Subgroups — C₂×C₃₇×#1569 #22639047

Order 9472.

758 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×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1569	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₇×#335	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₇×#334	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₃₇	—	2	1	no	—
8	C₂²×M₁₆×C₃₇	C₄	1	2	yes	—
9	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
10	C₃₇×#432	—	2	2	no	—
11	C₃₇×#432	C₂²	1	4	yes	—
12	C₃₇×#423	—	2	2	no	—
13	C₃₇×#423	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₃₇	—	2	1	no	—
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₃₇	—	2	1	no	—
19	C₄×M₁₆×C₃₇	C₄	1	4	yes	—
20	C₄×M₁₆×C₃₇	C₂²	1	4	yes	—
21	C₂×#97×C₃₇	—	2	1	no	—
22	C₂×#97×C₃₇	C₄	1	1	yes	characteristic
23	C₂×#97×C₃₇	C₄	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
25	C₃₇×#335	C₂²	1	8	yes	complemented
26	C₃₇×#334	C₂²	1	8	yes	complemented
27	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₇	Q₈	1	1	yes	characteristic
30	C₂×M₁₆×C₃₇	—	2	4	no	—
31	C₂×M₁₆×C₃₇	—	2	8	no	—
32	C₂×M₁₆×C₃₇	—	4	4	no	—
33	C₂×M₁₆×C₃₇	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₃₇	—	2	2	no	—
35	C₂²×C₈×C₃₇	—	2	1	no	—
36	C₂²×C₈×C₃₇	—	2	1	no	—
37	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×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	D₄	1	2	yes	—
42	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₇	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₃₇	Q₈	1	2	yes	—
45	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₃₇	—	2	4	no	—
48	C₄⋊C₈×C₃₇	—	2	2	no	—
49	C₄⋊C₈×C₃₇	—	2	1	no	—
50	C₄⋊C₈×C₃₇	—	2	1	no	—
51	C₄⋊C₈×C₃₇	C₂³	1	2	yes	—
52	C₄⋊C₈×C₃₇	C₂³	1	4	yes	—
53	C₄⋊C₈×C₃₇	C₂³	1	2	yes	—
54	C₂²⋊C₈×C₃₇	—	2	4	no	—
55	#97×C₃₇	—	2	4	no	—
56	#97×C₃₇	C₂×C₄	1	4	yes	—
57	#97×C₃₇	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₃₇	C₂³	1	4	yes	—
60	C₂⁴×C₃₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	—	2	4	no	—
62	C₂²×C₄×C₃₇	—	2	2	no	—
63	C₂²×C₄×C₃₇	—	2	2	no	—
64	C₂²×C₄×C₃₇	—	4	2	no	—
65	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
77	M₁₆×C₃₇	—	2	16	no	—
78	M₁₆×C₃₇	—	4	8	no	—
79	C₂×C₈×C₃₇	—	2	4	no	—
80	C₂×C₈×C₃₇	—	2	2	no	—
81	C₂×C₈×C₃₇	—	2	2	no	—
82	C₂×C₈×C₃₇	—	2	4	no	—
83	C₂×C₈×C₃₇	—	4	2	no	—
84	C₂×C₈×C₃₇	—	4	4	no	—
85	C₂×C₈×C₃₇	—	4	2	no	—
86	C₂×C₈×C₃₇	M₁₆	1	8	yes	complemented
87	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₃₇	C₄⋊C₄	1	2	yes	—
90	C₄²×C₃₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₃₇	C₄⋊C₄	1	2	yes	—
92	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₃₇	C₂×D₄	1	2	yes	—
95	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
97	C₄²×C₃₇	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
99	C₂³×C₃₇	—	2	2	no	—
100	C₂³×C₃₇	—	2	2	no	—
101	C₂³×C₃₇	—	2	2	no	—
102	C₂³×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₇	—	2	2	no	—
105	C₂×C₄×C₃₇	—	2	2	no	—
106	C₂×C₄×C₃₇	—	2	2	no	—
107	C₂×C₄×C₃₇	—	2	2	no	—
108	C₂×C₄×C₃₇	—	2	4	no	—
109	C₂×C₄×C₃₇	—	2	2	no	—
