Subgroups — C₃₇×#28265 #22649518

Order 9472.

782 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×C₈×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2582	C₂	1	2	yes	maximal
4	C₃₇×#2068	C₂	1	4	yes	maximal
5	C₃₇×#1969	C₂	1	2	yes	maximal
6	C₃₇×#1942	C₂	1	2	yes	maximal
7	C₃₇×#1845	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1803	C₂	1	2	yes	maximal
9	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₈×Q₈×C₃₇	—	2	4	no	—
13	C₄×Q₁₆×C₃₇	—	2	4	no	—
14	C₂×#107×C₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×#107×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×#103×C₃₇	C₂²	1	2	yes	—
20	C₂×#103×C₃₇	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#398	—	2	4	no	—
24	C₃₇×#391	—	2	2	no	—
25	C₃₇×#391	C₂²	1	4	yes	—
26	C₃₇×#389	—	2	2	no	—
27	C₃₇×#389	C₂²	1	4	yes	—
28	C₃₇×#384	C₂²	1	2	yes	—
29	C₃₇×#340	C₂²	1	4	yes	—
30	C₂²×Q₈×C₃₇	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂×Q₁₆×C₃₇	—	2	2	no	—
33	C₂×Q₁₆×C₃₇	—	2	2	no	—
34	C₂×Q₁₆×C₃₇	Q₈	1	4	yes	complemented
35	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₄×Q₈×C₃₇	—	2	2	no	—
39	C₄×Q₈×C₃₇	—	2	2	no	—
40	C₄×Q₈×C₃₇	—	2	4	no	—
41	C₄×Q₈×C₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
50	C₂×C₄²×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	4	no	—
55	#107×C₃₇	—	4	2	no	—
56	C₄⋊C₈×C₃₇	—	2	4	no	—
57	C₄⋊C₈×C₃₇	—	2	2	no	—
58	#103×C₃₇	—	2	4	no	—
59	#103×C₃₇	—	4	4	no	—
60	C₄×C₈×C₃₇	—	2	4	no	—
61	C₄×C₈×C₃₇	D₄	1	4	yes	—
62	#95×C₃₇	—	2	4	no	—
63	#95×C₃₇	—	2	4	no	—
64	#95×C₃₇	—	2	4	no	—
65	C₂×Q₈×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	2	2	no	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	2	2	no	—
70	C₂×Q₈×C₃₇	—	2	2	no	—
71	C₂×Q₈×C₃₇	—	2	2	no	—
72	C₂×Q₈×C₃₇	Q₁₆	1	4	yes	complemented
73	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₃₇	—	2	2	no	—
75	C₂²×C₄×C₃₇	—	2	4	no	—
76	C₂²×C₄×C₃₇	—	2	2	no	—
77	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
82	Q₁₆×C₃₇	—	2	8	no	—
83	C₂×C₈×C₃₇	—	2	1	no	—
84	C₂×C₈×C₃₇	—	2	1	no	—
85	C₂×C₈×C₃₇	—	2	2	no	—
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	—	2	2	no	—
88	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₃₇	#41	1	2	yes	—
92	C₄⋊C₄×C₃₇	—	2	4	no	—
93	C₄⋊C₄×C₃₇	—	2	2	no	—
94	C₄⋊C₄×C₃₇	—	2	2	no	—
95	C₄⋊C₄×C₃₇	—	2	4	no	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	2	4	no	—
98	C₄⋊C₄×C₃₇	—	4	4	no	—
99	C₄⋊C₄×C₃₇	—	4	4	no	—
100	C₄⋊C₄×C₃₇	—	8	2	no	—
101	C₄⋊C₄×C₃₇	—	8	4	no	—
102	C₄²×C₃₇	—	2	2	no	—
103	C₄²×C₃₇	—	2	2	no	—
104	C₄²×C₃₇	—	4	4	no	—
105	C₄²×C₃₇	C₂×D₄	1	2	yes	—
106	C₄²×C₃₇	#41	1	2	yes	—
107	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₃₇	—	2	8	no	—
109	Q₈×C₃₇	—	2	4	no	—
110	Q₈×C₃₇	—	2	4	no	—
111	Q₈×C₃₇	—	4	2	no	—
112	Q₈×C₃₇	—	4	2	no	—
113	C₂×C₄×C₃₇	—	2	2	no	—
114	C₂×C₄×C₃₇	—	2	2	no	—
115	C₂×C₄×C₃₇	—	2	1	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	—	2	2	no	—
