Subgroups — C₃₇×#28717 #22649970

Order 9472.

850 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×#487	C₂	1	2	yes	maximal
2	C₃₇×#2868	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2072	C₂	1	4	yes	maximal
5	C₃₇×#1990	C₂	1	2	yes	maximal
6	C₃₇×#1803	C₂	1	4	yes	maximal
7	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₃₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₃₇	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
13	C₃₇×#487	—	2	8	no	—
14	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
15	C₂×#103×C₃₇	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
17	C₃₇×#391	C₂²	1	2	yes	—
18	C₃₇×#389	—	2	1	no	—
19	C₃₇×#389	—	2	1	no	—
20	C₃₇×#384	—	2	1	no	—
21	C₃₇×#384	—	2	1	no	—
22	C₃₇×#384	C₂²	1	2	yes	—
23	C₃₇×#384	C₂²	1	2	yes	—
24	C₃₇×#384	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₃₇	—	2	1	no	—
26	C₄×C₄⋊C₄×C₃₇	—	2	1	no	—
27	C₄³×C₃₇	C₂²	1	1	yes	characteristic
28	C₃₇×#340	C₂²	1	8	yes	—
29	C₃₇×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₃₇	—	2	2	no	—
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
33	C₄⋊Q₈×C₃₇	—	2	4	no	—
34	C₄⋊Q₈×C₃₇	—	2	2	no	—
35	C₄⋊Q₈×C₃₇	—	2	4	no	—
36	C₄⋊Q₈×C₃₇	—	4	4	no	—
37	C₄⋊Q₈×C₃₇	—	4	4	no	—
38	C₄⋊Q₈×C₃₇	D₄	1	4	yes	—
39	C₄×Q₈×C₃₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₇	—	2	1	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	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₃₇	—	2	2	no	—
53	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₃₇	—	2	4	no	—
61	#103×C₃₇	—	2	16	no	—
62	C₄×C₈×C₃₇	—	2	4	no	—
63	#95×C₃₇	—	2	2	no	—
64	#95×C₃₇	—	2	2	no	—
65	C₂×Q₈×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	4	no	—
67	C₂×Q₈×C₃₇	—	2	4	no	—
68	C₂×Q₈×C₃₇	—	4	4	no	—
69	C₂×Q₈×C₃₇	—	4	4	no	—
70	C₂×Q₈×C₃₇	—	4	4	no	—
71	C₂×Q₈×C₃₇	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₃₇	—	2	2	no	—
73	C₂²×C₄×C₃₇	—	2	2	no	—
74	C₂²×C₄×C₃₇	—	2	4	no	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₃₇	—	2	4	no	—
83	C₂×C₈×C₃₇	—	2	4	no	—
84	C₂×C₈×C₃₇	—	2	4	no	—
85	C₄⋊C₄×C₃₇	—	2	4	no	—
86	C₄⋊C₄×C₃₇	—	2	4	no	—
87	C₄⋊C₄×C₃₇	—	2	4	no	—
88	C₄⋊C₄×C₃₇	—	2	4	no	—
89	C₄⋊C₄×C₃₇	—	2	4	no	—
90	C₄⋊C₄×C₃₇	—	4	4	no	—
91	C₄⋊C₄×C₃₇	—	4	4	no	—
92	C₄⋊C₄×C₃₇	—	4	4	no	—
93	C₄⋊C₄×C₃₇	—	4	4	no	—
94	C₄⋊C₄×C₃₇	—	4	4	no	—
95	C₄⋊C₄×C₃₇	—	4	4	no	—
96	C₄⋊C₄×C₃₇	—	4	2	no	—
97	C₄⋊C₄×C₃₇	—	4	2	no	—
98	C₄²×C₃₇	—	2	2	no	—
99	C₄²×C₃₇	—	2	2	no	—
100	C₄²×C₃₇	—	2	2	no	—
101	C₄²×C₃₇	—	4	4	no	—
102	C₄²×C₃₇	SD₁₆	1	4	yes	—
103	C₄²×C₃₇	Q₁₆	1	4	yes	—
104	C₄²×C₃₇	C₂×D₄	1	2	yes	—
105	C₄²×C₃₇	#41	1	2	yes	—
106	C₄²×C₃₇	#41	1	4	yes	—
107	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₃₇	—	2	8	no	—
109	Q₈×C₃₇	—	4	4	no	—
110	Q₈×C₃₇	—	8	8	no	—
111	C₂×C₄×C₃₇	—	2	1	no	—
112	C₂×C₄×C₃₇	—	2	1	no	—
113	C₂×C₄×C₃₇	—	2	1	no	—
114	C₂×C₄×C₃₇	—	2	1	no	—
115	C₂×C₄×C₃₇	—	2	1	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	—	2	4	no	—
118	C₂×C₄×C₃₇	—	4	2	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	4	4	no	—
