Subgroups — C₁₃₇×#20306 #46869692

Order 35072.

590 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3367×C₁₃₇	C₂	1	2	yes	maximal
2	#3328×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#1710×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1700×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1699×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1650×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1649×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1591×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1589×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	#556×C₁₃₇	C₂²	1	2	yes	—
11	D₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
12	#500×C₁₃₇	C₂²	1	1	yes	characteristic
13	#500×C₁₃₇	C₂²	1	1	yes	characteristic
14	#489×C₁₃₇	—	2	2	no	—
15	#487×C₁₃₇	C₂²	1	2	yes	—
16	#487×C₁₃₇	C₂²	1	2	yes	—
17	#485×C₁₃₇	—	2	2	no	—
18	#477×C₁₃₇	—	2	2	no	—
19	#475×C₁₃₇	C₂²	1	2	yes	—
20	#474×C₁₃₇	C₂²	1	2	yes	—
21	#470×C₁₃₇	—	2	2	no	—
22	#463×C₁₃₇	—	2	2	no	—
23	#462×C₁₃₇	C₂²	1	2	yes	—
24	#461×C₁₃₇	C₂²	1	2	yes	—
25	#451×C₁₃₇	—	2	2	no	—
26	#446×C₁₃₇	C₂²	1	2	yes	—
27	#440×C₁₃₇	—	2	2	no	—
28	C₄×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	#436×C₁₃₇	C₂²	1	2	yes	—
31	#419×C₁₃₇	—	2	2	no	—
32	#335×C₁₃₇	C₂²	1	1	yes	characteristic
33	#335×C₁₃₇	C₂²	1	1	yes	characteristic
34	#332×C₁₃₇	C₂²	1	2	yes	—
35	#332×C₁₃₇	C₂²	1	2	yes	—
36	Q₈⋊C₈×C₁₃₇	C₂²	1	2	yes	—
37	D₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
38	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₁₃₇	—	2	2	no	—
40	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
41	C₂×Q₁₆×C₁₃₇	D₄	1	1	yes	characteristic
42	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
43	C₂×D₈×C₁₃₇	D₄	1	1	yes	characteristic
44	C₂×M₁₆×C₁₃₇	—	2	2	no	—
45	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
46	C₄⋊Q₈×C₁₃₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₁₃₇	C₂³	1	2	yes	—
48	#126×C₁₃₇	—	4	2	no	—
49	#124×C₁₃₇	—	2	2	no	—
50	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
51	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
52	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
53	C₄×Q₈×C₁₃₇	—	2	2	no	—
54	C₄×Q₈×C₁₃₇	C₂³	1	2	yes	—
55	C₄×D₄×C₁₃₇	—	2	2	no	—
56	C₄×D₄×C₁₃₇	C₂³	1	2	yes	—
57	#117×C₁₃₇	—	2	2	no	—
58	#107×C₁₃₇	—	2	2	no	—
59	#107×C₁₃₇	—	4	1	no	—
60	#107×C₁₃₇	—	4	1	no	—
61	#107×C₁₃₇	D₄	1	1	yes	characteristic
62	#107×C₁₃₇	D₄	1	1	yes	characteristic
63	#106×C₁₃₇	—	2	2	no	—
64	C₄⋊C₈×C₁₃₇	—	2	2	no	—
65	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
66	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
67	#103×C₁₃₇	—	2	2	no	—
68	#103×C₁₃₇	—	2	2	no	—
69	#103×C₁₃₇	—	2	2	no	—
70	#103×C₁₃₇	—	2	2	no	—
71	#103×C₁₃₇	—	2	2	no	—
72	#103×C₁₃₇	—	2	2	no	—
73	#103×C₁₃₇	—	2	2	no	—
74	#103×C₁₃₇	D₄	1	2	yes	—
75	#102×C₁₃₇	—	2	2	no	—
76	#102×C₁₃₇	D₄	1	2	yes	—
77	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
78	#97×C₁₃₇	—	2	2	no	—
79	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₁₃₇	—	2	2	no	—
83	C₂×Q₈×C₁₃₇	—	2	2	no	—
84	C₂×Q₈×C₁₃₇	—	4	2	no	—
85	C₂×Q₈×C₁₃₇	—	4	4	no	—
86	C₂×Q₈×C₁₃₇	—	8	2	no	—
