Subgroups — C₃₇×#8160 #22629413

Order 9472.

582 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#99×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#1752	C₂	1	2	yes	maximal, complemented
4	C₃₇×#1459	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#1282	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
9	C₈×D₄×C₃₇	—	2	2	no	—
10	C₈×D₄×C₃₇	—	2	2	no	—
11	C₂×C₄⋊C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
12	C₂×#99×C₃₇	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×#97×C₃₇	—	2	2	no	—
17	C₂×C₄×C₈×C₃₇	C₄	1	1	yes	characteristic, complemented
18	C₄×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
19	C₃₇×#342	C₄	1	2	yes	—
20	C₃₇×#336	—	2	2	no	—
21	C₃₇×#324	C₄	1	2	yes	complemented
22	C₃₇×#323	C₄	1	2	yes	complemented
23	C₂²×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₃₇	—	2	2	no	—
27	C₂²×C₈×C₃₇	—	2	1	no	—
28	C₂²×C₈×C₃₇	—	2	1	no	—
29	C₂²×C₈×C₃₇	—	4	2	no	—
30	C₂²×C₈×C₃₇	—	4	2	no	—
31	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₇	D₄	1	2	yes	—
36	C₂²×C₈×C₃₇	D₄	1	2	yes	—
37	C₄×D₄×C₃₇	—	2	2	no	—
38	C₄×D₄×C₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₇	—	2	2	no	—
44	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₈×C₃₇	—	2	2	no	—
46	#99×C₃₇	—	2	4	no	—
47	C₂²⋊C₈×C₃₇	—	2	2	no	—
48	C₂²⋊C₈×C₃₇	—	2	2	no	—
49	C₂²⋊C₈×C₃₇	—	2	2	no	—
50	C₂²⋊C₈×C₃₇	C₂×C₄	1	4	yes	—
51	#97×C₃₇	—	4	4	no	—
52	C₄×C₈×C₃₇	—	2	1	no	—
53	C₄×C₈×C₃₇	—	2	1	no	—
54	#95×C₃₇	—	2	2	no	—
55	C₂⁴×C₃₇	C₄²	1	1	yes	characteristic
56	C₂⁴×C₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×D₄×C₃₇	—	2	2	no	—
58	C₂×D₄×C₃₇	—	2	2	no	—
59	C₂×D₄×C₃₇	M₁₆	1	4	yes	—
60	C₂²×C₄×C₃₇	—	2	1	no	—
61	C₂²×C₄×C₃₇	—	2	1	no	—
62	C₂²×C₄×C₃₇	—	2	4	no	—
63	C₂²×C₄×C₃₇	—	4	1	no	—
64	C₂²×C₄×C₃₇	—	4	1	no	—
65	C₂²×C₄×C₃₇	—	4	1	no	—
66	C₂²×C₄×C₃₇	—	4	1	no	—
67	C₂²×C₄×C₃₇	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₃₇	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₇	#41	1	2	yes	—
74	C₂×C₈×C₃₇	—	2	4	no	—
75	C₂×C₈×C₃₇	—	2	2	no	—
76	C₂×C₈×C₃₇	—	2	1	no	—
77	C₂×C₈×C₃₇	—	2	1	no	—
78	C₂×C₈×C₃₇	—	2	4	no	—
79	C₂×C₈×C₃₇	—	4	1	no	—
80	C₂×C₈×C₃₇	—	4	2	no	—
81	C₂×C₈×C₃₇	—	4	2	no	—
82	C₂×C₈×C₃₇	—	4	2	no	—
83	C₂×C₈×C₃₇	—	4	2	no	—
84	C₂×C₈×C₃₇	—	4	2	no	—
85	C₂×C₈×C₃₇	—	4	1	no	—
86	C₂×C₈×C₃₇	—	4	1	no	—
87	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₃₇	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₃₇	—	2	2	no	—
91	C₂²⋊C₄×C₃₇	—	2	2	no	—
92	C₂²⋊C₄×C₃₇	—	2	2	no	—
93	C₂²⋊C₄×C₃₇	—	2	4	no	—
94	C₂²⋊C₄×C₃₇	—	2	4	no	—
95	C₄²×C₃₇	—	2	1	no	—
96	C₄²×C₃₇	—	2	1	no	—
97	C₄²×C₃₇	—	4	2	no	—
98	C₄²×C₃₇	—	4	2	no	—
99	C₂³×C₃₇	—	2	1	no	—
100	C₂³×C₃₇	—	2	1	no	—
101	C₂³×C₃₇	—	4	1	no	—
102	C₂³×C₃₇	—	4	1	no	—
103	C₂³×C₃₇	—	4	1	no	—
104	C₂³×C₃₇	—	4	1	no	—
105	C₂³×C₃₇	#97	1	2	yes	complemented
106	C₂³×C₃₇	C₂×C₄²	1	1	yes	characteristic
107	C₂³×C₃₇	C₄×D₄	1	2	yes	—
108	C₂³×C₃₇	C₂×M₁₆	1	2	yes	—
109	C₂³×C₃₇	#131	1	2	yes	—
110	D₄×C₃₇	—	2	8	no	—
111	C₂×C₄×C₃₇	—	2	1	no	—
112	C₂×C₄×C₃₇	—	2	1	no	—
113	C₂×C₄×C₃₇	—	2	2	no	—
114	C₂×C₄×C₃₇	—	2	1	no	—
115	C₂×C₄×C₃₇	—	2	2	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	—	2	1	no	—
118	C₂×C₄×C₃₇	—	2	1	no	—
