Subgroups — C₁₃₇×#10799 #46860185

Order 35072.

650 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#338×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×H(3,ℤ/4)×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×H(3,ℤ/4)×C₁₃₇	C₂	1	2	yes	maximal
5	#1729×C₁₃₇	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
11	#405×C₁₃₇	—	2	4	no	—
12	C₂×#97×C₁₃₇	C₄	1	2	yes	complemented
13	C₂×C₄×C₈×C₁₃₇	C₄	1	2	yes	complemented
14	#338×C₁₃₇	—	2	2	no	—
15	H(3,ℤ/4)×C₁₃₇	—	2	2	no	—
16	H(3,ℤ/4)×C₁₃₇	—	2	4	no	—
17	#336×C₁₃₇	C₄	1	8	yes	complemented
18	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₁₃₇	—	2	2	no	—
20	C₂×M₁₆×C₁₃₇	—	2	2	no	—
21	C₂×M₁₆×C₁₃₇	—	2	2	no	—
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	2	no	—
26	C₂×M₁₆×C₁₃₇	—	2	2	no	—
27	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
28	C₂×M₁₆×C₁₃₇	Q₈	1	2	yes	—
29	C₂²×C₈×C₁₃₇	—	2	2	no	—
30	C₂²×C₈×C₁₃₇	—	2	1	no	—
31	C₂²×C₈×C₁₃₇	—	2	1	no	—
32	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
37	#117×C₁₃₇	—	2	2	no	—
38	#117×C₁₃₇	—	2	2	no	—
39	#117×C₁₃₇	—	2	2	no	—
40	#117×C₁₃₇	—	2	2	no	—
41	#117×C₁₃₇	—	2	2	no	—
42	#117×C₁₃₇	D₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
49	#97×C₁₃₇	—	2	4	no	—
50	C₄×C₈×C₁₃₇	—	2	4	no	—
51	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₇	—	2	2	no	—
53	C₂²×C₄×C₁₃₇	—	2	4	no	—
54	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
60	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
64	M₁₆×C₁₃₇	—	2	4	no	—
65	M₁₆×C₁₃₇	—	2	4	no	—
66	M₁₆×C₁₃₇	—	4	2	no	—
67	M₁₆×C₁₃₇	—	4	2	no	—
68	M₁₆×C₁₃₇	—	4	2	no	—
69	M₁₆×C₁₃₇	—	4	2	no	—
70	C₂×C₈×C₁₃₇	—	2	1	no	—
71	C₂×C₈×C₁₃₇	—	2	2	no	—
72	C₂×C₈×C₁₃₇	—	2	1	no	—
73	C₂×C₈×C₁₃₇	—	2	1	no	—
74	C₂×C₈×C₁₃₇	—	2	1	no	—
75	C₂×C₈×C₁₃₇	—	2	2	no	—
76	C₂×C₈×C₁₃₇	—	2	4	no	—
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	4	no	—
81	C₂×C₈×C₁₃₇	C₄²	1	2	yes	—
82	C₂×C₈×C₁₃₇	C₄²	1	2	yes	—
83	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	2	no	—
86	C₄⋊C₄×C₁₃₇	—	4	2	no	—
87	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
88	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
89	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
90	C₄²×C₁₃₇	—	2	2	no	—
91	C₄²×C₁₃₇	—	2	4	no	—
92	C₄²×C₁₃₇	—	2	4	no	—
93	C₄²×C₁₃₇	C₂²⋊C₄	1	4	yes	—
94	C₂³×C₁₃₇	—	4	2	no	—
95	C₂³×C₁₃₇	#117	1	1	yes	characteristic
96	C₂³×C₁₃₇	#117	1	2	yes	—
97	C₂³×C₁₃₇	#123	1	2	yes	—
98	C₂³×C₁₃₇	#124	1	1	yes	characteristic
99	C₂³×C₁₃₇	#125	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	—	2	4	no	—
101	C₂×C₄×C₁₃₇	—	4	4	no	—
102	C₂×C₄×C₁₃₇	—	4	8	no	—
103	C₂×C₄×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	#117	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₁₃₇	#121	1	2	yes	—
116	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₃₇	#123	1	2	yes	—
118	C₂×C₄×C₁₃₇	#124	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	#125	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	#126	1	2	yes	—
121	C₂×C₄×C₁₃₇	#126	1	2	yes	—
122	C₈×C₁₃₇	—	2	2	no	—
123	C₈×C₁₃₇	—	2	2	no	—
124	C₈×C₁₃₇	—	2	4	no	—
125	C₈×C₁₃₇	—	4	4	no	—
126	C₂²×C₁₃₇	—	2	2	no	—
127	C₂²×C₁₃₇	—	2	4	no	—
128	C₂²×C₁₃₇	—	4	2	no	—
129	C₂²×C₁₃₇	—	4	1	no	—
130	C₂²×C₁₃₇	—	4	1	no	—
131	C₂²×C₁₃₇	#376	1	1	yes	characteristic
132	C₂²×C₁₃₇	#381	1	1	yes	characteristic
133	C₂²×C₁₃₇	#382	1	1	yes	characteristic
134	C₂²×C₁₃₇	#387	1	2	yes	—
135	C₂²×C₁₃₇	#388	1	2	yes	—
136	C₄×C₁₃₇	—	4	4	no	—
137	C₄×C₁₃₇	—	8	4	no	—
138	C₄×C₁₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₁₃₇	#380	1	1	yes	characteristic
141	C₄×C₁₃₇	#383	1	1	yes	characteristic
142	C₄×C₁₃₇	#387	1	2	yes	—
143	C₄×C₁₃₇	#388	1	2	yes	—
144	C₂×C₁₃₇	—	2	1	no	—
145	C₂×C₁₃₇	—	2	2	no	—
146	C₂×C₁₃₇	—	2	1	no	—
147	C₂×C₁₃₇	—	2	2	no	—
148	C₂×C₁₃₇	#1434	1	1	yes	characteristic
149	C₂×C₁₃₇	#1747	1	2	yes	—
