Subgroups — C₁₃₇×#11331 #46860717

Order 35072.

722 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₁₃₇	C₂	1	2	yes	maximal
2	C₂²×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#343×C₁₃₇	C₂	1	2	yes	maximal
4	C₂×#341×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1729×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
11	#428×C₁₃₇	—	2	4	no	—
12	#428×C₁₃₇	—	2	4	no	—
13	C₂×#107×C₁₃₇	—	2	2	no	—
14	C₂×#107×C₁₃₇	—	2	4	no	—
15	C₂×#107×C₁₃₇	—	2	2	no	—
16	C₂×#106×C₁₃₇	—	2	2	no	—
17	#343×C₁₃₇	—	2	4	no	—
18	#341×C₁₃₇	C₄	1	4	yes	complemented
19	#336×C₁₃₇	—	2	2	no	—
20	C₂³×C₄×C₁₃₇	—	2	2	no	—
21	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₃₇	—	2	2	no	—
23	C₂×M₁₆×C₁₃₇	—	2	2	no	—
24	C₂×M₁₆×C₁₃₇	—	2	2	no	—
25	C₂×M₁₆×C₁₃₇	—	2	1	no	—
26	C₂×M₁₆×C₁₃₇	—	2	2	no	—
27	C₂×M₁₆×C₁₃₇	—	2	2	no	—
28	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₁₃₇	D₄	1	4	yes	—
30	C₂×M₁₆×C₁₃₇	Q₈	1	2	yes	—
31	C₂×M₁₆×C₁₃₇	Q₈	1	4	yes	—
32	C₂²×C₈×C₁₃₇	—	2	2	no	—
33	C₂²×C₈×C₁₃₇	—	2	2	no	—
34	C₂²×C₈×C₁₃₇	—	2	2	no	—
35	C₂²×C₈×C₁₃₇	—	2	1	no	—
36	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
38	C₂²×C₈×C₁₃₇	Q₈	1	2	yes	—
39	#117×C₁₃₇	—	2	2	no	—
40	#117×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	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₃₇	D₄	1	4	yes	—
49	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
50	C₂×C₄²×C₁₃₇	—	2	1	no	—
51	#107×C₁₃₇	—	2	8	no	—
52	#107×C₁₃₇	—	2	8	no	—
53	#106×C₁₃₇	—	2	8	no	—
54	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₇	—	2	1	no	—
56	C₂²×C₄×C₁₃₇	—	2	1	no	—
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	—	4	2	no	—
59	C₂²×C₄×C₁₃₇	—	4	2	no	—
60	C₂²×C₄×C₁₃₇	—	8	4	no	—
61	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
70	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
71	M₁₆×C₁₃₇	—	2	4	no	—
72	M₁₆×C₁₃₇	—	2	4	no	—
73	M₁₆×C₁₃₇	—	2	4	no	—
74	M₁₆×C₁₃₇	—	2	4	no	—
75	M₁₆×C₁₃₇	—	4	2	no	—
76	M₁₆×C₁₃₇	C₄⋊C₄	1	8	yes	—
77	C₂×C₈×C₁₃₇	—	2	4	no	—
78	C₂×C₈×C₁₃₇	—	2	4	no	—
79	C₂×C₈×C₁₃₇	—	2	4	no	—
80	C₂×C₈×C₁₃₇	—	2	2	no	—
81	C₂×C₈×C₁₃₇	—	2	4	no	—
82	C₂×C₈×C₁₃₇	—	2	4	no	—
83	C₂×C₈×C₁₃₇	—	2	2	no	—
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	—	4	1	no	—
86	C₂×C₈×C₁₃₇	—	4	1	no	—
87	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₁₃₇	—	2	2	no	—
89	C₄⋊C₄×C₁₃₇	—	2	4	no	—
90	C₄⋊C₄×C₁₃₇	—	2	4	no	—
91	C₄⋊C₄×C₁₃₇	—	2	2	no	—
92	C₄⋊C₄×C₁₃₇	—	4	2	no	—
93	C₄⋊C₄×C₁₃₇	—	4	2	no	—
94	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
95	C₄²×C₁₃₇	—	4	1	no	—
96	C₄²×C₁₃₇	—	4	1	no	—
97	C₂³×C₁₃₇	—	2	2	no	—
98	C₂³×C₁₃₇	—	2	2	no	—
99	C₂³×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
100	C₂³×C₁₃₇	#121	1	2	yes	—
101	C₂³×C₁₃₇	#121	1	1	yes	characteristic
102	C₂³×C₁₃₇	#125	1	2	yes	—
103	C₂³×C₁₃₇	#125	1	1	yes	characteristic
104	C₂×C₄×C₁₃₇	—	4	1	no	—
