Subgroups — C₃₇×#20667 #22641920

Order 9472.

1438 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#430	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₇×#407	C₂	1	2	yes	maximal
4	C₃₇×#1811	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1810	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1738	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₃₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₇	—	2	8	no	—
9	C₂²×M₁₆×C₃₇	C₄	1	8	yes	—
10	C₂²×M₁₆×C₃₇	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₃₇	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#430	—	2	8	no	—
16	C₂×#108×C₃₇	C₂²	1	8	yes	complemented
17	C₃₇×#407	—	2	4	no	—
18	C₃₇×#407	C₂²	1	4	yes	—
19	C₃₇×#407	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈×C₃₇	—	2	4	no	—
21	C₃₇×#341	C₂²	1	8	yes	complemented
22	C₂⁵×C₃₇	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₃₇	—	2	4	no	—
24	C₂³×C₄×C₃₇	—	2	4	no	—
25	C₂³×C₄×C₃₇	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₃₇	D₄	1	4	yes	—
27	C₂³×C₄×C₃₇	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₃₇	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₃₇	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₃₇	—	2	4	no	—
34	C₂×M₁₆×C₃₇	—	2	32	no	—
35	C₂×M₁₆×C₃₇	—	2	4	no	—
36	C₂×M₁₆×C₃₇	—	2	8	no	—
37	C₂×M₁₆×C₃₇	—	2	8	no	—
38	C₂×M₁₆×C₃₇	—	2	8	no	—
39	C₂×M₁₆×C₃₇	C₂×C₄	1	4	yes	—
40	C₂×M₁₆×C₃₇	C₂×C₄	1	8	yes	—
41	C₂×M₁₆×C₃₇	C₂×C₄	1	4	yes	—
42	C₂×M₁₆×C₃₇	D₄	1	8	yes	—
43	C₂×M₁₆×C₃₇	C₂³	1	8	yes	—
44	C₂²×C₈×C₃₇	—	2	8	no	—
45	C₂²×C₈×C₃₇	—	2	4	no	—
46	C₂²×C₈×C₃₇	C₂×C₄	1	8	yes	—
47	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
49	#108×C₃₇	—	2	16	no	—
50	C₂²⋊C₈×C₃₇	—	2	8	no	—
51	C₂²⋊C₈×C₃₇	—	2	8	no	—
52	C₂⁴×C₃₇	—	2	4	no	—
53	C₂⁴×C₃₇	—	4	4	no	—
54	C₂⁴×C₃₇	C₄⋊C₄	1	4	yes	—
55	C₂⁴×C₃₇	C₂×Q₈	1	1	yes	characteristic
56	C₂⁴×C₃₇	C₂×Q₈	1	1	yes	characteristic
57	C₂⁴×C₃₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₃₇	—	2	4	no	—
59	C₂²×C₄×C₃₇	—	2	8	no	—
60	C₂²×C₄×C₃₇	—	2	8	no	—
61	C₂²×C₄×C₃₇	—	2	16	no	—
62	C₂²×C₄×C₃₇	—	2	4	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	—	2	2	no	—
65	C₂²×C₄×C₃₇	—	2	2	no	—
66	C₂²×C₄×C₃₇	—	2	2	no	—
67	C₂²×C₄×C₃₇	—	2	2	no	—
68	C₂²×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₃₇	#41	1	4	yes	—
82	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₃₇	—	2	8	no	—
84	M₁₆×C₃₇	—	2	16	no	—
85	M₁₆×C₃₇	—	2	8	no	—
86	M₁₆×C₃₇	—	2	8	no	—
87	M₁₆×C₃₇	—	4	8	no	—
88	M₁₆×C₃₇	C₂²⋊C₄	1	16	yes	—
89	C₂×C₈×C₃₇	—	2	4	no	—
90	C₂×C₈×C₃₇	—	2	16	no	—
91	C₂×C₈×C₃₇	—	2	8	no	—
92	C₂×C₈×C₃₇	—	4	8	no	—
93	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
96	C₂³×C₃₇	—	2	2	no	—
97	C₂³×C₃₇	—	2	4	no	—
98	C₂³×C₃₇	—	2	2	no	—
99	C₂³×C₃₇	—	2	8	no	—
100	C₂³×C₃₇	—	2	8	no	—
101	C₂³×C₃₇	—	4	4	no	—
102	C₂³×C₃₇	—	4	2	no	—
103	C₂³×C₃₇	—	4	2	no	—
104	C₂³×C₃₇	—	4	16	no	—
105	C₂³×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₃₇	C₂²⋊Q₈	1	4	yes	—
109	C₂³×C₃₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	—	2	2	no	—
112	C₂×C₄×C₃₇	—	2	8	no	—
113	C₂×C₄×C₃₇	—	2	2	no	—
114	C₂×C₄×C₃₇	—	2	2	no	—
115	C₂×C₄×C₃₇	—	2	2	no	—
116	C₂×C₄×C₃₇	—	2	2	no	—
117	C₂×C₄×C₃₇	—	2	4	no	—
118	C₂×C₄×C₃₇	—	2	4	no	—
