Subgroups — C₂×#2047×C₁₃₇ #46872905

Order 35072.

1330 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#419×C₁₃₇	C₂	1	2	yes	maximal
4	C₂²×#101×C₁₃₇	C₂	1	2	yes	maximal
5	#2047×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#341×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃₇	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₁₃₇	—	2	8	no	—
9	C₂²×Q₁₆×C₁₃₇	—	2	2	no	—
10	C₂²×Q₁₆×C₁₃₇	—	2	2	no	—
11	C₂²×Q₁₆×C₁₃₇	C₂²	1	4	yes	—
12	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
16	#424×C₁₃₇	—	2	2	no	—
17	#424×C₁₃₇	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈×C₁₃₇	—	2	2	no	—
19	#419×C₁₃₇	—	2	4	no	—
20	#419×C₁₃₇	C₂²	1	8	yes	—
21	C₂×#103×C₁₃₇	—	2	4	no	—
22	C₂×#101×C₁₃₇	—	2	8	no	—
23	C₂×#101×C₁₃₇	C₂²	1	8	yes	—
24	#341×C₁₃₇	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₃₇	—	2	4	no	—
26	C₂²×Q₈×C₁₃₇	—	2	2	no	—
27	C₂²×Q₈×C₁₃₇	—	2	2	no	—
28	C₂²×Q₈×C₁₃₇	—	2	2	no	—
29	C₂²×Q₈×C₁₃₇	—	2	2	no	—
30	C₂²×Q₈×C₁₃₇	—	2	2	no	—
31	C₂²×Q₈×C₁₃₇	—	2	4	no	—
32	C₂²×Q₈×C₁₃₇	—	2	4	no	—
33	C₂²×Q₈×C₁₃₇	D₄	1	8	yes	—
34	C₂²×Q₈×C₁₃₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₃₇	—	2	2	no	—
36	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₁₃₇	—	2	16	no	—
38	C₂×Q₁₆×C₁₃₇	—	2	16	no	—
39	C₂×Q₁₆×C₁₃₇	—	2	16	no	—
40	C₂×Q₁₆×C₁₃₇	—	2	8	no	—
41	C₂×M₁₆×C₁₃₇	—	2	2	no	—
42	C₂×M₁₆×C₁₃₇	—	2	1	no	—
43	C₂×M₁₆×C₁₃₇	—	2	4	no	—
44	C₂×M₁₆×C₁₃₇	—	2	4	no	—
45	C₂×M₁₆×C₁₃₇	—	2	4	no	—
46	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
47	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
48	C₂²×C₈×C₁₃₇	—	2	4	no	—
49	C₂²×C₈×C₁₃₇	—	2	2	no	—
50	C₂²×C₈×C₁₃₇	—	2	1	no	—
51	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
52	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
53	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
54	#117×C₁₃₇	—	2	1	no	—
55	#117×C₁₃₇	—	2	1	no	—
56	#117×C₁₃₇	D₄	1	2	yes	—
57	#117×C₁₃₇	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
60	C₂×C₄²×C₁₃₇	—	2	1	no	—
61	C₄⋊C₈×C₁₃₇	—	2	8	no	—
62	#103×C₁₃₇	—	2	16	no	—
63	#101×C₁₃₇	—	2	16	no	—
64	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
65	C₂×Q₈×C₁₃₇	—	2	8	no	—
66	C₂×Q₈×C₁₃₇	—	2	4	no	—
67	C₂×Q₈×C₁₃₇	—	2	8	no	—
68	C₂×Q₈×C₁₃₇	—	2	8	no	—
69	C₂×Q₈×C₁₃₇	—	2	8	no	—
70	C₂×Q₈×C₁₃₇	—	2	8	no	—
71	C₂×Q₈×C₁₃₇	—	2	4	no	—
72	C₂×Q₈×C₁₃₇	—	2	4	no	—
73	C₂×Q₈×C₁₃₇	—	2	4	no	—
74	C₂×Q₈×C₁₃₇	—	2	4	no	—
75	C₂×Q₈×C₁₃₇	—	4	2	no	—
76	C₂×Q₈×C₁₃₇	—	4	2	no	—
77	C₂×Q₈×C₁₃₇	—	4	4	no	—
78	C₂×Q₈×C₁₃₇	—	4	4	no	—
79	C₂×Q₈×C₁₃₇	—	4	4	no	—
80	C₂×Q₈×C₁₃₇	—	4	4	no	—
81	C₂×Q₈×C₁₃₇	—	4	4	no	—
