Subgroups — C₁₃₇×#11346 #46860732

Order 35072.

878 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	C₂×C₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#336×C₁₃₇	C₂	1	4	yes	maximal
3	C₂×#336×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#336×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
6	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
12	C₄×C₄⋊C₄×C₁₃₇	—	2	2	no	—
13	C₂×#95×C₁₃₇	C₄	1	2	yes	—
14	#336×C₁₃₇	—	2	8	no	—
15	#336×C₁₃₇	—	2	2	no	—
16	#336×C₁₃₇	—	2	2	no	—
17	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₇	—	2	2	no	—
23	C₂²×C₈×C₁₃₇	—	2	4	no	—
24	C₂²×C₈×C₁₃₇	—	2	2	no	—
25	C₂²×C₈×C₁₃₇	—	2	2	no	—
26	C₂²×C₈×C₁₃₇	—	2	4	no	—
27	C₂²×C₈×C₁₃₇	—	2	2	no	—
28	C₂²×C₈×C₁₃₇	—	2	4	no	—
29	C₂²×C₈×C₁₃₇	—	2	4	no	—
30	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
31	C₂²×C₈×C₁₃₇	Q₈	1	4	yes	—
32	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	C₈	1	4	yes	complemented
37	C₂×C₄²×C₁₃₇	—	2	4	no	—
38	C₂×C₄²×C₁₃₇	—	2	2	no	—
39	C₂×C₄²×C₁₃₇	—	2	2	no	—
40	C₂×C₄²×C₁₃₇	—	2	4	no	—
41	C₂×C₄²×C₁₃₇	—	2	4	no	—
42	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
43	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
44	#95×C₁₃₇	—	2	4	no	—
45	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₇	—	2	2	no	—
47	C₂²×C₄×C₁₃₇	—	2	2	no	—
48	C₂²×C₄×C₁₃₇	—	2	2	no	—
49	C₂²×C₄×C₁₃₇	—	2	2	no	—
50	C₂²×C₄×C₁₃₇	—	2	2	no	—
51	C₂²×C₄×C₁₃₇	—	2	4	no	—
52	C₂²×C₄×C₁₃₇	—	2	4	no	—
53	C₂²×C₄×C₁₃₇	—	2	2	no	—
54	C₂²×C₄×C₁₃₇	—	2	2	no	—
55	C₂²×C₄×C₁₃₇	—	2	2	no	—
56	C₂²×C₄×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	—	2	4	no	—
59	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₃₇	M₁₆	1	4	yes	—
64	C₂²×C₄×C₁₃₇	M₁₆	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
70	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
71	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
72	C₂×C₈×C₁₃₇	—	2	4	no	—
73	C₂×C₈×C₁₃₇	—	2	8	no	—
74	C₂×C₈×C₁₃₇	—	2	4	no	—
75	C₂×C₈×C₁₃₇	—	2	8	no	—
76	C₂×C₈×C₁₃₇	—	4	2	no	—
77	C₂×C₈×C₁₃₇	—	4	4	no	—
78	C₂×C₈×C₁₃₇	—	4	2	no	—
79	C₂×C₈×C₁₃₇	—	4	2	no	—
80	C₂×C₈×C₁₃₇	—	4	4	no	—
81	C₂×C₈×C₁₃₇	—	4	2	no	—
82	C₄⋊C₄×C₁₃₇	—	2	8	no	—
83	C₄⋊C₄×C₁₃₇	—	2	8	no	—
84	C₄²×C₁₃₇	—	2	4	no	—
85	C₄²×C₁₃₇	—	2	4	no	—
86	C₄²×C₁₃₇	—	4	4	no	—
87	C₄²×C₁₃₇	—	4	2	no	—
88	C₄²×C₁₃₇	—	4	2	no	—
89	C₂³×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₁₃₇	#117	1	2	yes	—
91	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
92	C₂³×C₁₃₇	C₄×Q₈	1	2	yes	—
93	C₂³×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
94	C₂³×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂³×C₁₃₇	C₂×M₁₆	1	2	yes	—
96	C₂³×C₁₃₇	#131	1	2	yes	—
97	C₂³×C₁₃₇	#131	1	2	yes	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	4	no	—
101	C₂×C₄×C₁₃₇	—	2	4	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	8	no	—
108	C₂×C₄×C₁₃₇	—	2	8	no	—
109	C₂×C₄×C₁₃₇	—	4	2	no	—
110	C₂×C₄×C₁₃₇	—	4	2	no	—
111	C₂×C₄×C₁₃₇	—	4	2	no	—
112	C₂×C₄×C₁₃₇	—	4	2	no	—
113	C₂×C₄×C₁₃₇	—	4	4	no	—
114	C₂×C₄×C₁₃₇	—	4	4	no	—
115	C₂×C₄×C₁₃₇	—	4	4	no	—
116	C₂×C₄×C₁₃₇	C₂²⋊C₈	1	4	yes	—
