	2	2	no	—
68	#107×C₁₃₇	C₂³	1	1	yes	characteristic
69	#107×C₁₃₇	C₂³	1	1	yes	characteristic
70	#107×C₁₃₇	C₂³	1	2	yes	—
71	C₄⋊C₈×C₁₃₇	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
73	#103×C₁₃₇	—	2	2	no	—
74	#103×C₁₃₇	—	2	1	no	—
75	#103×C₁₃₇	—	2	2	no	—
76	#102×C₁₃₇	—	2	1	no	—
77	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
79	C₂×Q₈×C₁₃₇	—	2	1	no	—
80	C₂×Q₈×C₁₃₇	—	2	2	no	—
81	C₂×Q₈×C₁₃₇	—	4	2	no	—
82	C₂×Q₈×C₁₃₇	—	4	1	no	—
83	C₂×Q₈×C₁₃₇	D₈	1	1	yes	characteristic
84	C₂×D₄×C₁₃₇	—	2	1	no	—
85	C₂²×C₄×C₁₃₇	—	4	1	no	—
86	Q₁₆×C₁₃₇	—	2	2	no	—
87	Q₁₆×C₁₃₇	—	2	1	no	—
88	Q₁₆×C₁₃₇	—	2	2	no	—
89	Q₁₆×C₁₃₇	—	8	1	no	—
90	Q₁₆×C₁₃₇	#41	1	2	yes	—
91	SD₁₆×C₁₃₇	—	8	2	no	—
92	D₈×C₁₃₇	—	2	1	no	—
93	D₈×C₁₃₇	—	2	1	no	—
94	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₁₃₇	—	2	1	no	—
98	C₄⋊C₄×C₁₃₇	—	2	1	no	—
99	C₄⋊C₄×C₁₃₇	—	2	2	no	—
100	C₄⋊C₄×C₁₃₇	—	4	1	no	—
101	C₄⋊C₄×C₁₃₇	—	4	2	no	—
102	C₄⋊C₄×C₁₃₇	—	4	2	no	—
103	C₄⋊C₄×C₁₃₇	—	4	1	no	—
104	C₄⋊C₄×C₁₃₇	—	4	2	no	—
105	C₄⋊C₄×C₁₃₇	—	4	1	no	—
106	C₄⋊C₄×C₁₃₇	—	4	1	no	—
107	C₄⋊C₄×C₁₃₇	D₈	1	1	yes	characteristic
108	C₄⋊C₄×C₁₃₇	D₈	1	2	yes	—
109	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
110	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
111	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
112	C₄²×C₁₃₇	—	4	2	no	—
113	C₄²×C₁₃₇	—	4	2	no	—
114	C₄²×C₁₃₇	—	4	1	no	—
115	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
117	C₁₆×C₁₃₇	—	2	1	no	—
118	C₁₆×C₁₃₇	—	2	2	no	—
119	C₁₆×C₁₃₇	—	2	1	no	—
120	C₂³×C₁₃₇	—	4	1	no	—
121	Q₈×C₁₃₇	—	2	2	no	—
122	Q₈×C₁₃₇	—	2	2	no	—
123	Q₈×C₁₃₇	—	4	1	no	—
124	Q₈×C₁₃₇	—	4	2	no	—
125	Q₈×C₁₃₇	—	4	2	no	—
126	Q₈×C₁₃₇	—	16	2	no	—
127	Q₈×C₁₃₇	—	16	1	no	—
128	D₄×C₁₃₇	—	4	1	no	—
129	D₄×C₁₃₇	—	4	1	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	1	no	—
135	C₂×C₄×C₁₃₇	—	8	2	no	—
136	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₁₃₇	—	2	2	no	—
142	C₈×C₁₃₇	—	2	1	no	—
143	C₈×C₁₃₇	C₂×#41	1	1	yes	characteristic
144	C₈×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²×C₁₃₇	—	8	1	no	—
146	C₂²×C₁₃₇	—	8	1	no	—
147	C₂²×C₁₃₇	—	8	1	no	—
148	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₁₃₇	—	2	2	no	—
150	C₄×C₁₃₇	—	2	2	no	—
151	C₄×C₁₃₇	—	2	1	no	—
152	C₄×C₁₃₇	—	4	2	no	—
153	C₄×C₁₃₇	—	8	2	no	—
154	C₄×C₁₃₇	—	8	1	no	—
155	C₄×C₁₃₇	—	8	2	no	—
156	C₄×C₁₃₇	—	8	1	no	—
157	C₄×C₁₃₇	#510	1	2	yes	—
158	C₄×C₁₃₇	#548	1	1	yes	characteristic
159	C₄×C₁₃₇	#577	1	1	yes	characteristic
160	C₂×C₁₃₇	—	16	1	no	—
161	C₂×C₁₃₇	#3358	1	1	yes	characteristic
162	C₂×C₁₃₇	C₂×#510	1	1	yes	characteristic
163	C₂×C₁₃₇	#3416	1	1	yes	characteristic
164	#33936	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₁₃₇	#33936	1	1	yes	characteristic, direct product factor, Sylow
166	#3393	C₂×C₁₃₇	1	1	yes	characteristic
167	Q₈×Q₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
168	#2243	C₂×C₁₃₇	1	2	yes	—
169	#2241	C₂×C₁₃₇	1	1	yes	characteristic, complemented
170	#2226	C₂×C₁₃₇	1	1	yes	characteristic
171	#2225	C₂×C₁₃₇	1	2	yes	complemented
172	#2211	C₂×C₁₃₇	1	2	yes	—
173	#2209	C₂×C₁₃₇	1	1	yes	characteristic, complemented
174	#2183	C₂×C₁₃₇	1	1	yes	characteristic, complemented
175	#2176	C₂×C₁₃₇	1	2	yes	complemented
176	C₄×SD₃₂	C₂×C₁₃₇	1	1	yes	characteristic
177	Q₈²	—	2	1	no	—
178	#556	—	2	1	no	—
179	C₂×Q₃₂	C₂²×C₁₃₇	1	2	yes	—
180	C₂×SD₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
181	#500	C₂²×C₁₃₇	1	2	yes	—
182	#500	C₂²×C₁₃₇	1	1	yes	characteristic
183	#495	C₂²×C₁₃₇	1	2	yes	—
184	#494	C₂²×C₁₃₇	1	1	yes	characteristic
185	#477	—	2	1	no	—
186	#477	—	2	2	no	—
187	#464	—	2	2	no	—
188	#461	—	2	1	no	—
189	C₈×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
