Subgroups — C₃₇×#28271 #22649524

Order 9472.

746 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2581	C₂	1	2	yes	maximal
4	C₃₇×#2068	C₂	1	4	yes	maximal
5	C₃₇×#1971	C₂	1	2	yes	maximal
6	C₃₇×#1944	C₂	1	2	yes	maximal
7	C₃₇×#1846	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1805	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	C₃₇×#446	—	2	4	no	—
13	C₄×Q₁₆×C₃₇	—	2	4	no	—
14	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
15	C₂×#107×C₃₇	C₂²	1	2	yes	—
16	C₂×#106×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×#97×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#400	—	2	2	no	—
24	C₃₇×#395	—	2	2	no	—
25	C₃₇×#391	C₂²	1	4	yes	—
26	C₃₇×#389	C₂²	1	4	yes	—
27	C₃₇×#387	—	2	4	no	—
28	C₃₇×#382	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	4	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₃₇	—	2	2	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	—	4	2	no	—
53	#106×C₃₇	—	2	4	no	—
54	C₄⋊C₈×C₃₇	—	2	4	no	—
55	C₄⋊C₈×C₃₇	—	2	2	no	—
56	#103×C₃₇	—	2	4	no	—
57	#103×C₃₇	—	4	4	no	—
58	#97×C₃₇	—	2	4	no	—
59	C₄×C₈×C₃₇	D₄	1	4	yes	—
60	#95×C₃₇	—	2	2	no	—
61	#95×C₃₇	—	2	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	2	no	—
69	C₂×Q₈×C₃₇	—	2	2	no	—
70	C₂×Q₈×C₃₇	—	2	2	no	—
71	C₂×Q₈×C₃₇	—	4	2	no	—
72	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₃₇	—	2	2	no	—
74	C₂²×C₄×C₃₇	—	2	4	no	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
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	1	yes	characteristic
80	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₃₇	—	2	8	no	—
82	C₂×C₈×C₃₇	—	2	1	no	—
83	C₂×C₈×C₃₇	—	2	1	no	—
84	C₂×C₈×C₃₇	—	2	2	no	—
85	C₂×C₈×C₃₇	—	2	2	no	—
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₃₇	#41	1	2	yes	—
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	—	2	4	no	—
95	C₄⋊C₄×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	2	4	no	—
97	C₄⋊C₄×C₃₇	—	4	8	no	—
98	C₄⋊C₄×C₃₇	—	4	4	no	—
99	C₄²×C₃₇	—	2	2	no	—
100	C₄²×C₃₇	—	2	2	no	—
101	C₄²×C₃₇	—	4	4	no	—
102	C₄²×C₃₇	C₂×D₄	1	2	yes	—
103	C₄²×C₃₇	#41	1	2	yes	—
104	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₃₇	—	2	4	no	—
106	Q₈×C₃₇	—	2	4	no	—
107	Q₈×C₃₇	—	4	4	no	—
108	Q₈×C₃₇	—	4	2	no	—
109	Q₈×C₃₇	—	4	2	no	—
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	—	2	2	no	—
112	C₂×C₄×C₃₇	—	2	2	no	—
113	C₂×C₄×C₃₇	—	2	2	no	—
114	C₂×C₄×C₃₇	—	2	4	no	—
115	C₂×C₄×C₃₇	—	2	4	no	—
116	C₂×C₄×C₃₇	—	2	4	no	—
117	C₂×C₄×C₃₇	—	4	2	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	4	no	—
122	C₂×C₄×C₃₇	—	4	4	no	—
123	C₂×C₄×C₃₇	—	4	4	no	—
124	C₂×C₄×C₃₇	#121	1	2	yes	—
125	C₂×C₄×C₃₇	#121	1	2	yes	—
126	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈×C₃₇	—	2	2	no	—
133	C₈×C₃₇	—	4	2	no	—
134	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
135	#28271	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
137	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
138	C₂²×C₃₇	#550	1	1	yes	characteristic
139	C₂²×C₃₇	#575	1	2	yes	—
140	C₂²×C₃₇	#578	1	2	yes	—
141	C₄×C₃₇	—	2	2	no	—
142	C₄×C₃₇	—	2	2	no	—
143	C₄×C₃₇	—	4	4	no	—
144	C₄×C₃₇	—	4	2	no	—
145	C₄×C₃₇	—	4	2	no	—
146	C₄×C₃₇	—	8	4	no	—
147	C₄×C₃₇	—	8	2	no	—
148	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
149	C₄×C₃₇	#537	1	1	yes	characteristic
150	C₄×C₃₇	#554	1	1	yes	characteristic
151	C₄×C₃₇	#556	1	1	yes	characteristic
152	C₂×#446	C₂×C₃₇	1	1	yes	characteristic
153	C₂×C₄×Q₁₆	C₂×C₃₇	1	1	yes	characteristic
154	#2581	C₂×C₃₇	1	2	yes	—
155	#2068	C₂×C₃₇	1	4	yes	—
156	#1971	C₂×C₃₇	1	2	yes	—
157	#1944	C₂×C₃₇	1	2	yes	—
158	#1846	C₂×C₃₇	1	1	yes	characteristic
159	#1805	C₂×C₃₇	1	2	yes	—
160	C₂×C₃₇	#2538	1	1	yes	characteristic
161	C₂×C₃₇	#3054	1	1	yes	characteristic
