Subgroups — C₃₇×#28831 #22650084

Order 9472.

606 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	C₃₇×#2570	C₂	1	2	yes	maximal
2	C₃₇×#2085	C₂	1	2	yes	maximal
3	C₃₇×#2083	C₂	1	2	yes	maximal
4	C₃₇×#1949	C₂	1	2	yes	maximal
5	C₃₇×#1941	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#1941	C₂	1	1	yes	maximal, characteristic
7	C₃₇×#1846	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1846	C₂	1	1	yes	maximal, characteristic
9	C₃₇×#1844	C₂	1	1	yes	maximal, characteristic
10	C₄×#106×C₃₇	C₂	1	1	yes	maximal, characteristic
11	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#107×C₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₃₇	C₂²	1	2	yes	—
14	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×#106×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#400	—	2	2	no	—
24	C₃₇×#398	—	2	2	no	—
25	C₃₇×#389	C₂²	1	2	yes	—
26	C₃₇×#389	C₂²	1	2	yes	—
27	C₃₇×#387	—	2	2	no	—
28	C₃₇×#387	—	2	2	no	—
29	C₃₇×#384	C₂²	1	2	yes	—
30	C₃₇×#382	C₂²	1	2	yes	—
31	C₃₇×#382	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
34	C₃₇×#340	C₂²	1	2	yes	—
35	C₃₇×#340	C₂²	1	2	yes	—
36	C₃₇×#340	C₂²	1	2	yes	—
37	C₃₇×#336	C₂²	1	1	yes	characteristic
38	C₃₇×#336	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
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	2	yes	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₃₇	—	2	2	no	—
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
56	#107×C₃₇	—	4	2	no	—
57	#107×C₃₇	—	4	2	no	—
58	#106×C₃₇	—	2	2	no	—
59	#106×C₃₇	—	2	2	no	—
60	#106×C₃₇	—	2	2	no	—
61	#106×C₃₇	—	2	2	no	—
62	#106×C₃₇	—	4	2	no	—
63	C₄⋊C₈×C₃₇	Q₈	1	4	yes	—
64	C₄×C₈×C₃₇	—	2	2	no	—
65	C₄×C₈×C₃₇	—	2	2	no	—
66	C₄×C₈×C₃₇	Q₈	1	4	yes	—
67	#95×C₃₇	—	2	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	—	2	2	no	—
74	#95×C₃₇	D₄	1	2	yes	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	—	2	2	no	—
77	C₂²×C₄×C₃₇	—	2	2	no	—
78	C₂²×C₄×C₃₇	—	2	2	no	—
79	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₃₇	—	2	1	no	—
85	C₂×C₈×C₃₇	—	2	1	no	—
86	C₂×C₈×C₃₇	—	2	1	no	—
87	C₂×C₈×C₃₇	—	2	1	no	—
88	C₂×C₈×C₃₇	—	2	1	no	—
89	C₂×C₈×C₃₇	—	2	1	no	—
90	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₃₇	#41	1	2	yes	—
95	C₂×C₈×C₃₇	#41	1	2	yes	—
96	C₄⋊C₄×C₃₇	—	2	4	no	—
97	C₄⋊C₄×C₃₇	—	2	2	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	—	2	2	no	—
104	C₄⋊C₄×C₃₇	—	2	2	no	—
105	C₄⋊C₄×C₃₇	—	4	4	no	—
106	C₄⋊C₄×C₃₇	—	4	4	no	—
107	C₄⋊C₄×C₃₇	—	8	2	no	—
108	C₄⋊C₄×C₃₇	—	8	2	no	—
109	C₄²×C₃₇	—	2	1	no	—
110	C₄²×C₃₇	—	2	1	no	—
111	C₄²×C₃₇	—	2	1	no	—
112	C₄²×C₃₇	—	2	1	no	—
113	C₄²×C₃₇	—	8	2	no	—
114	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
115	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
116	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₇	—	2	2	no	—
118	C₂×C₄×C₃₇	—	2	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	—	4	2	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	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	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₃₇	#135	1	4	yes	—
139	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₈×C₃₇	—	2	2	no	—
146	C₈×C₃₇	—	2	2	no	—
147	C₈×C₃₇	—	2	2	no	—
148	C₈×C₃₇	#125	1	4	yes	—
149	#28831	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂²×C₃₇	#547	1	1	yes	characteristic
152	C₂²×C₃₇	#550	1	1	yes	characteristic
153	C₂²×C₃₇	C₂×#135	1	2	yes	—
154	C₂²×C₃₇	#577	1	2	yes	—
155	C₄×C₃₇	—	2	4	no	—
156	C₄×C₃₇	—	2	2	no	—
157	C₄×C₃₇	—	2	2	no	—
158	C₄×C₃₇	—	4	2	no	—
159	C₄×C₃₇	—	8	2	no	—
160	C₄×C₃₇	—	8	2	no	—
161	C₄×C₃₇	—	8	2	no	—
162	C₄×C₃₇	—	8	2	no	—
163	C₄×C₃₇	C₂×#125	1	1	yes	characteristic
164	C₄×C₃₇	#544	1	1	yes	characteristic
165	C₄×C₃₇	#554	1	1	yes	characteristic
166	C₄×C₃₇	#557	1	1	yes	characteristic
167	#2570	C₂×C₃₇	1	2	yes	—
168	#2085	C₂×C₃₇	1	2	yes	—
169	#2083	C₂×C₃₇	1	2	yes	—
170	#1949	C₂×C₃₇	1	2	yes	—
171	#1941	C₂×C₃₇	1	1	yes	characteristic
172	#1941	C₂×C₃₇	1	1	yes	characteristic
173	#1846	C₂×C₃₇	1	1	yes	characteristic
174	#1846	C₂×C₃₇	1	1	yes	characteristic
175	#1844	C₂×C₃₇	1	1	yes	characteristic
176	C₄×#106	C₂×C₃₇	1	1	yes	characteristic
