Subgroups — C₂×C₃₇×#1451 #22638570

Order 9472.

966 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1451	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₇×#322	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₇×#322	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₃₇	C₄	1	2	yes	—
8	C₂³×C₈×C₃₇	C₄	1	2	yes	—
9	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
10	C₃₇×#431	C₄	1	4	yes	—
11	C₃₇×#431	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₃₇	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈×C₃₇	C₄	1	2	yes	—
14	C₄×M₁₆×C₃₇	C₄	1	4	yes	—
15	C₄×M₁₆×C₃₇	C₂²	1	4	yes	—
16	C₂×#97×C₃₇	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₃₇	C₄	1	4	yes	—
18	C₂×C₄×C₈×C₃₇	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
24	C₃₇×#322	C₂²	1	8	yes	complemented
25	C₃₇×#322	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₃₇	—	2	8	no	—
30	C₂×M₁₆×C₃₇	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₃₇	—	2	8	no	—
32	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₃₇	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₃₇	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₃₇	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
46	C₂²⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
47	#97×C₃₇	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₃₇	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₃₇	C₂³	1	2	yes	—
54	C₄×C₈×C₃₇	C₂³	1	2	yes	—
55	C₄×C₈×C₃₇	C₂³	1	4	yes	—
56	C₄×C₈×C₃₇	C₂³	1	4	yes	—
57	C₂⁴×C₃₇	C₄²	1	1	yes	characteristic
58	C₂²×C₄×C₃₇	—	2	4	no	—
59	C₂²×C₄×C₃₇	—	2	4	no	—
60	C₂²×C₄×C₃₇	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	C₄²	1	2	yes	—
63	C₂²×C₄×C₃₇	C₄²	1	2	yes	—
64	C₂²×C₄×C₃₇	C₄²	1	2	yes	—
65	C₂²×C₄×C₃₇	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₃₇	M₁₆	1	8	yes	—
67	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
73	M₁₆×C₃₇	—	2	16	no	—
74	C₂×C₈×C₃₇	—	2	4	no	—
75	C₂×C₈×C₃₇	—	2	8	no	—
76	C₂×C₈×C₃₇	—	2	8	no	—
77	C₂×C₈×C₃₇	—	2	16	no	—
78	C₂×C₈×C₃₇	M₁₆	1	8	yes	complemented
79	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
81	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
83	C₄²×C₃₇	C₄²	1	2	yes	—
84	C₄²×C₃₇	C₄²	1	2	yes	—
85	C₄²×C₃₇	C₄²	1	2	yes	—
86	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
93	C₂³×C₃₇	—	2	2	no	—
94	C₂³×C₃₇	—	2	2	no	—
95	C₂³×C₃₇	#97	1	4	yes	—
96	C₂³×C₃₇	C₂×C₄²	1	2	yes	—
97	C₂³×C₃₇	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₃₇	—	2	2	no	—
99	C₂×C₄×C₃₇	—	2	2	no	—
100	C₂×C₄×C₃₇	—	2	4	no	—
101	C₂×C₄×C₃₇	—	2	4	no	—
102	C₂×C₄×C₃₇	—	2	4	no	—
103	C₂×C₄×C₃₇	—	2	4	no	—
104	C₂×C₄×C₃₇	#97	1	8	yes	—
105	C₂×C₄×C₃₇	#97	1	4	yes	—
106	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂×M₁₆	1	4	yes	—
113	C₂×C₄×C₃₇	C₂×M₁₆	1	8	yes	—
114	C₂×C₄×C₃₇	C₂×M₁₆	1	4	yes	—
115	C₂×C₄×C₃₇	#131	1	4	yes	—
116	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₃₇	—	2	4	no	—
120	C₈×C₃₇	—	2	8	no	—
121	C₈×C₃₇	—	2	8	no	—
122	C₈×C₃₇	C₂×M₁₆	1	8	yes	complemented
123	C₂×#1451	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₃₇	—	2	2	no	—
125	C₂²×C₃₇	—	2	2	no	—
126	C₂²×C₃₇	—	2	8	no	—
127	C₂²×C₃₇	C₂×#97	1	4	yes	—
128	C₂²×C₃₇	C₂×#97	1	2	yes	—
129	C₂²×C₃₇	C₄×M₁₆	1	2	yes	—
130	C₂²×C₃₇	#405	1	2	yes	—
