Subgroups — C₁₃₇×#25826 #46875212

Order 35072.

918 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	#3502×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#3165×C₁₃₇	C₂	1	2	yes	maximal
3	#2568×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2061×C₁₃₇	C₂	1	4	yes	maximal
5	#1986×C₁₃₇	C₂	1	2	yes	maximal
6	#1857×C₁₃₇	C₂	1	4	yes	maximal
7	C₄×#101×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₁₃₇	—	2	2	no	—
9	#557×C₁₃₇	—	2	2	no	—
10	#557×C₁₃₇	—	2	4	no	—
11	#544×C₁₃₇	—	2	2	no	—
12	#544×C₁₃₇	C₂²	1	2	yes	—
13	#533×C₁₃₇	—	2	2	no	—
14	#533×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	#519×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	#501×C₁₃₇	—	2	4	no	—
19	#500×C₁₃₇	—	2	2	no	—
20	#498×C₁₃₇	—	2	2	no	—
21	#428×C₁₃₇	C₂²	1	8	yes	—
22	C₂×#101×C₁₃₇	C₂²	1	2	yes	—
23	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
24	#400×C₁₃₇	—	2	2	no	—
25	#391×C₁₃₇	—	2	2	no	—
26	#391×C₁₃₇	C₂²	1	2	yes	—
27	#382×C₁₃₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
29	#343×C₁₃₇	C₂²	1	2	yes	—
30	#339×C₁₃₇	C₂²	1	8	yes	—
31	C₂²×Q₈×C₁₃₇	—	2	1	no	—
32	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
34	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
35	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
36	C₄⋊Q₈×C₁₃₇	—	2	8	no	—
37	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
38	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
39	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
40	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
41	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
42	#125×C₁₃₇	—	2	4	no	—
43	#125×C₁₃₇	—	2	8	no	—
44	#125×C₁₃₇	—	2	4	no	—
45	#125×C₁₃₇	D₄	1	4	yes	—
46	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
47	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
48	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
49	C₄×Q₈×C₁₃₇	—	2	4	no	—
50	C₄×Q₈×C₁₃₇	—	2	4	no	—
51	C₄×Q₈×C₁₃₇	—	2	4	no	—
52	C₄×Q₈×C₁₃₇	—	2	4	no	—
53	C₄×Q₈×C₁₃₇	—	2	2	no	—
54	C₄×Q₈×C₁₃₇	—	2	2	no	—
55	C₄×Q₈×C₁₃₇	—	2	2	no	—
56	C₄×Q₈×C₁₃₇	—	2	2	no	—
57	#117×C₁₃₇	—	2	4	no	—
58	#117×C₁₃₇	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
65	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
67	#107×C₁₃₇	—	2	8	no	—
68	#106×C₁₃₇	—	2	8	no	—
69	#101×C₁₃₇	Q₈	1	8	yes	complemented
70	#97×C₁₃₇	—	2	2	no	—
71	C₄×C₈×C₁₃₇	—	2	2	no	—
72	#95×C₁₃₇	—	2	1	no	—
73	#95×C₁₃₇	—	2	4	no	—
74	#95×C₁₃₇	—	2	4	no	—
75	C₂×Q₈×C₁₃₇	—	2	4	no	—
76	C₂×Q₈×C₁₃₇	—	2	2	no	—
77	C₂×Q₈×C₁₃₇	—	2	2	no	—
78	C₂×Q₈×C₁₃₇	—	2	2	no	—
79	C₂×Q₈×C₁₃₇	—	2	2	no	—
80	C₂×Q₈×C₁₃₇	—	4	4	no	—
81	C₂×Q₈×C₁₃₇	—	4	1	no	—
82	C₂×Q₈×C₁₃₇	—	4	1	no	—
83	C₂×Q₈×C₁₃₇	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₁₃₇	—	2	2	no	—
85	C₂²×C₄×C₁₃₇	—	2	2	no	—
86	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
91	M₁₆×C₁₃₇	—	2	4	no	—
92	M₁₆×C₁₃₇	C₂×Q₈	1	8	yes	—
93	C₂×C₈×C₁₃₇	—	2	4	no	—
94	C₄⋊C₄×C₁₃₇	—	2	4	no	—
95	C₄⋊C₄×C₁₃₇	—	2	4	no	—
96	C₄⋊C₄×C₁₃₇	—	2	4	no	—
97	C₄⋊C₄×C₁₃₇	—	2	8	no	—
98	C₄⋊C₄×C₁₃₇	—	2	4	no	—
99	C₄⋊C₄×C₁₃₇	—	2	4	no	—
100	C₄⋊C₄×C₁₃₇	—	2	2	no	—
101	C₄⋊C₄×C₁₃₇	—	2	2	no	—
102	C₄⋊C₄×C₁₃₇	—	2	4	no	—