177	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
178	C₂×C₃₇	#2552	1	1	yes	characteristic
179	C₂×C₃₇	#3160	1	1	yes	characteristic
180	C₂×C₃₇	#3163	1	1	yes	characteristic
181	C₂×C₃₇	#3337	1	2	yes	—
182	C₂×C₃₇	#3394	1	2	yes	—
183	C₂×#107	C₂²×C₃₇	1	2	yes	—
184	C₂×#107	C₂²×C₃₇	1	2	yes	—
185	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
186	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
187	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
188	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
189	C₂×#106	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×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
194	#400	—	2	2	no	—
195	#398	—	2	2	no	—
196	#389	C₂²×C₃₇	1	2	yes	—
197	#389	C₂²×C₃₇	1	2	yes	—
198	#387	—	2	2	no	—
199	#387	—	2	2	no	—
200	#384	C₂²×C₃₇	1	2	yes	—
201	#382	C₂²×C₃₇	1	2	yes	—
202	#382	C₂²×C₃₇	1	2	yes	—
203	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
204	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
205	#340	C₂²×C₃₇	1	2	yes	—
206	#340	C₂²×C₃₇	1	2	yes	—
207	#340	C₂²×C₃₇	1	2	yes	—
208	#336	C₂²×C₃₇	1	1	yes	characteristic
209	#336	C₂²×C₃₇	1	1	yes	characteristic
210	C₃₇	#28831	1	1	yes	characteristic, direct product factor, Sylow
211	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
228	#107	—	4	2	no	—
229	#107	—	4	2	no	—
230	#106	—	2	2	no	—
231	#106	—	2	2	no	—
232	#106	—	2	2	no	—
233	#106	—	2	2	no	—
234	#106	—	4	2	no	—
235	C₄⋊C₈	Q₈×C₃₇	1	4	yes	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	Q₈×C₃₇	1	4	yes	—
239	#95	—	2	2	no	—
240	#95	—	2	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	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
263	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
264	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
265	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
266	C₂×C₈	#41×C₃₇	1	2	yes	—
267	C₂×C₈	#41×C₃₇	1	2	yes	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	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	—	2	2	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	8	2	no	—
280	C₄⋊C₄	—	8	2	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	8	2	no	—
286	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
287	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
288	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	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	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	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	C₄⋊Q₈×C₃₇	1	2	yes	—
309	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
310	C₂×C₄	#135×C₃₇	1	4	yes	—
311	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
312	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
313	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
314	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
315	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
316	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	#125×C₃₇	1	4	yes	—
321	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
322	C₂²	C₃₇×#547	1	1	yes	characteristic
323	C₂²	C₃₇×#550	1	1	yes	characteristic
324	C₂²	C₂×#135×C₃₇	1	2	yes	—
325	C₂²	C₃₇×#577	1	2	yes	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	4	no	—
329	C₄	—	4	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	2	no	—
334	C₄	C₂×#125×C₃₇	1	1	yes	characteristic
335	C₄	C₃₇×#544	1	1	yes	characteristic
336	C₄	C₃₇×#554	1	1	yes	characteristic
337	C₄	C₃₇×#557	1	1	yes	characteristic
338	C₂	C₃₇×#2552	1	1	yes	characteristic
339	C₂	C₃₇×#3160	1	1	yes	characteristic
340	C₂	C₃₇×#3163	1	1	yes	characteristic
341	C₂	C₃₇×#3337	1	2	yes	—
342	C₂	C₃₇×#3394	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
149. #28831	C₃₇	direct (N × H)
210. C₃₇	#28831	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#2570	2
C₃₇×#2085	2
C₃₇×#2083	2
C₃₇×#1949	2
C₃₇×#1941	2
C₃₇×#1846	2
C₃₇×#1844	2
C₄×#106×C₃₇	2
C₈×C₄⋊C₄×C₃₇	2
#28831	37

Subgroups — C37×#28831 #22650084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28831 #22650084