190	C₄×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
191	C₄×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
192	C₄×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
193	C₄×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
194	#367	C₂²×C₁₃₇	1	1	yes	characteristic
195	#367	C₂²×C₁₃₇	1	2	yes	—
196	C₄⋊C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
197	C₄⋊C₁₆	C₂²×C₁₃₇	1	2	yes	—
198	#358	C₂²×C₁₃₇	1	1	yes	characteristic
199	#358	C₂²×C₁₃₇	1	2	yes	—
200	#358	C₂²×C₁₃₇	1	2	yes	—
201	#358	C₂²×C₁₃₇	1	2	yes	—
202	#357	C₂²×C₁₃₇	1	1	yes	characteristic
203	#357	C₂²×C₁₃₇	1	2	yes	—
204	#346	C₂²×C₁₃₇	1	2	yes	—
205	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	C₂³×C₁₃₇	1	2	yes	—
208	C₂×Q₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×D₈	C₂³×C₁₃₇	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	4	1	no	—
215	C₄⋊Q₈	—	4	2	no	—
216	#126	—	4	2	no	—
217	#124	—	2	2	no	—
218	C₂²⋊Q₈	—	4	1	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	4	1	no	—
223	C₄×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
224	C₄×D₄	—	2	1	no	—
225	#117	—	4	1	no	—
226	Q₃₂	—	2	4	no	—
227	SD₃₂	—	2	2	no	—
228	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₁₃₇	1	2	yes	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#107	C₂³×C₁₃₇	1	1	yes	characteristic
234	#107	C₂³×C₁₃₇	1	2	yes	—
235	#107	C₂³×C₁₃₇	1	1	yes	characteristic
236	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
237	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	1	no	—
240	#103	—	2	2	no	—
241	#102	—	2	1	no	—
242	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	D₈×C₁₃₇	1	1	yes	characteristic
249	C₂×D₄	—	2	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	2	1	no	—
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	8	1	no	—
255	Q₁₆	#41×C₁₃₇	1	2	yes	—
256	SD₁₆	—	8	2	no	—
257	D₈	—	2	1	no	—
258	D₈	—	2	1	no	—
259	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
261	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	D₈×C₁₃₇	1	1	yes	characteristic
273	C₄⋊C₄	D₈×C₁₃₇	1	2	yes	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
282	C₁₆	—	2	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₂³	—	4	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	16	2	no	—
292	Q₈	—	16	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
302	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
303	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×#41×C₁₃₇	1	1	yes	characteristic
309	C₈	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	1	no	—
322	C₄	#510×C₁₃₇	1	2	yes	—
323	C₄	#548×C₁₃₇	1	1	yes	characteristic
324	C₄	#577×C₁₃₇	1	1	yes	characteristic
325	C₂	—	16	1	no	—
326	C₂	#3358×C₁₃₇	1	1	yes	characteristic
327	C₂	C₂×#510×C₁₃₇	1	1	yes	characteristic
328	C₂	#3416×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
prime radical	C₂×D₈×C₁₃₇	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄×Q₈×C₁₃₇ ◁ C₈×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×Q₁₆×C₁₃₇	C₂	semidirect
4. #2241×C₁₃₇	C₂	semidirect
6. #2225×C₁₃₇	C₂	semidirect
8. #2209×C₁₃₇	C₂	semidirect
9. #2183×C₁₃₇	C₂	semidirect
10. #2176×C₁₃₇	C₂	semidirect
164. #33936	C₁₃₇	direct (N × H)
165. C₁₃₇	#33936	direct (N × H)
167. Q₈×Q₁₆	C₂×C₁₃₇	semidirect
169. #2241	C₂×C₁₃₇	semidirect
171. #2225	C₂×C₁₃₇	semidirect
173. #2209	C₂×C₁₃₇	semidirect
174. #2183	C₂×C₁₃₇	semidirect
175. #2176	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3393×C₁₃₇	2
Q₈×Q₁₆×C₁₃₇	2
#2243×C₁₃₇	2
#2241×C₁₃₇	2
#2226×C₁₃₇	2
#2225×C₁₃₇	2
#2211×C₁₃₇	2
#2209×C₁₃₇	2
#2183×C₁₃₇	2
#2176×C₁₃₇	2
C₄×SD₃₂×C₁₃₇	2
#33936	137

Subgroups — C137×#33936 #46883322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33936 #46883322