Subgroups — C₃₇×#28538 #22649791

Order 9472.

794 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#479	C₂	1	2	yes	maximal
3	C₂×C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2565	C₂	1	2	yes	maximal
5	C₃₇×#2097	C₂	1	2	yes	maximal
6	C₃₇×#2095	C₂	1	2	yes	maximal
7	C₃₇×#1971	C₂	1	2	yes	maximal
8	C₃₇×#1847	C₂	1	2	yes	maximal
9	C₃₇×#1770	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₃₇	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#499	—	2	4	no	—
14	C₃₇×#479	—	2	8	no	—
15	C₄×Q₁₆×C₃₇	—	2	4	no	—
16	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×#107×C₃₇	C₂²	1	2	yes	—
18	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×#106×C₃₇	C₂²	1	2	yes	—
21	C₂×#106×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₂×#103×C₃₇	C₂²	1	2	yes	—
24	C₂×#103×C₃₇	C₂²	1	2	yes	—
25	C₂×#103×C₃₇	C₂²	1	2	yes	—
26	C₂×#97×C₃₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
28	C₃₇×#398	—	2	4	no	—
29	C₃₇×#391	C₂²	1	2	yes	—
30	C₃₇×#391	C₂²	1	2	yes	—
31	C₃₇×#387	—	2	4	no	—
32	C₃₇×#384	C₂²	1	2	yes	—
33	C₃₇×#382	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
35	C₃₇×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
37	C₂×Q₁₆×C₃₇	—	2	2	no	—
38	C₂×Q₁₆×C₃₇	—	2	2	no	—
39	C₂×Q₁₆×C₃₇	Q₈	1	4	yes	—
40	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
43	#125×C₃₇	—	2	4	no	—
44	#125×C₃₇	—	2	4	no	—
45	C₄×Q₈×C₃₇	—	2	4	no	—
46	C₄×Q₈×C₃₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₃₇	—	2	2	no	—
56	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
58	#107×C₃₇	—	2	2	no	—
59	#107×C₃₇	—	2	4	no	—
60	#107×C₃₇	—	2	2	no	—
61	#107×C₃₇	—	2	2	no	—
62	#106×C₃₇	—	2	4	no	—
63	#106×C₃₇	—	2	4	no	—
64	C₄⋊C₈×C₃₇	D₄	1	4	yes	—
65	#103×C₃₇	—	2	4	no	—
66	#103×C₃₇	—	2	4	no	—
67	#103×C₃₇	—	2	4	no	—
68	#97×C₃₇	—	2	4	no	—
69	C₄×C₈×C₃₇	D₄	1	4	yes	—
70	#95×C₃₇	—	2	4	no	—
71	#95×C₃₇	—	2	4	no	—
72	#95×C₃₇	—	2	4	no	—
73	#95×C₃₇	D₄	1	2	yes	—
74	C₂×Q₈×C₃₇	—	2	2	no	—
75	C₂×Q₈×C₃₇	—	2	2	no	—
76	C₂×Q₈×C₃₇	—	2	2	no	—
77	C₂×Q₈×C₃₇	—	2	2	no	—
78	C₂×Q₈×C₃₇	—	2	2	no	—
79	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₃₇	—	2	2	no	—
81	C₂²×C₄×C₃₇	—	2	2	no	—
82	C₂²×C₄×C₃₇	—	2	2	no	—
83	C₂²×C₄×C₃₇	—	2	2	no	—
84	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
88	Q₁₆×C₃₇	—	2	8	no	—
89	C₂×C₈×C₃₇	—	2	2	no	—
90	C₂×C₈×C₃₇	—	2	1	no	—
91	C₂×C₈×C₃₇	—	2	1	no	—
92	C₂×C₈×C₃₇	—	2	2	no	—
93	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₃₇	#41	1	2	yes	—
98	C₄⋊C₄×C₃₇	—	2	2	no	—
99	C₄⋊C₄×C₃₇	—	2	2	no	—
100	C₄⋊C₄×C₃₇	—	2	2	no	—
101	C₄⋊C₄×C₃₇	—	2	2	no	—
102	C₄⋊C₄×C₃₇	—	2	2	no	—
103	C₄⋊C₄×C₃₇	—	2	2	no	—
104	C₄⋊C₄×C₃₇	—	4	2	no	—
105	C₄⋊C₄×C₃₇	—	4	8	no	—
106	C₄⋊C₄×C₃₇	—	4	4	no	—
107	C₄⋊C₄×C₃₇	—	4	4	no	—
108	C₄⋊C₄×C₃₇	—	4	4	no	—
109	C₄²×C₃₇	—	2	2	no	—
110	C₄²×C₃₇	—	2	2	no	—
111	C₄²×C₃₇	—	4	4	no	—