118	C₂×C₄×C₃₇	—	2	4	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	2	2	no	—
121	C₂×C₄×C₃₇	—	4	2	no	—
122	C₂×C₄×C₃₇	—	4	2	no	—
123	C₂×C₄×C₃₇	—	4	2	no	—
124	C₂×C₄×C₃₇	—	4	4	no	—
125	C₂×C₄×C₃₇	—	4	4	no	—
126	C₂×C₄×C₃₇	—	4	4	no	—
127	C₂×C₄×C₃₇	—	4	2	no	—
128	C₂×C₄×C₃₇	—	4	2	no	—
129	C₂×C₄×C₃₇	#121	1	2	yes	—
130	C₂×C₄×C₃₇	#121	1	2	yes	—
131	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
132	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
133	C₂×C₄×C₃₇	#135	1	2	yes	—
134	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
140	C₈×C₃₇	—	2	2	no	—
141	C₈×C₃₇	—	4	2	no	—
142	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
143	#28265	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
145	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
146	C₂²×C₃₇	#550	1	1	yes	characteristic
147	C₂²×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
148	C₂²×C₃₇	C₂×#135	1	1	yes	characteristic
149	C₂²×C₃₇	#576	1	1	yes	characteristic
150	C₂²×C₃₇	#578	1	1	yes	characteristic
151	C₄×C₃₇	—	2	4	no	—
152	C₄×C₃₇	—	2	2	no	—
153	C₄×C₃₇	—	4	4	no	—
154	C₄×C₃₇	—	4	2	no	—
155	C₄×C₃₇	—	8	4	no	—
156	C₄×C₃₇	—	8	2	no	—
157	C₄×C₃₇	#467	1	4	yes	—
158	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₄×C₃₇	#537	1	1	yes	characteristic
160	C₄×C₃₇	#554	1	1	yes	characteristic
161	C₄×C₃₇	#556	1	1	yes	characteristic
162	C₂×C₈×Q₈	C₂×C₃₇	1	1	yes	characteristic
163	C₂×C₄×Q₁₆	C₂×C₃₇	1	1	yes	characteristic
164	#2582	C₂×C₃₇	1	2	yes	—
165	#2068	C₂×C₃₇	1	4	yes	—
166	#1969	C₂×C₃₇	1	2	yes	—
167	#1942	C₂×C₃₇	1	2	yes	—
168	#1845	C₂×C₃₇	1	1	yes	characteristic
169	#1803	C₂×C₃₇	1	2	yes	—
170	C₂×C₃₇	#2538	1	1	yes	characteristic
171	C₂×C₃₇	C₂×#467	1	1	yes	characteristic
172	C₂×C₃₇	#3061	1	1	yes	characteristic
173	C₂×C₃₇	Q₈×Q₁₆	1	1	yes	characteristic
174	C₂×C₃₇	#3383	1	1	yes	characteristic
175	C₂×C₃₇	#3395	1	1	yes	characteristic
176	C₂×C₃₇	#3396	1	1	yes	characteristic
177	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
178	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
179	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
180	C₈×Q₈	—	2	4	no	—
181	C₄×Q₁₆	—	2	4	no	—
182	C₂×#107	C₂²×C₃₇	1	2	yes	—
183	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
184	C₂×#107	C₂²×C₃₇	1	2	yes	—
185	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
187	C₂×#103	C₂²×C₃₇	1	2	yes	—
188	C₂×#103	C₂²×C₃₇	1	4	yes	—
189	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
190	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
191	#398	—	2	4	no	—
192	#391	—	2	2	no	—
193	#391	C₂²×C₃₇	1	4	yes	—
194	#389	—	2	2	no	—
195	#389	C₂²×C₃₇	1	4	yes	—
196	#384	C₂²×C₃₇	1	2	yes	—
197	#340	C₂²×C₃₇	1	4	yes	—
198	C₃₇	#28265	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	D₄×C₃₇	1	1	yes	characteristic
200	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	Q₈×C₃₇	1	4	yes	complemented
204	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
