Subgroups — C₃₇×#28373 #22649626

Order 9472.

914 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#477	C₂	1	2	yes	maximal
2	C₂×C₃₇×#462	C₂	1	2	yes	maximal
3	C₃₇×#2862	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2095	C₂	1	2	yes	maximal
6	C₃₇×#2016	C₂	1	2	yes	maximal
7	C₃₇×#1969	C₂	1	4	yes	maximal
8	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₃₇	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#477	—	2	8	no	—
15	C₃₇×#462	—	2	8	no	—
16	C₂×#107×C₃₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×#103×C₃₇	C₂²	1	4	yes	—
19	C₂×#103×C₃₇	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
21	C₃₇×#391	—	2	2	no	—
22	C₃₇×#391	C₂²	1	2	yes	—
23	C₃₇×#391	C₂²	1	2	yes	—
24	C₃₇×#391	C₂²	1	2	yes	—
25	C₃₇×#389	—	2	2	no	—
26	C₃₇×#384	C₂²	1	1	yes	characteristic
27	C₃₇×#382	—	2	2	no	—
28	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
29	C₄³×C₃₇	C₂²	1	1	yes	characteristic
30	C₃₇×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	—	2	1	no	—
32	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
33	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
34	C₂×Q₁₆×C₃₇	—	2	8	no	—
35	C₂×Q₁₆×C₃₇	—	2	8	no	—
36	C₂×Q₁₆×C₃₇	—	4	4	no	—
37	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₃₇	—	2	4	no	—
39	C₄⋊Q₈×C₃₇	—	2	4	no	—
40	C₄⋊Q₈×C₃₇	—	2	4	no	—
41	C₄×Q₈×C₃₇	—	2	4	no	—
42	C₄×Q₈×C₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₃₇	—	2	2	no	—
52	C₂×C₄²×C₃₇	D₄	1	2	yes	—
53	C₂×C₄²×C₃₇	D₄	1	2	yes	—
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
57	#107×C₃₇	—	2	8	no	—
58	C₄⋊C₈×C₃₇	D₄	1	8	yes	—
59	#103×C₃₇	—	2	8	no	—
60	#103×C₃₇	—	2	8	no	—
61	C₄×C₈×C₃₇	—	4	2	no	—
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	4	no	—
69	C₂×Q₈×C₃₇	—	2	4	no	—
70	C₂×Q₈×C₃₇	—	2	2	no	—
71	C₂×Q₈×C₃₇	—	2	2	no	—
72	C₂×Q₈×C₃₇	—	2	2	no	—
73	C₂×Q₈×C₃₇	—	2	2	no	—
74	C₂×Q₈×C₃₇	—	4	4	no	—
75	C₂×Q₈×C₃₇	—	4	2	no	—
76	C₂×Q₈×C₃₇	#41	1	4	yes	—
77	C₂²×C₄×C₃₇	—	2	2	no	—
78	C₂²×C₄×C₃₇	—	2	2	no	—
79	C₂²×C₄×C₃₇	—	2	2	no	—
80	C₂²×C₄×C₃₇	—	2	2	no	—
81	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
86	Q₁₆×C₃₇	—	4	16	no	—
87	C₂×C₈×C₃₇	—	2	4	no	—
88	C₂×C₈×C₃₇	—	2	4	no	—
89	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
90	C₄⋊C₄×C₃₇	—	2	8	no	—
91	C₄⋊C₄×C₃₇	—	2	2	no	—
92	C₄⋊C₄×C₃₇	—	2	4	no	—
93	C₄⋊C₄×C₃₇	—	2	2	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	C₄⋊C₄×C₃₇	—	4	2	no	—
99	C₄²×C₃₇	—	2	2	no	—
100	C₄²×C₃₇	—	2	1	no	—
101	C₄²×C₃₇	—	2	1	no	—
102	C₄²×C₃₇	—	4	2	no	—
103	C₄²×C₃₇	—	4	4	no	—
104	C₄²×C₃₇	Q₁₆	1	8	yes	complemented
105	C₄²×C₃₇	C₂×D₄	1	4	yes	—
106	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₃₇	—	2	4	no	—
108	Q₈×C₃₇	—	2	4	no	—
109	Q₈×C₃₇	—	4	2	no	—
110	Q₈×C₃₇	—	4	2	no	—
111	Q₈×C₃₇	—	4	4	no	—
112	Q₈×C₃₇	—	4	4	no	—
113	Q₈×C₃₇	—	8	4	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	2	no	—
118	C₂×C₄×C₃₇	—	2	2	no	—
119	C₂×C₄×C₃₇	—	2	4	no	—
120	C₂×C₄×C₃₇	—	4	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	2	no	—
125	C₂×C₄×C₃₇	—	4	2	no	—
126	C₂×C₄×C₃₇	—	4	2	no	—
127	C₂×C₄×C₃₇	—	4	4	no	—
128	C₂×C₄×C₃₇	—	4	4	no	—
129	C₂×C₄×C₃₇	#127	1	2	yes	—
130	C₂×C₄×C₃₇	#127	1	2	yes	—
131	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
132	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
133	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
134	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
135	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
136	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
138	C₈×C₃₇	—	2	8	no	—
139	#28373	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₃₇	#525	1	1	yes	characteristic
141	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
142	C₂²×C₃₇	#537	1	1	yes	characteristic
143	C₂²×C₃₇	C₂²×Q₁₆	1	2	yes	—
144	C₂²×C₃₇	#575	1	2	yes	—
145	C₄×C₃₇	—	2	4	no	—
146	C₄×C₃₇	—	2	2	no	—
147	C₄×C₃₇	—	2	4	no	—
148	C₄×C₃₇	—	4	4	no	—
149	C₄×C₃₇	—	8	2	no	—
150	C₄×C₃₇	—	8	2	no	—
151	C₄×C₃₇	—	8	2	no	—
152	C₄×C₃₇	#494	1	4	yes	—
153	C₄×C₃₇	#546	1	2	yes	—
154	C₄×C₃₇	#548	1	2	yes	—
155	C₂×#477	C₂×C₃₇	1	2	yes	—
