Subgroups — C₃₇×#28294 #22649547

Order 9472.

738 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#479	C₂	1	2	yes	maximal
2	C₃₇×#2330	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2327	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2068	C₂	1	2	yes	maximal
5	C₃₇×#2066	C₂	1	2	yes	maximal
6	C₃₇×#1981	C₂	1	2	yes	maximal
7	C₃₇×#1949	C₂	1	2	yes	maximal
8	C₃₇×#1843	C₂	1	1	yes	maximal, characteristic
9	C₃₇×#1803	C₂	1	2	yes	maximal
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#479	—	2	8	no	—
14	C₂×#107×C₃₇	C₂²	1	2	yes	—
15	C₂×#106×C₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×#103×C₃₇	C₂²	1	2	yes	—
21	C₂×#103×C₃₇	C₂²	1	2	yes	—
22	C₂×#103×C₃₇	C₂²	1	2	yes	—
23	C₂×#103×C₃₇	C₂²	1	2	yes	—
24	C₃₇×#391	C₂²	1	2	yes	—
25	C₃₇×#389	—	2	2	no	—
26	C₃₇×#389	C₂²	1	2	yes	—
27	C₃₇×#387	—	2	2	no	—
28	C₃₇×#387	—	2	2	no	—
29	C₃₇×#387	—	2	2	no	—
30	C₃₇×#384	C₂²	1	1	yes	characteristic
31	C₃₇×#382	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
33	C₃₇×#376	C₂²	1	1	yes	characteristic
34	C₃₇×#340	C₂²	1	2	yes	—
35	C₃₇×#340	C₂²	1	2	yes	—
36	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₃₇	—	2	2	no	—
40	C₄⋊Q₈×C₃₇	Q₈	1	4	yes	—
41	#125×C₃₇	—	2	2	no	—
42	#125×C₃₇	D₄	1	4	yes	—
43	C₄×Q₈×C₃₇	—	2	4	no	—
44	C₄×Q₈×C₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₃₇	—	2	2	no	—
54	C₂×C₄²×C₃₇	—	2	2	no	—
55	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
58	#107×C₃₇	—	2	4	no	—
59	#106×C₃₇	—	2	4	no	—
60	C₄⋊C₈×C₃₇	—	2	4	no	—
61	C₄⋊C₈×C₃₇	—	2	4	no	—
62	C₄⋊C₈×C₃₇	—	4	1	no	—
63	C₄⋊C₈×C₃₇	—	4	1	no	—
64	#103×C₃₇	—	2	4	no	—
65	#103×C₃₇	—	2	4	no	—
66	#103×C₃₇	—	4	2	no	—
67	#103×C₃₇	—	4	2	no	—
68	#95×C₃₇	—	2	2	no	—
69	#95×C₃₇	—	2	2	no	—
70	#95×C₃₇	—	2	2	no	—
71	#95×C₃₇	—	2	2	no	—
72	#95×C₃₇	—	2	2	no	—
73	#95×C₃₇	D₄	1	2	yes	—
74	#95×C₃₇	D₄	1	2	yes	—
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	—	2	2	no	—
80	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
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	—	2	2	no	—
85	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₃₇	—	2	2	no	—
91	C₂×C₈×C₃₇	—	2	2	no	—
92	C₂×C₈×C₃₇	—	2	2	no	—
93	C₂×C₈×C₃₇	—	2	2	no	—
94	C₂×C₈×C₃₇	—	2	2	no	—
95	C₂×C₈×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	2	8	no	—
97	C₄⋊C₄×C₃₇	—	2	8	no	—
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	—	4	4	no	—
104	C₄⋊C₄×C₃₇	—	8	2	no	—
105	C₄⋊C₄×C₃₇	—	8	2	no	—
106	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
107	C₄²×C₃₇	—	2	1	no	—
108	C₄²×C₃₇	—	2	1	no	—
109	C₄²×C₃₇	—	4	4	no	—
110	C₄²×C₃₇	—	8	2	no	—
111	C₄²×C₃₇	C₂×D₄	1	2	yes	—
112	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
113	C₄²×C₃₇	#41	1	2	yes	—
114	C₄²×C₃₇	#41	1	2	yes	—
115	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₃₇	—	2	4	no	—
117	Q₈×C₃₇	—	2	4	no	—
118	Q₈×C₃₇	—	4	2	no	—
119	Q₈×C₃₇	—	4	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	4	no	—
125	C₂×C₄×C₃₇	—	2	4	no	—
126	C₂×C₄×C₃₇	—	2	1	no	—
127	C₂×C₄×C₃₇	—	2	1	no	—
128	C₂×C₄×C₃₇	—	2	2	no	—
129	C₂×C₄×C₃₇	—	2	2	no	—
130	C₂×C₄×C₃₇	—	4	2	no	—
131	C₂×C₄×C₃₇	—	4	2	no	—
132	C₂×C₄×C₃₇	—	4	2	no	—
133	C₂×C₄×C₃₇	—	4	2	no	—
134	C₂×C₄×C₃₇	—	4	2	no	—
135	C₂×C₄×C₃₇	—	4	2	no	—
136	C₂×C₄×C₃₇	—	4	2	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	#121	1	4	yes	—
141	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
142	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₃₇	—	4	2	no	—
149	C₈×C₃₇	—	4	2	no	—
150	#28294	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
152	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
153	C₂²×C₃₇	#537	1	1	yes	characteristic
154	C₂²×C₃₇	#577	1	2	yes	—
155	C₂²×C₃₇	#578	1	2	yes	—
156	C₄×C₃₇	—	2	4	no	—
157	C₄×C₃₇	—	2	4	no	—
158	C₄×C₃₇	—	4	1	no	—
159	C₄×C₃₇	—	4	1	no	—
160	C₄×C₃₇	—	4	4	no	—
161	C₄×C₃₇	—	4	4	no	—
162	C₄×C₃₇	—	8	2	no	—
163	C₄×C₃₇	—	8	2	no	—
164	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
165	C₄×C₃₇	#550	1	1	yes	characteristic
166	C₄×C₃₇	#554	1	1	yes	characteristic
167	C₄×C₃₇	#556	1	1	yes	characteristic
168	C₂×#479	C₂×C₃₇	1	2	yes	—
169	#2330	C₂×C₃₇	1	1	yes	characteristic
170	#2327	C₂×C₃₇	1	1	yes	characteristic
171	#2068	C₂×C₃₇	1	2	yes	—
172	#2066	C₂×C₃₇	1	2	yes	—
173	#1981	C₂×C₃₇	1	2	yes	—
174	#1949	C₂×C₃₇	1	2	yes	—
175	#1843	C₂×C₃₇	1	1	yes	characteristic