105	C₂×C₄×C₁₃₇	—	4	1	no	—
106	C₂×C₄×C₁₃₇	—	4	4	no	—
107	C₂×C₄×C₁₃₇	—	4	2	no	—
108	C₂×C₄×C₁₃₇	—	4	2	no	—
109	C₂×C₄×C₁₃₇	—	8	4	no	—
110	C₂×C₄×C₁₃₇	—	8	4	no	—
111	C₂×C₄×C₁₃₇	—	8	4	no	—
112	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃₇	#120	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	#120	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₁₃₇	#123	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	#123	1	2	yes	—
125	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
130	C₈×C₁₃₇	—	2	4	no	—
131	C₈×C₁₃₇	—	2	8	no	—
132	C₈×C₁₃₇	—	4	2	no	—
133	C₂²×C₁₃₇	—	2	4	no	—
134	C₂²×C₁₃₇	—	2	2	no	—
135	C₂²×C₁₃₇	—	2	2	no	—
136	C₂²×C₁₃₇	—	2	2	no	—
137	C₂²×C₁₃₇	—	2	2	no	—
138	C₂²×C₁₃₇	#389	1	2	yes	—
139	C₂²×C₁₃₇	#389	1	1	yes	characteristic
140	C₂²×C₁₃₇	#392	1	1	yes	characteristic
141	C₂²×C₁₃₇	#398	1	2	yes	—
142	C₂²×C₁₃₇	#398	1	1	yes	characteristic
143	C₂²×C₁₃₇	#472	1	2	yes	—
144	C₂²×C₁₃₇	#473	1	2	yes	—
145	C₄×C₁₃₇	—	8	1	no	—
146	C₄×C₁₃₇	—	8	4	no	—
147	C₄×C₁₃₇	—	8	1	no	—
148	C₄×C₁₃₇	#384	1	1	yes	characteristic
149	C₄×C₁₃₇	#385	1	1	yes	characteristic
150	C₄×C₁₃₇	#385	1	2	yes	—
151	C₄×C₁₃₇	#395	1	1	yes	characteristic
152	C₄×C₁₃₇	#395	1	2	yes	—
153	C₄×C₁₃₇	#399	1	1	yes	characteristic
154	C₂×C₁₃₇	—	2	2	no	—
155	C₂×C₁₃₇	—	2	2	no	—
156	C₂×C₁₃₇	#1444	1	1	yes	characteristic
157	C₂×C₁₃₇	#1952	1	2	yes	—
158	C₂×C₁₃₇	#2018	1	1	yes	characteristic
159	C₂×C₁₃₇	#2020	1	1	yes	characteristic
160	C₂×C₁₃₇	#2089	1	2	yes	—
161	#11331	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₁₃₇	#11331	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×#428	C₂×C₁₃₇	1	2	yes	—
164	C₂²×#107	C₂×C₁₃₇	1	1	yes	characteristic
165	C₂×#343	C₂×C₁₃₇	1	2	yes	—
166	C₂×#341	C₂×C₁₃₇	1	1	yes	characteristic
167	#1729	C₂×C₁₃₇	1	1	yes	characteristic
168	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
169	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
170	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
173	#428	—	2	4	no	—
174	#428	—	2	4	no	—
175	C₂×#107	—	2	2	no	—
176	C₂×#107	—	2	4	no	—
177	C₂×#107	—	2	2	no	—
178	C₂×#106	—	2	2	no	—
179	#343	—	2	4	no	—
180	#341	C₄×C₁₃₇	1	4	yes	complemented
181	#336	—	2	2	no	—
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
191	C₂×M₁₆	D₄×C₁₃₇	1	4	yes	—
192	C₂×M₁₆	Q₈×C₁₃₇	1	2	yes	—
193	C₂×M₁₆	Q₈×C₁₃₇	1	4	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
199	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
200	C₂²×C₈	Q₈×C₁₃₇	1	2	yes	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	D₄×C₁₃₇	1	4	yes	—
209	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
210	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	—	2	1	no	—
212	C₂×C₄²	—	2	1	no	—
213	#107	—	2	8	no	—
214	#107	—	2	8	no	—
215	#106	—	2	8	no	—
216	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	8	4	no	—
