Subgroups — C₁₃₇×#33924 #46883310

Order 35072.

482 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3341×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#3338×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2252×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2237×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2232×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2227×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#2226×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#2220×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#2169×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#2159×C₁₃₇	C₂	1	2	yes	maximal, complemented
12	#555×C₁₃₇	—	2	1	no	—
13	#554×C₁₃₇	—	2	1	no	—
14	C₂×SD₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
16	#500×C₁₃₇	C₂²	1	1	yes	characteristic
17	#500×C₁₃₇	C₂²	1	1	yes	characteristic
18	#484×C₁₃₇	—	2	2	no	—
19	#482×C₁₃₇	—	2	2	no	—
20	#477×C₁₃₇	—	2	1	no	—
21	#476×C₁₃₇	—	2	1	no	—
22	#467×C₁₃₇	C₂²	1	2	yes	—
23	#466×C₁₃₇	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
27	#427×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
29	#367×C₁₃₇	C₂²	1	1	yes	characteristic
30	#367×C₁₃₇	C₂²	1	2	yes	—
31	#366×C₁₃₇	C₂²	1	2	yes	—
32	#366×C₁₃₇	C₂²	1	2	yes	—
33	#366×C₁₃₇	C₂²	1	2	yes	—
34	C₄⋊C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
35	#358×C₁₃₇	C₂²	1	1	yes	characteristic
36	#358×C₁₃₇	C₂²	1	2	yes	—
37	#357×C₁₃₇	C₂²	1	1	yes	characteristic
38	#357×C₁₃₇	C₂²	1	2	yes	—
39	C₂²⋊C₁₆×C₁₃₇	C₂²	1	2	yes	—
40	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₂×D₈×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
45	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
46	#125×C₁₃₇	—	4	1	no	—
47	#124×C₁₃₇	—	4	2	no	—
48	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
49	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
50	#121×C₁₃₇	—	2	2	no	—
51	#121×C₁₃₇	—	4	1	no	—
52	C₄×Q₈×C₁₃₇	—	2	1	no	—
53	C₄×D₄×C₁₃₇	—	2	1	no	—
54	C₄×D₄×C₁₃₇	—	4	1	no	—
55	C₄×D₄×C₁₃₇	D₄	1	1	yes	characteristic
56	#117×C₁₃₇	—	2	2	no	—
57	#117×C₁₃₇	—	4	1	no	—
58	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
59	SD₃₂×C₁₃₇	—	2	2	no	—
60	M₃₂×C₁₃₇	—	2	4	no	—
61	C₂×C₁₆×C₁₃₇	C₂³	1	2	yes	—
62	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
64	#107×C₁₃₇	—	2	2	no	—
65	#107×C₁₃₇	—	2	1	no	—
66	#107×C₁₃₇	—	2	1	no	—
67	#107×C₁₃₇	C₂³	1	2	yes	—
68	#107×C₁₃₇	C₂³	1	1	yes	characteristic
69	#107×C₁₃₇	C₂³	1	2	yes	—
70	#107×C₁₃₇	C₂³	1	1	yes	characteristic
71	#106×C₁₃₇	—	2	2	no	—
72	C₄⋊C₈×C₁₃₇	D₄	1	1	yes	characteristic
73	#103×C₁₃₇	—	2	1	no	—
74	#103×C₁₃₇	—	2	2	no	—
75	#102×C₁₃₇	—	2	1	no	—
76	#102×C₁₃₇	—	2	2	no	—
77	C₂²⋊C₈×C₁₃₇	D₄	1	2	yes	—
78	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₃₇	—	2	1	no	—
80	C₂×Q₈×C₁₃₇	—	4	1	no	—
81	C₂×Q₈×C₁₃₇	—	4	1	no	—
82	C₂×D₄×C₁₃₇	—	2	1	no	—
83	C₂×D₄×C₁₃₇	—	4	2	no	—
84	C₂×D₄×C₁₃₇	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₇	—	4	2	no	—
86	C₂²×C₄×C₁₃₇	—	4	1	no	—
87	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
88	Q₁₆×C₁₃₇	—	2	1	no	—
89	Q₁₆×C₁₃₇	—	2	1	no	—
90	D₈×C₁₃₇	—	2	1	no	—
91	D₈×C₁₃₇	—	2	1	no	—
92	C₂×C₈×C₁₃₇	—	2	2	no	—
93	C₂×C₈×C₁₃₇	—	2	2	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	2	no	—
99	C₄⋊C₄×C₁₃₇	—	2	1	no	—
100	C₄⋊C₄×C₁₃₇	—	2	2	no	—
101	C₄⋊C₄×C₁₃₇	—	4	2	no	—
102	C₄⋊C₄×C₁₃₇	—	4	1	no	—
103	C₄⋊C₄×C₁₃₇	—	4	1	no	—
104	C₄⋊C₄×C₁₃₇	—	4	1	no	—
105	C₄⋊C₄×C₁₃₇	—	4	2	no	—
