Subgroups — C₃₇×#28527 #22649780

Order 9472.

1030 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₇×#474	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2542	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2093	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1970	C₂	1	2	yes	maximal
7	C₃₇×#1845	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊Q₈×C₃₇	—	2	4	no	—
12	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#500	—	2	4	no	—
14	C₃₇×#474	—	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	4	yes	—
19	C₂×#107×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×#102×C₃₇	C₂²	1	2	yes	—
22	C₂×#102×C₃₇	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
25	C₃₇×#399	—	2	4	no	—
26	C₃₇×#390	C₂²	1	4	yes	—
27	C₃₇×#384	C₂²	1	4	yes	—
28	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₃₇	—	2	2	no	—
31	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₃₇	—	2	2	no	—
33	C₂×D₈×C₃₇	—	2	2	no	—
34	C₂×D₈×C₃₇	—	2	2	no	—
35	C₂×D₈×C₃₇	Q₈	1	4	yes	complemented
36	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₃₇	—	2	4	no	—
40	C₄⋊Q₈×C₃₇	—	2	4	no	—
41	C₂²⋊Q₈×C₃₇	—	4	16	no	—
42	C₄×D₄×C₃₇	—	2	4	no	—
43	C₄×D₄×C₃₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	8	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	8	no	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
51	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	#107×C₃₇	—	2	2	no	—
54	#107×C₃₇	—	2	8	no	—
55	#107×C₃₇	—	2	4	no	—
56	#107×C₃₇	—	2	4	no	—
57	C₄⋊C₈×C₃₇	D₄	1	4	yes	—
58	#102×C₃₇	—	2	4	no	—
59	#102×C₃₇	—	2	8	no	—
60	C₄×C₈×C₃₇	—	2	4	no	—
61	C₄×C₈×C₃₇	D₄	1	4	yes	—
62	#95×C₃₇	D₄	1	2	yes	—
63	C₂⁴×C₃₇	—	2	2	no	—
64	C₂×Q₈×C₃₇	—	4	4	no	—
65	C₂×Q₈×C₃₇	—	4	8	no	—
66	C₂×D₄×C₃₇	—	2	2	no	—
67	C₂×D₄×C₃₇	—	2	2	no	—
68	C₂×D₄×C₃₇	—	2	2	no	—
69	C₂×D₄×C₃₇	—	2	2	no	—
70	C₂×D₄×C₃₇	—	2	2	no	—
71	C₂×D₄×C₃₇	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₃₇	—	2	4	no	—
73	C₂²×C₄×C₃₇	—	2	2	no	—
74	C₂²×C₄×C₃₇	—	4	4	no	—
75	C₂²×C₄×C₃₇	—	4	8	no	—
76	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
80	D₈×C₃₇	—	2	8	no	—
81	C₂×C₈×C₃₇	—	2	2	no	—
82	C₂×C₈×C₃₇	—	2	1	no	—
83	C₂×C₈×C₃₇	—	2	1	no	—
84	C₂×C₈×C₃₇	—	2	2	no	—
85	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₃₇	#41	1	2	yes	—
90	C₄⋊C₄×C₃₇	—	2	2	no	—
91	C₄⋊C₄×C₃₇	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	2	2	no	—
93	C₄⋊C₄×C₃₇	—	2	4	no	—
94	C₄⋊C₄×C₃₇	—	4	4	no	—
95	C₄⋊C₄×C₃₇	—	4	16	no	—
96	C₄⋊C₄×C₃₇	D₈	1	8	yes	complemented
97	C₂²⋊C₄×C₃₇	—	4	4	no	—
98	C₂²⋊C₄×C₃₇	—	4	16	no	—
99	C₄²×C₃₇	—	2	2	no	—
100	C₄²×C₃₇	—	2	2	no	—
101	C₄²×C₃₇	C₂×D₄	1	2	yes	—
102	C₄²×C₃₇	C₂×D₄	1	2	yes	—
103	C₂³×C₃₇	—	2	4	no	—
104	C₂³×C₃₇	—	4	2	no	—
105	C₂³×C₃₇	—	4	2	no	—
106	C₂³×C₃₇	—	4	4	no	—
107	C₂³×C₃₇	—	4	4	no	—
108	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₃₇	—	8	8	no	—
110	D₄×C₃₇	—	2	4	no	—
111	D₄×C₃₇	—	2	4	no	—
112	D₄×C₃₇	—	4	2	no	—
113	D₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	2	4	no	—
115	C₂×C₄×C₃₇	—	2	4	no	—
116	C₂×C₄×C₃₇	—	2	2	no	—
117	C₂×C₄×C₃₇	—	4	2	no	—
118	C₂×C₄×C₃₇	—	4	4	no	—
119	C₂×C₄×C₃₇	—	4	8	no	—
120	C₂×C₄×C₃₇	—	4	4	no	—
121	C₂×C₄×C₃₇	—	8	8	no	—
122	C₂×C₄×C₃₇	#127	1	2	yes	—
123	C₂×C₄×C₃₇	#127	1	2	yes	—
124	C₂×C₄×C₃₇	C₂×D₈	1	8	yes	—
