Subgroups — C₃₇×#28721 #22649974

Order 9472.

794 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#488	C₂	1	2	yes	maximal
2	C₃₇×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#124×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2074	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2073	C₂	1	2	yes	maximal
6	C₃₇×#1991	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1803	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1802	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#125×C₃₇	C₂²	1	2	yes	—
11	C₂×#124×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
14	C₃₇×#488	—	2	8	no	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×#103×C₃₇	C₂²	1	4	yes	—
17	C₂×#102×C₃₇	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
19	C₃₇×#391	C₂²	1	1	yes	characteristic
20	C₃₇×#390	C₂²	1	1	yes	characteristic
21	C₃₇×#389	—	2	2	no	—
22	C₃₇×#389	—	2	2	no	—
23	C₃₇×#388	—	2	2	no	—
24	C₃₇×#384	C₂²	1	2	yes	—
25	C₃₇×#382	C₂²	1	1	yes	characteristic
26	C₃₇×#382	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₃₇	—	2	2	no	—
28	C₄³×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#340	C₂²	1	4	yes	—
30	C₃₇×#340	C₂²	1	4	yes	—
31	C₃₇×#336	C₂²	1	2	yes	—
32	C₂²×Q₈×C₃₇	C₂³	1	1	yes	characteristic
33	C₂²×D₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₃₇	—	2	1	no	—
35	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
36	#125×C₃₇	—	2	4	no	—
37	#125×C₃₇	—	2	4	no	—
38	#124×C₃₇	—	2	2	no	—
39	#124×C₃₇	D₄	1	4	yes	—
40	C₄×Q₈×C₃₇	—	2	4	no	—
41	C₄×D₄×C₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₃₇	—	2	1	no	—
52	C₂×C₄²×C₃₇	—	2	1	no	—
53	C₂×C₄²×C₃₇	D₄	1	2	yes	—
54	C₂×C₄²×C₃₇	D₄	1	2	yes	—
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	C₄⋊C₈×C₃₇	—	2	4	no	—
59	#103×C₃₇	—	2	8	no	—
60	#102×C₃₇	—	2	8	no	—
61	C₄×C₈×C₃₇	—	2	4	no	—
62	#95×C₃₇	—	2	2	no	—
63	#95×C₃₇	—	2	2	no	—
64	#95×C₃₇	—	2	4	no	—
65	C₂⁴×C₃₇	—	2	1	no	—
66	C₂×Q₈×C₃₇	—	2	1	no	—
67	C₂×Q₈×C₃₇	—	2	2	no	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
70	C₂×D₄×C₃₇	—	2	2	no	—
71	C₂×D₄×C₃₇	—	2	1	no	—
72	C₂×D₄×C₃₇	—	2	2	no	—
73	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₇	—	2	1	no	—
75	C₂²×C₄×C₃₇	—	2	1	no	—
76	C₂²×C₄×C₃₇	—	2	1	no	—
77	C₂²×C₄×C₃₇	—	2	4	no	—
78	C₂²×C₄×C₃₇	—	4	2	no	—
79	C₂²×C₄×C₃₇	—	4	4	no	—
80	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₃₇	—	2	4	no	—
86	C₂×C₈×C₃₇	—	2	4	no	—
87	C₂×C₈×C₃₇	—	2	4	no	—
88	C₄⋊C₄×C₃₇	—	2	4	no	—
89	C₄⋊C₄×C₃₇	—	2	2	no	—
90	C₄⋊C₄×C₃₇	—	2	2	no	—
91	C₄⋊C₄×C₃₇	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	4	4	no	—
93	C₄⋊C₄×C₃₇	—	4	8	no	—
94	C₄⋊C₄×C₃₇	—	4	2	no	—
95	C₄⋊C₄×C₃₇	—	4	4	no	—
96	C₂²⋊C₄×C₃₇	—	2	8	no	—
97	C₂²⋊C₄×C₃₇	—	4	4	no	—
98	C₂²⋊C₄×C₃₇	—	4	2	no	—
99	C₄²×C₃₇	—	2	4	no	—
100	C₄²×C₃₇	—	2	2	no	—
101	C₄²×C₃₇	—	4	2	no	—
102	C₄²×C₃₇	—	4	2	no	—
103	C₄²×C₃₇	C₂×D₄	1	2	yes	—
104	C₄²×C₃₇	#41	1	2	yes	—
105	C₄²×C₃₇	#41	1	4	yes	—
106	C₂³×C₃₇	—	2	2	no	—
