Subgroups — C₁₃₇×#9815 #46859201

Order 35072.

548 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#466×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1810×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₂²⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1507×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1471×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×#121×C₁₃₇	C₂²	1	2	yes	—
9	#466×C₁₃₇	—	2	1	no	—
10	#466×C₁₃₇	—	2	1	no	—
11	#466×C₁₃₇	—	2	1	no	—
12	#466×C₁₃₇	—	2	1	no	—
13	C₂×#108×C₁₃₇	—	2	1	no	—
14	C₂×#107×C₁₃₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#102×C₁₃₇	C₄	1	2	yes	complemented
16	#407×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃₇	—	2	1	no	—
19	C₄×C₂²⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
20	#341×C₁₃₇	—	2	1	no	—
21	#336×C₁₃₇	—	2	1	no	—
22	#327×C₁₃₇	—	2	4	no	—
23	#323×C₁₃₇	—	2	4	no	—
24	C₂²×D₄×C₁₃₇	—	2	2	no	—
25	C₂²×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₂×D₈×C₁₃₇	—	2	1	no	—
28	C₂×D₈×C₁₃₇	—	2	1	no	—
29	C₂×D₈×C₁₃₇	—	4	2	no	—
30	C₂×M₁₆×C₁₃₇	—	2	2	no	—
31	C₂²×C₈×C₁₃₇	—	2	1	no	—
32	C₂²×C₈×C₁₃₇	—	4	2	no	—
33	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
39	#121×C₁₃₇	—	2	2	no	—
40	#121×C₁₃₇	—	2	2	no	—
41	#121×C₁₃₇	—	2	2	no	—
42	#121×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄²×C₁₃₇	—	2	1	no	—
47	#108×C₁₃₇	—	4	2	no	—
48	#107×C₁₃₇	—	2	1	no	—
49	#107×C₁₃₇	—	2	1	no	—
50	#102×C₁₃₇	—	2	2	no	—
51	#102×C₁₃₇	—	2	2	no	—
52	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
53	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
54	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
55	C₂²⋊C₈×C₁₃₇	D₄	1	2	yes	—
56	C₄×C₈×C₁₃₇	—	4	2	no	—
57	#95×C₁₃₇	D₄	1	2	yes	—
58	C₂⁴×C₁₃₇	—	2	2	no	—
59	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
60	C₂×D₄×C₁₃₇	—	2	2	no	—
61	C₂×D₄×C₁₃₇	—	2	2	no	—
62	C₂×D₄×C₁₃₇	—	4	2	no	—
63	C₂×D₄×C₁₃₇	—	4	2	no	—
64	C₂×D₄×C₁₃₇	—	4	2	no	—
65	C₂×D₄×C₁₃₇	—	4	2	no	—
66	C₂×D₄×C₁₃₇	—	8	4	no	—
67	C₂²×C₄×C₁₃₇	—	2	2	no	—
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	—	4	1	no	—
70	C₂²×C₄×C₁₃₇	—	4	1	no	—
71	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
77	D₈×C₁₃₇	—	4	2	no	—
78	D₈×C₁₃₇	—	4	2	no	—
79	M₁₆×C₁₃₇	—	4	2	no	—
80	M₁₆×C₁₃₇	—	4	2	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	1	no	—
86	C₂×C₈×C₁₃₇	—	4	1	no	—
87	C₂×C₈×C₁₃₇	—	4	1	no	—
88	C₂×C₈×C₁₃₇	—	4	1	no	—
89	C₂×C₈×C₁₃₇	—	4	1	no	—
90	C₂×C₈×C₁₃₇	—	4	2	no	—
91	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₄⋊C₄×C₁₃₇	—	2	2	no	—
96	C₄⋊C₄×C₁₃₇	—	2	2	no	—
97	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
98	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
99	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
100	C₄²×C₁₃₇	—	4	1	no	—
101	C₄²×C₁₃₇	—	4	1	no	—
102	C₂³×C₁₃₇	—	2	2	no	—
103	C₂³×C₁₃₇	—	2	2	no	—
104	C₂³×C₁₃₇	—	4	1	no	—
105	C₂³×C₁₃₇	—	4	2	no	—
106	C₂³×C₁₃₇	—	4	4	no	—
107	C₂³×C₁₃₇	—	8	2	no	—
108	C₂³×C₁₃₇	—	8	2	no	—
109	C₂³×C₁₃₇	C₄×D₄	1	1	yes	characteristic
