Subgroups — C₃₇×#28752 #22650005

Order 9472.

882 conjugacy classes of subgroups.

The subgroup lattice has 336 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, complemented
2	C₃₇×#2858	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2042	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1982	C₂	1	2	yes	maximal
6	C₃₇×#1981	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1958	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#124×C₃₇	C₂²	1	2	yes	—
13	C₂×#124×C₃₇	C₂²	1	2	yes	—
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×#103×C₃₇	C₂²	1	4	yes	—
18	C₂×#102×C₃₇	C₂²	1	4	yes	—
19	C₂×#102×C₃₇	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
21	C₃₇×#390	—	2	2	no	—
22	C₃₇×#390	—	2	2	no	—
23	C₃₇×#389	C₂²	1	2	yes	—
24	C₃₇×#387	—	2	2	no	—
25	C₃₇×#382	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
27	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
28	C₄³×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
31	C₂²×D₄×C₃₇	—	2	1	no	—
32	C₂²×D₄×C₃₇	—	2	1	no	—
33	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
35	C₄⋊Q₈×C₃₇	—	2	4	no	—
36	#127×C₃₇	—	2	4	no	—
37	#125×C₃₇	—	2	2	no	—
38	#125×C₃₇	D₄	1	4	yes	complemented
39	#124×C₃₇	—	2	4	no	—
40	#124×C₃₇	—	2	4	no	—
41	#124×C₃₇	—	4	4	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
50	C₂×C₄²×C₃₇	—	2	2	no	—
51	C₂×C₄²×C₃₇	D₄	1	2	yes	—
52	C₂×C₄²×C₃₇	D₄	1	2	yes	—
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	C₄⋊C₈×C₃₇	—	2	4	no	—
57	#103×C₃₇	—	2	8	no	—
58	#103×C₃₇	—	4	4	no	—
59	#102×C₃₇	—	2	8	no	—
60	#102×C₃₇	—	4	4	no	—
61	C₄×C₈×C₃₇	—	2	4	no	—
62	#95×C₃₇	—	2	2	no	—
63	#95×C₃₇	—	2	2	no	—
64	#95×C₃₇	—	2	2	no	—
65	C₂⁴×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	2	2	no	—
68	C₂×Q₈×C₃₇	—	2	4	no	—
69	C₂×Q₈×C₃₇	—	2	4	no	—
70	C₂×D₄×C₃₇	—	2	2	no	—
71	C₂×D₄×C₃₇	—	2	2	no	—
72	C₂×D₄×C₃₇	—	2	4	no	—
73	C₂×D₄×C₃₇	—	2	4	no	—
74	C₂×D₄×C₃₇	—	4	4	no	—
75	C₂×D₄×C₃₇	—	4	2	no	—
76	C₂×D₄×C₃₇	—	4	2	no	—
77	C₂×D₄×C₃₇	—	4	4	no	—
78	C₂²×C₄×C₃₇	—	2	2	no	—
79	C₂²×C₄×C₃₇	—	2	2	no	—
80	C₂²×C₄×C₃₇	—	2	2	no	—
81	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₃₇	—	2	4	no	—
87	C₂×C₈×C₃₇	—	2	4	no	—
88	C₂×C₈×C₃₇	—	2	4	no	—
89	C₄⋊C₄×C₃₇	—	2	8	no	—
90	C₄⋊C₄×C₃₇	—	2	2	no	—
91	C₄⋊C₄×C₃₇	—	2	2	no	—
92	C₄⋊C₄×C₃₇	—	2	2	no	—
93	C₄⋊C₄×C₃₇	—	4	4	no	—
94	C₄⋊C₄×C₃₇	—	4	2	no	—
95	C₄⋊C₄×C₃₇	—	4	2	no	—
96	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
97	C₂²⋊C₄×C₃₇	—	4	8	no	—
98	C₂²⋊C₄×C₃₇	—	4	8	no	—
99	C₄²×C₃₇	—	2	4	no	—
100	C₄²×C₃₇	—	2	2	no	—
101	C₄²×C₃₇	—	4	2	no	—
102	C₄²×C₃₇	—	4	4	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	—	4	2	no	—
107	C₂³×C₃₇	—	4	2	no	—
108	C₂³×C₃₇	—	4	2	no	—
109	C₂³×C₃₇	—	4	4	no	—
110	C₂³×C₃₇	—	4	4	no	—
111	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₃₇	—	4	4	no	—
113	Q₈×C₃₇	—	4	4	no	—
114	D₄×C₃₇	—	4	4	no	—
115	D₄×C₃₇	—	4	4	no	—
116	D₄×C₃₇	—	8	4	no	—
117	D₄×C₃₇	—	8	4	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	4	no	—
122	C₂×C₄×C₃₇	—	2	2	no	—
123	C₂×C₄×C₃₇	—	2	2	no	—
124	C₂×C₄×C₃₇	—	2	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	#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	H(5,𝔽₂)	1	2	yes	—
141	C₈×C₃₇	—	4	4	no	—
142	#28752	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₃₇	—	8	2	no	—
144	C₂²×C₃₇	—	8	2	no	—
145	C₂²×C₃₇	—	8	4	no	—
146	C₂²×C₃₇	—	8	4	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	#535	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	4	no	—
157	C₄×C₃₇	—	8	2	no	—
158	C₄×C₃₇	—	8	2	no	—
159	C₄×C₃₇	#488	1	4	yes	complemented
160	C₄×C₃₇	#550	1	2	yes	—
161	C₄×C₃₇	#559	1	2	yes	—
162	C₂×#488	C₂×C₃₇	1	2	yes	complemented
163	#2858	C₂×C₃₇	1	1	yes	characteristic
164	C₄×#125	C₂×C₃₇	1	1	yes	characteristic, complemented
165	#2042	C₂×C₃₇	1	4	yes	complemented
166	#1982	C₂×C₃₇	1	2	yes	—
167	#1981	C₂×C₃₇	1	2	yes	complemented
168	#1958	C₂×C₃₇	1	2	yes	complemented
169	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
170	C₂×C₃₇	—	16	2	no	—
171	C₂×C₃₇	#2548	1	1	yes	characteristic
172	C₂×C₃₇	#3035	1	2	yes	—
173	C₂×C₃₇	C₂×#488	1	1	yes	characteristic
174	C₂×C₃₇	#3140	1	1	yes	characteristic