82	C₂×Q₈×C₁₃₇	C₂×D₄	1	8	yes	—
83	C₂²×C₄×C₁₃₇	—	2	2	no	—
84	C₂²×C₄×C₁₃₇	—	2	8	no	—
85	C₂²×C₄×C₁₃₇	—	2	4	no	—
86	C₂²×C₄×C₁₃₇	—	4	2	no	—
87	C₂²×C₄×C₁₃₇	—	4	2	no	—
88	C₂²×C₄×C₁₃₇	—	4	4	no	—
89	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
90	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
95	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
96	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₁₃₇	—	2	32	no	—
98	M₁₆×C₁₃₇	—	2	4	no	—
99	M₁₆×C₁₃₇	—	2	4	no	—
100	M₁₆×C₁₃₇	—	2	4	no	—
101	M₁₆×C₁₃₇	—	4	2	no	—
102	M₁₆×C₁₃₇	—	4	4	no	—
103	C₂×C₈×C₁₃₇	—	2	4	no	—
104	C₂×C₈×C₁₃₇	—	2	4	no	—
105	C₂×C₈×C₁₃₇	—	2	4	no	—
106	C₂×C₈×C₁₃₇	—	2	4	no	—
107	C₂×C₈×C₁₃₇	—	2	2	no	—
108	C₂×C₈×C₁₃₇	—	4	2	no	—
109	C₂×C₈×C₁₃₇	—	4	4	no	—
110	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
111	C₄⋊C₄×C₁₃₇	—	2	2	no	—
112	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
113	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
114	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
115	C₄²×C₁₃₇	—	2	2	no	—
116	C₄²×C₁₃₇	—	2	2	no	—
117	C₂³×C₁₃₇	—	2	4	no	—
118	C₂³×C₁₃₇	#121	1	1	yes	characteristic
119	C₂³×C₁₃₇	#121	1	1	yes	characteristic
120	C₂³×C₁₃₇	#123	1	4	yes	—
121	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
122	Q₈×C₁₃₇	—	2	8	no	—
123	Q₈×C₁₃₇	—	2	8	no	—
124	Q₈×C₁₃₇	—	4	4	no	—
125	Q₈×C₁₃₇	—	4	4	no	—
126	Q₈×C₁₃₇	—	4	8	no	—
127	Q₈×C₁₃₇	—	4	8	no	—
128	C₂×C₄×C₁₃₇	—	2	4	no	—
129	C₂×C₄×C₁₃₇	—	2	8	no	—
130	C₂×C₄×C₁₃₇	—	4	4	no	—
131	C₂×C₄×C₁₃₇	—	4	4	no	—
132	C₂×C₄×C₁₃₇	—	4	8	no	—
133	C₂×C₄×C₁₃₇	—	4	8	no	—
134	C₂×C₄×C₁₃₇	—	4	4	no	—
135	C₂×C₄×C₁₃₇	#120	1	2	yes	—
136	C₂×C₄×C₁₃₇	#120	1	2	yes	—
137	C₂×C₄×C₁₃₇	#121	1	4	yes	—
138	C₂×C₄×C₁₃₇	#121	1	4	yes	—
139	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₃₇	#123	1	4	yes	—
142	C₂×C₄×C₁₃₇	#124	1	2	yes	—
143	C₂×C₄×C₁₃₇	#124	1	2	yes	—
144	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
145	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
148	C₈×C₁₃₇	—	2	4	no	—
149	C₈×C₁₃₇	—	4	2	no	—
150	C₈×C₁₃₇	—	4	4	no	—
151	C₂²×C₁₃₇	—	2	2	no	—
152	C₂²×C₁₃₇	—	2	2	no	—
153	C₂²×C₁₃₇	—	2	8	no	—
154	C₂²×C₁₃₇	#394	1	2	yes	—
155	C₂²×C₁₃₇	#394	1	2	yes	—
156	C₂²×C₁₃₇	#473	1	4	yes	—
157	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
158	C₂²×C₁₃₇	C₂×#123	1	2	yes	—
159	C₄×C₁₃₇	—	4	8	no	—
160	C₄×C₁₃₇	—	4	4	no	—
161	C₄×C₁₃₇	#394	1	2	yes	—
162	C₄×C₁₃₇	#394	1	2	yes	—
163	C₄×C₁₃₇	C₂×#120	1	1	yes	characteristic
164	C₄×C₁₃₇	C₂×#121	1	2	yes	—
165	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
166	C₂×C₁₃₇	—	2	2	no	—
