Subgroups — C₁₃₇×#28721 #46878107

Order 35072.

794 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#488×C₁₃₇	C₂	1	2	yes	maximal
2	#2859×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#124×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2074×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2073×C₁₃₇	C₂	1	2	yes	maximal
6	#1991×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1803×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1802×C₁₃₇	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#124×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
14	#488×C₁₃₇	—	2	8	no	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#103×C₁₃₇	C₂²	1	4	yes	—
17	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
19	#391×C₁₃₇	C₂²	1	1	yes	characteristic
20	#390×C₁₃₇	C₂²	1	1	yes	characteristic
21	#389×C₁₃₇	—	2	2	no	—
22	#389×C₁₃₇	—	2	2	no	—
23	#388×C₁₃₇	—	2	2	no	—
24	#384×C₁₃₇	C₂²	1	2	yes	—
25	#382×C₁₃₇	C₂²	1	1	yes	characteristic
26	#382×C₁₃₇	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
28	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
29	#340×C₁₃₇	C₂²	1	4	yes	—
30	#340×C₁₃₇	C₂²	1	4	yes	—
31	#336×C₁₃₇	C₂²	1	2	yes	—
32	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₃₇	—	2	1	no	—
35	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
36	#125×C₁₃₇	—	2	4	no	—
37	#125×C₁₃₇	—	2	4	no	—
38	#124×C₁₃₇	—	2	2	no	—
39	#124×C₁₃₇	D₄	1	4	yes	—
40	C₄×Q₈×C₁₃₇	—	2	4	no	—
41	C₄×D₄×C₁₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
52	C₂×C₄²×C₁₃₇	—	2	1	no	—
53	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
54	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
55	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₁₃₇	—	2	4	no	—
59	#103×C₁₃₇	—	2	8	no	—
60	#102×C₁₃₇	—	2	8	no	—
61	C₄×C₈×C₁₃₇	—	2	4	no	—
62	#95×C₁₃₇	—	2	2	no	—
63	#95×C₁₃₇	—	2	2	no	—
64	#95×C₁₃₇	—	2	4	no	—
65	C₂⁴×C₁₃₇	—	2	1	no	—
66	C₂×Q₈×C₁₃₇	—	2	1	no	—
67	C₂×Q₈×C₁₃₇	—	2	2	no	—
68	C₂×Q₈×C₁₃₇	—	2	2	no	—
69	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂×D₄×C₁₃₇	—	2	2	no	—
71	C₂×D₄×C₁₃₇	—	2	1	no	—
72	C₂×D₄×C₁₃₇	—	2	2	no	—
73	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₇	—	2	1	no	—
75	C₂²×C₄×C₁₃₇	—	2	1	no	—
76	C₂²×C₄×C₁₃₇	—	2	1	no	—
77	C₂²×C₄×C₁₃₇	—	2	4	no	—
78	C₂²×C₄×C₁₃₇	—	4	2	no	—
79	C₂²×C₄×C₁₃₇	—	4	4	no	—
80	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₁₃₇	—	2	4	no	—
86	C₂×C₈×C₁₃₇	—	2	4	no	—
87	C₂×C₈×C₁₃₇	—	2	4	no	—
88	C₄⋊C₄×C₁₃₇	—	2	4	no	—
89	C₄⋊C₄×C₁₃₇	—	2	2	no	—
90	C₄⋊C₄×C₁₃₇	—	2	2	no	—
91	C₄⋊C₄×C₁₃₇	—	2	4	no	—
92	C₄⋊C₄×C₁₃₇	—	4	4	no	—
93	C₄⋊C₄×C₁₃₇	—	4	8	no	—
94	C₄⋊C₄×C₁₃₇	—	4	2	no	—
95	C₄⋊C₄×C₁₃₇	—	4	4	no	—
96	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
97	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
98	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
99	C₄²×C₁₃₇	—	2	4	no	—
100	C₄²×C₁₃₇	—	2	2	no	—
101	C₄²×C₁₃₇	—	4	2	no	—
102	C₄²×C₁₃₇	—	4	2	no	—
103	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
104	C₄²×C₁₃₇	#41	1	2	yes	—
105	C₄²×C₁₃₇	#41	1	4	yes	—