125	C₂×C₄×C₃₇	C₂×D₈	1	4	yes	—
126	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈×C₃₇	—	2	2	no	—
133	C₈×C₃₇	—	2	4	no	—
134	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	complemented
135	#28527	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₃₇	—	4	8	no	—
137	C₂²×C₃₇	—	4	4	no	—
138	C₂²×C₃₇	—	8	4	no	—
139	C₂²×C₃₇	—	8	2	no	—
140	C₂²×C₃₇	—	8	2	no	—
141	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
142	C₂²×C₃₇	#547	1	1	yes	characteristic
143	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
144	C₂²×C₃₇	C₂²×D₈	1	2	yes	—
145	C₂²×C₃₇	#578	1	2	yes	—
146	C₄×C₃₇	—	2	8	no	—
147	C₄×C₃₇	—	2	2	no	—
148	C₄×C₃₇	—	8	2	no	—
149	C₄×C₃₇	—	8	4	no	—
150	C₄×C₃₇	#493	1	4	yes	—
151	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₄×C₃₇	#536	1	1	yes	characteristic
153	C₄×C₃₇	#547	1	1	yes	characteristic
154	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
155	C₂×#500	C₂×C₃₇	1	1	yes	characteristic, complemented
156	C₂×#474	C₂×C₃₇	1	2	yes	complemented
157	C₂×C₄×D₈	C₂×C₃₇	1	1	yes	characteristic
158	#2542	C₂×C₃₇	1	2	yes	complemented
159	#2093	C₂×C₃₇	1	4	yes	complemented
160	#1970	C₂×C₃₇	1	2	yes	—
161	#1845	C₂×C₃₇	1	2	yes	complemented
162	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic, complemented
163	C₂×C₃₇	—	8	4	no	—
164	C₂×C₃₇	#2542	1	1	yes	characteristic
165	C₂×C₃₇	C₂×#493	1	1	yes	characteristic
166	C₂×C₃₇	#3156	1	1	yes	characteristic
167	C₂×C₃₇	#3334	1	2	yes	—
168	C₂×C₃₇	Q₈×D₈	1	2	yes	—
169	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
170	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
171	C₂×C₂²⋊Q₈	—	2	4	no	—
172	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
173	#500	—	2	4	no	—
174	#474	—	2	8	no	—
175	C₄×D₈	—	2	4	no	—
176	C₂×#107	C₂²×C₃₇	1	2	yes	—
177	C₂×#107	C₂²×C₃₇	1	4	yes	—
178	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
179	C₂×#107	C₂²×C₃₇	1	2	yes	—
180	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
181	C₂×#102	C₂²×C₃₇	1	2	yes	—
182	C₂×#102	C₂²×C₃₇	1	4	yes	—
183	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
185	#399	—	2	4	no	—
186	#390	C₂²×C₃₇	1	4	yes	—
187	#384	C₂²×C₃₇	1	4	yes	—
188	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
189	#336	C₂²×C₃₇	1	2	yes	—
190	C₃₇	#28527	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²×Q₈	—	2	2	no	—
192	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
193	C₂³×C₄	—	2	2	no	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	Q₈×C₃₇	1	4	yes	complemented
197	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
199	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	4	no	—
201	C₄⋊Q₈	—	2	4	no	—
202	C₂²⋊Q₈	—	4	16	no	—
203	C₄×D₄	—	2	4	no	—
204	C₄×D₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	8	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	2	8	no	—
212	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
213	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
214	#107	—	2	2	no	—
215	#107	—	2	8	no	—
216	#107	—	2	4	no	—
217	#107	—	2	4	no	—
218	C₄⋊C₈	D₄×C₃₇	1	4	yes	—
219	#102	—	2	4	no	—
220	#102	—	2	8	no	—
221	C₄×C₈	—	2	4	no	—
222	C₄×C₈	D₄×C₃₇	1	4	yes	—
223	#95	D₄×C₃₇	1	2	yes	—
