Subgroups — C₃₇×#28536 #22649789

Order 9472.

890 conjugacy classes of subgroups.

The subgroup lattice has 354 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, complemented
2	C₂×C₃₇×#478	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2566	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2093	C₂	1	2	yes	maximal, complemented
6	C₃₇×#2092	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1972	C₂	1	2	yes	maximal
8	C₃₇×#1847	C₂	1	2	yes	maximal, complemented
9	C₃₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₃₇	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#499	—	2	4	no	—
14	C₃₇×#478	—	2	8	no	—
15	C₄×D₈×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×#102×C₃₇	C₂²	1	2	yes	—
24	C₂×#102×C₃₇	C₂²	1	2	yes	—
25	C₂×#102×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×#396	—	2	4	no	—
29	C₃₇×#390	C₂²	1	2	yes	—
30	C₃₇×#390	C₂²	1	2	yes	—
31	C₃₇×#385	—	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×D₄×C₃₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₃₇	—	2	2	no	—
38	C₂×D₈×C₃₇	—	2	2	no	—
39	C₂×D₈×C₃₇	—	2	2	no	—
40	C₂×D₈×C₃₇	Q₈	1	4	yes	—
41	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
44	#125×C₃₇	—	2	4	no	—
45	#125×C₃₇	—	2	4	no	—
46	C₄×D₄×C₃₇	—	2	4	no	—
47	C₄×D₄×C₃₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₇	—	2	4	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×C₂²⋊C₄×C₃₇	—	2	4	no	—
56	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
58	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
60	#107×C₃₇	—	2	2	no	—
61	#107×C₃₇	—	2	4	no	—
62	#107×C₃₇	—	2	2	no	—
63	#107×C₃₇	—	2	2	no	—
64	#106×C₃₇	—	2	4	no	—
65	#106×C₃₇	—	2	4	no	—
66	C₄⋊C₈×C₃₇	D₄	1	4	yes	—
67	#102×C₃₇	—	2	4	no	—
68	#102×C₃₇	—	2	4	no	—
69	#102×C₃₇	—	2	4	no	—
70	#97×C₃₇	—	2	4	no	—
71	C₄×C₈×C₃₇	D₄	1	4	yes	—
72	#95×C₃₇	—	2	4	no	—
73	#95×C₃₇	D₄	1	2	yes	—
74	C₂⁴×C₃₇	—	2	2	no	—
75	C₂×D₄×C₃₇	—	2	2	no	—
76	C₂×D₄×C₃₇	—	2	2	no	—
77	C₂×D₄×C₃₇	—	2	2	no	—
78	C₂×D₄×C₃₇	—	2	2	no	—
79	C₂×D₄×C₃₇	—	2	2	no	—
80	C₂×D₄×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	—	4	4	no	—
85	C₂²×C₄×C₃₇	—	8	4	no	—
86	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
90	D₈×C₃₇	—	2	8	no	—
91	C₂×C₈×C₃₇	—	2	2	no	—
92	C₂×C₈×C₃₇	—	2	1	no	—
93	C₂×C₈×C₃₇	—	2	1	no	—
94	C₂×C₈×C₃₇	—	2	2	no	—
95	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
97	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₃₇	#41	1	2	yes	—
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	—	2	2	no	—
105	C₄⋊C₄×C₃₇	—	2	2	no	—
106	C₄⋊C₄×C₃₇	—	4	2	no	—
107	C₄⋊C₄×C₃₇	—	4	4	no	—
108	C₄⋊C₄×C₃₇	—	4	4	no	—
109	C₄⋊C₄×C₃₇	—	8	4	no	—
110	C₂²⋊C₄×C₃₇	—	4	8	no	—
111	C₂²⋊C₄×C₃₇	—	4	4	no	—
112	C₂²⋊C₄×C₃₇	—	8	4	no	—
113	C₄²×C₃₇	—	2	2	no	—
114	C₄²×C₃₇	—	2	2	no	—
115	C₄²×C₃₇	C₂×D₄	1	2	yes	—
116	C₄²×C₃₇	C₂×D₄	1	2	yes	—
117	C₂³×C₃₇	—	2	4	no	—
118	C₂³×C₃₇	—	4	2	no	—
119	C₂³×C₃₇	—	4	2	no	—
120	C₂³×C₃₇	—	4	4	no	—
121	C₂³×C₃₇	—	4	4	no	—
122	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
123	D₄×C₃₇	—	2	4	no	—
124	D₄×C₃₇	—	2	4	no	—
125	D₄×C₃₇	—	4	2	no	—
126	D₄×C₃₇	—	4	2	no	—
127	C₂×C₄×C₃₇	—	2	2	no	—
128	C₂×C₄×C₃₇	—	2	2	no	—
129	C₂×C₄×C₃₇	—	2	4	no	—
130	C₂×C₄×C₃₇	—	2	4	no	—
131	C₂×C₄×C₃₇	—	2	4	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	4	no	—
136	C₂×C₄×C₃₇	—	4	4	no	—