112	C₄²×C₃₇	C₂×D₄	1	2	yes	—
113	C₄²×C₃₇	C₂×D₄	1	2	yes	—
114	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₃₇	—	2	4	no	—
116	Q₈×C₃₇	—	2	4	no	—
117	Q₈×C₃₇	—	4	2	no	—
118	Q₈×C₃₇	—	4	2	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	2	2	no	—
121	C₂×C₄×C₃₇	—	2	4	no	—
122	C₂×C₄×C₃₇	—	2	4	no	—
123	C₂×C₄×C₃₇	—	2	4	no	—
124	C₂×C₄×C₃₇	—	2	4	no	—
125	C₂×C₄×C₃₇	—	4	2	no	—
126	C₂×C₄×C₃₇	—	4	2	no	—
127	C₂×C₄×C₃₇	—	4	2	no	—
128	C₂×C₄×C₃₇	—	4	2	no	—
129	C₂×C₄×C₃₇	—	4	2	no	—
130	C₂×C₄×C₃₇	—	4	4	no	—
131	C₂×C₄×C₃₇	—	4	4	no	—
132	C₂×C₄×C₃₇	—	4	4	no	—
133	C₂×C₄×C₃₇	#127	1	2	yes	—
134	C₂×C₄×C₃₇	#127	1	2	yes	—
135	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
141	C₈×C₃₇	—	2	2	no	—
142	C₈×C₃₇	—	2	4	no	—
143	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
144	#28538	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
146	C₂²×C₃₇	#547	1	1	yes	characteristic
147	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
148	C₂²×C₃₇	#575	1	2	yes	—
149	C₂²×C₃₇	#578	1	2	yes	—
150	C₄×C₃₇	—	2	2	no	—
151	C₄×C₃₇	—	2	2	no	—
152	C₄×C₃₇	—	4	4	no	—
153	C₄×C₃₇	—	4	4	no	—
154	C₄×C₃₇	—	8	2	no	—
155	C₄×C₃₇	—	8	2	no	—
156	C₄×C₃₇	—	8	2	no	—
157	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₃₇	#536	1	1	yes	characteristic
159	C₄×C₃₇	#547	1	1	yes	characteristic
160	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
161	C₂×#499	C₂×C₃₇	1	1	yes	characteristic
162	C₂×#479	C₂×C₃₇	1	2	yes	—
163	C₂×C₄×Q₁₆	C₂×C₃₇	1	1	yes	characteristic
164	#2565	C₂×C₃₇	1	2	yes	—
165	#2097	C₂×C₃₇	1	2	yes	—
166	#2095	C₂×C₃₇	1	2	yes	—
167	#1971	C₂×C₃₇	1	2	yes	—
168	#1847	C₂×C₃₇	1	2	yes	—
169	#1770	C₂×C₃₇	1	1	yes	characteristic
170	C₂×C₃₇	#2542	1	1	yes	characteristic
171	C₂×C₃₇	#3150	1	1	yes	characteristic
172	C₂×C₃₇	#3156	1	1	yes	characteristic
173	C₂×C₃₇	#3352	1	2	yes	—
174	C₂×C₃₇	#3390	1	2	yes	—
175	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
176	C₂×#125	C₂²×C₃₇	1	2	yes	—
177	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
178	#499	—	2	4	no	—
179	#479	—	2	8	no	—
180	C₄×Q₁₆	—	2	4	no	—
181	C₂×#107	C₂²×C₃₇	1	2	yes	—
182	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
183	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
184	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
185	C₂×#106	C₂²×C₃₇	1	2	yes	—
186	C₂×#106	C₂²×C₃₇	1	2	yes	—
187	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
188	C₂×#103	C₂²×C₃₇	1	2	yes	—
189	C₂×#103	C₂²×C₃₇	1	2	yes	—
190	C₂×#103	C₂²×C₃₇	1	2	yes	—
191	C₂×#97	C₂²×C₃₇	1	2	yes	—
192	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
193	#398	—	2	4	no	—
194	#391	C₂²×C₃₇	1	2	yes	—
195	#391	C₂²×C₃₇	1	2	yes	—
196	#387	—	2	4	no	—
197	#384	C₂²×C₃₇	1	2	yes	—
198	#382	C₂²×C₃₇	1	2	yes	—
199	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