103	C₄⋊C₄×C₁₃₇	—	2	2	no	—
104	C₄⋊C₄×C₁₃₇	—	2	4	no	—
105	C₄⋊C₄×C₁₃₇	—	4	2	no	—
106	C₄⋊C₄×C₁₃₇	—	4	2	no	—
107	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
108	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
109	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
110	C₄²×C₁₃₇	—	2	8	no	—
111	C₄²×C₁₃₇	—	2	2	no	—
112	C₄²×C₁₃₇	—	2	2	no	—
113	C₄²×C₁₃₇	—	2	4	no	—
114	C₄²×C₁₃₇	—	2	4	no	—
115	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
116	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
117	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₁₃₇	—	2	8	no	—
119	Q₈×C₁₃₇	—	4	4	no	—
120	Q₈×C₁₃₇	—	8	2	no	—
121	Q₈×C₁₃₇	—	8	2	no	—
122	Q₈×C₁₃₇	—	8	4	no	—
123	Q₈×C₁₃₇	—	8	2	no	—
124	C₂×C₄×C₁₃₇	—	2	2	no	—
125	C₂×C₄×C₁₃₇	—	2	2	no	—
126	C₂×C₄×C₁₃₇	—	2	2	no	—
127	C₂×C₄×C₁₃₇	—	2	4	no	—
128	C₂×C₄×C₁₃₇	—	2	4	no	—
129	C₂×C₄×C₁₃₇	—	2	2	no	—
130	C₂×C₄×C₁₃₇	—	4	1	no	—
131	C₂×C₄×C₁₃₇	—	4	2	no	—
132	C₂×C₄×C₁₃₇	—	4	2	no	—
133	C₂×C₄×C₁₃₇	—	4	2	no	—
134	C₂×C₄×C₁₃₇	—	4	4	no	—
135	C₂×C₄×C₁₃₇	—	4	2	no	—
136	C₂×C₄×C₁₃₇	—	4	2	no	—
137	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
138	C₂×C₄×C₁₃₇	#120	1	2	yes	—
139	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	4	yes	—
141	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
143	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
144	C₈×C₁₃₇	—	2	8	no	—
145	C₂²×C₁₃₇	—	2	1	no	—
146	C₂²×C₁₃₇	—	2	1	no	—
147	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
148	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₁₃₇	#536	1	1	yes	characteristic
150	C₄×C₁₃₇	—	2	4	no	—
151	C₄×C₁₃₇	—	2	1	no	—
152	C₄×C₁₃₇	—	2	2	no	—
153	C₄×C₁₃₇	—	4	4	no	—
154	C₄×C₁₃₇	—	4	2	no	—
155	C₄×C₁₃₇	—	4	4	no	—
156	C₄×C₁₃₇	—	8	2	no	—
157	C₄×C₁₃₇	—	8	2	no	—
158	C₄×C₁₃₇	#454	1	2	yes	—
159	C₄×C₁₃₇	D₄×Q₈	1	4	yes	—
160	C₂×C₁₃₇	—	2	1	no	—
161	C₂×C₁₃₇	—	2	1	no	—
162	C₂×C₁₃₇	#2434	1	1	yes	characteristic
163	C₂×C₁₃₇	C₂×#454	1	1	yes	characteristic
164	C₂×C₁₃₇	#3022	1	1	yes	characteristic
165	#25826	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₇	#25826	1	1	yes	characteristic, direct product factor, Sylow
167	#3502	C₂×C₁₃₇	1	1	yes	characteristic
168	#3165	C₂×C₁₃₇	1	2	yes	—
169	#2568	C₂×C₁₃₇	1	1	yes	characteristic
170	#2061	C₂×C₁₃₇	1	4	yes	—
171	#1986	C₂×C₁₃₇	1	2	yes	—
172	#1857	C₂×C₁₃₇	1	4	yes	—
173	C₄×#101	C₂×C₁₃₇	1	1	yes	characteristic
174	Q₈²	—	2	2	no	—
175	#557	—	2	2	no	—
176	#557	—	2	4	no	—
177	#544	—	2	2	no	—
178	#544	C₂²×C₁₃₇	1	2	yes	—
179	#533	—	2	2	no	—
180	#533	C₂²×C₁₃₇	1	2	yes	—
181	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
182	#519	C₂²×C₁₃₇	1	2	yes	—
183	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
184	#501	—	2	4	no	—
185	#500	—	2	2	no	—
186	#498	—	2	2	no	—
187	#428	C₂²×C₁₃₇	1	8	yes	—
188	C₂×#101	C₂²×C₁₃₇	1	2	yes	—
189	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
190	#400	—	2	2	no	—
191	#391	—	2	2	no	—
192	#391	C₂²×C₁₃₇	1	2	yes	—
193	#382	C₂²×C₁₃₇	1	2	yes	—
194	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
195	#343	C₂²×C₁₃₇	1	2	yes	—
196	#339	C₂²×C₁₃₇	1	8	yes	—
197	C₂²×Q₈	—	2	1	no	—
198	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