224	C₂⁴	—	2	2	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂×Q₈	—	4	8	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	C₂×Q₈×C₃₇	1	4	yes	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	8	no	—
237	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
239	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
240	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
241	D₈	—	2	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
248	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
249	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
250	C₂×C₈	#41×C₃₇	1	2	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	16	no	—
257	C₄⋊C₄	D₈×C₃₇	1	8	yes	complemented
258	C₂²⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	—	4	16	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	C₂×D₄×C₃₇	1	2	yes	—
263	C₄²	C₂×D₄×C₃₇	1	2	yes	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	4	no	—
268	C₂³	—	4	4	no	—
269	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
270	Q₈	—	8	8	no	—
271	D₄	—	2	4	no	—
272	D₄	—	2	4	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	8	8	no	—
283	C₂×C₄	#127×C₃₇	1	2	yes	—
284	C₂×C₄	#127×C₃₇	1	2	yes	—
285	C₂×C₄	C₂×D₈×C₃₇	1	8	yes	—
286	C₂×C₄	C₂×D₈×C₃₇	1	4	yes	—
287	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
289	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
290	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
291	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
292	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	2	4	no	—
295	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	complemented
296	C₂²	—	4	8	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
302	C₂²	C₃₇×#547	1	1	yes	characteristic
303	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
304	C₂²	C₂²×D₈×C₃₇	1	2	yes	—
305	C₂²	C₃₇×#578	1	2	yes	—
306	C₄	—	2	2	no	—
307	C₄	—	2	8	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	4	no	—
310	C₄	C₃₇×#493	1	4	yes	—
311	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
312	C₄	C₃₇×#536	1	1	yes	characteristic
313	C₄	C₃₇×#547	1	1	yes	characteristic
314	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
315	C₂	—	8	4	no	—
316	C₂	C₃₇×#2542	1	1	yes	characteristic
317	C₂	C₂×C₃₇×#493	1	1	yes	characteristic
318	C₂	C₃₇×#3156	1	1	yes	characteristic
319	C₂	C₃₇×#3334	1	2	yes	—
320	C₂	Q₈×D₈×C₃₇	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×#500	C₂	semidirect
2. C₂×C₃₇×#474	C₂	semidirect
4. C₃₇×#2542	C₂	semidirect
5. C₃₇×#2093	C₂	semidirect
7. C₃₇×#1845	C₂	semidirect
8. C₈×C₄⋊C₄×C₃₇	C₂	semidirect
35. C₂×D₈×C₃₇	Q₈	semidirect
96. C₄⋊C₄×C₃₇	D₈	semidirect
134. C₈×C₃₇	C₂²⋊Q₈	semidirect
135. #28527	C₃₇	direct (N × H)
155. C₂×#500	C₂×C₃₇	semidirect
156. C₂×#474	C₂×C₃₇	semidirect
158. #2542	C₂×C₃₇	semidirect
159. #2093	C₂×C₃₇	semidirect
161. #1845	C₂×C₃₇	semidirect
162. C₈×C₄⋊C₄	C₂×C₃₇	semidirect
190. C₃₇	#28527	direct (N × H)
196. C₂×D₈	Q₈×C₃₇	semidirect
257. C₄⋊C₄	D₈×C₃₇	semidirect
295. C₈	C₂²⋊Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#500	2
C₂×C₃₇×#474	2
C₂×C₄×D₈×C₃₇	2
C₃₇×#2542	2
C₃₇×#2093	2
C₃₇×#1970	2
C₃₇×#1845	2
C₈×C₄⋊C₄×C₃₇	2
#28527	37

Subgroups — C37×#28527 #22649780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28527 #22649780