110	C₂³×C₁₃₇	#121	1	1	yes	characteristic
111	C₂³×C₁₃₇	#124	1	1	yes	characteristic
112	D₄×C₁₃₇	—	4	2	no	—
113	D₄×C₁₃₇	—	4	2	no	—
114	D₄×C₁₃₇	—	8	2	no	—
115	D₄×C₁₃₇	—	8	4	no	—
116	D₄×C₁₃₇	—	8	4	no	—
117	C₂×C₄×C₁₃₇	—	2	1	no	—
118	C₂×C₄×C₁₃₇	—	2	1	no	—
119	C₂×C₄×C₁₃₇	—	2	1	no	—
120	C₂×C₄×C₁₃₇	—	2	1	no	—
121	C₂×C₄×C₁₃₇	—	2	1	no	—
122	C₂×C₄×C₁₃₇	—	2	1	no	—
123	C₂×C₄×C₁₃₇	—	2	4	no	—
124	C₂×C₄×C₁₃₇	—	4	1	no	—
125	C₂×C₄×C₁₃₇	—	4	1	no	—
126	C₂×C₄×C₁₃₇	—	4	1	no	—
127	C₂×C₄×C₁₃₇	—	4	2	no	—
128	C₂×C₄×C₁₃₇	—	4	2	no	—
129	C₂×C₄×C₁₃₇	—	4	2	no	—
130	C₂×C₄×C₁₃₇	—	8	2	no	—
131	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
135	C₈×C₁₃₇	—	2	1	no	—
136	C₈×C₁₃₇	—	2	1	no	—
137	C₈×C₁₃₇	—	2	1	no	—
138	C₈×C₁₃₇	—	4	2	no	—
139	C₈×C₁₃₇	—	8	2	no	—
140	C₈×C₁₃₇	#99	1	2	yes	—
141	C₂²×C₁₃₇	—	2	1	no	—
142	C₂²×C₁₃₇	—	2	1	no	—
143	C₂²×C₁₃₇	—	2	1	no	—
144	C₂²×C₁₃₇	—	2	1	no	—
145	C₂²×C₁₃₇	—	2	1	no	—
146	C₂²×C₁₃₇	—	2	1	no	—
147	C₂²×C₁₃₇	—	2	1	no	—
148	C₂²×C₁₃₇	—	4	4	no	—
149	C₂²×C₁₃₇	—	8	2	no	—
150	C₂²×C₁₃₇	—	8	2	no	—
151	C₂²×C₁₃₇	—	8	2	no	—
152	C₂²×C₁₃₇	—	16	4	no	—
153	C₂²×C₁₃₇	#390	1	1	yes	characteristic
154	C₂²×C₁₃₇	#443	1	2	yes	—
155	C₂²×C₁₃₇	#472	1	2	yes	—
156	C₄×C₁₃₇	—	2	1	no	—
157	C₄×C₁₃₇	—	2	1	no	—
158	C₄×C₁₃₇	—	2	1	no	—
159	C₄×C₁₃₇	—	4	4	no	—
160	C₄×C₁₃₇	—	8	2	no	—
161	C₄×C₁₃₇	C₂×#99	1	1	yes	characteristic
162	C₄×C₁₃₇	#410	1	1	yes	characteristic
163	C₂×C₁₃₇	—	2	2	no	—
164	C₂×C₁₃₇	—	2	2	no	—
165	C₂×C₁₃₇	—	4	1	no	—
166	C₂×C₁₃₇	—	16	2	no	—
167	C₂×C₁₃₇	#1964	1	1	yes	characteristic
168	C₂×C₁₃₇	#1974	1	1	yes	characteristic
169	C₂×C₁₃₇	#1978	1	1	yes	characteristic
170	#9815	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₁₃₇	#9815	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#466	C₂×C₁₃₇	1	1	yes	characteristic
173	#1810	C₂×C₁₃₇	1	1	yes	characteristic, complemented
174	C₈×C₂²⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
175	#1507	C₂×C₁₃₇	1	2	yes	complemented
176	#1471	C₂×C₁₃₇	1	2	yes	complemented
177	C₂²×D₈	C₄×C₁₃₇	1	1	yes	characteristic, complemented
178	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
179	C₂×#121	C₂²×C₁₃₇	1	2	yes	—
180	#466	—	2	1	no	—
181	#466	—	2	1	no	—
182	#466	—	2	1	no	—
183	#466	—	2	1	no	—
184	C₂×#108	—	2	1	no	—
185	C₂×#107	C₄×C₁₃₇	1	1	yes	characteristic, complemented
186	C₂×#102	C₄×C₁₃₇	1	2	yes	complemented
187	#407	C₂²×C₁₃₇	1	2	yes	—
188	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
189	C₂×C₄×C₈	—	2	1	no	—
190	C₄×C₂²⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
191	#341	—	2	1	no	—
192	#336	—	2	1	no	—
193	#327	—	2	4	no	—
194	#323	—	2	4	no	—
195	C₂²×D₄	—	2	2	no	—
196	C₂²×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
197	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	4	2	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	4	2	no	—
204	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
205	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
206	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
