Subgroups — C₁₃₇×#33682 #46883068

Order 35072.

580 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3176×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#2948×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2234×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#2228×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2206×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2184×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2177×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×D₁₆×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#2164×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	#540×C₁₃₇	—	2	1	no	—
12	#518×C₁₃₇	—	2	1	no	—
13	C₂×D₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂×D₁₆×C₁₃₇	C₂²	1	2	yes	—
15	#499×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
16	#493×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
17	#490×C₁₃₇	—	2	1	no	—
18	#469×C₁₃₇	—	2	1	no	—
19	#466×C₁₃₇	C₂²	1	2	yes	—
20	#442×C₁₃₇	—	2	1	no	—
21	#442×C₁₃₇	—	2	1	no	—
22	#442×C₁₃₇	—	2	1	no	—
23	C₄×D₈×C₁₃₇	C₂²	1	2	yes	—
24	C₄×D₈×C₁₃₇	C₂²	1	2	yes	—
25	C₄×D₈×C₁₃₇	C₂²	1	2	yes	complemented
26	#427×C₁₃₇	C₂²	1	1	yes	characteristic
27	#418×C₁₃₇	—	2	1	no	—
28	#405×C₁₃₇	C₂²	1	1	yes	characteristic
29	#367×C₁₃₇	C₂²	1	2	yes	complemented
30	#366×C₁₃₇	C₂²	1	2	yes	complemented
31	C₄⋊C₁₆×C₁₃₇	C₂²	1	2	yes	complemented
32	#357×C₁₃₇	C₂²	1	2	yes	—
33	#357×C₁₃₇	C₂²	1	2	yes	—
34	#357×C₁₃₇	C₂²	1	2	yes	complemented
35	#357×C₁₃₇	C₂²	1	2	yes	complemented
36	C₂²⋊C₁₆×C₁₃₇	C₂²	1	2	yes	—
37	#346×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₁₃₇	—	2	1	no	—
40	C₂×D₈×C₁₃₇	—	2	4	no	—
41	C₂×D₈×C₁₃₇	—	2	1	no	—
42	C₂×D₈×C₁₃₇	—	2	1	no	—
43	C₂×D₈×C₁₃₇	C₂³	1	2	yes	—
44	C₂×D₈×C₁₃₇	C₂³	1	2	yes	—
45	C₂×M₁₆×C₁₃₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	#127×C₁₃₇	—	2	1	no	—
48	#125×C₁₃₇	—	2	1	no	—
49	#123×C₁₃₇	—	4	2	no	—
50	#121×C₁₃₇	—	2	2	no	—
51	#121×C₁₃₇	—	4	2	no	—
52	C₄×D₄×C₁₃₇	—	2	2	no	—
53	C₄×D₄×C₁₃₇	—	2	2	no	—
54	C₄×D₄×C₁₃₇	—	2	2	no	—
55	C₄×D₄×C₁₃₇	—	4	2	no	—
56	#117×C₁₃₇	—	2	1	no	—
57	#117×C₁₃₇	D₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₁₃₇	—	4	2	no	—
59	D₁₆×C₁₃₇	—	2	4	no	—
60	D₁₆×C₁₃₇	—	2	4	no	—
61	C₂×C₁₆×C₁₃₇	C₂³	1	2	yes	—
62	C₂×C₁₆×C₁₃₇	C₂³	1	2	yes	—
63	#107×C₁₃₇	C₂³	1	2	yes	—
64	#107×C₁₃₇	C₂³	1	2	yes	—
65	#106×C₁₃₇	—	2	1	no	—
66	#106×C₁₃₇	—	2	1	no	—
67	#102×C₁₃₇	—	2	2	no	—
68	#102×C₁₃₇	—	2	2	no	—
69	#102×C₁₃₇	—	2	2	no	—
70	#102×C₁₃₇	—	2	2	no	—
71	#97×C₁₃₇	D₄	1	1	yes	characteristic
72	#97×C₁₃₇	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
75	C₂⁴×C₁₃₇	—	4	1	no	—
