Subgroups — C₁₃₇×#9858 #46859244

Order 35072.

428 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3131×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1848×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#1493×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1493×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1492×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1492×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#533×C₁₃₇	C₂²	1	1	yes	characteristic
9	#532×C₁₃₇	C₂²	1	1	yes	characteristic
10	#488×C₁₃₇	—	2	2	no	—
11	#487×C₁₃₇	C₄	1	1	yes	characteristic
12	#487×C₁₃₇	C₄	1	1	yes	characteristic
13	#486×C₁₃₇	C₄	1	1	yes	characteristic
14	#486×C₁₃₇	C₄	1	1	yes	characteristic
15	C₂×#108×C₁₃₇	—	2	2	no	—
16	#423×C₁₃₇	C₂²	1	1	yes	characteristic
17	#423×C₁₃₇	C₂²	1	1	yes	characteristic
18	#416×C₁₃₇	—	2	2	no	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	Q₈⋊C₈×C₁₃₇	—	2	2	no	—
23	Q₈⋊C₈×C₁₃₇	—	2	2	no	—
24	D₄⋊C₈×C₁₃₇	—	2	2	no	—
25	D₄⋊C₈×C₁₃₇	—	2	2	no	—
26	#322×C₁₃₇	—	2	2	no	—
27	C₂×#41×C₁₃₇	—	2	1	no	—
28	C₂×M₁₆×C₁₃₇	—	2	1	no	—
29	C₂×M₁₆×C₁₃₇	—	2	1	no	—
30	C₂²×C₈×C₁₃₇	—	2	1	no	—
31	C₂²×C₈×C₁₃₇	—	2	1	no	—
32	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
33	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	#127×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	#125×C₁₃₇	—	2	1	no	—
38	#124×C₁₃₇	—	2	1	no	—
39	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
40	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
41	#121×C₁₃₇	—	2	1	no	—
42	#121×C₁₃₇	—	2	1	no	—
43	C₄×Q₈×C₁₃₇	—	2	1	no	—
44	C₄×Q₈×C₁₃₇	—	2	1	no	—
45	C₄×D₄×C₁₃₇	—	2	1	no	—
46	C₄×D₄×C₁₃₇	—	2	1	no	—
47	#117×C₁₃₇	—	2	1	no	—
48	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
49	#108×C₁₃₇	—	4	4	no	—
50	C₄⋊C₈×C₁₃₇	—	2	1	no	—
51	C₄⋊C₈×C₁₃₇	—	2	1	no	—
52	C₄⋊C₈×C₁₃₇	—	2	1	no	—
53	C₄⋊C₈×C₁₃₇	—	2	1	no	—
54	#103×C₁₃₇	—	2	2	no	—
55	#103×C₁₃₇	—	2	2	no	—
56	#102×C₁₃₇	—	2	2	no	—
57	#102×C₁₃₇	—	2	2	no	—
58	C₄×C₈×C₁₃₇	—	2	1	no	—
59	C₄×C₈×C₁₃₇	—	2	1	no	—
60	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
63	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
64	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
65	#41×C₁₃₇	—	4	2	no	—
66	#41×C₁₃₇	—	4	2	no	—
67	C₂×Q₈×C₁₃₇	—	2	1	no	—
68	C₂×Q₈×C₁₃₇	—	2	1	no	—
69	C₂×Q₈×C₁₃₇	—	2	1	no	—
70	C₂×D₄×C₁₃₇	—	2	1	no	—
71	C₂×D₄×C₁₃₇	—	2	1	no	—
72	C₂×D₄×C₁₃₇	—	2	1	no	—
73	C₂×D₄×C₁₃₇	—	4	1	no	—
74	C₂²×C₄×C₁₃₇	—	4	1	no	—
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
78	M₁₆×C₁₃₇	—	4	2	no	—
79	M₁₆×C₁₃₇	—	4	2	no	—
80	C₂×C₈×C₁₃₇	—	2	4	no	—
81	C₂×C₈×C₁₃₇	—	2	2	no	—
82	C₂×C₈×C₁₃₇	—	2	2	no	—
83	C₂×C₈×C₁₃₇	—	2	2	no	—
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	—	4	2	no	—
86	C₂×C₈×C₁₃₇	—	4	1	no	—
87	C₂×C₈×C₁₃₇	—	4	1	no	—
88	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
90	C₄⋊C₄×C₁₃₇	—	2	1	no	—
91	C₄⋊C₄×C₁₃₇	—	2	1	no	—
92	C₄⋊C₄×C₁₃₇	—	2	1	no	—
93	C₄⋊C₄×C₁₃₇	—	2	1	no	—