107	C₂³×C₃₇	—	4	1	no	—
108	C₂³×C₃₇	—	4	1	no	—
109	C₂³×C₃₇	—	4	2	no	—
110	C₂³×C₃₇	—	4	2	no	—
111	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₃₇	—	2	4	no	—
113	Q₈×C₃₇	—	4	2	no	—
114	D₄×C₃₇	—	2	4	no	—
115	D₄×C₃₇	—	4	2	no	—
116	C₂×C₄×C₃₇	—	2	2	no	—
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	—	4	1	no	—
124	C₂×C₄×C₃₇	—	4	1	no	—
125	C₂×C₄×C₃₇	—	4	4	no	—
126	C₂×C₄×C₃₇	—	4	1	no	—
127	C₂×C₄×C₃₇	—	4	1	no	—
128	C₂×C₄×C₃₇	—	4	4	no	—
129	C₂×C₄×C₃₇	—	4	2	no	—
130	C₂×C₄×C₃₇	—	4	2	no	—
131	C₂×C₄×C₃₇	—	4	4	no	—
132	C₂×C₄×C₃₇	—	8	4	no	—
133	C₂×C₄×C₃₇	#121	1	4	yes	—
134	C₂×C₄×C₃₇	#124	1	2	yes	—
135	C₂×C₄×C₃₇	#124	1	2	yes	—
136	C₂×C₄×C₃₇	#135	1	4	yes	—
137	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
141	C₈×C₃₇	—	4	4	no	—
142	#28721	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₃₇	—	4	4	no	—
144	C₂²×C₃₇	—	4	2	no	—
145	C₂²×C₃₇	—	8	1	no	—
146	C₂²×C₃₇	—	8	1	no	—
147	C₂²×C₃₇	—	8	2	no	—
148	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
149	C₂²×C₃₇	C₂×#124	1	1	yes	characteristic
150	C₂²×C₃₇	#536	1	1	yes	characteristic
151	C₂²×C₃₇	C₂×#135	1	2	yes	—
152	C₂²×C₃₇	#575	1	2	yes	—
153	C₄×C₃₇	—	2	2	no	—
154	C₄×C₃₇	—	2	4	no	—
155	C₄×C₃₇	—	4	2	no	—
156	C₄×C₃₇	—	4	2	no	—
157	C₄×C₃₇	—	8	1	no	—
158	C₄×C₃₇	—	8	1	no	—
159	C₄×C₃₇	—	8	4	no	—
160	C₄×C₃₇	#488	1	4	yes	complemented
161	C₄×C₃₇	#547	1	2	yes	—
162	C₄×C₃₇	#556	1	2	yes	—
163	C₂×#488	C₂×C₃₇	1	2	yes	—
164	#2859	C₂×C₃₇	1	1	yes	characteristic, complemented
165	C₄×#124	C₂×C₃₇	1	1	yes	characteristic
166	#2074	C₂×C₃₇	1	2	yes	complemented
167	#2073	C₂×C₃₇	1	2	yes	—
168	#1991	C₂×C₃₇	1	2	yes	complemented
169	#1803	C₂×C₃₇	1	2	yes	complemented
170	#1802	C₂×C₃₇	1	2	yes	—
171	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
172	C₂×C₃₇	—	8	2	no	—
173	C₂×C₃₇	#2547	1	1	yes	characteristic
174	C₂×C₃₇	#3035	1	2	yes	—
175	C₂×C₃₇	C₂×#488	1	1	yes	characteristic
176	C₂×C₃₇	#3142	1	1	yes	characteristic
177	C₂×C₃₇	#3233	1	2	yes	—
178	C₂×#125	C₂²×C₃₇	1	2	yes	—
179	C₂×#124	C₂²×C₃₇	1	1	yes	characteristic
180	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
181	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
182	#488	—	2	8	no	—
183	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
184	C₂×#103	C₂²×C₃₇	1	4	yes	—
185	C₂×#102	C₂²×C₃₇	1	4	yes	—
186	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
187	#391	C₂²×C₃₇	1	1	yes	characteristic
188	#390	C₂²×C₃₇	1	1	yes	characteristic
189	#389	—	2	2	no	—
190	#389	—	2	2	no	—
191	#388	—	2	2	no	—
192	#384	C₂²×C₃₇	1	2	yes	—
193	#382	C₂²×C₃₇	1	2	yes	—
194	#382	C₂²×C₃₇	1	1	yes	characteristic
195	C₄×C₂²⋊C₄	—	2	2	no	—
196	C₄³	C₂²×C₃₇	1	1	yes	characteristic
197	#340	C₂²×C₃₇	1	4	yes	—
198	#340	C₂²×C₃₇	1	4	yes	—
199	#336	C₂²×C₃₇	1	2	yes	—
200	C₃₇	#28721	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²×Q₈	C₂³×C₃₇	1	1	yes	characteristic
202	C₂²×D₄	C₂³×C₃₇	1	1	yes	characteristic
203	C₂³×C₄	—	2	1	no	—
