Subgroups — C₃₇×#19195 #22640448

Order 9472.

638 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3246	C₂	1	2	yes	maximal
2	C₃₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1691	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1675	C₂	1	2	yes	maximal, complemented
6	C₃₇×#1673	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1550	C₂	1	2	yes	maximal, complemented
9	C₂×#131×C₃₇	—	2	1	no	—
10	C₃₇×#533	C₂²	1	2	yes	—
11	C₃₇×#532	C₂²	1	1	yes	characteristic
12	C₃₇×#500	C₂²	1	1	yes	characteristic
13	C₃₇×#500	C₂²	1	1	yes	characteristic
14	C₃₇×#499	—	2	1	no	—
15	C₃₇×#499	—	2	1	no	—
16	C₃₇×#498	C₂²	1	2	yes	—
17	C₃₇×#476	C₂²	1	4	yes	—
18	C₃₇×#475	—	2	2	no	—
19	C₃₇×#465	—	2	4	no	—
20	C₃₇×#461	C₂²	1	4	yes	—
21	C₃₇×#460	—	2	2	no	—
22	C₃₇×#436	—	2	1	no	—
23	C₃₇×#436	C₂²	1	2	yes	—
24	C₈×D₄×C₃₇	—	2	1	no	—
25	C₈×D₄×C₃₇	C₂²	1	2	yes	—
26	C₃₇×#427	—	2	1	no	—
27	C₃₇×#427	—	2	1	no	—
28	C₃₇×#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
30	C₃₇×#408	—	2	1	no	—
31	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
32	C₃₇×#335	C₂²	1	2	yes	—
33	C₃₇×#335	C₂²	1	2	yes	—
34	C₃₇×#333	C₂²	1	4	yes	—
35	C₃₇×#332	C₂²	1	4	yes	—
36	C₂×#41×C₃₇	—	2	1	no	—
37	C₂×SD₁₆×C₃₇	—	2	4	no	—
38	#131×C₃₇	—	4	4	no	—
39	C₂×M₁₆×C₃₇	—	2	1	no	—
40	C₂×M₁₆×C₃₇	—	2	2	no	—
41	C₂²×C₈×C₃₇	—	2	2	no	—
42	C₂²×C₈×C₃₇	—	2	2	no	—
43	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
45	C₄⋊Q₈×C₃₇	D₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₃₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₃₇	C₂³	1	2	yes	—
48	#127×C₃₇	D₄	1	1	yes	characteristic
49	#125×C₃₇	—	2	2	no	—
50	#124×C₃₇	—	2	1	no	—
51	C₂²⋊Q₈×C₃₇	—	2	2	no	—
52	C₂²⋊Q₈×C₃₇	—	2	2	no	—
53	#121×C₃₇	—	2	1	no	—
54	#121×C₃₇	D₄	1	2	yes	—
55	C₄×Q₈×C₃₇	—	2	2	no	—
56	C₄×Q₈×C₃₇	—	2	2	no	—
57	C₄×D₄×C₃₇	—	2	1	no	—
58	C₄×D₄×C₃₇	C₂³	1	2	yes	—
59	#117×C₃₇	—	2	2	no	—
60	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
61	#107×C₃₇	—	2	1	no	—
62	#107×C₃₇	—	2	1	no	—
63	#107×C₃₇	—	2	1	no	—
64	#107×C₃₇	—	2	1	no	—
65	#106×C₃₇	—	2	2	no	—
66	#106×C₃₇	—	2	2	no	—
67	#106×C₃₇	—	2	4	no	—
68	C₄⋊C₈×C₃₇	—	2	2	no	—
69	C₄⋊C₈×C₃₇	—	2	1	no	—
70	C₄⋊C₈×C₃₇	C₂³	1	2	yes	—
71	C₄⋊C₈×C₃₇	C₂³	1	4	yes	—
72	#103×C₃₇	—	2	4	no	—
73	#102×C₃₇	—	2	4	no	—
74	C₂²⋊C₈×C₃₇	—	2	2	no	—
75	C₂²⋊C₈×C₃₇	—	2	2	no	—
76	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
80	#41×C₃₇	—	4	2	no	—
81	#41×C₃₇	—	4	2	no	—
82	C₂×Q₈×C₃₇	—	2	1	no	—
83	C₂×Q₈×C₃₇	—	2	2	no	—
84	C₂×Q₈×C₃₇	—	2	2	no	—
85	C₂×D₄×C₃₇	—	2	2	no	—
86	C₂×D₄×C₃₇	—	2	1	no	—
87	C₂×D₄×C₃₇	—	2	1	no	—
88	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₃₇	—	2	2	no	—
90	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
93	SD₁₆×C₃₇	—	8	4	no	—
94	M₁₆×C₃₇	—	4	2	no	—
95	M₁₆×C₃₇	—	4	4	no	—
96	C₂×C₈×C₃₇	—	2	2	no	—
97	C₂×C₈×C₃₇	—	2	2	no	—
98	C₂×C₈×C₃₇	—	2	4	no	—
99	C₂×C₈×C₃₇	—	4	4	no	—
100	C₂×C₈×C₃₇	—	8	2	no	—
101	C₂×C₈×C₃₇	SD₁₆	1	4	yes	complemented
102	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₃₇	—	2	1	no	—
107	C₄⋊C₄×C₃₇	—	2	2	no	—
108	C₄⋊C₄×C₃₇	—	2	2	no	—
109	C₄⋊C₄×C₃₇	—	2	2	no	—
110	C₄⋊C₄×C₃₇	—	2	2	no	—
111	C₄⋊C₄×C₃₇	—	4	2	no	—
112	C₄⋊C₄×C₃₇	—	4	2	no	—
113	C₄⋊C₄×C₃₇	C₂×D₄	1	2	yes	—
114	C₂²⋊C₄×C₃₇	—	2	2	no	—
115	C₂²⋊C₄×C₃₇	—	2	2	no	—
116	C₂²⋊C₄×C₃₇	—	4	2	no	—
117	C₄²×C₃₇	—	4	2	no	—
