Subgroups — C₃₇×#28274 #22649527

Order 9472.

762 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2582	C₂	1	2	yes	maximal
4	C₃₇×#2066	C₂	1	4	yes	maximal
5	C₃₇×#1971	C₂	1	2	yes	maximal
6	C₃₇×#1944	C₂	1	2	yes	maximal
7	C₃₇×#1844	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1803	C₂	1	2	yes	maximal
9	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#446	—	2	4	no	—
13	C₄×Q₁₆×C₃₇	—	2	4	no	—
14	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₃₇	C₂²	1	2	yes	—
16	C₂×#106×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×#103×C₃₇	C₂²	1	2	yes	—
20	C₂×#103×C₃₇	C₂²	1	4	yes	—
21	C₂×#97×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#398	—	2	4	no	—
24	C₃₇×#391	—	2	2	no	—
25	C₃₇×#391	C₂²	1	4	yes	—
26	C₃₇×#389	—	2	2	no	—
27	C₃₇×#389	C₂²	1	4	yes	—
28	C₃₇×#384	C₂²	1	2	yes	—
29	C₃₇×#340	C₂²	1	4	yes	—
30	C₂²×Q₈×C₃₇	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂×Q₁₆×C₃₇	—	2	2	no	—
33	C₂×Q₁₆×C₃₇	—	2	2	no	—
34	C₂×Q₁₆×C₃₇	Q₈	1	4	yes	complemented
35	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₄×Q₈×C₃₇	—	2	2	no	—
39	C₄×Q₈×C₃₇	—	2	2	no	—
40	C₄×Q₈×C₃₇	—	2	4	no	—
41	C₄×Q₈×C₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
50	C₂×C₄²×C₃₇	—	2	2	no	—
51	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	#107×C₃₇	—	2	2	no	—
54	#106×C₃₇	—	2	4	no	—
55	#106×C₃₇	—	4	2	no	—
56	C₄⋊C₈×C₃₇	—	2	4	no	—
57	C₄⋊C₈×C₃₇	—	2	2	no	—
58	#103×C₃₇	—	2	4	no	—
59	#103×C₃₇	—	4	4	no	—
60	#97×C₃₇	—	2	4	no	—
61	C₄×C₈×C₃₇	D₄	1	4	yes	—
62	#95×C₃₇	—	2	4	no	—
63	#95×C₃₇	—	2	4	no	—
64	#95×C₃₇	—	2	4	no	—
65	C₂×Q₈×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	4	no	—
67	C₂×Q₈×C₃₇	—	2	2	no	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	2	2	no	—
70	C₂×Q₈×C₃₇	—	2	2	no	—
71	C₂×Q₈×C₃₇	—	2	2	no	—
72	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₃₇	—	2	2	no	—
74	C₂²×C₄×C₃₇	—	2	4	no	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₃₇	—	2	8	no	—
82	C₂×C₈×C₃₇	—	2	1	no	—
83	C₂×C₈×C₃₇	—	2	1	no	—
84	C₂×C₈×C₃₇	—	2	2	no	—
85	C₂×C₈×C₃₇	—	2	2	no	—
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₃₇	#41	1	2	yes	—
91	C₄⋊C₄×C₃₇	—	2	2	no	—
92	C₄⋊C₄×C₃₇	—	2	4	no	—
93	C₄⋊C₄×C₃₇	—	2	2	no	—
94	C₄⋊C₄×C₃₇	—	2	4	no	—
95	C₄⋊C₄×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	2	4	no	—
97	C₄⋊C₄×C₃₇	—	4	4	no	—
98	C₄⋊C₄×C₃₇	—	4	4	no	—
99	C₄⋊C₄×C₃₇	—	8	2	no	—
100	C₄⋊C₄×C₃₇	—	8	4	no	—
101	C₄²×C₃₇	—	2	2	no	—
102	C₄²×C₃₇	—	2	2	no	—
103	C₄²×C₃₇	—	4	4	no	—
104	C₄²×C₃₇	C₂×D₄	1	2	yes	—
105	C₄²×C₃₇	#41	1	2	yes	—
106	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₃₇	—	2	4	no	—
108	Q₈×C₃₇	—	2	4	no	—
109	Q₈×C₃₇	—	4	2	no	—
110	Q₈×C₃₇	—	4	2	no	—
111	Q₈×C₃₇	—	4	4	no	—
112	C₂×C₄×C₃₇	—	2	2	no	—
113	C₂×C₄×C₃₇	—	2	2	no	—
114	C₂×C₄×C₃₇	—	2	2	no	—
115	C₂×C₄×C₃₇	—	2	4	no	—
116	C₂×C₄×C₃₇	—	2	4	no	—
117	C₂×C₄×C₃₇	—	2	4	no	—
118	C₂×C₄×C₃₇	—	4	2	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	—	4	4	no	—
122	C₂×C₄×C₃₇	—	4	4	no	—
123	C₂×C₄×C₃₇	—	4	4	no	—
124	C₂×C₄×C₃₇	—	4	2	no	—
125	C₂×C₄×C₃₇	—	4	2	no	—
126	C₂×C₄×C₃₇	#121	1	2	yes	—
127	C₂×C₄×C₃₇	#121	1	2	yes	—
128	C₂×C₄×C₃₇	#137	1	4	yes	—
129	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
135	C₈×C₃₇	—	2	2	no	—
136	C₈×C₃₇	—	4	2	no	—
137	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
138	#28274	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
140	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
141	C₂²×C₃₇	#550	1	1	yes	characteristic
142	C₂²×C₃₇	C₂×#137	1	2	yes	—
143	C₂²×C₃₇	#577	1	2	yes	—
144	C₄×C₃₇	—	2	2	no	—
145	C₄×C₃₇	—	4	4	no	—
146	C₄×C₃₇	—	4	2	no	—
147	C₄×C₃₇	—	4	2	no	—
148	C₄×C₃₇	—	8	4	no	—
149	C₄×C₃₇	—	8	2	no	—
150	C₄×C₃₇	#470	1	4	yes	—
151	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₄×C₃₇	#537	1	1	yes	characteristic
153	C₄×C₃₇	#554	1	1	yes	characteristic
154	C₄×C₃₇	#556	1	1	yes	characteristic
155	C₂×#446	C₂×C₃₇	1	1	yes	characteristic
156	C₂×C₄×Q₁₆	C₂×C₃₇	1	1	yes	characteristic
157	#2582	C₂×C₃₇	1	2	yes	—
158	#2066	C₂×C₃₇	1	4	yes	—
159	#1971	C₂×C₃₇	1	2	yes	—