121	C₂×C₄×C₃₇	—	4	2	no	—
122	C₂×C₄×C₃₇	—	4	4	no	—
123	C₂×C₄×C₃₇	—	4	4	no	—
124	C₂×C₄×C₃₇	—	4	2	no	—
125	C₂×C₄×C₃₇	—	4	2	no	—
126	C₂×C₄×C₃₇	#121	1	2	yes	—
127	C₂×C₄×C₃₇	#121	1	2	yes	—
128	C₂×C₄×C₃₇	#124	1	2	yes	—
129	C₂×C₄×C₃₇	#124	1	2	yes	—
130	C₂×C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
135	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₄×C₃₇	#137	1	2	yes	—
138	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
142	C₈×C₃₇	—	4	4	no	—
143	#28717	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
145	C₂²×C₃₇	C₂×#124	1	1	yes	characteristic
146	C₂²×C₃₇	#536	1	1	yes	characteristic
147	C₂²×C₃₇	C₂²×SD₁₆	1	1	yes	characteristic
148	C₂²×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
149	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂²×C₃₇	C₂×#137	1	1	yes	characteristic
151	C₄×C₃₇	—	2	2	no	—
152	C₄×C₃₇	—	2	2	no	—
153	C₄×C₃₇	—	2	2	no	—
154	C₄×C₃₇	—	4	4	no	—
155	C₄×C₃₇	—	8	2	no	—
156	C₄×C₃₇	—	8	4	no	—
157	C₄×C₃₇	#460	1	4	yes	—
158	C₄×C₃₇	#462	1	4	yes	—
159	C₄×C₃₇	#487	1	4	yes	complemented
160	C₄×C₃₇	#547	1	2	yes	—
161	C₄×C₃₇	#556	1	2	yes	—
162	C₂×#487	C₂×C₃₇	1	2	yes	—
163	#2868	C₂×C₃₇	1	1	yes	characteristic
164	C₄×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic
165	#2072	C₂×C₃₇	1	4	yes	—
166	#1990	C₂×C₃₇	1	2	yes	—
167	#1803	C₂×C₃₇	1	4	yes	—
168	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic
169	C₂×C₃₇	#2547	1	1	yes	characteristic
170	C₂×C₃₇	C₂×#460	1	1	yes	characteristic
171	C₂×C₃₇	C₂×#462	1	1	yes	characteristic
172	C₂×C₃₇	C₂×#487	1	1	yes	characteristic
173	C₂×C₃₇	#3141	1	1	yes	characteristic
174	C₂×C₃₇	#3229	1	1	yes	characteristic
175	C₂×C₃₇	#3230	1	1	yes	characteristic
176	C₂×C₄⋊Q₈	—	2	1	no	—
177	C₂×C₄⋊Q₈	—	2	1	no	—
178	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
179	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
180	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
181	#487	—	2	8	no	—
182	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
183	C₂×#103	C₂²×C₃₇	1	8	yes	—
184	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
185	#391	C₂²×C₃₇	1	2	yes	—
186	#389	—	2	1	no	—
187	#389	—	2	1	no	—
188	#384	—	2	1	no	—
189	#384	—	2	1	no	—
190	#384	C₂²×C₃₇	1	1	yes	characteristic
191	#384	C₂²×C₃₇	1	2	yes	—
192	#384	C₂²×C₃₇	1	2	yes	—
193	C₄×C₄⋊C₄	—	2	1	no	—
194	C₄×C₄⋊C₄	—	2	1	no	—
195	C₄³	C₂²×C₃₇	1	1	yes	characteristic
196	#340	C₂²×C₃₇	1	8	yes	—
197	#336	C₂²×C₃₇	1	2	yes	—
198	C₃₇	#28717	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	—	2	2	no	—
200	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
201	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
202	C₄⋊Q₈	—	2	4	no	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄⋊Q₈	—	4	4	no	—
206	C₄⋊Q₈	—	4	4	no	—
207	C₄⋊Q₈	D₄×C₃₇	1	4	yes	—
208	C₄×Q₈	—	2	8	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	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	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