76	C₂×D₄×C₁₃₇	—	2	4	no	—
77	C₂×D₄×C₁₃₇	—	2	2	no	—
78	C₂×D₄×C₁₃₇	—	2	2	no	—
79	C₂×D₄×C₁₃₇	—	4	2	no	—
80	C₂×D₄×C₁₃₇	—	4	2	no	—
81	C₂×D₄×C₁₃₇	—	4	1	no	—
82	C₂×D₄×C₁₃₇	—	4	2	no	—
83	C₂²×C₄×C₁₃₇	—	4	2	no	—
84	C₂²×C₄×C₁₃₇	—	4	2	no	—
85	C₂²×C₄×C₁₃₇	—	4	2	no	—
86	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
87	D₈×C₁₃₇	—	2	4	no	—
88	D₈×C₁₃₇	—	2	2	no	—
89	D₈×C₁₃₇	—	2	2	no	—
90	D₈×C₁₃₇	—	2	2	no	—
91	D₈×C₁₃₇	—	4	2	no	—
92	D₈×C₁₃₇	#41	1	4	yes	—
93	M₁₆×C₁₃₇	—	4	1	no	—
94	C₂×C₈×C₁₃₇	—	2	2	no	—
95	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₁₃₇	—	2	2	no	—
100	C₄⋊C₄×C₁₃₇	—	2	2	no	—
101	C₄⋊C₄×C₁₃₇	—	4	2	no	—
102	C₄⋊C₄×C₁₃₇	D₈	1	2	yes	—
103	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
104	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
105	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
106	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
107	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
108	C₂²⋊C₄×C₁₃₇	D₈	1	2	yes	—
109	C₄²×C₁₃₇	—	4	1	no	—
110	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₁₃₇	—	2	2	no	—
113	C₁₆×C₁₃₇	—	2	2	no	—
114	C₂³×C₁₃₇	—	4	4	no	—
115	C₂³×C₁₃₇	—	4	2	no	—
116	C₂³×C₁₃₇	—	4	2	no	—
117	C₂³×C₁₃₇	—	16	2	no	—
118	C₂³×C₁₃₇	C₂×D₈	1	1	yes	characteristic
119	D₄×C₁₃₇	—	2	4	no	—
120	D₄×C₁₃₇	—	4	4	no	—
121	D₄×C₁₃₇	—	4	2	no	—
122	D₄×C₁₃₇	—	4	2	no	—
123	D₄×C₁₃₇	—	4	2	no	—
124	D₄×C₁₃₇	—	16	2	no	—
125	D₄×C₁₃₇	—	16	2	no	—
126	C₂×C₄×C₁₃₇	—	2	1	no	—
127	C₂×C₄×C₁₃₇	—	2	1	no	—
128	C₂×C₄×C₁₃₇	—	4	2	no	—
129	C₂×C₄×C₁₃₇	—	4	2	no	—
130	C₂×C₄×C₁₃₇	—	16	2	no	—
131	C₂×C₄×C₁₃₇	—	16	2	no	—
132	C₂×C₄×C₁₃₇	—	16	2	no	—
133	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₁₃₇	—	2	1	no	—
138	C₈×C₁₃₇	—	2	1	no	—
139	C₈×C₁₃₇	—	2	1	no	—
140	C₈×C₁₃₇	C₂×#41	1	2	yes	—
141	C₂²×C₁₃₇	—	2	1	no	—
142	C₂²×C₁₃₇	—	2	2	no	—
143	C₂²×C₁₃₇	—	4	4	no	—
144	C₂²×C₁₃₇	—	8	2	no	—
145	C₂²×C₁₃₇	—	8	2	no	—
146	C₂²×C₁₃₇	—	8	2	no	—
147	C₂²×C₁₃₇	—	8	2	no	—
148	C₂²×C₁₃₇	—	8	2	no	—
149	C₂²×C₁₃₇	—	16	4	no	—
150	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₁₃₇	—	2	1	no	—
152	C₄×C₁₃₇	—	2	2	no	—
153	C₄×C₁₃₇	—	2	2	no	—
154	C₄×C₁₃₇	—	8	2	no	—
155	C₄×C₁₃₇	—	8	2	no	—
156	C₄×C₁₃₇	#525	1	1	yes	characteristic
157	C₄×C₁₃₇	#575	1	1	yes	characteristic
158	C₂×C₁₃₇	—	4	1	no	—
159	C₂×C₁₃₇	—	8	4	no	—
160	C₂×C₁₃₇	—	16	2	no	—
161	C₂×C₁₃₇	#3100	1	1	yes	characteristic
162	C₂×C₁₃₇	#3415	1	2	yes	—
163	#33682	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₁₃₇	#33682	1	1	yes	characteristic, direct product factor, Sylow
165	#3176	C₂×C₁₃₇	1	1	yes	characteristic, complemented