167	C₂×C₁₃₇	—	2	2	no	—
168	C₂×C₁₃₇	#2047	1	4	yes	direct product factor
169	C₂×C₁₃₇	C₂×#394	1	1	yes	characteristic
170	C₂×C₁₃₇	C₂×#473	1	2	yes	—
171	C₂×#2047	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₁₃₇	C₂×#2047	1	1	yes	characteristic, direct product factor, Sylow
173	C₂³×Q₁₆	C₂×C₁₃₇	1	1	yes	characteristic
174	C₂×#424	C₂×C₁₃₇	1	1	yes	characteristic
175	C₂×#419	C₂×C₁₃₇	1	2	yes	—
176	C₂²×#101	C₂×C₁₃₇	1	2	yes	—
177	#2047	C₂×C₁₃₇	1	8	yes	direct product factor
178	C₂×#341	C₂×C₁₃₇	1	1	yes	characteristic
179	C₂³×Q₈	C₂²×C₁₃₇	1	2	yes	—
180	C₂²×Q₁₆	—	2	8	no	—
181	C₂²×Q₁₆	—	2	2	no	—
182	C₂²×Q₁₆	—	2	2	no	—
183	C₂²×Q₁₆	C₂²×C₁₃₇	1	4	yes	—
184	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
185	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
186	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
187	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
188	#424	—	2	2	no	—
189	#424	C₂²×C₁₃₇	1	4	yes	—
190	C₂×C₄⋊C₈	—	2	2	no	—
191	#419	—	2	4	no	—
192	#419	C₂²×C₁₃₇	1	8	yes	—
193	C₂×#103	—	2	4	no	—
194	C₂×#101	—	2	8	no	—
195	C₂×#101	C₂²×C₁₃₇	1	8	yes	—
196	#341	C₂²×C₁₃₇	1	4	yes	—
197	C₂²×Q₈	—	2	4	no	—
198	C₂²×Q₈	—	2	2	no	—
199	C₂²×Q₈	—	2	2	no	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	2	4	no	—
202	C₂²×Q₈	—	2	2	no	—
203	C₂²×Q₈	—	2	2	no	—
204	C₂²×Q₈	—	2	4	no	—
205	C₂²×Q₈	D₄×C₁₃₇	1	8	yes	—
206	C₂²×Q₈	C₂³×C₁₃₇	1	4	yes	—
207	C₂³×C₄	—	2	2	no	—
208	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
209	C₂×Q₁₆	—	2	16	no	—
210	C₂×Q₁₆	—	2	16	no	—
211	C₂×Q₁₆	—	2	16	no	—
212	C₂×Q₁₆	—	2	8	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	2	4	no	—
216	C₂×M₁₆	—	2	4	no	—
217	C₂×M₁₆	—	2	4	no	—
218	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
219	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	2	2	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
224	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
225	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
226	#117	—	2	1	no	—
227	#117	—	2	1	no	—
228	#117	D₄×C₁₃₇	1	2	yes	—
229	#117	C₂³×C₁₃₇	1	2	yes	—
230	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
231	C₂×C₂²⋊C₄	—	2	1	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₄⋊C₈	—	2	8	no	—
234	#103	—	2	16	no	—
235	#101	—	2	16	no	—
236	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
237	C₂×Q₈	—	2	8	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	2	8	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	4	no	—
253	C₂×Q₈	—	4	4	no	—