223	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
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	2	yes	—
228	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
230	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
231	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
232	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	4	2	no	—
238	M₁₆	C₄⋊C₄×C₁₃₇	1	8	yes	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
262	C₂³	#121×C₁₃₇	1	2	yes	—
263	C₂³	#121×C₁₃₇	1	1	yes	characteristic
264	C₂³	#125×C₁₃₇	1	2	yes	—
265	C₂³	#125×C₁₃₇	1	1	yes	characteristic
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	—	8	4	no	—
273	C₂×C₄	—	8	4	no	—
274	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
276	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
277	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
278	C₂×C₄	#120×C₁₃₇	1	1	yes	characteristic
279	C₂×C₄	#120×C₁₃₇	1	2	yes	—
280	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
281	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
282	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
283	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
285	C₂×C₄	#123×C₁₃₇	1	1	yes	characteristic
286	C₂×C₄	#123×C₁₃₇	1	2	yes	—
287	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
288	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
289	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
290	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
291	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	2	8	no	—
294	C₈	—	4	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	#389×C₁₃₇	1	2	yes	—
301	C₂²	#389×C₁₃₇	1	1	yes	characteristic
302	C₂²	#392×C₁₃₇	1	1	yes	characteristic
303	C₂²	#398×C₁₃₇	1	2	yes	—
304	C₂²	#398×C₁₃₇	1	1	yes	characteristic
305	C₂²	#472×C₁₃₇	1	2	yes	—
306	C₂²	#473×C₁₃₇	1	2	yes	—
307	C₄	—	8	1	no	—
308	C₄	—	8	4	no	—
309	C₄	—	8	1	no	—
310	C₄	#384×C₁₃₇	1	1	yes	characteristic
311	C₄	#385×C₁₃₇	1	1	yes	characteristic
312	C₄	#385×C₁₃₇	1	2	yes	—
313	C₄	#395×C₁₃₇	1	1	yes	characteristic
314	C₄	#395×C₁₃₇	1	2	yes	—
315	C₄	#399×C₁₃₇	1	1	yes	characteristic
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	#1444×C₁₃₇	1	1	yes	characteristic
319	C₂	#1952×C₁₃₇	1	2	yes	—
320	C₂	#2018×C₁₃₇	1	1	yes	characteristic
321	C₂	#2020×C₁₃₇	1	1	yes	characteristic
322	C₂	#2089×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	C₂⁴×C₁₃₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
18. #341×C₁₃₇	C₄	semidirect
161. #11331	C₁₃₇	direct (N × H)
162. C₁₃₇	#11331	direct (N × H)
180. #341	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×#428×C₁₃₇	2
C₂²×#107×C₁₃₇	2
C₂×#343×C₁₃₇	2
C₂×#341×C₁₃₇	2
#1729×C₁₃₇	2
#11331	137

Subgroups — C137×#11331 #46860717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#11331 #46860717