176	#1803	C₂×C₃₇	1	2	yes	—
177	C₂×C₃₇	#2538	1	1	yes	characteristic
178	C₂×C₃₇	#3046	1	2	yes	—
179	C₂×C₃₇	#3082	1	2	yes	—
180	C₂×C₃₇	#3260	1	1	yes	characteristic
181	C₂×C₃₇	#3261	1	1	yes	characteristic
182	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
183	C₂×#125	C₂²×C₃₇	1	1	yes	characteristic
184	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
185	#479	—	2	8	no	—
186	C₂×#107	C₂²×C₃₇	1	2	yes	—
187	C₂×#106	C₂²×C₃₇	1	2	yes	—
188	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
189	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
190	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
191	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
192	C₂×#103	C₂²×C₃₇	1	2	yes	—
193	C₂×#103	C₂²×C₃₇	1	2	yes	—
194	C₂×#103	C₂²×C₃₇	1	2	yes	—
195	C₂×#103	C₂²×C₃₇	1	2	yes	—
196	#391	C₂²×C₃₇	1	2	yes	—
197	#389	—	2	2	no	—
198	#389	C₂²×C₃₇	1	2	yes	—
199	#387	—	2	2	no	—
200	#387	—	2	2	no	—
201	#387	—	2	2	no	—
202	#384	C₂²×C₃₇	1	1	yes	characteristic
203	#382	C₂²×C₃₇	1	1	yes	characteristic
204	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
205	#376	C₂²×C₃₇	1	1	yes	characteristic
206	#340	C₂²×C₃₇	1	2	yes	—
207	#340	C₂²×C₃₇	1	2	yes	—
208	C₃₇	#28294	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
210	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
211	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
212	C₄⋊Q₈	—	2	2	no	—
213	C₄⋊Q₈	Q₈×C₃₇	1	4	yes	—
214	#125	—	2	2	no	—
215	#125	D₄×C₃₇	1	4	yes	—
216	C₄×Q₈	—	2	4	no	—
217	C₄×Q₈	—	2	4	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	—	2	2	no	—
228	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
229	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
230	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
231	#107	—	2	4	no	—
232	#106	—	2	4	no	—
233	C₄⋊C₈	—	2	4	no	—
234	C₄⋊C₈	—	2	4	no	—
235	C₄⋊C₈	—	4	1	no	—
236	C₄⋊C₈	—	4	1	no	—
237	#103	—	2	4	no	—
238	#103	—	2	4	no	—
239	#103	—	4	2	no	—
240	#103	—	4	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	—	2	2	no	—
246	#95	D₄×C₃₇	1	2	yes	—
247	#95	D₄×C₃₇	1	2	yes	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
262	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	8	2	no	—
278	C₄⋊C₄	—	8	2	no	—
279	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	4	4	no	—
283	C₄²	—	8	2	no	—
284	C₄²	C₂×D₄×C₃₇	1	2	yes	—
285	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
286	C₄²	#41×C₃₇	1	2	yes	—
287	C₄²	#41×C₃₇	1	2	yes	—
288	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
289	Q₈	—	2	4	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	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	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	#121×C₃₇	1	4	yes	—
314	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
315	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
316	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
317	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
318	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
319	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
320	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
321	C₈	—	4	2	no	—
322	C₈	—	4	2	no	—
323	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
324	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
325	C₂²	C₃₇×#537	1	1	yes	characteristic
326	C₂²	C₃₇×#577	1	2	yes	—
327	C₂²	C₃₇×#578	1	2	yes	—
328	C₄	—	2	4	no	—
329	C₄	—	2	4	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	4	4	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
337	C₄	C₃₇×#550	1	1	yes	characteristic
338	C₄	C₃₇×#554	1	1	yes	characteristic
339	C₄	C₃₇×#556	1	1	yes	characteristic
340	C₂	C₃₇×#2538	1	1	yes	characteristic
341	C₂	C₃₇×#3046	1	2	yes	—
342	C₂	C₃₇×#3082	1	2	yes	—
343	C₂	C₃₇×#3260	1	1	yes	characteristic
344	C₂	C₃₇×#3261	1	1	yes	characteristic

	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 ◁ C₃₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
150. #28294	C₃₇	direct (N × H)
208. C₃₇	#28294	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×#479	2
C₃₇×#2330	2
C₃₇×#2327	2
C₃₇×#2068	2
C₃₇×#2066	2
C₃₇×#1981	2
C₃₇×#1949	2
C₃₇×#1843	2
C₃₇×#1803	2
#28294	37

Subgroups — C37×#28294 #22649547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28294 #22649547