137	C₂×C₄×C₃₇	—	4	4	no	—
138	C₂×C₄×C₃₇	—	8	8	no	—
139	C₂×C₄×C₃₇	#127	1	2	yes	—
140	C₂×C₄×C₃₇	#127	1	2	yes	—
141	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
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	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₈×C₃₇	—	2	2	no	—
148	C₈×C₃₇	—	2	4	no	—
149	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
150	#28536	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₃₇	—	4	8	no	—
152	C₂²×C₃₇	—	4	4	no	—
153	C₂²×C₃₇	—	8	4	no	—
154	C₂²×C₃₇	—	8	2	no	—
155	C₂²×C₃₇	—	8	2	no	—
156	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
157	C₂²×C₃₇	#547	1	1	yes	characteristic
158	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
159	C₂²×C₃₇	#575	1	2	yes	—
160	C₂²×C₃₇	#576	1	2	yes	—
161	C₄×C₃₇	—	2	2	no	—
162	C₄×C₃₇	—	2	2	no	—
163	C₄×C₃₇	—	4	4	no	—
164	C₄×C₃₇	—	8	2	no	—
165	C₄×C₃₇	—	8	2	no	—
166	C₄×C₃₇	—	8	2	no	—
167	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₄×C₃₇	#536	1	1	yes	characteristic
169	C₄×C₃₇	#547	1	1	yes	characteristic
170	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
171	C₂×#499	C₂×C₃₇	1	1	yes	characteristic, complemented
172	C₂×#478	C₂×C₃₇	1	2	yes	complemented
173	C₂×C₄×D₈	C₂×C₃₇	1	1	yes	characteristic
174	#2566	C₂×C₃₇	1	2	yes	complemented
175	#2093	C₂×C₃₇	1	2	yes	complemented
176	#2092	C₂×C₃₇	1	2	yes	complemented
177	#1972	C₂×C₃₇	1	2	yes	—
178	#1847	C₂×C₃₇	1	2	yes	complemented
179	#1770	C₂×C₃₇	1	1	yes	characteristic, complemented
180	C₂×C₃₇	—	8	4	no	—
181	C₂×C₃₇	#2542	1	1	yes	characteristic
182	C₂×C₃₇	#3150	1	1	yes	characteristic
183	C₂×C₃₇	#3155	1	1	yes	characteristic
184	C₂×C₃₇	#3355	1	2	yes	—
185	C₂×C₃₇	#3389	1	2	yes	—
186	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
187	C₂×#125	C₂²×C₃₇	1	2	yes	—
188	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
189	#499	—	2	4	no	—
190	#478	—	2	8	no	—
191	C₄×D₈	—	2	4	no	—
192	C₂×#107	C₂²×C₃₇	1	2	yes	—
193	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
194	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
195	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
196	C₂×#106	C₂²×C₃₇	1	2	yes	—
197	C₂×#106	C₂²×C₃₇	1	2	yes	—
198	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
199	C₂×#102	C₂²×C₃₇	1	2	yes	—
200	C₂×#102	C₂²×C₃₇	1	2	yes	—
201	C₂×#102	C₂²×C₃₇	1	2	yes	—
202	C₂×#97	C₂²×C₃₇	1	2	yes	—
203	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
204	#396	—	2	4	no	—
205	#390	C₂²×C₃₇	1	2	yes	—
206	#390	C₂²×C₃₇	1	2	yes	—
207	#385	—	2	4	no	—
208	#384	C₂²×C₃₇	1	2	yes	—
209	#382	C₂²×C₃₇	1	2	yes	—
210	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
211	#336	C₂²×C₃₇	1	2	yes	—
212	C₃₇	#28536	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
214	C₂³×C₄	—	2	2	no	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	Q₈×C₃₇	1	4	yes	—
218	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
220	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
221	#125	—	2	4	no	—
222	#125	—	2	4	no	—
223	C₄×D₄	—	2	4	no	—
224	C₄×D₄	—	2	4	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	4	no	—
228	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₂²⋊C₄	—	2	4	no	—
234	C₂×C₂²⋊C₄	—	2	4	no	—
235	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
236	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