254	C₂×Q₈	C₂×D₄×C₁₃₇	1	8	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	8	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
262	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
266	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
267	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
268	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
269	Q₁₆	—	2	32	no	—
270	M₁₆	—	2	4	no	—
271	M₁₆	—	2	4	no	—
272	M₁₆	—	2	4	no	—
273	M₁₆	—	4	2	no	—
274	M₁₆	—	4	4	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	4	4	no	—
282	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₂³	—	2	4	no	—
290	C₂³	#121×C₁₃₇	1	1	yes	characteristic
291	C₂³	#121×C₁₃₇	1	1	yes	characteristic
292	C₂³	#123×C₁₃₇	1	4	yes	—
293	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
294	Q₈	—	2	8	no	—
295	Q₈	—	2	8	no	—
296	Q₈	—	4	8	no	—
297	Q₈	—	4	8	no	—
298	Q₈	—	4	4	no	—
299	Q₈	—	4	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	#120×C₁₃₇	1	2	yes	—
308	C₂×C₄	#120×C₁₃₇	1	2	yes	—
309	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
310	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
311	C₂×C₄	#121×C₁₃₇	1	4	yes	—
312	C₂×C₄	#121×C₁₃₇	1	4	yes	—
313	C₂×C₄	#123×C₁₃₇	1	4	yes	—
314	C₂×C₄	#124×C₁₃₇	1	2	yes	—
315	C₂×C₄	#124×C₁₃₇	1	2	yes	—
316	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
317	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
319	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	4	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	8	no	—
326	C₂²	#394×C₁₃₇	1	2	yes	—
327	C₂²	#394×C₁₃₇	1	2	yes	—
328	C₂²	#473×C₁₃₇	1	4	yes	—
329	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
330	C₂²	C₂×#123×C₁₃₇	1	2	yes	—
331	C₄	—	4	4	no	—
332	C₄	—	4	8	no	—
333	C₄	#394×C₁₃₇	1	2	yes	—
334	C₄	#394×C₁₃₇	1	2	yes	—
335	C₄	C₂×#120×C₁₃₇	1	1	yes	characteristic
336	C₄	C₂×#121×C₁₃₇	1	2	yes	—
337	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	2	2	no	—
340	C₂	#2047×C₁₃₇	1	4	yes	direct product factor
341	C₂	C₂×#394×C₁₃₇	1	1	yes	characteristic
342	C₂	C₂×#473×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂⁴×C₁₃₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. #2047×C₁₃₇	C₂	direct (N × H)
168. C₂×C₁₃₇	#2047	direct (N × H)
171. C₂×#2047	C₁₃₇	direct (N × H)
172. C₁₃₇	C₂×#2047	direct (N × H)
177. #2047	C₂×C₁₃₇	direct (N × H)
340. C₂	#2047×C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₁₃₇	2
C₂×#424×C₁₃₇	2
C₂×#419×C₁₃₇	2
C₂²×#101×C₁₃₇	2
#2047×C₁₃₇	2
C₂×#341×C₁₃₇	2
C₂×#2047	137

Subgroups — C2×#2047×C137 #46872905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2047×C₁₃₇ #46872905