106	C₄⋊C₄×C₁₃₇	D₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
108	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
109	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
110	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
111	C₂²⋊C₄×C₁₃₇	D₈	1	2	yes	—
112	C₄²×C₁₃₇	—	4	2	no	—
113	C₄²×C₁₃₇	—	4	1	no	—
114	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₁₃₇	—	2	2	no	—
116	C₁₆×C₁₃₇	—	2	1	no	—
117	C₁₆×C₁₃₇	#41	1	2	yes	—
118	C₂³×C₁₃₇	—	4	1	no	—
119	C₂³×C₁₃₇	C₂×D₈	1	2	yes	—
120	Q₈×C₁₃₇	—	4	1	no	—
121	Q₈×C₁₃₇	—	4	1	no	—
122	Q₈×C₁₃₇	—	16	1	no	—
123	Q₈×C₁₃₇	—	16	1	no	—
124	D₄×C₁₃₇	—	2	2	no	—
125	D₄×C₁₃₇	—	4	1	no	—
126	D₄×C₁₃₇	—	4	1	no	—
127	D₄×C₁₃₇	—	16	2	no	—
128	C₂×C₄×C₁₃₇	—	2	2	no	—
129	C₂×C₄×C₁₃₇	—	2	2	no	—
130	C₂×C₄×C₁₃₇	—	4	1	no	—
131	C₂×C₄×C₁₃₇	—	4	2	no	—
132	C₂×C₄×C₁₃₇	—	4	1	no	—
133	C₂×C₄×C₁₃₇	—	4	1	no	—
134	C₂×C₄×C₁₃₇	—	4	2	no	—
135	C₂×C₄×C₁₃₇	—	8	2	no	—
136	C₂×C₄×C₁₃₇	—	16	2	no	—
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	—	2	2	no	—
146	C₂²×C₁₃₇	—	2	2	no	—
147	C₂²×C₁₃₇	—	2	2	no	—
148	C₂²×C₁₃₇	—	8	1	no	—
149	C₂²×C₁₃₇	—	8	1	no	—
150	C₂²×C₁₃₇	—	8	1	no	—
151	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₁₃₇	—	2	2	no	—
153	C₄×C₁₃₇	—	2	1	no	—
154	C₄×C₁₃₇	—	8	1	no	—
155	C₄×C₁₃₇	—	8	1	no	—
156	C₄×C₁₃₇	—	8	1	no	—
157	C₄×C₁₃₇	—	8	2	no	—
158	C₄×C₁₃₇	—	8	2	no	—
159	C₄×C₁₃₇	#510	1	2	yes	—
160	C₄×C₁₃₇	#547	1	1	yes	characteristic
161	C₄×C₁₃₇	#578	1	1	yes	characteristic
162	C₂×C₁₃₇	—	4	2	no	—
163	C₂×C₁₃₇	—	16	1	no	—
164	C₂×C₁₃₇	#3334	1	1	yes	characteristic
165	C₂×C₁₃₇	C₂×#510	1	1	yes	characteristic
166	C₂×C₁₃₇	#3415	1	1	yes	characteristic
167	#33924	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₁₃₇	#33924	1	1	yes	characteristic, direct product factor, Sylow
169	#3341	C₂×C₁₃₇	1	1	yes	characteristic
170	#3338	C₂×C₁₃₇	1	1	yes	characteristic, complemented
171	#2252	C₂×C₁₃₇	1	1	yes	characteristic, complemented
172	#2237	C₂×C₁₃₇	1	2	yes	complemented
173	#2232	C₂×C₁₃₇	1	2	yes	complemented
174	#2227	C₂×C₁₃₇	1	1	yes	characteristic, complemented
175	#2226	C₂×C₁₃₇	1	1	yes	characteristic, complemented
176	#2220	C₂×C₁₃₇	1	2	yes	complemented
177	C₄×SD₃₂	C₂×C₁₃₇	1	1	yes	characteristic, complemented
178	#2169	C₂×C₁₃₇	1	1	yes	characteristic, complemented
179	#2159	C₂×C₁₃₇	1	2	yes	complemented
180	#555	—	2	1	no	—
181	#554	—	2	1	no	—
182	C₂×SD₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
183	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
184	#500	C₂²×C₁₃₇	1	1	yes	characteristic
185	#500	C₂²×C₁₃₇	1	1	yes	characteristic
186	#484	—	2	2	no	—
187	#482	—	2	2	no	—
188	#477	—	2	1	no	—
189	#476	—	2	1	no	—
190	#467	C₂²×C₁₃₇	1	2	yes	—
191	#466	C₂²×C₁₃₇	1	2	yes	—
192	C₄×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
193	C₄×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
194	C₈×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
195	#427	C₂²×C₁₃₇	1	2	yes	—
196	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
197	#367	C₂²×C₁₃₇	1	1	yes	characteristic
198	#367	C₂²×C₁₃₇	1	2	yes	—
199	#366	C₂²×C₁₃₇	1	2	yes	—
200	#366	C₂²×C₁₃₇	1	2	yes	—
201	#366	C₂²×C₁₃₇	1	2	yes	—
202	C₄⋊C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
203	#358	C₂²×C₁₃₇	1	1	yes	characteristic
204	#358	C₂²×C₁₃₇	1	2	yes	—
205	#357	C₂²×C₁₃₇	1	1	yes	characteristic
206	#357	C₂²×C₁₃₇	1	2	yes	—
207	C₂²⋊C₁₆	C₂²×C₁₃₇	1	2	yes	—
208	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