119	C₂×C₄×C₃₇	—	4	1	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	—	4	2	no	—
122	C₂×C₄×C₃₇	—	4	1	no	—
123	C₂×C₄×C₃₇	—	4	1	no	—
124	C₂×C₄×C₃₇	—	4	1	no	—
125	C₂×C₄×C₃₇	—	4	1	no	—
126	C₂×C₄×C₃₇	—	4	4	no	—
127	C₂×C₄×C₃₇	—	4	4	no	—
128	C₂×C₄×C₃₇	—	4	4	no	—
129	C₂×C₄×C₃₇	#97	1	2	yes	complemented
130	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₃₇	#117	1	1	yes	characteristic
132	C₂×C₄×C₃₇	C₄×D₄	1	2	yes	—
133	C₈×C₃₇	—	2	1	no	—
134	C₈×C₃₇	—	2	2	no	—
135	C₈×C₃₇	—	4	4	no	—
136	C₈×C₃₇	—	4	1	no	—
137	C₈×C₃₇	—	4	1	no	—
138	C₈×C₃₇	#99	1	2	yes	complemented
139	#8160	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₃₇	—	2	2	no	—
141	C₂²×C₃₇	—	2	1	no	—
142	C₂²×C₃₇	—	2	2	no	—
143	C₂²×C₃₇	—	2	1	no	—
144	C₂²×C₃₇	—	2	2	no	—
145	C₂²×C₃₇	—	4	2	no	—
146	C₂²×C₃₇	—	4	2	no	—
147	C₂²×C₃₇	—	4	2	no	—
148	C₂²×C₃₇	—	4	2	no	—
149	C₂²×C₃₇	—	4	1	no	—
150	C₂²×C₃₇	—	4	1	no	—
151	C₂²×C₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₃₇	C₂×#97	1	1	yes	characteristic
153	C₂²×C₃₇	#405	1	1	yes	characteristic
154	C₂²×C₃₇	#435	1	4	yes	—
155	C₄×C₃₇	—	2	2	no	—
156	C₄×C₃₇	—	2	2	no	—
157	C₄×C₃₇	—	2	1	no	—
158	C₄×C₃₇	—	4	1	no	—
159	C₄×C₃₇	—	4	1	no	—
160	C₄×C₃₇	—	8	4	no	—
161	C₄×C₃₇	C₂×#99	1	1	yes	characteristic
162	C₄×C₃₇	#410	1	1	yes	characteristic
163	C₂×C₈×D₄	C₂×C₃₇	1	1	yes	characteristic
164	C₄×#99	C₂×C₃₇	1	1	yes	characteristic
165	#1752	C₂×C₃₇	1	2	yes	complemented
166	#1459	C₂×C₃₇	1	1	yes	characteristic
167	#1282	C₂×C₃₇	1	2	yes	complemented
168	C₂×C₃₇	—	2	4	no	—
169	C₂×C₃₇	—	2	1	no	—
170	C₂×C₃₇	—	2	1	no	—
171	C₂×C₃₇	—	4	2	no	—
172	C₂×C₃₇	#1752	1	1	yes	characteristic
173	C₂×C₃₇	C₄×#99	1	1	yes	characteristic
174	C₂×C₃₇	#1757	1	1	yes	characteristic
175	C₂³×C₈	C₄×C₃₇	1	1	yes	characteristic, complemented
176	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
177	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
178	C₈×D₄	—	2	2	no	—
179	C₈×D₄	—	2	2	no	—
180	C₂×C₄⋊C₈	C₄×C₃₇	1	1	yes	characteristic, complemented
181	C₂×#99	C₄×C₃₇	1	2	yes	—
182	C₂×C₂²⋊C₈	C₄×C₃₇	1	1	yes	characteristic, complemented
183	C₂×C₂²⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
184	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
185	C₂×#97	—	2	2	no	—
186	C₂×C₄×C₈	C₄×C₃₇	1	1	yes	characteristic, complemented
187	C₄×C₂²⋊C₄	C₂²×C₃₇	1	2	yes	—
188	#342	C₄×C₃₇	1	2	yes	—
189	#336	—	2	2	no	—
190	#324	C₄×C₃₇	1	2	yes	complemented
191	#323	C₄×C₃₇	1	2	yes	complemented
192	C₃₇	#8160	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²×D₄	C₂×C₄×C₃₇	1	1	yes	characteristic
194	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
195	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	4	2	no	—
200	C₂²×C₈	—	4	2	no	—
201	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
202	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
203	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
204	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
205	C₂²×C₈	D₄×C₃₇	1	2	yes	—
206	C₂²×C₈	D₄×C₃₇	1	2	yes	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
211	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
215	C₄⋊C₈	—	2	2	no	—