118	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₃₇	#41	1	1	yes	characteristic
121	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
122	C₂³×C₃₇	—	2	2	no	—
123	C₂³×C₃₇	#121	1	1	yes	characteristic
124	Q₈×C₃₇	—	4	2	no	—
125	Q₈×C₃₇	—	4	2	no	—
126	Q₈×C₃₇	—	4	2	no	—
127	Q₈×C₃₇	—	8	2	no	—
128	D₄×C₃₇	—	2	2	no	—
129	D₄×C₃₇	—	2	2	no	—
130	D₄×C₃₇	—	4	2	no	—
131	D₄×C₃₇	—	4	4	no	—
132	D₄×C₃₇	—	4	2	no	—
133	C₂×C₄×C₃₇	—	2	1	no	—
134	C₂×C₄×C₃₇	—	2	1	no	—
135	C₂×C₄×C₃₇	—	2	2	no	—
136	C₂×C₄×C₃₇	—	4	1	no	—
137	C₂×C₄×C₃₇	—	4	2	no	—
138	C₂×C₄×C₃₇	—	4	2	no	—
139	C₂×C₄×C₃₇	—	4	2	no	—
140	C₂×C₄×C₃₇	—	4	2	no	—
141	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
145	C₂×C₄×C₃₇	#137	1	2	yes	—
146	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
149	C₈×C₃₇	—	2	1	no	—
150	C₈×C₃₇	—	2	1	no	—
151	C₈×C₃₇	—	4	2	no	—
152	C₈×C₃₇	—	4	4	no	—
153	C₈×C₃₇	C₂×SD₁₆	1	2	yes	—
154	C₈×C₃₇	C₂×SD₁₆	1	2	yes	—
155	#19195	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂²×C₃₇	—	2	1	no	—
157	C₂²×C₃₇	—	2	1	no	—
158	C₂²×C₃₇	—	4	2	no	—
159	C₂²×C₃₇	—	4	2	no	—
160	C₂²×C₃₇	—	4	2	no	—
161	C₂²×C₃₇	#461	1	2	yes	—
162	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
163	C₄×C₃₇	—	2	1	no	—
164	C₄×C₃₇	—	2	1	no	—
165	C₄×C₃₇	—	2	1	no	—
166	C₄×C₃₇	—	4	2	no	—
167	C₄×C₃₇	—	8	2	no	—
168	C₄×C₃₇	—	8	2	no	—
169	C₄×C₃₇	#461	1	2	yes	—
170	C₄×C₃₇	C₂²×SD₁₆	1	1	yes	characteristic
171	C₄×C₃₇	C₂×#137	1	1	yes	characteristic
172	C₄×C₃₇	#576	1	1	yes	characteristic
173	C₄×C₃₇	#578	1	1	yes	characteristic
174	#3246	C₂×C₃₇	1	2	yes	—
175	#3160	C₂×C₃₇	1	1	yes	characteristic, complemented
176	#2931	C₂×C₃₇	1	1	yes	characteristic
177	#1691	C₂×C₃₇	1	4	yes	complemented
178	#1675	C₂×C₃₇	1	2	yes	complemented
179	#1673	C₂×C₃₇	1	2	yes	complemented
180	C₂×#335	C₂×C₃₇	1	1	yes	characteristic, complemented
181	#1550	C₂×C₃₇	1	2	yes	complemented
182	C₂×C₃₇	—	2	2	no	—
183	C₂×C₃₇	—	8	2	no	—
184	C₂×C₃₇	C₂×#461	1	1	yes	characteristic
185	C₂×C₃₇	#3061	1	1	yes	characteristic
186	C₂×C₃₇	#3067	1	1	yes	characteristic
187	C₂×#131	—	2	1	no	—
188	#533	C₂²×C₃₇	1	2	yes	—
189	#532	C₂²×C₃₇	1	1	yes	characteristic
190	#500	C₂²×C₃₇	1	1	yes	characteristic
191	#500	C₂²×C₃₇	1	1	yes	characteristic
192	#499	—	2	1	no	—
193	#499	—	2	1	no	—
194	#498	C₂²×C₃₇	1	2	yes	—
195	#476	C₂²×C₃₇	1	4	yes	—
196	#475	—	2	2	no	—
197	#465	—	2	4	no	—
198	#461	C₂²×C₃₇	1	4	yes	—
199	#460	—	2	2	no	—
200	#436	—	2	1	no	—
201	#436	C₂²×C₃₇	1	2	yes	—
202	C₈×D₄	—	2	1	no	—
203	C₈×D₄	C₂²×C₃₇	1	2	yes	—
204	#427	—	2	1	no	—
205	#427	—	2	1	no	—
206	#423	C₂²×C₃₇	1	1	yes	characteristic
207	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
208	#408	—	2	1	no	—
209	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
210	#335	C₂²×C₃₇	1	2	yes	—
211	#335	C₂²×C₃₇	1	2	yes	—
212	#333	C₂²×C₃₇	1	4	yes	—
213	#332	C₂²×C₃₇	1	4	yes	—
214	C₃₇	#19195	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×#41	—	2	1	no	—
216	C₂×SD₁₆	—	2	4	no	—
217	#131	—	4	4	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	—	2	2	no	—
222	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
223	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
224	C₄⋊Q₈	D₄×C₃₇	1	1	yes	characteristic
225	C₄⋊Q₈	C₂³×C₃₇	1	2	yes	—
226	C₄⋊Q₈	C₂³×C₃₇	1	2	yes	—
227	#127	D₄×C₃₇	1	1	yes	characteristic