209	C₂×Q₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
210	C₂×D₈	C₂³×C₁₃₇	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	#125	—	4	1	no	—
215	#124	—	4	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	#121	—	2	2	no	—
219	#121	—	4	1	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	4	1	no	—
223	C₄×D₄	D₄×C₁₃₇	1	1	yes	characteristic
224	#117	—	2	2	no	—
225	#117	—	4	1	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	SD₃₂	—	2	2	no	—
228	M₃₂	—	2	4	no	—
229	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₁₃₇	1	2	yes	—
231	C₂×C₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
232	#107	—	2	2	no	—
233	#107	—	2	1	no	—
234	#107	—	2	1	no	—
235	#107	C₂³×C₁₃₇	1	1	yes	characteristic
236	#107	C₂³×C₁₃₇	1	2	yes	—
237	#107	C₂³×C₁₃₇	1	2	yes	—
238	#107	C₂³×C₁₃₇	1	1	yes	characteristic
239	#106	—	2	2	no	—
240	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
241	#103	—	2	1	no	—
242	#103	—	2	2	no	—
243	#102	—	2	1	no	—
244	#102	—	2	2	no	—
245	C₂²⋊C₈	D₄×C₁₃₇	1	2	yes	—
246	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	D₈×C₁₃₇	1	1	yes	characteristic
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	1	no	—
258	D₈	—	2	1	no	—
259	D₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
264	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	D₈×C₁₃₇	1	1	yes	characteristic
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	D₈×C₁₃₇	1	2	yes	—
280	C₄²	—	4	2	no	—
281	C₄²	—	4	1	no	—
282	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₁₆	#41×C₁₃₇	1	2	yes	—
286	C₂³	—	4	1	no	—
287	C₂³	C₂×D₈×C₁₃₇	1	2	yes	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	16	1	no	—
292	D₄	—	2	2	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	D₄	—	16	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
306	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
307	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	1	no	—
311	C₈	C₂×#41×C₁₃₇	1	1	yes	characteristic
312	C₈	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	#510×C₁₃₇	1	2	yes	—
328	C₄	#547×C₁₃₇	1	1	yes	characteristic
329	C₄	#578×C₁₃₇	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	16	1	no	—
332	C₂	#3334×C₁₃₇	1	1	yes	characteristic
333	C₂	C₂×#510×C₁₃₇	1	1	yes	characteristic
334	C₂	#3415×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
prime radical	#3341×C₁₃₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄×D₄×C₁₃₇ ◁ C₈×D₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #3338×C₁₃₇	C₂	semidirect
3. #2252×C₁₃₇	C₂	semidirect
4. #2237×C₁₃₇	C₂	semidirect
5. #2232×C₁₃₇	C₂	semidirect
6. #2227×C₁₃₇	C₂	semidirect
7. #2226×C₁₃₇	C₂	semidirect
8. #2220×C₁₃₇	C₂	semidirect
9. C₄×SD₃₂×C₁₃₇	C₂	semidirect
10. #2169×C₁₃₇	C₂	semidirect
11. #2159×C₁₃₇	C₂	semidirect
167. #33924	C₁₃₇	direct (N × H)
168. C₁₃₇	#33924	direct (N × H)
170. #3338	C₂×C₁₃₇	semidirect
171. #2252	C₂×C₁₃₇	semidirect
172. #2237	C₂×C₁₃₇	semidirect
173. #2232	C₂×C₁₃₇	semidirect
174. #2227	C₂×C₁₃₇	semidirect
175. #2226	C₂×C₁₃₇	semidirect
176. #2220	C₂×C₁₃₇	semidirect
177. C₄×SD₃₂	C₂×C₁₃₇	semidirect
178. #2169	C₂×C₁₃₇	semidirect
179. #2159	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3341×C₁₃₇	2
#3338×C₁₃₇	2
#2252×C₁₃₇	2
#2237×C₁₃₇	2
#2232×C₁₃₇	2
#2227×C₁₃₇	2
#2226×C₁₃₇	2
#2220×C₁₃₇	2
C₄×SD₃₂×C₁₃₇	2
#2169×C₁₃₇	2
#2159×C₁₃₇	2
#33924	137

Subgroups — C137×#33924 #46883310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33924 #46883310