237	#107	—	2	2	no	—
238	#107	—	2	4	no	—
239	#107	—	2	2	no	—
240	#107	—	2	2	no	—
241	#106	—	2	4	no	—
242	#106	—	2	4	no	—
243	C₄⋊C₈	D₄×C₃₇	1	4	yes	—
244	#102	—	2	4	no	—
245	#102	—	2	4	no	—
246	#102	—	2	4	no	—
247	#97	—	2	4	no	—
248	C₄×C₈	D₄×C₃₇	1	4	yes	—
249	#95	—	2	4	no	—
250	#95	D₄×C₃₇	1	2	yes	—
251	C₂⁴	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	C₂×Q₈×C₃₇	1	4	yes	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	—	8	4	no	—
263	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
264	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
266	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
267	D₈	—	2	8	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
274	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
275	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
276	C₂×C₈	#41×C₃₇	1	2	yes	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₄⋊C₄	—	8	4	no	—
287	C₂²⋊C₄	—	4	8	no	—
288	C₂²⋊C₄	—	4	4	no	—
289	C₂²⋊C₄	—	8	4	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	C₂×D₄×C₃₇	1	2	yes	—
293	C₄²	C₂×D₄×C₃₇	1	2	yes	—
294	C₂³	—	2	4	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	4	no	—
298	C₂³	—	4	4	no	—
299	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
300	D₄	—	2	4	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	8	8	no	—
316	C₂×C₄	#127×C₃₇	1	2	yes	—
317	C₂×C₄	#127×C₃₇	1	2	yes	—
318	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
321	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
322	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
323	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	4	no	—
326	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
327	C₂²	—	4	8	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	4	no	—
332	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
333	C₂²	C₃₇×#547	1	1	yes	characteristic
334	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
335	C₂²	C₃₇×#575	1	2	yes	—
336	C₂²	C₃₇×#576	1	2	yes	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	4	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
344	C₄	C₃₇×#536	1	1	yes	characteristic
345	C₄	C₃₇×#547	1	1	yes	characteristic
346	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
347	C₂	—	8	4	no	—
348	C₂	C₃₇×#2542	1	1	yes	characteristic
349	C₂	C₃₇×#3150	1	1	yes	characteristic
350	C₂	C₃₇×#3155	1	1	yes	characteristic
351	C₂	C₃₇×#3355	1	2	yes	—
352	C₂	C₃₇×#3389	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₈×C₃₇	C₂²
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
1. C₂×C₃₇×#499	C₂	semidirect
2. C₂×C₃₇×#478	C₂	semidirect
4. C₃₇×#2566	C₂	semidirect
5. C₃₇×#2093	C₂	semidirect
6. C₃₇×#2092	C₂	semidirect
8. C₃₇×#1847	C₂	semidirect
9. C₃₇×#1770	C₂	semidirect
150. #28536	C₃₇	direct (N × H)
171. C₂×#499	C₂×C₃₇	semidirect
172. C₂×#478	C₂×C₃₇	semidirect
174. #2566	C₂×C₃₇	semidirect
175. #2093	C₂×C₃₇	semidirect
176. #2092	C₂×C₃₇	semidirect
178. #1847	C₂×C₃₇	semidirect
179. #1770	C₂×C₃₇	semidirect
212. C₃₇	#28536	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×#478	2
C₂×C₄×D₈×C₃₇	2
C₃₇×#2566	2
C₃₇×#2093	2
C₃₇×#2092	2
C₃₇×#1972	2
C₃₇×#1847	2
C₃₇×#1770	2
#28536	37

Subgroups — C37×#28536 #22649789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28536 #22649789