204	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
205	#125	—	2	4	no	—
206	#125	—	2	4	no	—
207	#124	—	2	2	no	—
208	#124	D₄×C₃₇	1	4	yes	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×D₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₃₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂³×C₃₇	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
218	C₂×C₂²⋊C₄	—	2	2	no	—
219	C₂×C₂²⋊C₄	—	2	1	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	D₄×C₃₇	1	2	yes	—
223	C₂×C₄²	D₄×C₃₇	1	2	yes	—
224	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
225	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
227	C₄⋊C₈	—	2	4	no	—
228	#103	—	2	8	no	—
229	#102	—	2	8	no	—
230	C₄×C₈	—	2	4	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	4	no	—
234	C₂⁴	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
253	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	8	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	2	8	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×D₄×C₃₇	1	2	yes	—
273	C₄²	#41×C₃₇	1	2	yes	—
274	C₄²	#41×C₃₇	1	4	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	4	1	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	Q₈	—	4	2	no	—
283	D₄	—	2	4	no	—
284	D₄	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	#121×C₃₇	1	4	yes	—
303	C₂×C₄	#124×C₃₇	1	2	yes	—
304	C₂×C₄	#124×C₃₇	1	2	yes	—
305	C₂×C₄	#135×C₃₇	1	4	yes	—
306	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
308	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
309	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
310	C₈	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
317	C₂²	C₂×#124×C₃₇	1	1	yes	characteristic
318	C₂²	C₃₇×#536	1	1	yes	characteristic
319	C₂²	C₂×#135×C₃₇	1	2	yes	—
320	C₂²	C₃₇×#575	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	4	no	—
328	C₄	C₃₇×#488	1	4	yes	complemented
329	C₄	C₃₇×#547	1	2	yes	—
330	C₄	C₃₇×#556	1	2	yes	—
331	C₂	—	8	2	no	—
332	C₂	C₃₇×#2547	1	1	yes	characteristic
333	C₂	C₃₇×#3035	1	2	yes	—
334	C₂	C₂×C₃₇×#488	1	1	yes	characteristic
335	C₂	C₃₇×#3142	1	1	yes	characteristic
336	C₂	C₃₇×#3233	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#2859	C₂	semidirect
4. C₃₇×#2074	C₂	semidirect
6. C₃₇×#1991	C₂	semidirect
7. C₃₇×#1803	C₂	semidirect
9. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
142. #28721	C₃₇	direct (N × H)
160. C₄×C₃₇	#488	semidirect
164. #2859	C₂×C₃₇	semidirect
166. #2074	C₂×C₃₇	semidirect
168. #1991	C₂×C₃₇	semidirect
169. #1803	C₂×C₃₇	semidirect
171. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
200. C₃₇	#28721	direct (N × H)
328. C₄	C₃₇×#488	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#488	2
C₃₇×#2859	2
C₄×#124×C₃₇	2
C₃₇×#2074	2
C₃₇×#2073	2
C₃₇×#1991	2
C₃₇×#1803	2
C₃₇×#1802	2
C₄×C₄⋊C₈×C₃₇	2
#28721	37

Subgroups — C37×#28721 #22649974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28721 #22649974