119	C₂×C₄×C₃₇	—	2	8	no	—
120	C₂×C₄×C₃₇	—	2	8	no	—
121	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	4	yes	—
122	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₃₇	#117	1	4	yes	—
126	C₂×C₄×C₃₇	#121	1	8	yes	—
127	C₂×C₄×C₃₇	C₂²⋊Q₈	1	4	yes	—
128	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
132	C₈×C₃₇	—	2	4	no	—
133	C₈×C₃₇	—	4	8	no	—
134	C₈×C₃₇	C₂×C₂²⋊C₄	1	8	yes	—
135	#20667	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₃₇	—	2	8	no	—
137	C₂²×C₃₇	—	2	2	no	—
138	C₂²×C₃₇	—	2	4	no	—
139	C₂²×C₃₇	—	2	2	no	—
140	C₂²×C₃₇	—	2	2	no	—
141	C₂²×C₃₇	—	2	4	no	—
142	C₂²×C₃₇	—	2	2	no	—
143	C₂²×C₃₇	—	4	4	no	—
144	C₂²×C₃₇	—	4	16	no	—
145	C₂²×C₃₇	—	4	4	no	—
146	C₂²×C₃₇	#380	1	4	yes	—
147	C₂²×C₃₇	#430	1	4	yes	complemented
148	C₂²×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	2	yes	—
150	C₄×C₃₇	—	2	2	no	—
151	C₄×C₃₇	—	2	2	no	—
152	C₄×C₃₇	#380	1	4	yes	—
153	C₄×C₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₃₇	C₂×#117	1	1	yes	characteristic
155	C₄×C₃₇	C₂×#121	1	2	yes	—
156	C₂³×M₁₆	C₂×C₃₇	1	1	yes	characteristic
157	C₂×#430	C₂×C₃₇	1	2	yes	complemented
158	C₂×#407	C₂×C₃₇	1	2	yes	—
159	#1811	C₂×C₃₇	1	4	yes	complemented
160	#1810	C₂×C₃₇	1	4	yes	complemented
161	#1738	C₂×C₃₇	1	2	yes	complemented
162	C₂×C₃₇	—	2	2	no	—
163	C₂×C₃₇	—	2	4	no	—
164	C₂×C₃₇	—	2	4	no	—
165	C₂×C₃₇	—	4	2	no	—
166	C₂×C₃₇	C₂×#380	1	1	yes	characteristic
167	C₂×C₃₇	C₂×#430	1	2	yes	—
168	C₂⁴×C₄	C₂²×C₃₇	1	1	yes	characteristic
169	C₂²×M₁₆	—	2	8	no	—
170	C₂²×M₁₆	C₄×C₃₇	1	8	yes	—
171	C₂²×M₁₆	C₂²×C₃₇	1	4	yes	—
172	C₂²×M₁₆	C₂²×C₃₇	1	2	yes	—
173	C₂²×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
174	C₂²×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
175	C₂³×C₈	C₂²×C₃₇	1	2	yes	—
176	#430	—	2	8	no	—
177	C₂×#108	C₂²×C₃₇	1	8	yes	complemented
178	#407	—	2	4	no	—
179	#407	C₂²×C₃₇	1	4	yes	—
180	#407	C₂²×C₃₇	1	4	yes	—
181	C₂×C₂²⋊C₈	—	2	4	no	—
182	#341	C₂²×C₃₇	1	8	yes	complemented
183	C₃₇	#20667	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁵	Q₈×C₃₇	1	1	yes	characteristic
185	C₂³×C₄	—	2	4	no	—
186	C₂³×C₄	—	2	4	no	—
187	C₂³×C₄	C₂×C₄×C₃₇	1	4	yes	—
188	C₂³×C₄	D₄×C₃₇	1	4	yes	—
189	C₂³×C₄	Q₈×C₃₇	1	1	yes	characteristic
190	C₂³×C₄	Q₈×C₃₇	1	1	yes	characteristic
191	C₂³×C₄	Q₈×C₃₇	1	1	yes	characteristic
192	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
193	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
194	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	32	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	2	8	no	—
199	C₂×M₁₆	—	2	8	no	—
200	C₂×M₁₆	—	2	8	no	—
201	C₂×M₁₆	C₂×C₄×C₃₇	1	4	yes	—
202	C₂×M₁₆	C₂×C₄×C₃₇	1	8	yes	—
203	C₂×M₁₆	C₂×C₄×C₃₇	1	4	yes	—
204	C₂×M₁₆	D₄×C₃₇	1	8	yes	—
205	C₂×M₁₆	C₂³×C₃₇	1	8	yes	—
206	C₂²×C₈	—	2	8	no	—
207	C₂²×C₈	—	2	4	no	—
208	C₂²×C₈	C₂×C₄×C₃₇	1	8	yes	—
209	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
210	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
211	#108	—	2	16	no	—
212	C₂²⋊C₈	—	2	8	no	—
213	C₂²⋊C₈	—	2	8	no	—
214	C₂⁴	—	2	4	no	—
215	C₂⁴	—	4	4	no	—
216	C₂⁴	C₄⋊C₄×C₃₇	1	4	yes	—
217	C₂⁴	C₂×Q₈×C₃₇	1	1	yes	characteristic
218	C₂⁴	C₂×Q₈×C₃₇	1	1	yes	characteristic