166	#2948	C₂×C₁₃₇	1	1	yes	characteristic, complemented
167	#2234	C₂×C₁₃₇	1	2	yes	complemented
168	#2228	C₂×C₁₃₇	1	2	yes	complemented
169	#2206	C₂×C₁₃₇	1	2	yes	complemented
170	#2184	C₂×C₁₃₇	1	2	yes	complemented
171	#2177	C₂×C₁₃₇	1	2	yes	complemented
172	C₄×D₁₆	C₂×C₁₃₇	1	2	yes	complemented
173	#2164	C₂×C₁₃₇	1	1	yes	characteristic, complemented
174	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
175	#540	—	2	1	no	—
176	#518	—	2	1	no	—
177	C₂×D₁₆	C₂²×C₁₃₇	1	2	yes	—
178	C₂×D₁₆	C₂²×C₁₃₇	1	2	yes	—
179	#499	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
180	#493	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
181	#490	—	2	1	no	—
182	#469	—	2	1	no	—
183	#466	C₂²×C₁₃₇	1	2	yes	—
184	#442	—	2	1	no	—
185	#442	—	2	1	no	—
186	#442	—	2	1	no	—
187	C₄×D₈	C₂²×C₁₃₇	1	2	yes	—
188	C₄×D₈	C₂²×C₁₃₇	1	2	yes	—
189	C₄×D₈	C₂²×C₁₃₇	1	2	yes	complemented
190	#427	C₂²×C₁₃₇	1	1	yes	characteristic
191	#418	—	2	1	no	—
192	#405	C₂²×C₁₃₇	1	1	yes	characteristic
193	#367	C₂²×C₁₃₇	1	2	yes	complemented
194	#366	C₂²×C₁₃₇	1	2	yes	complemented
195	C₄⋊C₁₆	C₂²×C₁₃₇	1	2	yes	complemented
196	#357	C₂²×C₁₃₇	1	2	yes	—
197	#357	C₂²×C₁₃₇	1	2	yes	—
198	#357	C₂²×C₁₃₇	1	2	yes	complemented
199	#357	C₂²×C₁₃₇	1	2	yes	complemented
200	C₂²⋊C₁₆	C₂²×C₁₃₇	1	2	yes	—
201	#346	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
202	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
203	C₂²×D₄	—	2	1	no	—
204	C₂×D₈	—	2	4	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	C₂³×C₁₃₇	1	2	yes	—
208	C₂×D₈	C₂³×C₁₃₇	1	2	yes	—
209	C₂×M₁₆	D₄×C₁₃₇	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
211	#127	—	2	1	no	—
212	#125	—	2	1	no	—
213	#123	—	4	2	no	—
214	#121	—	2	2	no	—
215	#121	—	4	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	4	2	no	—
220	#117	—	2	1	no	—
221	#117	D₄×C₁₃₇	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	4	2	no	—
223	D₁₆	—	2	4	no	—
224	D₁₆	—	2	4	no	—
225	C₂×C₁₆	C₂³×C₁₃₇	1	2	yes	—
226	C₂×C₁₆	C₂³×C₁₃₇	1	2	yes	—
227	#107	C₂³×C₁₃₇	1	2	yes	—
228	#107	C₂³×C₁₃₇	1	2	yes	—
229	#106	—	2	1	no	—
230	#106	—	2	1	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#97	D₄×C₁₃₇	1	1	yes	characteristic
236	#97	D₄×C₁₃₇	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
239	C₂⁴	—	4	1	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
251	D₈	—	2	4	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	4	2	no	—
256	D₈	#41×C₁₃₇	1	4	yes	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
262	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	D₈×C₁₃₇	1	2	yes	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	D₈×C₁₃₇	1	2	yes	—
