Subgroups — C₃₇×#28727 #22649980

Order 9472.

798 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#489	C₂	1	2	yes	maximal
2	C₃₇×#2862	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#124×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2074	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2073	C₂	1	2	yes	maximal
6	C₃₇×#1993	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1805	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1804	C₂	1	2	yes	maximal
9	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×#124×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
14	C₃₇×#489	—	2	8	no	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×#103×C₃₇	C₂²	1	4	yes	—
17	C₂×#102×C₃₇	C₂²	1	4	yes	—
18	C₂×#97×C₃₇	C₂²	1	2	yes	—
19	C₃₇×#391	—	2	2	no	—
20	C₃₇×#391	—	2	2	no	—
21	C₃₇×#391	C₂²	1	1	yes	characteristic
22	C₃₇×#390	C₂²	1	1	yes	characteristic
23	C₃₇×#388	—	2	2	no	—
24	C₃₇×#382	C₂²	1	2	yes	—
25	C₃₇×#382	C₂²	1	2	yes	—
26	C₃₇×#382	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₃₇	—	2	2	no	—
28	C₄³×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#340	C₂²	1	4	yes	—
30	C₃₇×#340	C₂²	1	4	yes	—
31	C₃₇×#336	C₂²	1	2	yes	—
32	C₂²×Q₈×C₃₇	—	2	2	no	—
33	C₂²×Q₈×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²×D₄×C₃₇	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₃₇	—	2	1	no	—
36	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
37	C₄⋊Q₈×C₃₇	—	2	4	no	—
38	C₄⋊Q₈×C₃₇	—	2	4	no	—
39	#124×C₃₇	—	2	2	no	—
40	#124×C₃₇	D₄	1	4	yes	—
41	C₄×Q₈×C₃₇	—	2	4	no	—
42	C₄×D₄×C₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	1	no	—
51	C₂×C₄²×C₃₇	—	2	1	no	—
52	C₂×C₄²×C₃₇	D₄	1	2	yes	—
53	C₂×C₄²×C₃₇	D₄	1	2	yes	—
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₃₇	—	2	4	no	—
58	#103×C₃₇	—	2	8	no	—
59	#102×C₃₇	—	2	8	no	—
60	#97×C₃₇	—	2	4	no	—
61	#95×C₃₇	—	2	2	no	—
62	#95×C₃₇	—	2	2	no	—
63	#95×C₃₇	—	2	4	no	—
64	#95×C₃₇	—	2	2	no	—
65	#95×C₃₇	—	2	2	no	—
66	C₂⁴×C₃₇	—	2	1	no	—
67	C₂×Q₈×C₃₇	—	2	1	no	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	2	2	no	—
70	C₂×Q₈×C₃₇	—	4	4	no	—
71	C₂×Q₈×C₃₇	—	4	4	no	—
72	C₂×Q₈×C₃₇	—	4	4	no	—
73	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
74	C₂×D₄×C₃₇	—	2	2	no	—
75	C₂×D₄×C₃₇	—	2	1	no	—
76	C₂×D₄×C₃₇	—	2	2	no	—
77	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₇	—	2	1	no	—
79	C₂²×C₄×C₃₇	—	2	1	no	—
80	C₂²×C₄×C₃₇	—	2	4	no	—
81	C₂²×C₄×C₃₇	—	2	1	no	—
82	C₂²×C₄×C₃₇	—	4	4	no	—
83	C₂²×C₄×C₃₇	—	4	2	no	—
84	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₃₇	—	2	4	no	—
90	C₂×C₈×C₃₇	—	2	4	no	—
91	C₂×C₈×C₃₇	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	2	4	no	—
93	C₄⋊C₄×C₃₇	—	2	2	no	—
94	C₄⋊C₄×C₃₇	—	2	2	no	—
95	C₄⋊C₄×C₃₇	—	2	4	no	—
96	C₄⋊C₄×C₃₇	—	4	4	no	—
97	C₄⋊C₄×C₃₇	—	4	2	no	—
98	C₂²⋊C₄×C₃₇	—	2	8	no	—
99	C₂²⋊C₄×C₃₇	—	4	4	no	—
100	C₂²⋊C₄×C₃₇	—	4	2	no	—
101	C₄²×C₃₇	—	2	2	no	—
102	C₄²×C₃₇	—	4	2	no	—
103	C₄²×C₃₇	—	4	2	no	—
104	C₄²×C₃₇	—	4	2	no	—
105	C₄²×C₃₇	C₂×D₄	1	2	yes	—
106	C₄²×C₃₇	#41	1	2	yes	—