94	C₄⋊C₄×C₁₃₇	—	2	1	no	—
95	C₄⋊C₄×C₁₃₇	—	2	1	no	—
96	C₄⋊C₄×C₁₃₇	—	4	1	no	—
97	C₄⋊C₄×C₁₃₇	—	4	1	no	—
98	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
99	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
100	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
101	C₄²×C₁₃₇	—	4	1	no	—
102	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₁₃₇	—	4	1	no	—
107	C₂³×C₁₃₇	#124	1	1	yes	characteristic
108	Q₈×C₁₃₇	—	4	2	no	—
109	Q₈×C₁₃₇	—	4	2	no	—
110	Q₈×C₁₃₇	—	4	2	no	—
111	D₄×C₁₃₇	—	4	2	no	—
112	D₄×C₁₃₇	—	4	2	no	—
113	D₄×C₁₃₇	—	4	2	no	—
114	D₄×C₁₃₇	—	8	2	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	4	1	no	—
118	C₂×C₄×C₁₃₇	—	4	1	no	—
119	C₂×C₄×C₁₃₇	—	4	1	no	—
120	C₂×C₄×C₁₃₇	—	8	1	no	—
121	C₂×C₄×C₁₃₇	—	8	1	no	—
122	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₁₃₇	#137	1	2	yes	—
130	C₈×C₁₃₇	—	2	4	no	—
131	C₈×C₁₃₇	—	2	2	no	—
132	C₈×C₁₃₇	—	2	2	no	—
133	C₈×C₁₃₇	—	4	2	no	—
134	C₈×C₁₃₇	—	8	1	no	—
135	C₈×C₁₃₇	—	8	1	no	—
136	C₂²×C₁₃₇	—	2	1	no	—
137	C₂²×C₁₃₇	—	2	1	no	—
138	C₂²×C₁₃₇	—	8	1	no	—
139	C₂²×C₁₃₇	—	8	1	no	—
140	C₂²×C₁₃₇	—	8	1	no	—
141	C₂²×C₁₃₇	#390	1	1	yes	characteristic
142	C₂²×C₁₃₇	#489	1	2	yes	—
143	C₄×C₁₃₇	—	2	1	no	—
144	C₄×C₁₃₇	—	2	1	no	—
145	C₄×C₁₃₇	—	8	1	no	—
146	C₄×C₁₃₇	—	8	1	no	—
147	C₄×C₁₃₇	—	8	1	no	—
148	C₄×C₁₃₇	#417	1	2	yes	—
149	C₄×C₁₃₇	#443	1	2	yes	—
150	C₄×C₁₃₇	#444	1	2	yes	—
151	C₄×C₁₃₇	#496	1	1	yes	characteristic
152	C₄×C₁₃₇	#497	1	1	yes	characteristic
153	C₂×C₁₃₇	—	2	2	no	—
154	C₂×C₁₃₇	—	16	1	no	—
155	C₂×C₁₃₇	#1959	1	1	yes	characteristic
156	C₂×C₁₃₇	#1974	1	1	yes	characteristic
157	C₂×C₁₃₇	#1975	1	1	yes	characteristic
158	#9858	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₁₃₇	#9858	1	1	yes	characteristic, direct product factor, Sylow
160	#3131	C₂×C₁₃₇	1	1	yes	characteristic
161	#1848	C₂×C₁₃₇	1	1	yes	characteristic, complemented
162	#1493	C₂×C₁₃₇	1	1	yes	characteristic, complemented
163	#1493	C₂×C₁₃₇	1	1	yes	characteristic, complemented
164	#1492	C₂×C₁₃₇	1	1	yes	characteristic
165	#1492	C₂×C₁₃₇	1	1	yes	characteristic
166	C₂×#322	C₂×C₁₃₇	1	1	yes	characteristic, complemented
167	#533	C₂²×C₁₃₇	1	1	yes	characteristic
168	#532	C₂²×C₁₃₇	1	1	yes	characteristic
169	#488	—	2	2	no	—
170	#487	C₄×C₁₃₇	1	1	yes	characteristic
171	#487	C₄×C₁₃₇	1	1	yes	characteristic
172	#486	C₄×C₁₃₇	1	1	yes	characteristic
173	#486	C₄×C₁₃₇	1	1	yes	characteristic
174	C₂×#108	—	2	2	no	—
175	#423	C₂²×C₁₃₇	1	1	yes	characteristic
176	#423	C₂²×C₁₃₇	1	1	yes	characteristic
177	#416	—	2	2	no	—
178	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
181	Q₈⋊C₈	—	2	2	no	—
182	Q₈⋊C₈	—	2	2	no	—
183	D₄⋊C₈	—	2	2	no	—
184	D₄⋊C₈	—	2	2	no	—
185	#322	—	2	2	no	—
186	C₂×#41	—	2	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
192	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
193	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
194	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
195	#127	C₂×C₄×C₁₃₇	1	1	yes	characteristic