156	C₂×#462	C₂×C₃₇	1	2	yes	—
157	#2862	C₂×C₃₇	1	1	yes	characteristic
158	C₄×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic
159	#2095	C₂×C₃₇	1	2	yes	—
160	#2016	C₂×C₃₇	1	2	yes	—
161	#1969	C₂×C₃₇	1	4	yes	—
162	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic
163	C₂×C₃₇	#2540	1	1	yes	characteristic
164	C₂×C₃₇	#3104	1	2	yes	—
165	C₂×C₃₇	C₂×#494	1	1	yes	characteristic
166	C₂×C₃₇	#3153	1	1	yes	characteristic
167	C₂×C₃₇	#3249	1	2	yes	—
168	C₂²×Q₁₆	C₂²×C₃₇	1	4	yes	—
169	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
170	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
171	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
172	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
173	#477	—	2	8	no	—
174	#462	—	2	8	no	—
175	C₂×#107	C₂²×C₃₇	1	4	yes	—
176	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
177	C₂×#103	C₂²×C₃₇	1	4	yes	—
178	C₂×#103	C₂²×C₃₇	1	4	yes	—
179	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
180	#391	—	2	2	no	—
181	#391	C₂²×C₃₇	1	2	yes	—
182	#391	C₂²×C₃₇	1	2	yes	—
183	#391	C₂²×C₃₇	1	2	yes	—
184	#389	—	2	2	no	—
185	#384	C₂²×C₃₇	1	1	yes	characteristic
186	#382	—	2	2	no	—
187	C₄×C₄⋊C₄	—	2	2	no	—
188	C₄³	C₂²×C₃₇	1	1	yes	characteristic
189	#336	C₂²×C₃₇	1	2	yes	—
190	C₃₇	#28373	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²×Q₈	—	2	1	no	—
192	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
193	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
194	C₂×Q₁₆	—	2	8	no	—
195	C₂×Q₁₆	—	2	8	no	—
196	C₂×Q₁₆	—	4	4	no	—
197	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
198	C₄⋊Q₈	—	2	4	no	—
199	C₄⋊Q₈	—	2	4	no	—
200	C₄⋊Q₈	—	2	4	no	—
201	C₄×Q₈	—	2	4	no	—
202	C₄×Q₈	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	D₄×C₃₇	1	2	yes	—
213	C₂×C₄²	D₄×C₃₇	1	2	yes	—
214	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
217	#107	—	2	8	no	—
218	C₄⋊C₈	D₄×C₃₇	1	8	yes	—
219	#103	—	2	8	no	—
220	#103	—	2	8	no	—
221	C₄×C₈	—	4	2	no	—
222	#95	—	2	4	no	—
223	#95	—	2	4	no	—
224	#95	—	2	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×Q₈	#41×C₃₇	1	4	yes	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
245	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
246	Q₁₆	—	4	16	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	4	4	no	—
264	C₄²	Q₁₆×C₃₇	1	8	yes	complemented
265	C₄²	C₂×D₄×C₃₇	1	4	yes	—
266	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	2	4	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	4	4	no	—
272	Q₈	—	4	4	no	—
273	Q₈	—	8	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	#127×C₃₇	1	2	yes	—
290	C₂×C₄	#127×C₃₇	1	2	yes	—
291	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
292	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
293	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
294	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
295	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
296	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
297	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
298	C₈	—	2	8	no	—
299	C₂²	C₃₇×#525	1	1	yes	characteristic
300	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
301	C₂²	C₃₇×#537	1	1	yes	characteristic
302	C₂²	C₂²×Q₁₆×C₃₇	1	2	yes	—
303	C₂²	C₃₇×#575	1	2	yes	—
304	C₄	—	2	2	no	—
305	C₄	—	2	4	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₃₇×#494	1	4	yes	—
312	C₄	C₃₇×#546	1	2	yes	—
313	C₄	C₃₇×#548	1	2	yes	—
314	C₂	C₃₇×#2540	1	1	yes	characteristic
315	C₂	C₃₇×#3104	1	2	yes	—
316	C₂	C₂×C₃₇×#494	1	1	yes	characteristic
317	C₂	C₃₇×#3153	1	1	yes	characteristic
318	C₂	C₃₇×#3249	1	2	yes	—

	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
104. C₄²×C₃₇	Q₁₆	semidirect
139. #28373	C₃₇	direct (N × H)
190. C₃₇	#28373	direct (N × H)
264. C₄²	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×C₃₇×#477	2
C₂×C₃₇×#462	2
C₃₇×#2862	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2095	2
C₃₇×#2016	2
C₃₇×#1969	2
C₄×C₄⋊C₈×C₃₇	2
#28373	37

Subgroups — C37×#28373 #22649626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28373 #22649626