Subgroups — C₁₃₇×#21657 #46871043

Order 35072.

802 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#491×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1992×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1960×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1933×C₁₃₇	C₂	1	4	yes	maximal
7	#1836×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1763×C₁₃₇	C₂	1	4	yes	maximal
9	C₄×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
13	#491×C₁₃₇	C₄	1	8	yes	complemented
14	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
15	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
16	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
20	#391×C₁₃₇	C₂²	1	2	yes	—
21	#389×C₁₃₇	—	2	2	no	—
22	#389×C₁₃₇	C₂²	1	2	yes	—
23	#387×C₁₃₇	—	2	2	no	—
24	#384×C₁₃₇	C₂²	1	1	yes	characteristic
25	#382×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
27	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
28	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
29	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
30	#340×C₁₃₇	C₂²	1	4	yes	—
31	#340×C₁₃₇	C₂²	1	4	yes	—
32	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
36	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
37	#125×C₁₃₇	—	2	2	no	—
38	#125×C₁₃₇	C₂×C₄	1	4	yes	—
39	C₄×Q₈×C₁₃₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₃₇	—	2	2	no	—
49	C₂×C₄²×C₁₃₇	—	2	2	no	—
50	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
52	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
55	#103×C₁₃₇	—	2	8	no	—
56	#103×C₁₃₇	C₂×C₄	1	16	yes	—
57	#97×C₁₃₇	—	2	2	no	—
58	#97×C₁₃₇	—	2	4	no	—
59	#97×C₁₃₇	C₂×C₄	1	4	yes	—
60	C₄×C₈×C₁₃₇	—	2	4	no	—
61	#95×C₁₃₇	—	2	2	no	—
62	#95×C₁₃₇	—	2	2	no	—
63	#95×C₁₃₇	—	2	4	no	—
64	#95×C₁₃₇	—	2	2	no	—
65	C₂×Q₈×C₁₃₇	—	2	2	no	—
66	C₂×Q₈×C₁₃₇	—	2	4	no	—
67	C₂×Q₈×C₁₃₇	—	2	4	no	—
68	C₂×Q₈×C₁₃₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃₇	—	2	2	no	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	2	no	—
72	C₂²×C₄×C₁₃₇	—	2	2	no	—
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	2	2	no	—
80	C₂×C₈×C₁₃₇	—	2	2	no	—
81	C₂×C₈×C₁₃₇	—	2	2	no	—
82	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₁₃₇	#41	1	4	yes	—
84	C₄⋊C₄×C₁₃₇	—	2	8	no	—
85	C₄⋊C₄×C₁₃₇	—	2	8	no	—
86	C₄⋊C₄×C₁₃₇	—	2	2	no	—
87	C₄⋊C₄×C₁₃₇	—	2	4	no	—
88	C₄⋊C₄×C₁₃₇	—	2	4	no	—
89	C₄⋊C₄×C₁₃₇	—	4	4	no	—
90	C₄⋊C₄×C₁₃₇	—	4	4	no	—
91	C₄⋊C₄×C₁₃₇	—	4	4	no	—
92	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
93	C₄²×C₁₃₇	—	2	4	no	—
94	C₄²×C₁₃₇	—	2	2	no	—
95	C₄²×C₁₃₇	—	4	2	no	—
96	C₄²×C₁₃₇	—	4	4	no	—
97	C₄²×C₁₃₇	—	4	4	no	—
98	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₃₇	#41	1	2	yes	—
100	C₄²×C₁₃₇	#41	1	4	yes	—
101	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₁₃₇	—	2	8	no	—
103	Q₈×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	—	2	2	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	2	2	no	—
110	C₂×C₄×C₁₃₇	—	2	4	no	—
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	4	2	no	—
113	C₂×C₄×C₁₃₇	—	4	2	no	—
114	C₂×C₄×C₁₃₇	—	4	2	no	—
115	C₂×C₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	4	2	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	—	4	2	no	—
119	C₂×C₄×C₁₃₇	—	4	2	no	—
120	C₂×C₄×C₁₃₇	—	4	4	no	—
121	C₂×C₄×C₁₃₇	C₄×D₄	1	4	yes	—
122	C₂×C₄×C₁₃₇	#124	1	2	yes	—
123	C₂×C₄×C₁₃₇	#124	1	2	yes	—
124	C₂×C₄×C₁₃₇	#137	1	4	yes	—
125	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
127	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
129	C₈×C₁₃₇	—	2	4	no	—
130	C₈×C₁₃₇	—	2	4	no	—
131	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
133	C₂²×C₁₃₇	#532	1	1	yes	characteristic
134	C₂²×C₁₃₇	C₂×#137	1	2	yes	—
135	C₂²×C₁₃₇	#575	1	2	yes	—
136	C₄×C₁₃₇	—	2	2	no	—
137	C₄×C₁₃₇	—	2	2	no	—
138	C₄×C₁₃₇	—	2	4	no	—
139	C₄×C₁₃₇	—	2	4	no	—
140	C₄×C₁₃₇	—	4	4	no	—
141	C₄×C₁₃₇	—	4	4	no	—
142	C₄×C₁₃₇	—	8	2	no	—
143	C₄×C₁₃₇	—	8	2	no	—
144	C₄×C₁₃₇	C₄×#41	1	2	yes	—
145	C₄×C₁₃₇	#529	1	2	yes	—