160	#1944	C₂×C₃₇	1	2	yes	—
161	#1844	C₂×C₃₇	1	1	yes	characteristic
162	#1803	C₂×C₃₇	1	2	yes	—
163	C₂×C₃₇	#2538	1	1	yes	characteristic
164	C₂×C₃₇	C₂×#470	1	1	yes	characteristic
165	C₂×C₃₇	#3060	1	1	yes	characteristic
166	C₂×C₃₇	#3387	1	2	yes	—
167	C₂×C₃₇	#3400	1	2	yes	—
168	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
169	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
170	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
171	#446	—	2	4	no	—
172	C₄×Q₁₆	—	2	4	no	—
173	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
174	C₂×#106	C₂²×C₃₇	1	2	yes	—
175	C₂×#106	C₂²×C₃₇	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
178	C₂×#103	C₂²×C₃₇	1	2	yes	—
179	C₂×#103	C₂²×C₃₇	1	4	yes	—
180	C₂×#97	C₂²×C₃₇	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
182	#398	—	2	4	no	—
183	#391	—	2	2	no	—
184	#391	C₂²×C₃₇	1	4	yes	—
185	#389	—	2	2	no	—
186	#389	C₂²×C₃₇	1	4	yes	—
187	#384	C₂²×C₃₇	1	2	yes	—
188	#340	C₂²×C₃₇	1	4	yes	—
189	C₃₇	#28274	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²×Q₈	D₄×C₃₇	1	1	yes	characteristic
191	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	—	2	2	no	—
194	C₂×Q₁₆	Q₈×C₃₇	1	4	yes	complemented
195	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
196	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
198	C₄×Q₈	—	2	2	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	2	4	no	—
201	C₄×Q₈	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
212	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
213	#107	—	2	2	no	—
214	#106	—	2	4	no	—
215	#106	—	4	2	no	—
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	2	2	no	—
218	#103	—	2	4	no	—
219	#103	—	4	4	no	—
220	#97	—	2	4	no	—
221	C₄×C₈	D₄×C₃₇	1	4	yes	—
222	#95	—	2	4	no	—
223	#95	—	2	4	no	—
224	#95	—	2	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
240	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
241	Q₁₆	—	2	8	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
249	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
250	C₂×C₈	#41×C₃₇	1	2	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	8	2	no	—
260	C₄⋊C₄	—	8	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	4	4	no	—
264	C₄²	C₂×D₄×C₃₇	1	2	yes	—
265	C₄²	#41×C₃₇	1	2	yes	—
266	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	2	4	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	4	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	#121×C₃₇	1	2	yes	—
287	C₂×C₄	#121×C₃₇	1	2	yes	—
288	C₂×C₄	#137×C₃₇	1	4	yes	—
289	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
292	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
293	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
294	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	2	no	—
297	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
298	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
299	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
300	C₂²	C₃₇×#550	1	1	yes	characteristic
301	C₂²	C₂×#137×C₃₇	1	2	yes	—
302	C₂²	C₃₇×#577	1	2	yes	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	4	no	—
308	C₄	—	8	2	no	—
309	C₄	C₃₇×#470	1	4	yes	—
310	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
311	C₄	C₃₇×#537	1	1	yes	characteristic
312	C₄	C₃₇×#554	1	1	yes	characteristic
313	C₄	C₃₇×#556	1	1	yes	characteristic
314	C₂	C₃₇×#2538	1	1	yes	characteristic
315	C₂	C₂×C₃₇×#470	1	1	yes	characteristic
316	C₂	C₃₇×#3060	1	1	yes	characteristic
317	C₂	C₃₇×#3387	1	2	yes	—
318	C₂	C₃₇×#3400	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 ◁ C₂×C₄×Q₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
34. C₂×Q₁₆×C₃₇	Q₈	semidirect
138. #28274	C₃₇	direct (N × H)
189. C₃₇	#28274	direct (N × H)
194. C₂×Q₁₆	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#446	2
C₂×C₄×Q₁₆×C₃₇	2
C₃₇×#2582	2
C₃₇×#2066	2
C₃₇×#1971	2
C₃₇×#1944	2
C₃₇×#1844	2
C₃₇×#1803	2
#28274	37

Subgroups — C37×#28274 #22649527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28274 #22649527