Subgroups — C₃₇×#20537 #22641790

Order 9472.

894 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#431	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1771	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1770	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊C₄×C₃₇	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1750	C₂	1	4	yes	maximal
8	C₂×#117×C₃₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
10	C₃₇×#432	C₄	1	8	yes	complemented
11	C₃₇×#431	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×#97×C₃₇	C₂²	1	4	yes	—
17	C₂×#97×C₃₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₇	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#384	C₄	1	2	yes	—
22	C₃₇×#382	C₄	1	2	yes	—
23	C₃₇×#380	C₄	1	2	yes	—
24	C₃₇×#380	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
27	C₃₇×#336	C₂²	1	4	yes	—
28	C₃₇×#336	C₂²	1	4	yes	—
29	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
34	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
35	#117×C₃₇	—	4	4	no	—
36	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₃₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
48	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
49	C₂²⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
50	C₂²⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
51	#97×C₃₇	—	4	4	no	—
52	#97×C₃₇	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₃₇	—	4	4	no	—
54	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
56	#95×C₃₇	C₂×C₄	1	4	yes	—
57	#95×C₃₇	C₂×C₄	1	4	yes	—
58	C₂⁴×C₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₇	—	2	4	no	—
60	C₂²×C₄×C₃₇	—	2	2	no	—
61	C₂²×C₄×C₃₇	—	2	2	no	—
62	C₂²×C₄×C₃₇	—	4	2	no	—
63	C₂²×C₄×C₃₇	C₄²	1	4	yes	—
64	C₂²×C₄×C₃₇	C₄²	1	4	yes	—
65	C₂²×C₄×C₃₇	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₃₇	—	2	2	no	—
75	C₂×C₈×C₃₇	—	2	4	no	—
76	C₂×C₈×C₃₇	—	2	2	no	—
77	C₂×C₈×C₃₇	—	2	2	no	—
78	C₂×C₈×C₃₇	—	2	4	no	—
79	C₂×C₈×C₃₇	—	2	2	no	—
80	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
83	C₄⋊C₄×C₃₇	—	2	8	no	—
84	C₄⋊C₄×C₃₇	—	4	4	no	—
85	C₄⋊C₄×C₃₇	—	4	4	no	—
86	C₂²⋊C₄×C₃₇	—	4	4	no	—
87	C₄²×C₃₇	—	2	4	no	—
88	C₄²×C₃₇	—	2	4	no	—
89	C₄²×C₃₇	C₄²	1	4	yes	—
90	C₄²×C₃₇	C₄²	1	4	yes	—
91	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
95	C₂³×C₃₇	—	2	1	no	—
96	C₂³×C₃₇	—	2	1	no	—
97	C₂³×C₃₇	—	2	2	no	—
98	C₂³×C₃₇	—	2	2	no	—
99	C₂³×C₃₇	C₂×C₄²	1	2	yes	—
100	C₂³×C₃₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₇	—	2	1	no	—
102	C₂×C₄×C₃₇	—	2	1	no	—
103	C₂×C₄×C₃₇	—	2	2	no	—
104	C₂×C₄×C₃₇	—	2	2	no	—
105	C₂×C₄×C₃₇	—	2	2	no	—
106	C₂×C₄×C₃₇	—	2	4	no	—
107	C₂×C₄×C₃₇	—	2	4	no	—
108	C₂×C₄×C₃₇	—	2	8	no	—
109	C₂×C₄×C₃₇	—	2	8	no	—
110	C₂×C₄×C₃₇	—	2	8	no	—
111	C₂×C₄×C₃₇	—	2	8	no	—
112	C₂×C₄×C₃₇	—	4	4	no	—
113	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₃₇	#131	1	4	yes	—
119	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
122	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
123	C₈×C₃₇	—	2	4	no	—
124	C₈×C₃₇	—	2	8	no	—
125	C₈×C₃₇	—	2	4	no	—
126	#20537	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₃₇	—	2	2	no	—
128	C₂²×C₃₇	—	2	2	no	—
129	C₂²×C₃₇	—	2	2	no	—
130	C₂²×C₃₇	—	4	2	no	—
131	C₂²×C₃₇	—	4	2	no	—
132	C₂²×C₃₇	C₂²×C₄²	1	1	yes	characteristic
133	C₂²×C₃₇	#518	1	1	yes	characteristic
134	C₂²×C₃₇	#519	1	1	yes	characteristic
135	C₂²×C₃₇	C₂×#131	1	2	yes	—
136	C₂²×C₃₇	#568	1	2	yes	—
137	C₄×C₃₇	—	2	2	no	—
138	C₄×C₃₇	—	2	2	no	—
139	C₄×C₃₇	—	2	2	no	—
140	C₄×C₃₇	—	2	2	no	—
141	C₄×C₃₇	—	4	8	no	—
142	C₄×C₃₇	#405	1	4	yes	—
143	C₄×C₃₇	#520	1	2	yes	—
144	C₄×C₃₇	#520	1	2	yes	—
145	C₂×#432	C₂×C₃₇	1	1	yes	characteristic
146	C₂×#431	C₂×C₃₇	1	1	yes	characteristic
147	#2285	C₂×C₃₇	1	1	yes	characteristic
148	#1771	C₂×C₃₇	1	4	yes	complemented
149	#1770	C₂×C₃₇	1	2	yes	complemented
150	C₈×C₄⋊C₄	C₂×C₃₇	1	2	yes	complemented
151	#1750	C₂×C₃₇	1	4	yes	—
152	C₂×C₃₇	—	4	2	no	—
153	C₂×C₃₇	#2274	1	1	yes	characteristic
154	C₂×C₃₇	C₂×#405	1	1	yes	characteristic
155	C₂×C₃₇	#2873	1	1	yes	characteristic
156	C₂×C₃₇	#2981	1	2	yes	—
157	C₂×C₃₇	#2985	1	2	yes	—
158	C₂×#117	C₂²×C₃₇	1	2	yes	—
159	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
160	#432	C₄×C₃₇	1	8	yes	complemented
161	#431	C₄×C₃₇	1	8	yes	complemented
162	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
