Subgroups — C₃₇×#28623 #22649876

Order 9472.

874 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#492	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#491	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#2641	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2327	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2055	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1991	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#1961	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1880	C₂	1	4	yes	maximal, complemented
9	C₃₇×#1768	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₃₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#492	—	2	4	no	—
16	C₃₇×#491	—	2	4	no	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×#103×C₃₇	C₂²	1	4	yes	—
19	C₂×#102×C₃₇	C₂²	1	4	yes	—
20	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₃₇×#394	—	2	4	no	—
23	C₃₇×#391	C₂²	1	2	yes	—
24	C₃₇×#390	C₂²	1	2	yes	—
25	C₃₇×#390	C₂²	1	2	yes	—
26	C₃₇×#387	—	2	4	no	—
27	C₃₇×#382	C₂²	1	1	yes	characteristic
28	C₃₇×#376	C₂²	1	1	yes	characteristic
29	C₃₇×#340	C₂²	1	4	yes	—
30	C₃₇×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×D₄×C₃₇	—	2	2	no	—
33	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
34	C₂×SD₁₆×C₃₇	—	2	8	no	—
35	C₂×SD₁₆×C₃₇	—	2	8	no	—
36	C₂×SD₁₆×C₃₇	—	4	4	no	—
37	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₃₇	—	2	2	no	—
40	C₄⋊Q₈×C₃₇	D₄	1	4	yes	—
41	#127×C₃₇	—	2	2	no	—
42	#127×C₃₇	—	2	2	no	—
43	#125×C₃₇	—	2	2	no	—
44	#125×C₃₇	—	2	2	no	—
45	C₄×Q₈×C₃₇	—	4	4	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	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×C₂²⋊C₄×C₃₇	—	2	4	no	—
52	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
53	C₂×C₄²×C₃₇	—	2	2	no	—
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	C₄⋊C₈×C₃₇	—	2	4	no	—
58	#103×C₃₇	—	2	8	no	—
59	#102×C₃₇	—	4	4	no	—
60	#97×C₃₇	—	2	2	no	—
61	C₄×C₈×C₃₇	—	2	2	no	—
62	#95×C₃₇	—	2	2	no	—
63	#95×C₃₇	—	2	4	no	—
64	#95×C₃₇	D₄	1	4	yes	—
65	C₂⁴×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	2	4	no	—
68	C₂×Q₈×C₃₇	—	2	4	no	—
69	C₂×Q₈×C₃₇	C₂×D₄	1	4	yes	—
70	C₂×D₄×C₃₇	—	2	2	no	—
71	C₂×D₄×C₃₇	—	2	2	no	—
72	C₂×D₄×C₃₇	—	2	4	no	—
73	C₂×D₄×C₃₇	—	2	4	no	—
74	C₂×D₄×C₃₇	—	4	4	no	—
75	C₂×D₄×C₃₇	—	8	4	no	—
76	C₂²×C₄×C₃₇	—	2	2	no	—
77	C₂²×C₄×C₃₇	—	2	2	no	—
78	C₂²×C₄×C₃₇	—	2	2	no	—
79	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
83	SD₁₆×C₃₇	—	4	16	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	—	2	2	no	—
88	C₂×C₈×C₃₇	—	2	2	no	—
89	C₂×C₈×C₃₇	#41	1	4	yes	—
90	C₄⋊C₄×C₃₇	—	2	8	no	—
91	C₄⋊C₄×C₃₇	—	2	2	no	—
92	C₄⋊C₄×C₃₇	—	2	2	no	—
93	C₄⋊C₄×C₃₇	—	2	4	no	—
94	C₄⋊C₄×C₃₇	—	4	4	no	—
95	C₄⋊C₄×C₃₇	—	4	4	no	—
96	C₂²⋊C₄×C₃₇	—	8	4	no	—
97	C₂²⋊C₄×C₃₇	—	8	4	no	—
98	C₄²×C₃₇	—	2	2	no	—
99	C₄²×C₃₇	—	2	2	no	—
100	C₄²×C₃₇	—	4	4	no	—
