Subgroups — C₁₃₇×#28638 #46878024

Order 35072.

750 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#487×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2568×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2567×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#2068×C₁₃₇	C₂	1	4	yes	maximal
6	#1987×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1981×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1933×C₁₃₇	C₂	1	4	yes	maximal
9	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	#487×C₁₃₇	—	2	4	no	—
14	C₈×Q₈×C₁₃₇	—	2	4	no	—
15	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
20	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
23	#400×C₁₃₇	—	2	2	no	—
24	#395×C₁₃₇	—	2	2	no	—
25	#391×C₁₃₇	—	2	1	no	—
26	#391×C₁₃₇	—	2	1	no	—
27	#391×C₁₃₇	C₂²	1	4	yes	—
28	#389×C₁₃₇	C₂²	1	4	yes	—
29	#384×C₁₃₇	—	2	1	no	—
30	#382×C₁₃₇	—	2	1	no	—
31	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
33	#340×C₁₃₇	C₂²	1	2	yes	—
34	#340×C₁₃₇	C₂²	1	4	yes	—
35	C₂²×Q₈×C₁₃₇	—	2	1	no	—
36	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
37	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
42	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
43	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
44	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
45	C₄×Q₈×C₁₃₇	—	2	2	no	—
46	C₄×Q₈×C₁₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
49	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
51	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₁₃₇	—	2	1	no	—
57	C₂×C₄²×C₁₃₇	—	2	1	no	—
58	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
60	#107×C₁₃₇	—	2	4	no	—
61	C₄⋊C₈×C₁₃₇	—	2	4	no	—
62	C₄⋊C₈×C₁₃₇	—	2	2	no	—
63	#103×C₁₃₇	—	2	4	no	—
64	#103×C₁₃₇	—	2	4	no	—
65	#103×C₁₃₇	—	2	8	no	—
66	C₄×C₈×C₁₃₇	—	2	4	no	—
67	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
68	#95×C₁₃₇	—	2	1	no	—
69	#95×C₁₃₇	—	2	1	no	—
70	#95×C₁₃₇	—	2	4	no	—
71	#95×C₁₃₇	—	2	4	no	—
72	#95×C₁₃₇	—	2	4	no	—
73	C₂×Q₈×C₁₃₇	—	2	2	no	—
74	C₂×Q₈×C₁₃₇	—	2	2	no	—
75	C₂×Q₈×C₁₃₇	—	2	2	no	—
76	C₂×Q₈×C₁₃₇	—	2	4	no	—
77	C₂×Q₈×C₁₃₇	—	2	2	no	—
78	C₂×Q₈×C₁₃₇	—	2	4	no	—
79	C₂×Q₈×C₁₃₇	—	4	2	no	—
80	C₂×Q₈×C₁₃₇	—	8	2	no	—
81	C₂×Q₈×C₁₃₇	Q₁₆	1	4	yes	—
82	C₂²×C₄×C₁₃₇	—	2	2	no	—
83	C₂²×C₄×C₁₃₇	—	2	2	no	—
84	C₂²×C₄×C₁₃₇	—	2	2	no	—
85	C₂²×C₄×C₁₃₇	—	2	2	no	—
86	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
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	C₂×C₈×C₁₃₇	—	2	2	no	—
92	C₂×C₈×C₁₃₇	—	2	2	no	—
93	C₂×C₈×C₁₃₇	—	2	2	no	—
94	C₂×C₈×C₁₃₇	—	2	4	no	—
95	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
97	C₄⋊C₄×C₁₃₇	—	2	4	no	—
98	C₄⋊C₄×C₁₃₇	—	2	2	no	—
99	C₄⋊C₄×C₁₃₇	—	2	2	no	—
100	C₄⋊C₄×C₁₃₇	—	2	2	no	—
101	C₄⋊C₄×C₁₃₇	—	2	2	no	—
102	C₄⋊C₄×C₁₃₇	—	2	2	no	—
103	C₄⋊C₄×C₁₃₇	—	2	2	no	—
104	C₄⋊C₄×C₁₃₇	—	4	2	no	—
105	C₄⋊C₄×C₁₃₇	—	4	2	no	—
106	C₄⋊C₄×C₁₃₇	—	4	2	no	—
107	C₄⋊C₄×C₁₃₇	—	8	4	no	—
108	C₄⋊C₄×C₁₃₇	#41	1	4	yes	—
109	C₄²×C₁₃₇	—	2	2	no	—
110	C₄²×C₁₃₇	—	2	2	no	—
111	C₄²×C₁₃₇	—	4	2	no	—
112	C₄²×C₁₃₇	—	4	2	no	—
113	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
114	C₄²×C₁₃₇	#41	1	2	yes	—
115	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₁₃₇	—	2	8	no	—
117	Q₈×C₁₃₇	—	4	2	no	—
118	Q₈×C₁₃₇	—	4	2	no	—
119	Q₈×C₁₃₇	—	4	4	no	—
120	Q₈×C₁₃₇	—	8	4	no	—
121	C₂×C₄×C₁₃₇	—	2	2	no	—
122	C₂×C₄×C₁₃₇	—	2	2	no	—
123	C₂×C₄×C₁₃₇	—	2	2	no	—
124	C₂×C₄×C₁₃₇	—	2	2	no	—
125	C₂×C₄×C₁₃₇	—	2	4	no	—
126	C₂×C₄×C₁₃₇	—	2	4	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	—	4	2	no	—
131	C₂×C₄×C₁₃₇	—	4	2	no	—
132	C₂×C₄×C₁₃₇	—	4	4	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	#121	1	2	yes	—
137	C₂×C₄×C₁₃₇	#121	1	2	yes	—
138	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	4	yes	—
139	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
140	C₂×C₄×C₁₃₇	#135	1	2	yes	—
141	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
143	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
145	C₈×C₁₃₇	—	2	2	no	—
146	C₈×C₁₃₇	—	2	2	no	—
147	C₈×C₁₃₇	—	4	2	no	—
148	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
149	C₂²×C₁₃₇	#548	1	2	yes	—
150	C₂²×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
151	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
152	C₂²×C₁₃₇	#577	1	2	yes	—
153	C₄×C₁₃₇	—	2	4	no	—
154	C₄×C₁₃₇	—	2	2	no	—
155	C₄×C₁₃₇	—	4	4	no	—
156	C₄×C₁₃₇	—	4	2	no	—
157	C₄×C₁₃₇	—	8	2	no	—
158	C₄×C₁₃₇	—	8	2	no	—
159	C₄×C₁₃₇	—	8	2	no	—
160	C₄×C₁₃₇	#467	1	4	yes	—
161	C₄×C₁₃₇	#532	1	1	yes	characteristic
162	C₄×C₁₃₇	#536	1	1	yes	characteristic
163	C₄×C₁₃₇	#556	1	2	yes	—
164	C₂×C₁₃₇	#2545	1	1	yes	characteristic