175	C₂×C₃₇	#3212	1	2	yes	—
176	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
177	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
178	C₂×#125	C₂²×C₃₇	1	1	yes	characteristic
179	C₂×#124	C₂²×C₃₇	1	2	yes	—
180	C₂×#124	C₂²×C₃₇	1	2	yes	—
181	#488	—	2	8	no	—
182	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
183	C₂×#103	C₂²×C₃₇	1	4	yes	—
184	C₂×#103	C₂²×C₃₇	1	4	yes	—
185	C₂×#102	C₂²×C₃₇	1	4	yes	—
186	C₂×#102	C₂²×C₃₇	1	4	yes	—
187	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
188	#390	—	2	2	no	—
189	#390	—	2	2	no	—
190	#389	C₂²×C₃₇	1	2	yes	—
191	#387	—	2	2	no	—
192	#382	C₂²×C₃₇	1	1	yes	characteristic
193	C₄×C₄⋊C₄	—	2	2	no	—
194	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
195	C₄³	C₂²×C₃₇	1	1	yes	characteristic
196	#336	C₂²×C₃₇	1	2	yes	—
197	C₃₇	#28752	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
199	C₂²×D₄	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
202	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
203	C₄⋊Q₈	—	2	4	no	—
204	#127	—	2	4	no	—
205	#125	—	2	2	no	—
206	#125	D₄×C₃₇	1	4	yes	complemented
207	#124	—	2	4	no	—
208	#124	—	2	4	no	—
209	#124	—	4	4	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
216	C₂×C₂²⋊C₄	—	2	4	no	—
217	C₂×C₂²⋊C₄	—	2	4	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	D₄×C₃₇	1	2	yes	—
220	C₂×C₄²	D₄×C₃₇	1	2	yes	—
221	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
224	C₄⋊C₈	—	2	4	no	—
225	#103	—	2	8	no	—
226	#103	—	4	4	no	—
227	#102	—	2	8	no	—
228	#102	—	4	4	no	—
229	C₄×C₈	—	2	4	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	C₂⁴	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	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	8	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
265	C₂²⋊C₄	—	4	8	no	—
266	C₂²⋊C₄	—	4	8	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	4	4	no	—
271	C₄²	C₂×D₄×C₃₇	1	2	yes	—
272	C₄²	#41×C₃₇	1	2	yes	—
273	C₄²	#41×C₃₇	1	4	yes	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	4	4	no	—
279	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
280	Q₈	—	4	4	no	—
281	Q₈	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	8	4	no	—
285	D₄	—	8	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	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	#121×C₃₇	1	4	yes	—
302	C₂×C₄	#124×C₃₇	1	2	yes	—
303	C₂×C₄	#124×C₃₇	1	2	yes	—
304	C₂×C₄	#135×C₃₇	1	4	yes	—
305	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
307	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
308	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
309	C₈	—	4	4	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
316	C₂²	C₂×#124×C₃₇	1	1	yes	characteristic
317	C₂²	C₃₇×#535	1	1	yes	characteristic
318	C₂²	C₂×#135×C₃₇	1	2	yes	—
319	C₂²	C₃₇×#575	1	2	yes	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	4	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₃₇×#488	1	4	yes	complemented
327	C₄	C₃₇×#550	1	2	yes	—
328	C₄	C₃₇×#559	1	2	yes	—
329	C₂	—	16	2	no	—
330	C₂	C₃₇×#2548	1	1	yes	characteristic
331	C₂	C₃₇×#3035	1	2	yes	—
332	C₂	C₂×C₃₇×#488	1	1	yes	characteristic
333	C₂	C₃₇×#3140	1	1	yes	characteristic
334	C₂	C₃₇×#3212	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂²
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
1. C₂×C₃₇×#488	C₂	semidirect
3. C₄×#125×C₃₇	C₂	semidirect
4. C₃₇×#2042	C₂	semidirect
6. C₃₇×#1981	C₂	semidirect
7. C₃₇×#1958	C₂	semidirect
8. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
38. #125×C₃₇	D₄	semidirect
142. #28752	C₃₇	direct (N × H)
159. C₄×C₃₇	#488	semidirect
162. C₂×#488	C₂×C₃₇	semidirect
164. C₄×#125	C₂×C₃₇	semidirect
165. #2042	C₂×C₃₇	semidirect
167. #1981	C₂×C₃₇	semidirect
168. #1958	C₂×C₃₇	semidirect
169. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
197. C₃₇	#28752	direct (N × H)
206. #125	D₄×C₃₇	semidirect
326. 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×#2858	2
C₄×#125×C₃₇	2
C₃₇×#2042	2
C₃₇×#1982	2
C₃₇×#1981	2
C₃₇×#1958	2
C₄×C₄⋊C₈×C₃₇	2
#28752	37

Subgroups — C37×#28752 #22650005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28752 #22650005