199	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	—	2	8	no	—
203	C₄⋊Q₈	—	2	4	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
207	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
208	#125	—	2	4	no	—
209	#125	—	2	8	no	—
210	#125	—	2	4	no	—
211	#125	D₄×C₁₃₇	1	4	yes	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	2	4	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	4	no	—
216	C₄×Q₈	—	2	4	no	—
217	C₄×Q₈	—	2	4	no	—
218	C₄×Q₈	—	2	4	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	#117	—	2	4	no	—
224	#117	C₂³×C₁₃₇	1	4	yes	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
230	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
231	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
232	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
233	#107	—	2	8	no	—
234	#106	—	2	8	no	—
235	#101	Q₈×C₁₃₇	1	8	yes	complemented
236	#97	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	#95	—	2	1	no	—
239	#95	—	2	4	no	—
240	#95	—	2	4	no	—
241	C₂×Q₈	—	2	4	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	4	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	C₂×Q₈×C₁₃₇	1	4	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
256	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
257	M₁₆	—	2	4	no	—
258	M₁₆	C₂×Q₈×C₁₃₇	1	8	yes	—
259	C₂×C₈	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
274	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₄²	—	2	8	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	4	no	—
281	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
282	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
283	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
284	Q₈	—	2	8	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	Q₈	—	8	4	no	—
289	Q₈	—	8	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
304	C₂×C₄	#120×C₁₃₇	1	2	yes	—
305	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
306	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	4	yes	—
307	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
308	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
309	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
310	C₈	—	2	8	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
314	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
315	C₂²	#536×C₁₃₇	1	1	yes	characteristic
316	C₄	—	2	4	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	#454×C₁₃₇	1	2	yes	—
325	C₄	D₄×Q₈×C₁₃₇	1	4	yes	—
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	#2434×C₁₃₇	1	1	yes	characteristic
329	C₂	C₂×#454×C₁₃₇	1	1	yes	characteristic
330	C₂	#3022×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂³×C₁₃₇	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
69. #101×C₁₃₇	Q₈	semidirect
165. #25826	C₁₃₇	direct (N × H)
166. C₁₃₇	#25826	direct (N × H)
235. #101	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3502×C₁₃₇	2
#3165×C₁₃₇	2
#2568×C₁₃₇	2
#2061×C₁₃₇	2
#1986×C₁₃₇	2
#1857×C₁₃₇	2
C₄×#101×C₁₃₇	2
#25826	137

Subgroups — C137×#25826 #46875212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25826 #46875212