219	C₂⁴	C₂×Q₈×C₃₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	16	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	C₄⋊C₄×C₃₇	1	4	yes	—
231	C₂²×C₄	C₄⋊C₄×C₃₇	1	4	yes	—
232	C₂²×C₄	C₄⋊C₄×C₃₇	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
234	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
239	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
240	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
243	C₂²×C₄	#41×C₃₇	1	4	yes	—
244	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
245	M₁₆	—	2	8	no	—
246	M₁₆	—	2	16	no	—
247	M₁₆	—	2	8	no	—
248	M₁₆	—	2	8	no	—
249	M₁₆	—	4	8	no	—
250	M₁₆	C₂²⋊C₄×C₃₇	1	16	yes	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	16	no	—
253	C₂×C₈	—	2	8	no	—
254	C₂×C₈	—	4	8	no	—
255	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
256	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
257	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	8	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	16	no	—
267	C₂³	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
268	C₂³	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
269	C₂³	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
270	C₂³	C₂²⋊Q₈×C₃₇	1	4	yes	—
271	C₂³	C₂²×Q₈×C₃₇	1	1	yes	characteristic
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	4	yes	—
284	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
286	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
287	C₂×C₄	#117×C₃₇	1	4	yes	—
288	C₂×C₄	#121×C₃₇	1	8	yes	—
289	C₂×C₄	C₂²⋊Q₈×C₃₇	1	4	yes	—
290	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
293	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
294	C₈	—	2	4	no	—
295	C₈	—	4	8	no	—
296	C₈	C₂×C₂²⋊C₄×C₃₇	1	8	yes	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	16	no	—
306	C₂²	—	4	4	no	—
307	C₂²	C₃₇×#380	1	4	yes	—
308	C₂²	C₃₇×#430	1	4	yes	complemented
309	C₂²	C₂²×C₄⋊C₄×C₃₇	1	1	yes	characteristic
310	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	C₃₇×#380	1	4	yes	—
314	C₄	C₂²×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
315	C₄	C₂×#117×C₃₇	1	1	yes	characteristic
316	C₄	C₂×#121×C₃₇	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	2	4	no	—
319	C₂	—	2	4	no	—
320	C₂	—	4	2	no	—
321	C₂	C₂×C₃₇×#380	1	1	yes	characteristic
322	C₂	C₂×C₃₇×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂²×C₄⋊C₄
prime radical	C₂⁵×C₃₇	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂⁴×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#430	C₂	semidirect
4. C₃₇×#1811	C₂	semidirect
5. C₃₇×#1810	C₂	semidirect
6. C₃₇×#1738	C₂	semidirect
16. C₂×#108×C₃₇	C₂²	semidirect
21. C₃₇×#341	C₂²	semidirect
135. #20667	C₃₇	direct (N × H)
147. C₂²×C₃₇	#430	semidirect
157. C₂×#430	C₂×C₃₇	semidirect
159. #1811	C₂×C₃₇	semidirect
160. #1810	C₂×C₃₇	semidirect
161. #1738	C₂×C₃₇	semidirect
177. C₂×#108	C₂²×C₃₇	semidirect
182. #341	C₂²×C₃₇	semidirect
183. C₃₇	#20667	direct (N × H)
308. C₂²	C₃₇×#430	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₃₇	2
C₂×C₃₇×#430	2
C₂×C₃₇×#407	2
C₃₇×#1811	2
C₃₇×#1810	2
C₃₇×#1738	2
#20667	37

Subgroups — C37×#20667 #22641920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#20667 #22641920