Subgroups — C₃₇×#28528 #22649781

Order 9472.

918 conjugacy classes of subgroups.

The subgroup lattice has 318 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×#476	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2563	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2092	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1970	C₂	1	2	yes	maximal
7	C₃₇×#1846	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×C₄×D₄×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#500	—	2	4	no	—
13	C₃₇×#476	—	2	8	no	—
14	C₄×D₈×C₃₇	—	2	4	no	—
15	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×#107×C₃₇	C₂²	1	2	yes	—
17	C₂×#107×C₃₇	C₂²	1	2	yes	—
18	C₂×#106×C₃₇	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×#102×C₃₇	C₂²	1	2	yes	—
21	C₂×#102×C₃₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₃₇×#393	—	2	4	no	—
25	C₃₇×#390	C₂²	1	4	yes	—
26	C₃₇×#382	C₂²	1	4	yes	—
27	C₃₇×#380	—	2	4	no	—
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	C₄×D₄×C₃₇	—	2	4	no	—
42	C₄×D₄×C₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
47	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₃₇	—	2	8	no	—
49	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
51	#107×C₃₇	—	2	2	no	—
52	#107×C₃₇	—	2	4	no	—
53	#107×C₃₇	—	2	4	no	—
54	#106×C₃₇	—	2	8	no	—
55	C₄⋊C₈×C₃₇	D₄	1	4	yes	—
56	#102×C₃₇	—	2	4	no	—
57	#102×C₃₇	—	2	8	no	—
58	C₄×C₈×C₃₇	—	2	4	no	—
59	C₄×C₈×C₃₇	D₄	1	4	yes	—
60	#95×C₃₇	—	2	8	no	—
61	#95×C₃₇	D₄	1	2	yes	—
62	C₂⁴×C₃₇	—	2	2	no	—
63	C₂×Q₈×C₃₇	—	4	4	no	—
64	C₂×Q₈×C₃₇	—	8	4	no	—
65	C₂×D₄×C₃₇	—	2	2	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	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₃₇	—	2	2	no	—
72	C₂²×C₄×C₃₇	—	2	4	no	—
73	C₂²×C₄×C₃₇	—	4	4	no	—
74	C₂²×C₄×C₃₇	—	4	8	no	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
79	D₈×C₃₇	—	2	8	no	—
80	C₂×C₈×C₃₇	—	2	2	no	—
81	C₂×C₈×C₃₇	—	2	1	no	—
82	C₂×C₈×C₃₇	—	2	1	no	—
83	C₂×C₈×C₃₇	—	2	2	no	—
84	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₃₇	#41	1	2	yes	—
89	C₄⋊C₄×C₃₇	—	2	4	no	—
90	C₄⋊C₄×C₃₇	—	2	4	no	—
91	C₄⋊C₄×C₃₇	—	2	2	no	—
92	C₄⋊C₄×C₃₇	—	2	2	no	—
93	C₄⋊C₄×C₃₇	—	2	4	no	—
94	C₄⋊C₄×C₃₇	—	4	4	no	—
95	C₂²⋊C₄×C₃₇	—	4	4	no	—
96	C₂²⋊C₄×C₃₇	—	8	8	no	—
97	C₄²×C₃₇	—	2	2	no	—
98	C₄²×C₃₇	—	2	2	no	—
99	C₄²×C₃₇	C₂×D₄	1	2	yes	—
100	C₄²×C₃₇	C₂×D₄	1	2	yes	—
101	C₂³×C₃₇	—	2	4	no	—
102	C₂³×C₃₇	—	4	2	no	—
103	C₂³×C₃₇	—	4	2	no	—
104	C₂³×C₃₇	—	4	4	no	—
105	C₂³×C₃₇	—	4	4	no	—
106	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₃₇	—	8	8	no	—
108	D₄×C₃₇	—	2	4	no	—
109	D₄×C₃₇	—	2	4	no	—
110	D₄×C₃₇	—	4	2	no	—
111	D₄×C₃₇	—	4	2	no	—
112	C₂×C₄×C₃₇	—	2	2	no	—
113	C₂×C₄×C₃₇	—	2	4	no	—
114	C₂×C₄×C₃₇	—	2	4	no	—
115	C₂×C₄×C₃₇	—	2	4	no	—
116	C₂×C₄×C₃₇	—	4	2	no	—
117	C₂×C₄×C₃₇	—	4	4	no	—
118	C₂×C₄×C₃₇	—	4	4	no	—
119	C₂×C₄×C₃₇	—	4	8	no	—
120	C₂×C₄×C₃₇	—	8	8	no	—
121	C₂×C₄×C₃₇	#127	1	2	yes	—
122	C₂×C₄×C₃₇	#127	1	2	yes	—
123	C₂×C₄×C₃₇	#135	1	4	yes	—
124	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
130	C₈×C₃₇	—	2	2	no	—
131	C₈×C₃₇	—	2	4	no	—
132	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
133	#28528	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₃₇	—	4	8	no	—
135	C₂²×C₃₇	—	4	4	no	—
136	C₂²×C₃₇	—	8	4	no	—
137	C₂²×C₃₇	—	8	2	no	—
138	C₂²×C₃₇	—	8	2	no	—
139	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
140	C₂²×C₃₇	#547	1	1	yes	characteristic
141	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
142	C₂²×C₃₇	C₂×#135	1	2	yes	—
143	C₂²×C₃₇	#576	1	2	yes	—
144	C₄×C₃₇	—	2	2	no	—
145	C₄×C₃₇	—	4	4	no	—
146	C₄×C₃₇	—	8	2	no	—
147	C₄×C₃₇	—	8	4	no	—
148	C₄×C₃₇	#495	1	4	yes	—
149	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
150	C₄×C₃₇	#536	1	1	yes	characteristic
151	C₄×C₃₇	#547	1	1	yes	characteristic
152	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
153	C₂×#500	C₂×C₃₇	1	1	yes	characteristic, complemented
154	C₂×#476	C₂×C₃₇	1	2	yes	complemented
155	C₂×C₄×D₈	C₂×C₃₇	1	1	yes	characteristic
156	#2563	C₂×C₃₇	1	2	yes	complemented
157	#2092	C₂×C₃₇	1	4	yes	complemented
158	#1970	C₂×C₃₇	1	2	yes	—
159	#1846	C₂×C₃₇	1	2	yes	complemented
160	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic, complemented
161	C₂×C₃₇	—	8	4	no	—
162	C₂×C₃₇	#2542	1	1	yes	characteristic
163	C₂×C₃₇	C₂×#495	1	1	yes	characteristic
164	C₂×C₃₇	#3155	1	1	yes	characteristic
165	C₂×C₃₇	#3341	1	2	yes	—
166	C₂×C₃₇	#3380	1	2	yes	—