87	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₃₇	—	2	2	no	—
90	C₂²×C₄×C₁₃₇	—	4	2	no	—
91	Q₁₆×C₁₃₇	—	4	1	no	—
92	Q₁₆×C₁₃₇	—	8	2	no	—
93	SD₁₆×C₁₃₇	—	4	4	no	—
94	D₈×C₁₃₇	—	4	1	no	—
95	M₁₆×C₁₃₇	—	8	2	no	—
96	C₂×C₈×C₁₃₇	—	2	1	no	—
97	C₂×C₈×C₁₃₇	—	2	2	no	—
98	C₂×C₈×C₁₃₇	—	2	2	no	—
99	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₇	—	2	2	no	—
104	C₄⋊C₄×C₁₃₇	—	2	2	no	—
105	C₄⋊C₄×C₁₃₇	—	2	2	no	—
106	C₄⋊C₄×C₁₃₇	—	2	2	no	—
107	C₄⋊C₄×C₁₃₇	—	2	2	no	—
108	C₄⋊C₄×C₁₃₇	—	4	2	no	—
109	C₄⋊C₄×C₁₃₇	—	4	2	no	—
110	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
113	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
114	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
115	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
116	C₄²×C₁₃₇	—	2	2	no	—
117	C₄²×C₁₃₇	—	4	2	no	—
118	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
119	C₂³×C₁₃₇	—	2	2	no	—
120	Q₈×C₁₃₇	—	2	2	no	—
121	Q₈×C₁₃₇	—	2	2	no	—
122	Q₈×C₁₃₇	—	4	2	no	—
123	Q₈×C₁₃₇	—	4	2	no	—
124	Q₈×C₁₃₇	—	4	2	no	—
125	Q₈×C₁₃₇	—	4	2	no	—
126	Q₈×C₁₃₇	—	8	4	no	—
127	D₄×C₁₃₇	—	2	2	no	—
128	D₄×C₁₃₇	—	2	2	no	—
129	C₂×C₄×C₁₃₇	—	2	2	no	—
130	C₂×C₄×C₁₃₇	—	2	2	no	—
131	C₂×C₄×C₁₃₇	—	2	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	—	8	4	no	—
137	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
140	C₈×C₁₃₇	—	2	2	no	—
141	C₈×C₁₃₇	—	2	1	no	—
142	C₈×C₁₃₇	—	2	1	no	—
143	C₈×C₁₃₇	—	2	1	no	—
144	C₈×C₁₃₇	—	2	1	no	—
145	C₈×C₁₃₇	—	4	2	no	—
146	C₈×C₁₃₇	—	4	2	no	—
147	C₂²×C₁₃₇	—	4	2	no	—
148	C₂²×C₁₃₇	—	4	2	no	—
149	C₂²×C₁₃₇	—	4	2	no	—
150	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
151	C₄×C₁₃₇	—	2	1	no	—
152	C₄×C₁₃₇	—	4	2	no	—
153	C₄×C₁₃₇	—	4	2	no	—
154	C₄×C₁₃₇	—	4	2	no	—
155	C₄×C₁₃₇	—	8	2	no	—
156	C₄×C₁₃₇	—	8	2	no	—
157	C₄×C₁₃₇	#575	1	1	yes	characteristic
158	C₄×C₁₃₇	#575	1	1	yes	characteristic
159	C₄×C₁₃₇	#576	1	1	yes	characteristic
160	C₄×C₁₃₇	#578	1	1	yes	characteristic
161	C₂×C₁₃₇	—	8	2	no	—
162	C₂×C₁₃₇	#3273	1	1	yes	characteristic
163	C₂×C₁₃₇	#3277	1	1	yes	characteristic
164	C₂×C₁₃₇	#3293	1	1	yes	characteristic
165	#20306	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₇	#20306	1	1	yes	characteristic, direct product factor, Sylow
167	#3367	C₂×C₁₃₇	1	2	yes	—
168	#3328	C₂×C₁₃₇	1	2	yes	complemented
169	#1710	C₂×C₁₃₇	1	1	yes	characteristic, complemented
170	#1700	C₂×C₁₃₇	1	2	yes	complemented
171	#1699	C₂×C₁₃₇	1	2	yes	complemented
172	#1650	C₂×C₁₃₇	1	2	yes	complemented
173	#1649	C₂×C₁₃₇	1	2	yes	complemented
174	#1591	C₂×C₁₃₇	1	1	yes	characteristic, complemented
175	#1589	C₂×C₁₃₇	1	1	yes	characteristic
176	#556	C₂²×C₁₃₇	1	2	yes	—
177	D₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
178	#500	C₂²×C₁₃₇	1	1	yes	characteristic
179	#500	C₂²×C₁₃₇	1	1	yes	characteristic
180	#489	—	2	2	no	—
181	#487	C₂²×C₁₃₇	1	2	yes	—
182	#487	C₂²×C₁₃₇	1	2	yes	—
183	#485	—	2	2	no	—
184	#477	—	2	2	no	—
185	#475	C₂²×C₁₃₇	1	2	yes	—
186	#474	C₂²×C₁₃₇	1	2	yes	—
187	#470	—	2	2	no	—
188	#463	—	2	2	no	—
189	#462	C₂²×C₁₃₇	1	2	yes	—
190	#461	C₂²×C₁₃₇	1	2	yes	—
191	#451	—	2	2	no	—
192	#446	C₂²×C₁₃₇	1	2	yes	—
193	#440	—	2	2	no	—
194	C₄×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