106	C₂³×C₁₃₇	—	2	2	no	—
107	C₂³×C₁₃₇	—	4	1	no	—
108	C₂³×C₁₃₇	—	4	1	no	—
109	C₂³×C₁₃₇	—	4	2	no	—
110	C₂³×C₁₃₇	—	4	2	no	—
111	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₁₃₇	—	2	4	no	—
113	Q₈×C₁₃₇	—	4	2	no	—
114	D₄×C₁₃₇	—	2	4	no	—
115	D₄×C₁₃₇	—	4	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	2	2	no	—
118	C₂×C₄×C₁₃₇	—	2	2	no	—
119	C₂×C₄×C₁₃₇	—	2	2	no	—
120	C₂×C₄×C₁₃₇	—	2	2	no	—
121	C₂×C₄×C₁₃₇	—	2	2	no	—
122	C₂×C₄×C₁₃₇	—	2	2	no	—
123	C₂×C₄×C₁₃₇	—	4	1	no	—
124	C₂×C₄×C₁₃₇	—	4	1	no	—
125	C₂×C₄×C₁₃₇	—	4	4	no	—
126	C₂×C₄×C₁₃₇	—	4	1	no	—
127	C₂×C₄×C₁₃₇	—	4	1	no	—
128	C₂×C₄×C₁₃₇	—	4	4	no	—
129	C₂×C₄×C₁₃₇	—	4	2	no	—
130	C₂×C₄×C₁₃₇	—	4	2	no	—
131	C₂×C₄×C₁₃₇	—	4	4	no	—
132	C₂×C₄×C₁₃₇	—	8	4	no	—
133	C₂×C₄×C₁₃₇	#121	1	4	yes	—
134	C₂×C₄×C₁₃₇	#124	1	2	yes	—
135	C₂×C₄×C₁₃₇	#124	1	2	yes	—
136	C₂×C₄×C₁₃₇	#135	1	4	yes	—
137	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
141	C₈×C₁₃₇	—	4	4	no	—
142	C₂²×C₁₃₇	—	4	4	no	—
143	C₂²×C₁₃₇	—	4	2	no	—
144	C₂²×C₁₃₇	—	8	1	no	—
145	C₂²×C₁₃₇	—	8	1	no	—
146	C₂²×C₁₃₇	—	8	2	no	—
147	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
148	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
149	C₂²×C₁₃₇	#536	1	1	yes	characteristic
150	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
151	C₂²×C₁₃₇	#575	1	2	yes	—
152	C₄×C₁₃₇	—	2	2	no	—
153	C₄×C₁₃₇	—	2	4	no	—
154	C₄×C₁₃₇	—	4	2	no	—
155	C₄×C₁₃₇	—	4	2	no	—
156	C₄×C₁₃₇	—	8	1	no	—
157	C₄×C₁₃₇	—	8	1	no	—
158	C₄×C₁₃₇	—	8	4	no	—
159	C₄×C₁₃₇	#488	1	4	yes	complemented
160	C₄×C₁₃₇	#547	1	2	yes	—
161	C₄×C₁₃₇	#556	1	2	yes	—
162	C₂×C₁₃₇	—	8	2	no	—
163	C₂×C₁₃₇	#2547	1	1	yes	characteristic
164	C₂×C₁₃₇	#3035	1	2	yes	—
165	C₂×C₁₃₇	C₂×#488	1	1	yes	characteristic
166	C₂×C₁₃₇	#3142	1	1	yes	characteristic
167	C₂×C₁₃₇	#3233	1	2	yes	—
168	#28721	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₁₃₇	#28721	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×#488	C₂×C₁₃₇	1	2	yes	—
171	#2859	C₂×C₁₃₇	1	1	yes	characteristic, complemented
172	C₄×#124	C₂×C₁₃₇	1	1	yes	characteristic
173	#2074	C₂×C₁₃₇	1	2	yes	complemented
174	#2073	C₂×C₁₃₇	1	2	yes	—
175	#1991	C₂×C₁₃₇	1	2	yes	complemented
176	#1803	C₂×C₁₃₇	1	2	yes	complemented
177	#1802	C₂×C₁₃₇	1	2	yes	—
178	C₄×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
179	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
180	C₂×#124	C₂²×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
183	#488	—	2	8	no	—
184	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
185	C₂×#103	C₂²×C₁₃₇	1	4	yes	—
186	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
187	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
188	#391	C₂²×C₁₃₇	1	1	yes	characteristic
189	#390	C₂²×C₁₃₇	1	1	yes	characteristic
190	#389	—	2	2	no	—
191	#389	—	2	2	no	—
192	#388	—	2	2	no	—
193	#384	C₂²×C₁₃₇	1	2	yes	—
194	#382	C₂²×C₁₃₇	1	2	yes	—
195	#382	C₂²×C₁₃₇	1	1	yes	characteristic
196	C₄×C₂²⋊C₄	—	2	2	no	—
197	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
198	#340	C₂²×C₁₃₇	1	4	yes	—
199	#340	C₂²×C₁₃₇	1	4	yes	—
200	#336	C₂²×C₁₃₇	1	2	yes	—