117	C₂×C₄×C₁₃₇	#121	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
119	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₃₇	#123	1	2	yes	—
121	C₂×C₄×C₁₃₇	#123	1	4	yes	—
122	C₂×C₄×C₁₃₇	#123	1	2	yes	—
123	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
125	C₂×C₄×C₁₃₇	#125	1	4	yes	—
126	C₂×C₄×C₁₃₇	#126	1	4	yes	—
127	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
129	C₈×C₁₃₇	—	4	4	no	—
130	C₈×C₁₃₇	—	4	4	no	—
131	C₂²×C₁₃₇	#381	1	1	yes	characteristic
132	C₂²×C₁₃₇	#387	1	4	yes	—
133	C₂²×C₁₃₇	#390	1	1	yes	characteristic
134	C₂²×C₁₃₇	#391	1	1	yes	characteristic
135	C₂²×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂²×C₁₃₇	#407	1	1	yes	characteristic
137	C₂²×C₁₃₇	#408	1	1	yes	characteristic
138	C₂²×C₁₃₇	#408	1	1	yes	characteristic
139	C₂²×C₁₃₇	#431	1	2	yes	—
140	C₂²×C₁₃₇	#432	1	2	yes	—
141	C₂²×C₁₃₇	#433	1	4	yes	—
142	C₂²×C₁₃₇	C₈×D₄	1	2	yes	—
143	C₂²×C₁₃₇	#435	1	4	yes	—
144	C₂²×C₁₃₇	#436	1	2	yes	—
145	C₂²×C₁₃₇	C₈×Q₈	1	2	yes	—
146	C₂²×C₁₃₇	#446	1	4	yes	—
147	C₂²×C₁₃₇	#446	1	2	yes	—
148	C₄×C₁₃₇	—	2	4	no	—
149	C₄×C₁₃₇	—	4	2	no	—
150	C₄×C₁₃₇	—	4	4	no	—
151	C₄×C₁₃₇	—	4	4	no	—
152	C₄×C₁₃₇	#397	1	2	yes	—
153	C₄×C₁₃₇	#398	1	2	yes	—
154	C₄×C₁₃₇	#399	1	2	yes	—
155	C₄×C₁₃₇	#400	1	2	yes	—
156	C₂×C₁₃₇	#1445	1	1	yes	characteristic
157	C₂×C₁₃₇	#1817	1	1	yes	characteristic
158	C₂×C₁₃₇	#1818	1	1	yes	characteristic
159	C₂×C₁₃₇	#1916	1	4	yes	—
160	C₂×C₁₃₇	#1917	1	4	yes	—
161	C₂×C₁₃₇	#1954	1	1	yes	characteristic
162	C₂×C₁₃₇	#1955	1	1	yes	characteristic
163	C₂×C₁₃₇	#1984	1	1	yes	characteristic
164	C₂×C₁₃₇	#1985	1	1	yes	characteristic
165	#11346	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₇	#11346	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×C₄×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
168	C₂×#336	C₂×C₁₃₇	1	1	yes	characteristic
169	C₂×#336	C₂×C₁₃₇	1	4	yes	—
170	C₂×#336	C₂×C₁₃₇	1	1	yes	characteristic
171	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
172	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
173	C₂²×C₄⋊C₄	C₄×C₁₃₇	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	C₄×C₁₃₇	1	1	yes	characteristic
175	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
176	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
177	C₄×C₄⋊C₄	—	2	2	no	—
178	C₄×C₄⋊C₄	—	2	2	no	—
179	C₂×#95	C₄×C₁₃₇	1	2	yes	—
180	#336	—	2	8	no	—
181	#336	—	2	2	no	—
182	#336	—	2	2	no	—
183	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
185	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
186	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	4	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
197	C₂²×C₈	Q₈×C₁₃₇	1	4	yes	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	C₈×C₁₃₇	1	4	yes	complemented
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
209	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
210	#95	—	2	4	no	—
211	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
226	C₂²×C₄	C₂²⋊C₄×C₁₃₇	1	2	yes	—
227	C₂²×C₄	C₂×C₈×C₁₃₇	1	4	yes	—
228	C₂²×C₄	C₂×C₈×C₁₃₇	1	2	yes	—
229	C₂²×C₄	M₁₆×C₁₃₇	1	4	yes	—