207	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
208	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
209	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
210	#121	—	2	2	no	—
211	#121	—	2	2	no	—
212	#121	—	2	2	no	—
213	#121	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	1	no	—
218	#108	—	4	2	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#102	—	2	2	no	—
222	#102	—	2	2	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	D₄×C₁₃₇	1	2	yes	—
227	C₄×C₈	—	4	2	no	—
228	#95	D₄×C₁₃₇	1	2	yes	—
229	C₂⁴	—	2	2	no	—
230	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	8	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
247	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
248	D₈	—	4	2	no	—
249	D₈	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	M₁₆	—	4	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
263	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
265	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	2	4	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	4	1	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	8	2	no	—
279	C₂³	—	8	2	no	—
280	C₂³	C₄×D₄×C₁₃₇	1	1	yes	characteristic
281	C₂³	#121×C₁₃₇	1	1	yes	characteristic
282	C₂³	#124×C₁₃₇	1	1	yes	characteristic
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	8	2	no	—
286	D₄	—	8	4	no	—
287	D₄	—	8	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
303	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
304	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
305	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	—	8	2	no	—
311	C₈	#99×C₁₃₇	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	4	no	—
324	C₂²	#390×C₁₃₇	1	1	yes	characteristic
325	C₂²	#443×C₁₃₇	1	2	yes	—
326	C₂²	#472×C₁₃₇	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	4	no	—
331	C₄	—	8	2	no	—
332	C₄	C₂×#99×C₁₃₇	1	1	yes	characteristic
333	C₄	#410×C₁₃₇	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	2	2	no	—
336	C₂	—	4	1	no	—
337	C₂	—	16	2	no	—
338	C₂	#1964×C₁₃₇	1	1	yes	characteristic
339	C₂	#1974×C₁₃₇	1	1	yes	characteristic
340	C₂	#1978×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#390
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	C₂×#466×C₁₃₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1810×C₁₃₇	C₂	semidirect
3. C₈×C₂²⋊C₄×C₁₃₇	C₂	semidirect
4. #1507×C₁₃₇	C₂	semidirect
5. #1471×C₁₃₇	C₂	semidirect
6. C₂²×D₈×C₁₃₇	C₄	semidirect
14. C₂×#107×C₁₃₇	C₄	semidirect
15. C₂×#102×C₁₃₇	C₄	semidirect
170. #9815	C₁₃₇	direct (N × H)
171. C₁₃₇	#9815	direct (N × H)
173. #1810	C₂×C₁₃₇	semidirect
174. C₈×C₂²⋊C₄	C₂×C₁₃₇	semidirect
175. #1507	C₂×C₁₃₇	semidirect
176. #1471	C₂×C₁₃₇	semidirect
177. C₂²×D₈	C₄×C₁₃₇	semidirect
185. C₂×#107	C₄×C₁₃₇	semidirect
186. C₂×#102	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#466×C₁₃₇	2
#1810×C₁₃₇	2
C₈×C₂²⋊C₄×C₁₃₇	2
#1507×C₁₃₇	2
#1471×C₁₃₇	2
#9815	137

Subgroups — C137×#9815 #46859201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9815 #46859201