101	C₄²×C₃₇	C₂×D₄	1	2	yes	—
102	C₄²×C₃₇	#41	1	2	yes	—
103	C₂³×C₃₇	—	4	2	no	—
104	C₂³×C₃₇	—	4	2	no	—
105	C₂³×C₃₇	—	4	2	no	—
106	C₂³×C₃₇	—	4	4	no	—
107	C₂³×C₃₇	—	4	4	no	—
108	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₃₇	—	2	8	no	—
110	Q₈×C₃₇	—	4	4	no	—
111	D₄×C₃₇	—	4	4	no	—
112	D₄×C₃₇	—	4	4	no	—
113	D₄×C₃₇	—	8	8	no	—
114	C₂×C₄×C₃₇	—	2	4	no	—
115	C₂×C₄×C₃₇	—	2	4	no	—
116	C₂×C₄×C₃₇	—	2	2	no	—
117	C₂×C₄×C₃₇	—	2	2	no	—
118	C₂×C₄×C₃₇	—	2	4	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	4	no	—
125	C₂×C₄×C₃₇	—	4	4	no	—
126	C₂×C₄×C₃₇	#121	1	4	yes	—
127	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
129	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
131	C₈×C₃₇	—	2	4	no	—
132	C₈×C₃₇	—	4	2	no	—
133	#28623	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₃₇	—	8	4	no	—
135	C₂²×C₃₇	—	8	4	no	—
136	C₂²×C₃₇	—	8	2	no	—
137	C₂²×C₃₇	—	8	2	no	—
138	C₂²×C₃₇	—	8	2	no	—
139	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
140	C₂²×C₃₇	#532	1	1	yes	characteristic
141	C₂²×C₃₇	#535	1	1	yes	characteristic
142	C₂²×C₃₇	#577	1	4	yes	—
143	C₄×C₃₇	—	2	4	no	—
144	C₄×C₃₇	—	4	2	no	—
145	C₄×C₃₇	—	4	2	no	—
146	C₄×C₃₇	—	4	4	no	—
147	C₄×C₃₇	—	8	2	no	—
148	C₄×C₃₇	—	8	2	no	—
149	C₄×C₃₇	#546	1	2	yes	—
150	C₄×C₃₇	#559	1	2	yes	—
151	C₂×#492	C₂×C₃₇	1	1	yes	characteristic
152	C₂×#491	C₂×C₃₇	1	1	yes	characteristic, complemented
153	#2641	C₂×C₃₇	1	1	yes	characteristic
154	#2327	C₂×C₃₇	1	1	yes	characteristic, complemented
155	#2055	C₂×C₃₇	1	4	yes	complemented
156	#1991	C₂×C₃₇	1	1	yes	characteristic, complemented
157	#1961	C₂×C₃₇	1	1	yes	characteristic
158	#1880	C₂×C₃₇	1	4	yes	complemented
159	#1768	C₂×C₃₇	1	1	yes	characteristic, complemented
160	C₂×C₃₇	—	16	2	no	—
161	C₂×C₃₇	#2544	1	1	yes	characteristic
162	C₂×C₃₇	#3043	1	2	yes	—
163	C₂×C₃₇	#3174	1	2	yes	—
164	C₂×C₃₇	#3221	1	2	yes	—
165	C₂²×SD₁₆	C₂²×C₃₇	1	4	yes	—
166	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
167	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
168	C₂×#125	C₂²×C₃₇	1	1	yes	characteristic
169	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
170	#492	—	2	4	no	—
171	#491	—	2	4	no	—
172	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
173	C₂×#103	C₂²×C₃₇	1	4	yes	—
174	C₂×#102	C₂²×C₃₇	1	4	yes	—
175	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
177	#394	—	2	4	no	—
178	#391	C₂²×C₃₇	1	2	yes	—
179	#390	C₂²×C₃₇	1	2	yes	—
180	#390	C₂²×C₃₇	1	2	yes	—
181	#387	—	2	4	no	—
182	#382	C₂²×C₃₇	1	1	yes	characteristic
183	#376	C₂²×C₃₇	1	1	yes	characteristic
184	#340	C₂²×C₃₇	1	4	yes	—
185	#336	C₂²×C₃₇	1	2	yes	—
186	C₃₇	#28623	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
188	C₂²×D₄	—	2	2	no	—
189	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
190	C₂×SD₁₆	—	2	8	no	—
191	C₂×SD₁₆	—	2	8	no	—
192	C₂×SD₁₆	—	4	4	no	—
193	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