201	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
202	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
203	C₂³×C₄	—	2	1	no	—
204	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
205	#125	—	2	4	no	—
206	#125	—	2	4	no	—
207	#124	—	2	2	no	—
208	#124	D₄×C₁₃₇	1	4	yes	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×D₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
218	C₂×C₂²⋊C₄	—	2	2	no	—
219	C₂×C₂²⋊C₄	—	2	1	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
223	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
224	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
225	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
227	C₄⋊C₈	—	2	4	no	—
228	#103	—	2	8	no	—
229	#102	—	2	8	no	—
230	C₄×C₈	—	2	4	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	4	no	—
234	C₂⁴	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
253	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	8	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	2	8	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
273	C₄²	#41×C₁₃₇	1	2	yes	—
274	C₄²	#41×C₁₃₇	1	4	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	4	1	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	Q₈	—	4	2	no	—
283	D₄	—	2	4	no	—
284	D₄	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	#121×C₁₃₇	1	4	yes	—
303	C₂×C₄	#124×C₁₃₇	1	2	yes	—
304	C₂×C₄	#124×C₁₃₇	1	2	yes	—
305	C₂×C₄	#135×C₁₃₇	1	4	yes	—
306	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
308	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
309	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
310	C₈	—	4	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
317	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
318	C₂²	#536×C₁₃₇	1	1	yes	characteristic
319	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
320	C₂²	#575×C₁₃₇	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	4	no	—
328	C₄	#488×C₁₃₇	1	4	yes	complemented
329	C₄	#547×C₁₃₇	1	2	yes	—
330	C₄	#556×C₁₃₇	1	2	yes	—
331	C₂	—	8	2	no	—
332	C₂	#2547×C₁₃₇	1	1	yes	characteristic
333	C₂	#3035×C₁₃₇	1	2	yes	—
334	C₂	C₂×#488×C₁₃₇	1	1	yes	characteristic
335	C₂	#3142×C₁₃₇	1	1	yes	characteristic
336	C₂	#3233×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	C₂³
LCS	G ▷ C₂²×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
2. #2859×C₁₃₇	C₂	semidirect
4. #2074×C₁₃₇	C₂	semidirect
6. #1991×C₁₃₇	C₂	semidirect
7. #1803×C₁₃₇	C₂	semidirect
9. C₄×C₄⋊C₈×C₁₃₇	C₂	semidirect
159. C₄×C₁₃₇	#488	semidirect
168. #28721	C₁₃₇	direct (N × H)
169. C₁₃₇	#28721	direct (N × H)
171. #2859	C₂×C₁₃₇	semidirect
173. #2074	C₂×C₁₃₇	semidirect
175. #1991	C₂×C₁₃₇	semidirect
176. #1803	C₂×C₁₃₇	semidirect
178. C₄×C₄⋊C₈	C₂×C₁₃₇	semidirect
328. C₄	#488×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#488×C₁₃₇	2
#2859×C₁₃₇	2
C₄×#124×C₁₃₇	2
#2074×C₁₃₇	2
#2073×C₁₃₇	2
#1991×C₁₃₇	2
#1803×C₁₃₇	2
#1802×C₁₃₇	2
C₄×C₄⋊C₈×C₁₃₇	2
#28721	137

Subgroups — C137×#28721 #46878107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28721 #46878107