107	C₄²×C₃₇	#41	1	4	yes	—
108	C₂³×C₃₇	—	2	2	no	—
109	C₂³×C₃₇	—	4	1	no	—
110	C₂³×C₃₇	—	4	1	no	—
111	C₂³×C₃₇	—	4	2	no	—
112	C₂³×C₃₇	—	4	2	no	—
113	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₃₇	—	2	4	no	—
115	Q₈×C₃₇	—	4	2	no	—
116	Q₈×C₃₇	—	8	8	no	—
117	D₄×C₃₇	—	2	4	no	—
118	D₄×C₃₇	—	4	2	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	2	2	no	—
121	C₂×C₄×C₃₇	—	2	2	no	—
122	C₂×C₄×C₃₇	—	2	2	no	—
123	C₂×C₄×C₃₇	—	2	2	no	—
124	C₂×C₄×C₃₇	—	2	2	no	—
125	C₂×C₄×C₃₇	—	2	2	no	—
126	C₂×C₄×C₃₇	—	2	2	no	—
127	C₂×C₄×C₃₇	—	4	1	no	—
128	C₂×C₄×C₃₇	—	4	1	no	—
129	C₂×C₄×C₃₇	—	4	4	no	—
130	C₂×C₄×C₃₇	—	4	1	no	—
131	C₂×C₄×C₃₇	—	4	1	no	—
132	C₂×C₄×C₃₇	—	4	2	no	—
133	C₂×C₄×C₃₇	—	4	2	no	—
134	C₂×C₄×C₃₇	—	4	4	no	—
135	C₂×C₄×C₃₇	—	4	4	no	—
136	C₂×C₄×C₃₇	—	8	4	no	—
137	C₂×C₄×C₃₇	#121	1	4	yes	—
138	C₂×C₄×C₃₇	#124	1	2	yes	—
139	C₂×C₄×C₃₇	#124	1	2	yes	—
140	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
142	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
144	C₈×C₃₇	—	4	4	no	—
145	#28727	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₃₇	—	4	4	no	—
147	C₂²×C₃₇	—	4	2	no	—
148	C₂²×C₃₇	—	8	1	no	—
149	C₂²×C₃₇	—	8	1	no	—
150	C₂²×C₃₇	—	8	2	no	—
151	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
152	C₂²×C₃₇	C₂×#124	1	1	yes	characteristic
153	C₂²×C₃₇	#536	1	1	yes	characteristic
154	C₂²×C₃₇	#575	1	2	yes	—
155	C₂²×C₃₇	#575	1	2	yes	—
156	C₄×C₃₇	—	2	2	no	—
157	C₄×C₃₇	—	2	2	no	—
158	C₄×C₃₇	—	2	4	no	—
159	C₄×C₃₇	—	4	2	no	—
160	C₄×C₃₇	—	4	2	no	—
161	C₄×C₃₇	—	8	1	no	—
162	C₄×C₃₇	—	8	1	no	—
163	C₄×C₃₇	—	8	4	no	—
164	C₄×C₃₇	#547	1	2	yes	—
165	C₄×C₃₇	#556	1	2	yes	—
166	C₂×#489	C₂×C₃₇	1	2	yes	—
167	#2862	C₂×C₃₇	1	1	yes	characteristic, complemented
168	C₄×#124	C₂×C₃₇	1	1	yes	characteristic
169	#2074	C₂×C₃₇	1	2	yes	complemented
170	#2073	C₂×C₃₇	1	2	yes	—
171	#1993	C₂×C₃₇	1	2	yes	complemented
172	#1805	C₂×C₃₇	1	2	yes	complemented
173	#1804	C₂×C₃₇	1	2	yes	—
174	#1767	C₂×C₃₇	1	1	yes	characteristic, complemented
175	C₂×C₃₇	—	8	2	no	—
176	C₂×C₃₇	#2547	1	1	yes	characteristic
177	C₂×C₃₇	#3040	1	2	yes	—
178	C₂×C₃₇	#3135	1	1	yes	characteristic
179	C₂×C₃₇	#3140	1	1	yes	characteristic
180	C₂×C₃₇	#3240	1	2	yes	—
181	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
182	C₂×#124	C₂²×C₃₇	1	1	yes	characteristic
183	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
184	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
185	#489	—	2	8	no	—
186	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
187	C₂×#103	C₂²×C₃₇	1	4	yes	—
188	C₂×#102	C₂²×C₃₇	1	4	yes	—
189	C₂×#97	C₂²×C₃₇	1	2	yes	—
190	#391	—	2	2	no	—
191	#391	—	2	2	no	—
192	#391	C₂²×C₃₇	1	1	yes	characteristic
193	#390	C₂²×C₃₇	1	1	yes	characteristic
194	#388	—	2	2	no	—
195	#382	C₂²×C₃₇	1	2	yes	—
196	#382	C₂²×C₃₇	1	2	yes	—
197	#382	C₂²×C₃₇	1	1	yes	characteristic
198	C₄×C₂²⋊C₄	—	2	2	no	—
199	C₄³	C₂²×C₃₇	1	1	yes	characteristic
200	#340	C₂²×C₃₇	1	4	yes	—
201	#340	C₂²×C₃₇	1	4	yes	—
202	#336	C₂²×C₃₇	1	2	yes	—