230	C₂²×C₄	M₁₆×C₁₃₇	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
235	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
236	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
237	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	8	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	4	no	—
247	C₂×C₈	—	4	2	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	4	4	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₂³	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
256	C₂³	#117×C₁₃₇	1	2	yes	—
257	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
258	C₂³	C₄×Q₈×C₁₃₇	1	2	yes	—
259	C₂³	C₂²×C₈×C₁₃₇	1	1	yes	characteristic
260	C₂³	C₂×M₁₆×C₁₃₇	1	1	yes	characteristic
261	C₂³	C₂×M₁₆×C₁₃₇	1	2	yes	—
262	C₂³	#131×C₁₃₇	1	2	yes	—
263	C₂³	#131×C₁₃₇	1	2	yes	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	C₂²⋊C₈×C₁₃₇	1	4	yes	—
283	C₂×C₄	#121×C₁₃₇	1	2	yes	—
284	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
285	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
286	C₂×C₄	#123×C₁₃₇	1	2	yes	—
287	C₂×C₄	#123×C₁₃₇	1	4	yes	—
288	C₂×C₄	#123×C₁₃₇	1	2	yes	—
289	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
290	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
291	C₂×C₄	#125×C₁₃₇	1	4	yes	—
292	C₂×C₄	#126×C₁₃₇	1	4	yes	—
293	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
294	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
295	C₈	—	4	4	no	—
296	C₈	—	4	4	no	—
297	C₂²	#381×C₁₃₇	1	1	yes	characteristic
298	C₂²	#387×C₁₃₇	1	4	yes	—
299	C₂²	#390×C₁₃₇	1	1	yes	characteristic
300	C₂²	#391×C₁₃₇	1	1	yes	characteristic
301	C₂²	C₂×C₂²⋊C₈×C₁₃₇	1	1	yes	characteristic
302	C₂²	#407×C₁₃₇	1	1	yes	characteristic
303	C₂²	#408×C₁₃₇	1	1	yes	characteristic
304	C₂²	#408×C₁₃₇	1	1	yes	characteristic
305	C₂²	#431×C₁₃₇	1	2	yes	—
306	C₂²	#432×C₁₃₇	1	2	yes	—
307	C₂²	#433×C₁₃₇	1	4	yes	—
308	C₂²	C₈×D₄×C₁₃₇	1	2	yes	—
309	C₂²	#435×C₁₃₇	1	4	yes	—
310	C₂²	#436×C₁₃₇	1	2	yes	—
311	C₂²	C₈×Q₈×C₁₃₇	1	2	yes	—
312	C₂²	#446×C₁₃₇	1	4	yes	—
313	C₂²	#446×C₁₃₇	1	2	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	#397×C₁₃₇	1	2	yes	—
319	C₄	#398×C₁₃₇	1	2	yes	—
320	C₄	#399×C₁₃₇	1	2	yes	—
321	C₄	#400×C₁₃₇	1	2	yes	—
322	C₂	#1445×C₁₃₇	1	1	yes	characteristic
323	C₂	#1817×C₁₃₇	1	1	yes	characteristic
324	C₂	#1818×C₁₃₇	1	1	yes	characteristic
325	C₂	#1916×C₁₃₇	1	4	yes	—
326	C₂	#1917×C₁₃₇	1	4	yes	—
327	C₂	#1954×C₁₃₇	1	1	yes	characteristic
328	C₂	#1955×C₁₃₇	1	1	yes	characteristic
329	C₂	#1984×C₁₃₇	1	1	yes	characteristic
330	C₂	#1985×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₁₃₇	C₂³
commutator	C₂³	C₂²×C₈×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
socle, prime radical	C₂⁴×C₁₃₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
36. C₂×C₄⋊C₄×C₁₃₇	C₈	semidirect
165. #11346	C₁₃₇	direct (N × H)
166. C₁₃₇	#11346	direct (N × H)
202. C₂×C₄⋊C₄	C₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊C₄×C₁₃₇	2
C₂×#336×C₁₃₇	2
#11346	137

Subgroups — C137×#11346 #46860732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11346 #46860732