131	C₂²×C₃₇	C₂²×C₄²	1	1	yes	characteristic
132	C₄×C₃₇	—	2	2	no	—
133	C₄×C₃₇	—	2	2	no	—
134	C₄×C₃₇	C₂×#97	1	2	yes	—
135	C₄×C₃₇	C₂×#97	1	4	yes	—
136	C₄×C₃₇	C₄×M₁₆	1	2	yes	—
137	C₄×C₃₇	#405	1	2	yes	—
138	C₄×C₃₇	C₂²×M₁₆	1	2	yes	—
139	C₄×C₃₇	C₂²×M₁₆	1	2	yes	—
140	C₄×C₃₇	C₂×#131	1	2	yes	—
141	C₂×#431	C₂×C₃₇	1	1	yes	characteristic
142	C₂×C₄×M₁₆	C₂×C₃₇	1	1	yes	characteristic
143	C₂²×C₄×C₈	C₂×C₃₇	1	1	yes	characteristic
144	#1451	C₂×C₃₇	1	8	yes	direct product factor
145	C₂×#322	C₂×C₃₇	1	2	yes	complemented
146	C₂×#322	C₂×C₃₇	1	2	yes	complemented
147	C₂×C₃₇	—	2	4	no	—
148	C₂×C₃₇	#1451	1	4	yes	direct product factor
149	C₂×C₃₇	C₂²×#97	1	1	yes	characteristic
150	C₂×C₃₇	C₂×C₄×M₁₆	1	1	yes	characteristic
151	C₂×C₃₇	C₂×#405	1	1	yes	characteristic
152	C₂²×M₁₆	C₄×C₃₇	1	2	yes	—
153	C₂³×C₈	C₄×C₃₇	1	2	yes	—
154	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
155	#431	C₄×C₃₇	1	4	yes	—
156	#431	C₂²×C₃₇	1	4	yes	—
157	C₂×C₄⋊C₈	C₄×C₃₇	1	2	yes	—
158	C₂×C₂²⋊C₈	C₄×C₃₇	1	2	yes	—
159	C₄×M₁₆	C₄×C₃₇	1	4	yes	—
160	C₄×M₁₆	C₂²×C₃₇	1	4	yes	—
161	C₂×#97	C₄×C₃₇	1	2	yes	—
162	C₂×C₄×C₈	C₄×C₃₇	1	4	yes	—
163	C₂×C₄×C₈	C₄×C₃₇	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₃₇	1	4	yes	—
167	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
169	#322	C₂²×C₃₇	1	8	yes	complemented
170	#322	C₂²×C₃₇	1	8	yes	complemented
171	C₃₇	C₂×#1451	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
173	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
174	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	8	no	—
176	C₂×M₁₆	C₂×C₄×C₃₇	1	8	yes	—
177	C₂²×C₈	—	2	8	no	—
178	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
179	C₂²×C₈	C₂×C₄×C₃₇	1	8	yes	—
180	C₂²×C₈	C₂×C₄×C₃₇	1	4	yes	—
181	C₂²×C₈	C₂×C₄×C₃₇	1	4	yes	—
182	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
183	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
184	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
185	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
186	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
187	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
189	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
191	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
192	C₂²⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
193	#97	C₂×C₄×C₃₇	1	8	yes	—
194	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
195	C₄×C₈	C₂×C₄×C₃₇	1	8	yes	—
196	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
197	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
198	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
199	C₄×C₈	C₂³×C₃₇	1	2	yes	—
200	C₄×C₈	C₂³×C₃₇	1	2	yes	—
201	C₄×C₈	C₂³×C₃₇	1	4	yes	—
202	C₄×C₈	C₂³×C₃₇	1	4	yes	—
203	C₂⁴	C₄²×C₃₇	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	C₄²×C₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₄²×C₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₄²×C₃₇	1	2	yes	—
209	C₂²×C₄	C₄²×C₃₇	1	2	yes	—
210	C₂²×C₄	C₄²×C₃₇	1	2	yes	—
211	C₂²×C₄	C₄²×C₃₇	1	1	yes	characteristic
212	C₂²×C₄	M₁₆×C₃₇	1	8	yes	—
213	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
219	M₁₆	—	2	16	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	2	16	no	—
224	C₂×C₈	M₁₆×C₃₇	1	8	yes	complemented
225	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