164	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
165	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
166	C₂×#97	C₂²×C₃₇	1	4	yes	—
167	C₂×#97	C₂²×C₃₇	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₃₇	1	4	yes	—
169	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
171	#384	C₄×C₃₇	1	2	yes	—
172	#382	C₄×C₃₇	1	2	yes	—
173	#380	C₄×C₃₇	1	2	yes	—
174	#380	C₄×C₃₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
177	#336	C₂²×C₃₇	1	4	yes	—
178	#336	C₂²×C₃₇	1	4	yes	—
179	C₃₇	#20537	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
181	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
184	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
185	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
186	#117	—	4	4	no	—
187	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
189	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
190	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
191	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
194	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
195	C₂×C₄²	C₂³×C₃₇	1	4	yes	—
196	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
198	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
199	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
200	C₂²⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
201	C₂²⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
202	#97	—	4	4	no	—
203	#97	C₂×C₄×C₃₇	1	8	yes	—
204	C₄×C₈	—	4	4	no	—
205	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
206	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
207	#95	C₂×C₄×C₃₇	1	4	yes	—
208	#95	C₂×C₄×C₃₇	1	4	yes	—
209	C₂⁴	C₂²×C₄×C₃₇	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₄²×C₃₇	1	4	yes	—
215	C₂²×C₄	C₄²×C₃₇	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₃₇	1	8	yes	—
217	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
224	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
232	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
233	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	C₄²×C₃₇	1	4	yes	—
241	C₄²	C₄²×C₃₇	1	4	yes	—
242	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
243	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
244	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
245	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	1	no	—
250	C₂³	C₂×C₄²×C₃₇	1	2	yes	—
251	C₂³	C₂³×C₄×C₃₇	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
265	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
266	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
267	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
268	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
269	C₂×C₄	#131×C₃₇	1	4	yes	—
270	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
272	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
273	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	2	8	no	—
276	C₈	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₂²×C₄²×C₃₇	1	1	yes	characteristic
283	C₂²	C₃₇×#518	1	1	yes	characteristic
284	C₂²	C₃₇×#519	1	1	yes	characteristic
285	C₂²	C₂×#131×C₃₇	1	2	yes	—
286	C₂²	C₃₇×#568	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	8	no	—
292	C₄	C₃₇×#405	1	4	yes	—
293	C₄	C₃₇×#520	1	2	yes	—
294	C₄	C₃₇×#520	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	C₃₇×#2274	1	1	yes	characteristic
297	C₂	C₂×C₃₇×#405	1	1	yes	characteristic
298	C₂	C₃₇×#2873	1	1	yes	characteristic
299	C₂	C₃₇×#2981	1	2	yes	—
300	C₂	C₃₇×#2985	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂³×C₄
prime radical	C₂⁴×C₃₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₇×#1771	C₂	semidirect
5. C₃₇×#1770	C₂	semidirect
6. C₈×C₄⋊C₄×C₃₇	C₂	semidirect
10. C₃₇×#432	C₄	semidirect
11. C₃₇×#431	C₄	semidirect
126. #20537	C₃₇	direct (N × H)
148. #1771	C₂×C₃₇	semidirect
149. #1770	C₂×C₃₇	semidirect
150. C₈×C₄⋊C₄	C₂×C₃₇	semidirect
160. #432	C₄×C₃₇	semidirect
161. #431	C₄×C₃₇	semidirect
179. C₃₇	#20537	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×#432	2
C₂×C₃₇×#431	2
C₃₇×#2285	2
C₃₇×#1771	2
C₃₇×#1770	2
C₈×C₄⋊C₄×C₃₇	2
C₃₇×#1750	2
#20537	37

Subgroups — C37×#20537 #22641790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#20537 #22641790