216	#99	—	2	4	no	—
217	C₂²⋊C₈	—	2	2	no	—
218	C₂²⋊C₈	—	2	2	no	—
219	C₂²⋊C₈	—	2	2	no	—
220	C₂²⋊C₈	C₂×C₄×C₃₇	1	4	yes	—
221	#97	—	4	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	#95	—	2	2	no	—
225	C₂⁴	C₄²×C₃₇	1	1	yes	characteristic
226	C₂⁴	C₂²×C₄×C₃₇	1	1	yes	characteristic
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	M₁₆×C₃₇	1	4	yes	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₄²×C₃₇	1	1	yes	characteristic
238	C₂²×C₄	C₄²×C₃₇	1	1	yes	characteristic
239	C₂²×C₄	C₄²×C₃₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
243	C₂²×C₄	#41×C₃₇	1	2	yes	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
258	C₂×C₈	C₂²⋊C₄×C₃₇	1	2	yes	—
259	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	2	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	2	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	4	1	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	#97×C₃₇	1	2	yes	complemented
276	C₂³	C₂×C₄²×C₃₇	1	1	yes	characteristic
277	C₂³	C₄×D₄×C₃₇	1	2	yes	—
278	C₂³	C₂×M₁₆×C₃₇	1	2	yes	—
279	C₂³	#131×C₃₇	1	2	yes	—
280	D₄	—	2	8	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	#97×C₃₇	1	2	yes	complemented
300	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
301	C₂×C₄	#117×C₃₇	1	1	yes	characteristic
302	C₂×C₄	C₄×D₄×C₃₇	1	2	yes	—
303	C₈	—	2	1	no	—
304	C₈	—	2	2	no	—
305	C₈	—	4	4	no	—
306	C₈	—	4	1	no	—
307	C₈	—	4	1	no	—
308	C₈	#99×C₃₇	1	2	yes	complemented
309	C₂²	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	C₄×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
321	C₂²	C₂×#97×C₃₇	1	1	yes	characteristic
322	C₂²	C₃₇×#405	1	1	yes	characteristic
323	C₂²	C₃₇×#435	1	4	yes	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	8	4	no	—
330	C₄	C₂×#99×C₃₇	1	1	yes	characteristic
331	C₄	C₃₇×#410	1	1	yes	characteristic
332	C₂	—	2	4	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	4	2	no	—
336	C₂	C₃₇×#1752	1	1	yes	characteristic
337	C₂	C₄×#99×C₃₇	1	1	yes	characteristic
338	C₂	C₃₇×#1757	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₃₇
socle	C₂²×C₃₇	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₃₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂³×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1752	C₂	semidirect
5. C₃₇×#1282	C₂	semidirect
6. C₂³×C₈×C₃₇	C₄	semidirect
11. C₂×C₄⋊C₈×C₃₇	C₄	semidirect
13. C₂×C₂²⋊C₈×C₃₇	C₄	semidirect
17. C₂×C₄×C₈×C₃₇	C₄	semidirect
21. C₃₇×#324	C₄	semidirect
22. C₃₇×#323	C₄	semidirect
105. C₂³×C₃₇	#97	semidirect
129. C₂×C₄×C₃₇	#97	semidirect
138. C₈×C₃₇	#99	semidirect
139. #8160	C₃₇	direct (N × H)
165. #1752	C₂×C₃₇	semidirect
167. #1282	C₂×C₃₇	semidirect
175. C₂³×C₈	C₄×C₃₇	semidirect
180. C₂×C₄⋊C₈	C₄×C₃₇	semidirect
182. C₂×C₂²⋊C₈	C₄×C₃₇	semidirect
186. C₂×C₄×C₈	C₄×C₃₇	semidirect
190. #324	C₄×C₃₇	semidirect
191. #323	C₄×C₃₇	semidirect
192. C₃₇	#8160	direct (N × H)
275. C₂³	#97×C₃₇	semidirect
299. C₂×C₄	#97×C₃₇	semidirect
308. C₈	#99×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₃₇	2
C₄×#99×C₃₇	2
C₃₇×#1752	2
C₃₇×#1459	2
C₃₇×#1282	2
#8160	37

Subgroups — C37×#8160 #22629413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#8160 #22629413