273	C₄²	—	4	1	no	—
274	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
276	C₁₆	—	2	2	no	—
277	C₁₆	—	2	2	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	16	2	no	—
282	C₂³	C₂×D₈×C₁₃₇	1	1	yes	characteristic
283	D₄	—	2	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	16	2	no	—
289	D₄	—	16	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	2	no	—
295	C₂×C₄	—	16	2	no	—
296	C₂×C₄	—	16	2	no	—
297	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
298	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
299	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
300	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	C₂×#41×C₁₃₇	1	2	yes	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	16	4	no	—
314	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	#525×C₁₃₇	1	1	yes	characteristic
321	C₄	#575×C₁₃₇	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	8	4	no	—
324	C₂	—	16	2	no	—
325	C₂	#3100×C₁₃₇	1	1	yes	characteristic
326	C₂	#3415×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ #117×C₁₃₇ ◁ #405×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3176×C₁₃₇	C₂	semidirect
2. #2948×C₁₃₇	C₂	semidirect
3. #2234×C₁₃₇	C₂	semidirect
4. #2228×C₁₃₇	C₂	semidirect
5. #2206×C₁₃₇	C₂	semidirect
6. #2184×C₁₃₇	C₂	semidirect
7. #2177×C₁₃₇	C₂	semidirect
8. C₄×D₁₆×C₁₃₇	C₂	semidirect
9. #2164×C₁₃₇	C₂	semidirect
15. #499×C₁₃₇	C₂²	semidirect
16. #493×C₁₃₇	C₂²	semidirect
25. C₄×D₈×C₁₃₇	C₂²	semidirect
29. #367×C₁₃₇	C₂²	semidirect
30. #366×C₁₃₇	C₂²	semidirect
31. C₄⋊C₁₆×C₁₃₇	C₂²	semidirect
34. #357×C₁₃₇	C₂²	semidirect
35. #357×C₁₃₇	C₂²	semidirect
37. #346×C₁₃₇	C₂²	semidirect
38. C₄×C₁₆×C₁₃₇	C₂²	semidirect
163. #33682	C₁₃₇	direct (N × H)
164. C₁₃₇	#33682	direct (N × H)
165. #3176	C₂×C₁₃₇	semidirect
166. #2948	C₂×C₁₃₇	semidirect
167. #2234	C₂×C₁₃₇	semidirect
168. #2228	C₂×C₁₃₇	semidirect
169. #2206	C₂×C₁₃₇	semidirect
170. #2184	C₂×C₁₃₇	semidirect
171. #2177	C₂×C₁₃₇	semidirect
172. C₄×D₁₆	C₂×C₁₃₇	semidirect
173. #2164	C₂×C₁₃₇	semidirect
179. #499	C₂²×C₁₃₇	semidirect
180. #493	C₂²×C₁₃₇	semidirect
189. C₄×D₈	C₂²×C₁₃₇	semidirect
193. #367	C₂²×C₁₃₇	semidirect
194. #366	C₂²×C₁₃₇	semidirect
195. C₄⋊C₁₆	C₂²×C₁₃₇	semidirect
198. #357	C₂²×C₁₃₇	semidirect
199. #357	C₂²×C₁₃₇	semidirect
201. #346	C₂²×C₁₃₇	semidirect
202. C₄×C₁₆	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3176×C₁₃₇	2
#2948×C₁₃₇	2
#2234×C₁₃₇	2
#2228×C₁₃₇	2
#2206×C₁₃₇	2
#2184×C₁₃₇	2
#2177×C₁₃₇	2
C₄×D₁₆×C₁₃₇	2
#2164×C₁₃₇	2
#33682	137

Subgroups — C137×#33682 #46883068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33682 #46883068