162	C₂×C₃₇	#3059	1	1	yes	characteristic
163	C₂×C₃₇	#3390	1	2	yes	—
164	C₂×C₃₇	#3402	1	2	yes	—
165	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
166	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
167	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
168	#446	—	2	4	no	—
169	C₄×Q₁₆	—	2	4	no	—
170	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
171	C₂×#107	C₂²×C₃₇	1	2	yes	—
172	C₂×#106	C₂²×C₃₇	1	2	yes	—
173	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
174	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
175	C₂×#103	C₂²×C₃₇	1	2	yes	—
176	C₂×#103	C₂²×C₃₇	1	4	yes	—
177	C₂×#97	C₂²×C₃₇	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
179	#400	—	2	2	no	—
180	#395	—	2	2	no	—
181	#391	C₂²×C₃₇	1	4	yes	—
182	#389	C₂²×C₃₇	1	4	yes	—
183	#387	—	2	4	no	—
184	#382	C₂²×C₃₇	1	2	yes	—
185	#340	C₂²×C₃₇	1	4	yes	—
186	C₃₇	#28271	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²×Q₈	D₄×C₃₇	1	1	yes	characteristic
188	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
189	C₂×Q₁₆	—	2	2	no	—
190	C₂×Q₁₆	—	2	2	no	—
191	C₂×Q₁₆	Q₈×C₃₇	1	4	yes	complemented
192	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
193	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
195	C₄×Q₈	—	2	2	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
202	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
203	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
207	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
208	#107	—	2	2	no	—
209	#107	—	4	2	no	—
210	#106	—	2	4	no	—
211	C₄⋊C₈	—	2	4	no	—
212	C₄⋊C₈	—	2	2	no	—
213	#103	—	2	4	no	—
214	#103	—	4	4	no	—
215	#97	—	2	4	no	—
216	C₄×C₈	D₄×C₃₇	1	4	yes	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	4	no	—
220	#95	—	2	4	no	—
221	#95	—	2	4	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₃₇	1	2	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	Q₁₆	—	2	8	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂×D₄×C₃₇	1	2	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	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	8	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	4	4	no	—
259	C₄²	C₂×D₄×C₃₇	1	2	yes	—
260	C₄²	#41×C₃₇	1	2	yes	—
261	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
262	Q₈	—	2	4	no	—
263	Q₈	—	2	4	no	—
264	Q₈	—	4	4	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	#121×C₃₇	1	2	yes	—
282	C₂×C₄	#121×C₃₇	1	2	yes	—
283	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
287	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
288	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
292	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
293	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
294	C₂²	C₃₇×#550	1	1	yes	characteristic
295	C₂²	C₃₇×#575	1	2	yes	—
296	C₂²	C₃₇×#578	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	8	4	no	—
303	C₄	—	8	2	no	—
304	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
305	C₄	C₃₇×#537	1	1	yes	characteristic
306	C₄	C₃₇×#554	1	1	yes	characteristic
307	C₄	C₃₇×#556	1	1	yes	characteristic
308	C₂	C₃₇×#2538	1	1	yes	characteristic
309	C₂	C₃₇×#3054	1	1	yes	characteristic
310	C₂	C₃₇×#3059	1	1	yes	characteristic
311	C₂	C₃₇×#3390	1	2	yes	—
312	C₂	C₃₇×#3402	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 ◁ 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
135. #28271	C₃₇	direct (N × H)
186. C₃₇	#28271	direct (N × H)
191. C₂×Q₁₆	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×#446	2
C₂×C₄×Q₁₆×C₃₇	2
C₃₇×#2581	2
C₃₇×#2068	2
C₃₇×#1971	2
C₃₇×#1944	2
C₃₇×#1846	2
C₃₇×#1805	2
#28271	37

Subgroups — C37×#28271 #22649524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28271 #22649524