205	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	4	no	—
215	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
221	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
222	#107	—	2	2	no	—
223	#107	—	2	4	no	—
224	#107	—	4	2	no	—
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	—	2	2	no	—
227	#103	—	2	4	no	—
228	#103	—	4	4	no	—
229	C₄×C₈	—	2	4	no	—
230	C₄×C₈	D₄×C₃₇	1	4	yes	—
231	#95	—	2	4	no	—
232	#95	—	2	4	no	—
233	#95	—	2	4	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	Q₁₆×C₃₇	1	4	yes	complemented
242	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
251	Q₁₆	—	2	8	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
259	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
260	C₂×C₈	#41×C₃₇	1	2	yes	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	8	2	no	—
270	C₄⋊C₄	—	8	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	4	no	—
274	C₄²	C₂×D₄×C₃₇	1	2	yes	—
275	C₄²	#41×C₃₇	1	2	yes	—
276	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	Q₈	—	2	4	no	—
279	Q₈	—	2	4	no	—
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	#121×C₃₇	1	2	yes	—
299	C₂×C₄	#121×C₃₇	1	2	yes	—
300	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
301	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
302	C₂×C₄	#135×C₃₇	1	2	yes	—
303	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
307	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
308	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
312	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
313	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
314	C₂²	C₃₇×#550	1	1	yes	characteristic
315	C₂²	C₂²×Q₁₆×C₃₇	1	1	yes	characteristic
316	C₂²	C₂×#135×C₃₇	1	1	yes	characteristic
317	C₂²	C₃₇×#576	1	1	yes	characteristic
318	C₂²	C₃₇×#578	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	8	4	no	—
324	C₄	—	8	2	no	—
325	C₄	C₃₇×#467	1	4	yes	—
326	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
327	C₄	C₃₇×#537	1	1	yes	characteristic
328	C₄	C₃₇×#554	1	1	yes	characteristic
329	C₄	C₃₇×#556	1	1	yes	characteristic
330	C₂	C₃₇×#2538	1	1	yes	characteristic
331	C₂	C₂×C₃₇×#467	1	1	yes	characteristic
332	C₂	C₃₇×#3061	1	1	yes	characteristic
333	C₂	Q₈×Q₁₆×C₃₇	1	1	yes	characteristic
334	C₂	C₃₇×#3383	1	1	yes	characteristic
335	C₂	C₃₇×#3395	1	1	yes	characteristic
336	C₂	C₃₇×#3396	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄×Q₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
34. C₂×Q₁₆×C₃₇	Q₈	semidirect
72. C₂×Q₈×C₃₇	Q₁₆	semidirect
143. #28265	C₃₇	direct (N × H)
198. C₃₇	#28265	direct (N × H)
203. C₂×Q₁₆	Q₈×C₃₇	semidirect
241. C₂×Q₈	Q₁₆×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₃₇	2
C₂×C₄×Q₁₆×C₃₇	2
C₃₇×#2582	2
C₃₇×#2068	2
C₃₇×#1969	2
C₃₇×#1942	2
C₃₇×#1845	2
C₃₇×#1803	2
#28265	37

Subgroups — C37×#28265 #22649518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28265 #22649518