150	#10799	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₁₃₇	#10799	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×#405	C₂×C₁₃₇	1	1	yes	characteristic
153	C₂×#338	C₂×C₁₃₇	1	1	yes	characteristic
154	C₂×H(3,ℤ/4)	C₂×C₁₃₇	1	1	yes	characteristic
155	C₂×H(3,ℤ/4)	C₂×C₁₃₇	1	2	yes	—
156	#1729	C₂×C₁₃₇	1	2	yes	—
157	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
158	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
159	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
161	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
162	#405	—	2	4	no	—
163	C₂×#97	C₄×C₁₃₇	1	2	yes	complemented
164	C₂×C₄×C₈	C₄×C₁₃₇	1	2	yes	complemented
165	#338	—	2	2	no	—
166	H(3,ℤ/4)	—	2	2	no	—
167	H(3,ℤ/4)	—	2	4	no	—
168	#336	C₄×C₁₃₇	1	8	yes	complemented
169	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
179	C₂×M₁₆	Q₈×C₁₃₇	1	2	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
184	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
185	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	D₄×C₁₃₇	1	4	yes	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
199	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
200	#97	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
210	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
211	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
212	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
213	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
214	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₄²×C₁₃₇	1	2	yes	—
233	C₂×C₈	C₄²×C₁₃₇	1	2	yes	—
234	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	C₂²⋊C₄×C₁₃₇	1	4	yes	—
245	C₂³	—	4	2	no	—
246	C₂³	#117×C₁₃₇	1	1	yes	characteristic
247	C₂³	#117×C₁₃₇	1	2	yes	—
248	C₂³	#123×C₁₃₇	1	2	yes	—
249	C₂³	#124×C₁₃₇	1	1	yes	characteristic
250	C₂³	#125×C₁₃₇	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
258	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
259	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
260	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
261	C₂×C₄	#117×C₁₃₇	1	2	yes	—
262	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
263	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
264	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
265	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
266	C₂×C₄	#121×C₁₃₇	1	2	yes	—
267	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
268	C₂×C₄	#123×C₁₃₇	1	2	yes	—
269	C₂×C₄	#124×C₁₃₇	1	1	yes	characteristic
270	C₂×C₄	#125×C₁₃₇	1	1	yes	characteristic
271	C₂×C₄	#126×C₁₃₇	1	2	yes	—
272	C₂×C₄	#126×C₁₃₇	1	2	yes	—
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	#376×C₁₃₇	1	1	yes	characteristic
283	C₂²	#381×C₁₃₇	1	1	yes	characteristic
284	C₂²	#382×C₁₃₇	1	1	yes	characteristic
285	C₂²	#387×C₁₃₇	1	2	yes	—
286	C₂²	#388×C₁₃₇	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	8	4	no	—
289	C₄	C₄×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
290	C₄	C₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
291	C₄	#380×C₁₃₇	1	1	yes	characteristic
292	C₄	#383×C₁₃₇	1	1	yes	characteristic
293	C₄	#387×C₁₃₇	1	2	yes	—
294	C₄	#388×C₁₃₇	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#1434×C₁₃₇	1	1	yes	characteristic
300	C₂	#1747×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#376
prime radical	C₂⁴×C₁₃₇	#41
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×#97×C₁₃₇	C₄	semidirect
13. C₂×C₄×C₈×C₁₃₇	C₄	semidirect
17. #336×C₁₃₇	C₄	semidirect
150. #10799	C₁₃₇	direct (N × H)
151. C₁₃₇	#10799	direct (N × H)
163. C₂×#97	C₄×C₁₃₇	semidirect
164. C₂×C₄×C₈	C₄×C₁₃₇	semidirect
168. #336	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×#405×C₁₃₇	2
C₂×#338×C₁₃₇	2
C₂×H(3,ℤ/4)×C₁₃₇	2
#1729×C₁₃₇	2
#10799	137

Subgroups — C137×#10799 #46860185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#10799 #46860185