195	C₄×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
196	#436	C₂²×C₁₃₇	1	2	yes	—
197	#419	—	2	2	no	—
198	#335	C₂²×C₁₃₇	1	1	yes	characteristic
199	#335	C₂²×C₁₃₇	1	1	yes	characteristic
200	#332	C₂²×C₁₃₇	1	2	yes	—
201	#332	C₂²×C₁₃₇	1	2	yes	—
202	Q₈⋊C₈	C₂²×C₁₃₇	1	2	yes	—
203	D₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
204	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
205	C₂²×Q₈	—	2	2	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	D₄×C₁₃₇	1	1	yes	characteristic
208	C₂×SD₁₆	—	2	2	no	—
209	C₂×D₈	D₄×C₁₃₇	1	1	yes	characteristic
210	C₂×M₁₆	—	2	2	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	C₂³×C₁₃₇	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₁₃₇	1	2	yes	—
214	#126	—	4	2	no	—
215	#124	—	2	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	4	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	C₂³×C₁₃₇	1	2	yes	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	C₂³×C₁₃₇	1	2	yes	—
223	#117	—	2	2	no	—
224	#107	—	2	2	no	—
225	#107	—	4	1	no	—
226	#107	—	4	1	no	—
227	#107	D₄×C₁₃₇	1	1	yes	characteristic
228	#107	D₄×C₁₃₇	1	1	yes	characteristic
229	#106	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
232	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	#103	D₄×C₁₃₇	1	2	yes	—
241	#102	—	2	2	no	—
242	#102	D₄×C₁₃₇	1	2	yes	—
243	C₂²⋊C₈	—	2	2	no	—
244	#97	—	2	2	no	—
245	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	8	2	no	—
253	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
254	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	Q₁₆	—	4	1	no	—
258	Q₁₆	—	8	2	no	—
259	SD₁₆	—	4	4	no	—
260	D₈	—	4	1	no	—
261	M₁₆	—	8	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
277	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	4	no	—
293	D₄	—	2	2	no	—
294	D₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	4	no	—
303	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
305	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	#575×C₁₃₇	1	1	yes	characteristic
324	C₄	#575×C₁₃₇	1	1	yes	characteristic
325	C₄	#576×C₁₃₇	1	1	yes	characteristic
326	C₄	#578×C₁₃₇	1	1	yes	characteristic
327	C₂	—	8	2	no	—
328	C₂	#3273×C₁₃₇	1	1	yes	characteristic
329	C₂	#3277×C₁₃₇	1	1	yes	characteristic
330	C₂	#3293×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	C₂×D₈×C₁₃₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #3328×C₁₃₇	C₂	semidirect
3. #1710×C₁₃₇	C₂	semidirect
4. #1700×C₁₃₇	C₂	semidirect
5. #1699×C₁₃₇	C₂	semidirect
6. #1650×C₁₃₇	C₂	semidirect
7. #1649×C₁₃₇	C₂	semidirect
8. #1591×C₁₃₇	C₂	semidirect
165. #20306	C₁₃₇	direct (N × H)
166. C₁₃₇	#20306	direct (N × H)
168. #3328	C₂×C₁₃₇	semidirect
169. #1710	C₂×C₁₃₇	semidirect
170. #1700	C₂×C₁₃₇	semidirect
171. #1699	C₂×C₁₃₇	semidirect
172. #1650	C₂×C₁₃₇	semidirect
173. #1649	C₂×C₁₃₇	semidirect
174. #1591	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3367×C₁₃₇	2
#3328×C₁₃₇	2
#1710×C₁₃₇	2
#1700×C₁₃₇	2
#1699×C₁₃₇	2
#1650×C₁₃₇	2
#1649×C₁₃₇	2
#1591×C₁₃₇	2
#1589×C₁₃₇	2
#20306	137

Subgroups — C137×#20306 #46869692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20306 #46869692