194	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	D₄×C₃₇	1	4	yes	—
197	#127	—	2	2	no	—
198	#127	—	2	2	no	—
199	#125	—	2	2	no	—
200	#125	—	2	2	no	—
201	C₄×Q₈	—	4	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
207	C₂×C₂²⋊C₄	—	2	4	no	—
208	C₂×C₂²⋊C₄	—	2	4	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
213	C₄⋊C₈	—	2	4	no	—
214	#103	—	2	8	no	—
215	#102	—	4	4	no	—
216	#97	—	2	2	no	—
217	C₄×C₈	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	4	no	—
220	#95	D₄×C₃₇	1	4	yes	—
221	C₂⁴	—	2	2	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	C₂×D₄×C₃₇	1	4	yes	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	8	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	C₂×D₄×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	C₂⁴×C₃₇	1	1	yes	characteristic
239	SD₁₆	—	4	16	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	#41×C₃₇	1	4	yes	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	8	4	no	—
253	C₂²⋊C₄	—	8	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	4	no	—
257	C₄²	C₂×D₄×C₃₇	1	2	yes	—
258	C₄²	#41×C₃₇	1	2	yes	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
265	Q₈	—	2	8	no	—
266	Q₈	—	4	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	8	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	#121×C₃₇	1	4	yes	—
283	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
285	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
286	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
287	C₈	—	2	4	no	—
288	C₈	—	4	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	4	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
295	C₂²	C₃₇×#532	1	1	yes	characteristic
296	C₂²	C₃₇×#535	1	1	yes	characteristic
297	C₂²	C₃₇×#577	1	4	yes	—
298	C₄	—	2	4	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	4	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	C₃₇×#546	1	2	yes	—
305	C₄	C₃₇×#559	1	2	yes	—
306	C₂	—	16	2	no	—
307	C₂	C₃₇×#2544	1	1	yes	characteristic
308	C₂	C₃₇×#3043	1	2	yes	—
309	C₂	C₃₇×#3174	1	2	yes	—
310	C₂	C₃₇×#3221	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₃₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#491	C₂	semidirect
4. C₃₇×#2327	C₂	semidirect
5. C₃₇×#2055	C₂	semidirect
6. C₃₇×#1991	C₂	semidirect
8. C₃₇×#1880	C₂	semidirect
9. C₃₇×#1768	C₂	semidirect
133. #28623	C₃₇	direct (N × H)
152. C₂×#491	C₂×C₃₇	semidirect
154. #2327	C₂×C₃₇	semidirect
155. #2055	C₂×C₃₇	semidirect
156. #1991	C₂×C₃₇	semidirect
158. #1880	C₂×C₃₇	semidirect
159. #1768	C₂×C₃₇	semidirect
186. C₃₇	#28623	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#492	2
C₂×C₃₇×#491	2
C₃₇×#2641	2
C₃₇×#2327	2
C₃₇×#2055	2
C₃₇×#1991	2
C₃₇×#1961	2
C₃₇×#1880	2
C₃₇×#1768	2
#28623	37

Subgroups — C37×#28623 #22649876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28623 #22649876