228	#125	—	2	2	no	—
229	#124	—	2	1	no	—
230	C₂²⋊Q₈	—	2	2	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	#121	—	2	1	no	—
233	#121	D₄×C₃₇	1	2	yes	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	C₂³×C₃₇	1	2	yes	—
238	#117	—	2	2	no	—
239	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
240	#107	—	2	1	no	—
241	#107	—	2	1	no	—
242	#107	—	2	1	no	—
243	#107	—	2	1	no	—
244	#106	—	2	2	no	—
245	#106	—	2	2	no	—
246	#106	—	2	4	no	—
247	C₄⋊C₈	—	2	2	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	C₂³×C₃₇	1	2	yes	—
250	C₄⋊C₈	C₂³×C₃₇	1	4	yes	—
251	#103	—	2	4	no	—
252	#102	—	2	4	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
256	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
259	#41	—	4	2	no	—
260	#41	—	4	2	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
271	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
272	SD₁₆	—	8	4	no	—
273	M₁₆	—	4	2	no	—
274	M₁₆	—	4	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	4	4	no	—
279	C₂×C₈	—	8	2	no	—
280	C₂×C₈	SD₁₆×C₃₇	1	4	yes	complemented
281	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	C₂×D₄×C₃₇	1	2	yes	—
293	C₂²⋊C₄	—	2	2	no	—
294	C₂²⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₄²	—	4	2	no	—
297	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
298	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
299	C₄²	#41×C₃₇	1	1	yes	characteristic
300	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
301	C₂³	—	2	2	no	—
302	C₂³	#121×C₃₇	1	1	yes	characteristic
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
321	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
322	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
323	C₂×C₄	C₂×SD₁₆×C₃₇	1	2	yes	—
324	C₂×C₄	#137×C₃₇	1	2	yes	—
325	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
327	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	4	2	no	—
331	C₈	—	4	4	no	—
332	C₈	C₂×SD₁₆×C₃₇	1	2	yes	—
333	C₈	C₂×SD₁₆×C₃₇	1	2	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	C₃₇×#461	1	2	yes	—
340	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	C₃₇×#461	1	2	yes	—
348	C₄	C₂²×SD₁₆×C₃₇	1	1	yes	characteristic
349	C₄	C₂×#137×C₃₇	1	1	yes	characteristic
350	C₄	C₃₇×#576	1	1	yes	characteristic
351	C₄	C₃₇×#578	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	8	2	no	—
354	C₂	C₂×C₃₇×#461	1	1	yes	characteristic
355	C₂	C₃₇×#3061	1	1	yes	characteristic
356	C₂	C₃₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₇
prime radical	C₃₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#3160	C₂	semidirect
4. C₃₇×#1691	C₂	semidirect
5. C₃₇×#1675	C₂	semidirect
6. C₃₇×#1673	C₂	semidirect
7. C₂×C₃₇×#335	C₂	semidirect
8. C₃₇×#1550	C₂	semidirect
101. C₂×C₈×C₃₇	SD₁₆	semidirect
155. #19195	C₃₇	direct (N × H)
175. #3160	C₂×C₃₇	semidirect
177. #1691	C₂×C₃₇	semidirect
178. #1675	C₂×C₃₇	semidirect
179. #1673	C₂×C₃₇	semidirect
180. C₂×#335	C₂×C₃₇	semidirect
181. #1550	C₂×C₃₇	semidirect
214. C₃₇	#19195	direct (N × H)
280. C₂×C₈	SD₁₆×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3246	2
C₃₇×#3160	2
C₃₇×#2931	2
C₃₇×#1691	2
C₃₇×#1675	2
C₃₇×#1673	2
C₂×C₃₇×#335	2
C₃₇×#1550	2
#19195	37

Subgroups — C37×#19195 #22640448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#19195 #22640448