223	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
224	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
225	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
228	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
229	C₄⋊C₈	—	2	4	no	—
230	#103	—	2	16	no	—
231	C₄×C₈	—	2	4	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	4	4	no	—
238	C₂×Q₈	—	4	4	no	—
239	C₂×Q₈	—	4	4	no	—
240	C₂×Q₈	C₂×D₄×C₃₇	1	4	yes	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	4	4	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	4	no	—
271	C₄²	SD₁₆×C₃₇	1	4	yes	—
272	C₄²	Q₁₆×C₃₇	1	4	yes	—
273	C₄²	C₂×D₄×C₃₇	1	2	yes	—
274	C₄²	#41×C₃₇	1	2	yes	—
275	C₄²	#41×C₃₇	1	4	yes	—
276	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
277	Q₈	—	2	8	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	8	8	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	#121×C₃₇	1	2	yes	—
296	C₂×C₄	#121×C₃₇	1	2	yes	—
297	C₂×C₄	#124×C₃₇	1	2	yes	—
298	C₂×C₄	#124×C₃₇	1	2	yes	—
299	C₂×C₄	C₂×SD₁₆×C₃₇	1	2	yes	—
300	C₂×C₄	C₂×SD₁₆×C₃₇	1	2	yes	—
301	C₂×C₄	C₂×SD₁₆×C₃₇	1	2	yes	—
302	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
303	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
304	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
305	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	2	yes	—
306	C₂×C₄	#137×C₃₇	1	2	yes	—
307	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
309	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
310	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
311	C₈	—	4	4	no	—
312	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
313	C₂²	C₂×#124×C₃₇	1	1	yes	characteristic
314	C₂²	C₃₇×#536	1	1	yes	characteristic
315	C₂²	C₂²×SD₁₆×C₃₇	1	1	yes	characteristic
316	C₂²	C₂²×Q₁₆×C₃₇	1	1	yes	characteristic
317	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	1	yes	characteristic
318	C₂²	C₂×#137×C₃₇	1	1	yes	characteristic
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	4	no	—
325	C₄	C₃₇×#460	1	4	yes	—
326	C₄	C₃₇×#462	1	4	yes	—
327	C₄	C₃₇×#487	1	4	yes	complemented
328	C₄	C₃₇×#547	1	2	yes	—
329	C₄	C₃₇×#556	1	2	yes	—
330	C₂	C₃₇×#2547	1	1	yes	characteristic
331	C₂	C₂×C₃₇×#460	1	1	yes	characteristic
332	C₂	C₂×C₃₇×#462	1	1	yes	characteristic
333	C₂	C₂×C₃₇×#487	1	1	yes	characteristic
334	C₂	C₃₇×#3141	1	1	yes	characteristic
335	C₂	C₃₇×#3229	1	1	yes	characteristic
336	C₂	C₃₇×#3230	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 ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
143. #28717	C₃₇	direct (N × H)
159. C₄×C₃₇	#487	semidirect
198. C₃₇	#28717	direct (N × H)
327. C₄	C₃₇×#487	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#487	2
C₃₇×#2868	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2072	2
C₃₇×#1990	2
C₃₇×#1803	2
C₄×C₄⋊C₈×C₃₇	2
#28717	37

Subgroups — C37×#28717 #22649970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28717 #22649970