167	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
168	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
169	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
170	#500	—	2	4	no	—
171	#476	—	2	8	no	—
172	C₄×D₈	—	2	4	no	—
173	C₂×#107	C₂²×C₃₇	1	2	yes	—
174	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
175	C₂×#107	C₂²×C₃₇	1	2	yes	—
176	C₂×#106	C₂²×C₃₇	1	4	yes	—
177	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
178	C₂×#102	C₂²×C₃₇	1	2	yes	—
179	C₂×#102	C₂²×C₃₇	1	4	yes	—
180	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
182	#393	—	2	4	no	—
183	#390	C₂²×C₃₇	1	4	yes	—
184	#382	C₂²×C₃₇	1	4	yes	—
185	#380	—	2	4	no	—
186	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
187	#336	C₂²×C₃₇	1	2	yes	—
188	C₃₇	#28528	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×Q₈	—	2	2	no	—
190	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	Q₈×C₃₇	1	4	yes	complemented
195	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
197	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	4	no	—
199	C₄⋊Q₈	—	2	4	no	—
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	2	8	no	—
208	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
209	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
210	#107	—	2	2	no	—
211	#107	—	2	4	no	—
212	#107	—	2	4	no	—
213	#106	—	2	8	no	—
214	C₄⋊C₈	D₄×C₃₇	1	4	yes	—
215	#102	—	2	4	no	—
216	#102	—	2	8	no	—
217	C₄×C₈	—	2	4	no	—
218	C₄×C₈	D₄×C₃₇	1	4	yes	—
219	#95	—	2	8	no	—
220	#95	D₄×C₃₇	1	2	yes	—
221	C₂⁴	—	2	2	no	—
222	C₂×Q₈	—	4	4	no	—
223	C₂×Q₈	—	8	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂×Q₈×C₃₇	1	4	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	4	8	no	—
234	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
235	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
237	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
238	D₈	—	2	8	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
246	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
247	C₂×C₈	#41×C₃₇	1	2	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	8	8	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	C₂×D₄×C₃₇	1	2	yes	—
259	C₄²	C₂×D₄×C₃₇	1	2	yes	—
260	C₂³	—	2	4	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	4	no	—
265	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
266	Q₈	—	8	8	no	—
267	D₄	—	2	4	no	—
268	D₄	—	2	4	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	—	8	8	no	—
280	C₂×C₄	#127×C₃₇	1	2	yes	—
281	C₂×C₄	#127×C₃₇	1	2	yes	—
282	C₂×C₄	#135×C₃₇	1	4	yes	—
283	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
285	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
287	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
288	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	4	no	—
291	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
292	C₂²	—	4	8	no	—
293	C₂²	—	4	4	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	4	no	—
297	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
298	C₂²	C₃₇×#547	1	1	yes	characteristic
299	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
300	C₂²	C₂×#135×C₃₇	1	2	yes	—
301	C₂²	C₃₇×#576	1	2	yes	—
302	C₄	—	2	2	no	—
303	C₄	—	4	4	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	4	no	—
306	C₄	C₃₇×#495	1	4	yes	—
307	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
308	C₄	C₃₇×#536	1	1	yes	characteristic
309	C₄	C₃₇×#547	1	1	yes	characteristic
310	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
311	C₂	—	8	4	no	—
312	C₂	C₃₇×#2542	1	1	yes	characteristic
313	C₂	C₂×C₃₇×#495	1	1	yes	characteristic
314	C₂	C₃₇×#3155	1	1	yes	characteristic
315	C₂	C₃₇×#3341	1	2	yes	—
316	C₂	C₃₇×#3380	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×#476	C₂	semidirect
4. C₃₇×#2563	C₂	semidirect
5. C₃₇×#2092	C₂	semidirect
7. C₃₇×#1846	C₂	semidirect
8. C₈×C₄⋊C₄×C₃₇	C₂	semidirect
35. C₂×D₈×C₃₇	Q₈	semidirect
133. #28528	C₃₇	direct (N × H)
153. C₂×#500	C₂×C₃₇	semidirect
154. C₂×#476	C₂×C₃₇	semidirect
156. #2563	C₂×C₃₇	semidirect
157. #2092	C₂×C₃₇	semidirect
159. #1846	C₂×C₃₇	semidirect
160. C₈×C₄⋊C₄	C₂×C₃₇	semidirect
188. C₃₇	#28528	direct (N × H)
194. C₂×D₈	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×#476	2
C₂×C₄×D₈×C₃₇	2
C₃₇×#2563	2
C₃₇×#2092	2
C₃₇×#1970	2
C₃₇×#1846	2
C₈×C₄⋊C₄×C₃₇	2
#28528	37

Subgroups — C37×#28528 #22649781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28528 #22649781