146	C₂×C₁₃₇	C₄×#124	1	1	yes	characteristic
147	C₂×C₁₃₇	C₄×#137	1	2	yes	—
148	C₂×C₁₃₇	#3139	1	1	yes	characteristic
149	C₂×C₁₃₇	#3142	1	1	yes	characteristic
150	C₂×C₁₃₇	#3181	1	2	yes	—
151	#21657	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₁₃₇	#21657	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×#491	C₂×C₁₃₇	1	1	yes	characteristic
154	C₄×C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic
155	C₄×#125	C₂×C₁₃₇	1	1	yes	characteristic
156	#1992	C₂×C₁₃₇	1	1	yes	characteristic
157	#1960	C₂×C₁₃₇	1	1	yes	characteristic
158	#1933	C₂×C₁₃₇	1	4	yes	—
159	#1836	C₂×C₁₃₇	1	1	yes	characteristic
160	#1763	C₂×C₁₃₇	1	4	yes	—
161	C₄×#97	C₂×C₁₃₇	1	1	yes	characteristic
162	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
163	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
164	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
165	#491	C₄×C₁₃₇	1	8	yes	complemented
166	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
167	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
168	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
169	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
170	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
172	#391	C₂²×C₁₃₇	1	2	yes	—
173	#389	—	2	2	no	—
174	#389	C₂²×C₁₃₇	1	2	yes	—
175	#387	—	2	2	no	—
176	#384	C₂²×C₁₃₇	1	1	yes	characteristic
177	#382	C₂²×C₁₃₇	1	1	yes	characteristic
178	C₄×C₄⋊C₄	—	2	2	no	—
179	C₄×C₄⋊C₄	—	2	2	no	—
180	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
181	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
182	#340	C₂²×C₁₃₇	1	4	yes	—
183	#340	C₂²×C₁₃₇	1	4	yes	—
184	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	4	yes	—
189	#125	—	2	2	no	—
190	#125	C₂×C₄×C₁₃₇	1	4	yes	—
191	C₄×Q₈	—	2	8	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
203	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
204	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
207	#103	—	2	8	no	—
208	#103	C₂×C₄×C₁₃₇	1	16	yes	—
209	#97	—	2	2	no	—
210	#97	—	2	4	no	—
211	#97	C₂×C₄×C₁₃₇	1	4	yes	—
212	C₄×C₈	—	2	4	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	4	no	—
216	#95	—	2	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	4	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	C₂²×C₄×C₁₃₇	1	4	yes	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
235	C₂×C₈	#41×C₁₃₇	1	4	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	4	no	—
249	C₄²	—	4	4	no	—
250	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
251	C₄²	#41×C₁₃₇	1	2	yes	—
252	C₄²	#41×C₁₃₇	1	4	yes	—
253	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
254	Q₈	—	2	8	no	—
255	Q₈	—	4	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₄×D₄×C₁₃₇	1	4	yes	—
274	C₂×C₄	#124×C₁₃₇	1	2	yes	—
275	C₂×C₄	#124×C₁₃₇	1	2	yes	—
276	C₂×C₄	#137×C₁₃₇	1	4	yes	—
277	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
279	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
280	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
284	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
285	C₂²	#532×C₁₃₇	1	1	yes	characteristic
286	C₂²	C₂×#137×C₁₃₇	1	2	yes	—
287	C₂²	#575×C₁₃₇	1	2	yes	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	4	4	no	—
293	C₄	—	4	4	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	C₄×#41×C₁₃₇	1	2	yes	—
297	C₄	#529×C₁₃₇	1	2	yes	—
298	C₂	C₄×#124×C₁₃₇	1	1	yes	characteristic
299	C₂	C₄×#137×C₁₃₇	1	2	yes	—
300	C₂	#3139×C₁₃₇	1	1	yes	characteristic
301	C₂	#3142×C₁₃₇	1	1	yes	characteristic
302	C₂	#3181×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. #491×C₁₃₇	C₄	semidirect
151. #21657	C₁₃₇	direct (N × H)
152. C₁₃₇	#21657	direct (N × H)
165. #491	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×#491×C₁₃₇	2
C₄×C₄⋊Q₈×C₁₃₇	2
C₄×#125×C₁₃₇	2
#1992×C₁₃₇	2
#1960×C₁₃₇	2
#1933×C₁₃₇	2
#1836×C₁₃₇	2
#1763×C₁₃₇	2
C₄×#97×C₁₃₇	2
#21657	137

Subgroups — C137×#21657 #46871043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21657 #46871043