200	#336	C₂²×C₃₇	1	2	yes	—
201	C₃₇	#28538	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	Q₈×C₃₇	1	4	yes	—
206	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
208	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
209	#125	—	2	4	no	—
210	#125	—	2	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
223	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
224	#107	—	2	2	no	—
225	#107	—	2	4	no	—
226	#107	—	2	2	no	—
227	#107	—	2	2	no	—
228	#106	—	2	4	no	—
229	#106	—	2	4	no	—
230	C₄⋊C₈	D₄×C₃₇	1	4	yes	—
231	#103	—	2	4	no	—
232	#103	—	2	4	no	—
233	#103	—	2	4	no	—
234	#97	—	2	4	no	—
235	C₄×C₈	D₄×C₃₇	1	4	yes	—
236	#95	—	2	4	no	—
237	#95	—	2	4	no	—
238	#95	—	2	4	no	—
239	#95	D₄×C₃₇	1	2	yes	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
251	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
253	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
254	Q₁₆	—	2	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
260	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
261	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
262	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
263	C₂×C₈	#41×C₃₇	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	8	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	4	4	no	—
278	C₄²	C₂×D₄×C₃₇	1	2	yes	—
279	C₄²	C₂×D₄×C₃₇	1	2	yes	—
280	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	#127×C₃₇	1	2	yes	—
300	C₂×C₄	#127×C₃₇	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
305	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
306	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	—	2	4	no	—
309	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
310	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
311	C₂²	C₃₇×#547	1	1	yes	characteristic
312	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
313	C₂²	C₃₇×#575	1	2	yes	—
314	C₂²	C₃₇×#578	1	2	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	4	4	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
323	C₄	C₃₇×#536	1	1	yes	characteristic
324	C₄	C₃₇×#547	1	1	yes	characteristic
325	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
326	C₂	C₃₇×#2542	1	1	yes	characteristic
327	C₂	C₃₇×#3150	1	1	yes	characteristic
328	C₂	C₃₇×#3156	1	1	yes	characteristic
329	C₂	C₃₇×#3352	1	2	yes	—
330	C₂	C₃₇×#3390	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₄×C₄⋊C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
144. #28538	C₃₇	direct (N × H)
201. C₃₇	#28538	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#499	2
C₂×C₃₇×#479	2
C₂×C₄×Q₁₆×C₃₇	2
C₃₇×#2565	2
C₃₇×#2097	2
C₃₇×#2095	2
C₃₇×#1971	2
C₃₇×#1847	2
C₃₇×#1770	2
#28538	37

Subgroups — C37×#28538 #22649791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28538 #22649791