196	#125	—	2	1	no	—
197	#124	—	2	1	no	—
198	C₂²⋊Q₈	—	2	1	no	—
199	C₂²⋊Q₈	—	2	1	no	—
200	#121	—	2	1	no	—
201	#121	—	2	1	no	—
202	C₄×Q₈	—	2	1	no	—
203	C₄×Q₈	—	2	1	no	—
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	#117	—	2	1	no	—
207	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
208	#108	—	4	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	#103	—	2	2	no	—
214	#103	—	2	2	no	—
215	#102	—	2	2	no	—
216	#102	—	2	2	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
220	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
221	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
222	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
223	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
224	#41	—	4	2	no	—
225	#41	—	4	2	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
235	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
236	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
237	M₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
248	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₂²⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
265	C₂³	—	4	1	no	—
266	C₂³	#124×C₁₃₇	1	1	yes	characteristic
267	Q₈	—	4	2	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	8	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
284	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
285	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
286	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
287	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
288	C₂×C₄	#137×C₁₃₇	1	2	yes	—
289	C₈	—	2	4	no	—
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	8	1	no	—
294	C₈	—	8	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	8	1	no	—
298	C₂²	—	8	1	no	—
299	C₂²	—	8	1	no	—
300	C₂²	#390×C₁₃₇	1	1	yes	characteristic
301	C₂²	#489×C₁₃₇	1	2	yes	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₄	—	8	1	no	—
307	C₄	#417×C₁₃₇	1	2	yes	—
308	C₄	#443×C₁₃₇	1	2	yes	—
309	C₄	#444×C₁₃₇	1	2	yes	—
310	C₄	#496×C₁₃₇	1	1	yes	characteristic
311	C₄	#497×C₁₃₇	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	16	1	no	—
314	C₂	#1959×C₁₃₇	1	1	yes	characteristic
315	C₂	#1974×C₁₃₇	1	1	yes	characteristic
316	C₂	#1975×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#127
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#390
prime radical	#532×C₁₃₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1848×C₁₃₇	C₂	semidirect
3. #1493×C₁₃₇	C₂	semidirect
4. #1493×C₁₃₇	C₂	semidirect
7. C₂×#322×C₁₃₇	C₂	semidirect
158. #9858	C₁₃₇	direct (N × H)
159. C₁₃₇	#9858	direct (N × H)
161. #1848	C₂×C₁₃₇	semidirect
162. #1493	C₂×C₁₃₇	semidirect
163. #1493	C₂×C₁₃₇	semidirect
166. C₂×#322	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3131×C₁₃₇	2
#1848×C₁₃₇	2
#1493×C₁₃₇	2
#1492×C₁₃₇	2
C₂×#322×C₁₃₇	2
#9858	137

Subgroups — C137×#9858 #46859244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9858 #46859244