226	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
227	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
228	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
229	C₄²	C₄²×C₃₇	1	2	yes	—
230	C₄²	C₄²×C₃₇	1	2	yes	—
231	C₄²	C₄²×C₃₇	1	2	yes	—
232	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
233	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
234	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
235	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
236	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
237	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
238	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	#97×C₃₇	1	4	yes	—
242	C₂³	C₂×C₄²×C₃₇	1	2	yes	—
243	C₂³	C₂×C₄²×C₃₇	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	#97×C₃₇	1	8	yes	—
251	C₂×C₄	#97×C₃₇	1	4	yes	—
252	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
253	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
254	C₂×C₄	C₂×C₄²×C₃₇	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
256	C₂×C₄	C₂×C₄²×C₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄²×C₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂×M₁₆×C₃₇	1	4	yes	—
259	C₂×C₄	C₂×M₁₆×C₃₇	1	8	yes	—
260	C₂×C₄	C₂×M₁₆×C₃₇	1	4	yes	—
261	C₂×C₄	#131×C₃₇	1	4	yes	—
262	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
263	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
264	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	8	no	—
267	C₈	—	2	8	no	—
268	C₈	C₂×M₁₆×C₃₇	1	8	yes	complemented
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	8	no	—
272	C₂²	C₂×#97×C₃₇	1	4	yes	—
273	C₂²	C₂×#97×C₃₇	1	2	yes	—
274	C₂²	C₄×M₁₆×C₃₇	1	2	yes	—
275	C₂²	C₃₇×#405	1	2	yes	—
276	C₂²	C₂²×C₄²×C₃₇	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	C₂×#97×C₃₇	1	2	yes	—
280	C₄	C₂×#97×C₃₇	1	4	yes	—
281	C₄	C₄×M₁₆×C₃₇	1	2	yes	—
282	C₄	C₃₇×#405	1	2	yes	—
283	C₄	C₂²×M₁₆×C₃₇	1	2	yes	—
284	C₄	C₂²×M₁₆×C₃₇	1	2	yes	—
285	C₄	C₂×#131×C₃₇	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	C₃₇×#1451	1	4	yes	direct product factor
288	C₂	C₂²×#97×C₃₇	1	1	yes	characteristic
289	C₂	C₂×C₄×M₁₆×C₃₇	1	1	yes	characteristic
290	C₂	C₂×C₃₇×#405	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₃₇	C₂³
commutator	C₂²	C₂²×C₄²×C₃₇
Frattini, cosocle	C₄²	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂×C₄²
prime radical	C₂⁴×C₃₇	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₇×#1451	C₂	direct (N × H)
5. C₂×C₃₇×#322	C₂	semidirect
6. C₂×C₃₇×#322	C₂	semidirect
24. C₃₇×#322	C₂²	semidirect
25. C₃₇×#322	C₂²	semidirect
78. C₂×C₈×C₃₇	M₁₆	semidirect
122. C₈×C₃₇	C₂×M₁₆	semidirect
123. C₂×#1451	C₃₇	direct (N × H)
144. #1451	C₂×C₃₇	direct (N × H)
145. C₂×#322	C₂×C₃₇	semidirect
146. C₂×#322	C₂×C₃₇	semidirect
148. C₂×C₃₇	#1451	direct (N × H)
169. #322	C₂²×C₃₇	semidirect
170. #322	C₂²×C₃₇	semidirect
171. C₃₇	C₂×#1451	direct (N × H)
224. C₂×C₈	M₁₆×C₃₇	semidirect
268. C₈	C₂×M₁₆×C₃₇	semidirect
287. C₂	C₃₇×#1451	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×#431	2
C₂×C₄×M₁₆×C₃₇	2
C₂²×C₄×C₈×C₃₇	2
C₃₇×#1451	2
C₂×C₃₇×#322	2
C₂×#1451	37

Subgroups — C2×C37×#1451 #22638570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#1451 #22638570