203	C₃₇	#28727	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²×Q₈	—	2	2	no	—
205	C₂²×Q₈	C₂³×C₃₇	1	1	yes	characteristic
206	C₂²×D₄	C₂³×C₃₇	1	1	yes	characteristic
207	C₂³×C₄	—	2	1	no	—
208	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	#124	—	2	2	no	—
212	#124	D₄×C₃₇	1	4	yes	—
213	C₄×Q₈	—	2	4	no	—
214	C₄×D₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	C₂³×C₃₇	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂³×C₃₇	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	D₄×C₃₇	1	2	yes	—
225	C₂×C₄²	D₄×C₃₇	1	2	yes	—
226	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
228	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
229	C₄⋊C₈	—	2	4	no	—
230	#103	—	2	8	no	—
231	#102	—	2	8	no	—
232	#97	—	2	4	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	4	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	C₂⁴	—	2	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	—	4	4	no	—
245	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	4	4	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
260	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	2	8	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	C₂×D₄×C₃₇	1	2	yes	—
278	C₄²	#41×C₃₇	1	2	yes	—
279	C₄²	#41×C₃₇	1	4	yes	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	8	no	—
289	D₄	—	2	4	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	#121×C₃₇	1	4	yes	—
310	C₂×C₄	#124×C₃₇	1	2	yes	—
311	C₂×C₄	#124×C₃₇	1	2	yes	—
312	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
313	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
314	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
315	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
316	C₈	—	4	4	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	2	no	—
322	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
323	C₂²	C₂×#124×C₃₇	1	1	yes	characteristic
324	C₂²	C₃₇×#536	1	1	yes	characteristic
325	C₂²	C₃₇×#575	1	2	yes	—
326	C₂²	C₃₇×#575	1	2	yes	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	4	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	4	no	—
335	C₄	C₃₇×#547	1	2	yes	—
336	C₄	C₃₇×#556	1	2	yes	—
337	C₂	—	8	2	no	—
338	C₂	C₃₇×#2547	1	1	yes	characteristic
339	C₂	C₃₇×#3040	1	2	yes	—
340	C₂	C₃₇×#3135	1	1	yes	characteristic
341	C₂	C₃₇×#3140	1	1	yes	characteristic
342	C₂	C₃₇×#3240	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#2862	C₂	semidirect
4. C₃₇×#2074	C₂	semidirect
6. C₃₇×#1993	C₂	semidirect
7. C₃₇×#1805	C₂	semidirect
9. C₃₇×#1767	C₂	semidirect
145. #28727	C₃₇	direct (N × H)
167. #2862	C₂×C₃₇	semidirect
169. #2074	C₂×C₃₇	semidirect
171. #1993	C₂×C₃₇	semidirect
172. #1805	C₂×C₃₇	semidirect
174. #1767	C₂×C₃₇	semidirect
203. C₃₇	#28727	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×#489	2
C₃₇×#2862	2
C₄×#124×C₃₇	2
C₃₇×#2074	2
C₃₇×#2073	2
C₃₇×#1993	2
C₃₇×#1805	2
C₃₇×#1804	2
C₃₇×#1767	2
#28727	37

Subgroups — C37×#28727 #22649980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28727 #22649980