165	C₂×C₁₃₇	C₂×#467	1	1	yes	characteristic
166	C₂×C₁₃₇	#3057	1	1	yes	characteristic
167	C₂×C₁₃₇	#3360	1	2	yes	—
168	C₂×C₁₃₇	#3363	1	2	yes	—
169	#28638	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₁₃₇	#28638	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×#487	C₂×C₁₃₇	1	1	yes	characteristic
172	C₂×C₈×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
173	#2568	C₂×C₁₃₇	1	1	yes	characteristic
174	#2567	C₂×C₁₃₇	1	1	yes	characteristic
175	#2068	C₂×C₁₃₇	1	4	yes	—
176	#1987	C₂×C₁₃₇	1	1	yes	characteristic
177	#1981	C₂×C₁₃₇	1	1	yes	characteristic
178	#1933	C₂×C₁₃₇	1	4	yes	—
179	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
183	#487	—	2	4	no	—
184	C₈×Q₈	—	2	4	no	—
185	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
186	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
187	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
188	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
189	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
190	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
191	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
192	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
193	#400	—	2	2	no	—
194	#395	—	2	2	no	—
195	#391	—	2	1	no	—
196	#391	—	2	1	no	—
197	#391	C₂²×C₁₃₇	1	4	yes	—
198	#389	C₂²×C₁₃₇	1	4	yes	—
199	#384	—	2	1	no	—
200	#382	—	2	1	no	—
201	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
202	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
203	#340	C₂²×C₁₃₇	1	4	yes	—
204	#340	C₂²×C₁₃₇	1	2	yes	—
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
207	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
208	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
209	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	—	2	2	no	—
214	C₄⋊Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	4	no	—
221	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
222	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
226	C₂×C₄²	—	2	1	no	—
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
229	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
230	#107	—	2	4	no	—
231	C₄⋊C₈	—	2	4	no	—
232	C₄⋊C₈	—	2	2	no	—
233	#103	—	2	4	no	—
234	#103	—	2	4	no	—
235	#103	—	2	8	no	—
236	C₄×C₈	—	2	4	no	—
237	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
238	#95	—	2	1	no	—
239	#95	—	2	1	no	—
240	#95	—	2	4	no	—
241	#95	—	2	4	no	—
242	#95	—	2	4	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	—	8	2	no	—
251	C₂×Q₈	Q₁₆×C₁₃₇	1	4	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
259	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
260	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	8	4	no	—
278	C₄⋊C₄	#41×C₁₃₇	1	4	yes	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	4	2	no	—
282	C₄²	—	4	2	no	—
283	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
284	C₄²	#41×C₁₃₇	1	2	yes	—
285	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
286	Q₈	—	2	8	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	4	no	—
290	Q₈	—	8	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	#121×C₁₃₇	1	2	yes	—
307	C₂×C₄	#121×C₁₃₇	1	2	yes	—
308	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	4	yes	—
309	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
310	C₂×C₄	#135×C₁₃₇	1	2	yes	—
311	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
312	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
313	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
314	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
315	C₈	—	2	2	no	—
316	C₈	—	2	2	no	—
317	C₈	—	4	2	no	—
318	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
319	C₂²	#548×C₁₃₇	1	2	yes	—
320	C₂²	C₂²×Q₁₆×C₁₃₇	1	1	yes	characteristic
321	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
322	C₂²	#577×C₁₃₇	1	2	yes	—
323	C₄	—	2	4	no	—
324	C₄	—	2	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	#467×C₁₃₇	1	4	yes	—
331	C₄	#532×C₁₃₇	1	1	yes	characteristic
332	C₄	#536×C₁₃₇	1	1	yes	characteristic
333	C₄	#556×C₁₃₇	1	2	yes	—
334	C₂	#2545×C₁₃₇	1	1	yes	characteristic
335	C₂	C₂×#467×C₁₃₇	1	1	yes	characteristic
336	C₂	#3057×C₁₃₇	1	1	yes	characteristic
337	C₂	#3360×C₁₃₇	1	2	yes	—
338	C₂	#3363×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
169. #28638	C₁₃₇	direct (N × H)
170. C₁₃₇	#28638	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#487×C₁₃₇	2
C₂×C₈×Q₈×C₁₃₇	2
#2568×C₁₃₇	2
#2567×C₁₃₇	2
#2068×C₁₃₇	2
#1987×C₁₃₇	2
#1981×C₁₃₇	2
#1933×C₁₃₇	2
C₄×#107×C₁₃₇	2
#28638	137

Subgroups — C137×#28638 #46878024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28638 #46878024