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	—	2	4	no	—
112	C₂×C₄×C₃₇	—	2	4	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	2	no	—
115	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₃₇	#137	1	2	yes	—
124	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₃₇	—	2	4	no	—
128	C₈×C₃₇	—	4	2	no	—
129	C₈×C₃₇	—	4	4	no	—
130	C₈×C₃₇	—	4	2	no	—
131	C₈×C₃₇	C₂×M₁₆	1	8	yes	complemented
132	C₂×#1569	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₃₇	—	2	2	no	—
134	C₂²×C₃₇	—	2	2	no	—
135	C₂²×C₃₇	—	2	2	no	—
136	C₂²×C₃₇	—	4	4	no	—
137	C₂²×C₃₇	#423	1	2	yes	—
138	C₂²×C₃₇	#428	1	2	yes	—
139	C₂²×C₃₇	#430	1	2	yes	—
140	C₂²×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₃₇	—	2	2	no	—
142	C₄×C₃₇	—	2	2	no	—
143	C₄×C₃₇	—	2	2	no	—
144	C₄×C₃₇	—	2	1	no	—
145	C₄×C₃₇	—	2	1	no	—
146	C₄×C₃₇	#423	1	2	yes	—
147	C₄×C₃₇	#428	1	2	yes	—
148	C₄×C₃₇	C₂²×M₁₆	1	2	yes	—
149	C₄×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₄×C₃₇	C₂×#137	1	1	yes	characteristic
151	C₂×#432	C₂×C₃₇	1	1	yes	characteristic
152	C₂×#423	C₂×C₃₇	1	1	yes	characteristic
153	C₂×C₄×M₁₆	C₂×C₃₇	1	1	yes	characteristic
154	#1569	C₂×C₃₇	1	8	yes	direct product factor
155	C₂×#335	C₂×C₃₇	1	2	yes	complemented
156	C₂×#334	C₂×C₃₇	1	2	yes	complemented
157	C₂×C₃₇	—	4	2	no	—
158	C₂×C₃₇	#1569	1	4	yes	direct product factor
159	C₂×C₃₇	C₂×#423	1	1	yes	characteristic
160	C₂×C₃₇	C₂×#428	1	1	yes	characteristic
161	C₂×C₃₇	C₂×#430	1	1	yes	characteristic
162	C₂²×M₁₆	—	2	1	no	—
163	C₂²×M₁₆	C₄×C₃₇	1	2	yes	—
164	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
165	#432	—	2	2	no	—
166	#432	C₂²×C₃₇	1	4	yes	—
167	#423	—	2	2	no	—
168	#423	C₂²×C₃₇	1	4	yes	—
169	C₂×C₄⋊C₈	—	2	1	no	—
170	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
173	C₂×C₂²⋊C₈	—	2	1	no	—
174	C₄×M₁₆	C₄×C₃₇	1	4	yes	—
175	C₄×M₁₆	C₂²×C₃₇	1	4	yes	—
176	C₂×#97	—	2	1	no	—
177	C₂×#97	C₄×C₃₇	1	1	yes	characteristic
178	C₂×#97	C₄×C₃₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
180	#335	C₂²×C₃₇	1	8	yes	complemented
181	#334	C₂²×C₃₇	1	8	yes	complemented
182	C₃₇	C₂×#1569	1	1	yes	characteristic, direct product factor, Sylow
183	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
184	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
185	C₂³×C₄	Q₈×C₃₇	1	1	yes	characteristic
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	2	8	no	—
188	C₂×M₁₆	—	4	4	no	—
189	C₂×M₁₆	C₂×C₄×C₃₇	1	8	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
194	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
197	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₃₇	1	2	yes	—
199	C₂×C₄²	Q₈×C₃₇	1	1	yes	characteristic
200	C₂×C₄²	Q₈×C₃₇	1	2	yes	—
201	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
203	C₄⋊C₈	—	2	4	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	C₂³×C₃₇	1	2	yes	—
208	C₄⋊C₈	C₂³×C₃₇	1	4	yes	—
209	C₄⋊C₈	C₂³×C₃₇	1	2	yes	—
210	C₂²⋊C₈	—	2	4	no	—
211	#97	—	2	4	no	—
212	#97	C₂×C₄×C₃₇	1	4	yes	—
213	#97	C₂×C₄×C₃₇	1	4	yes	—
214	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
215	C₄×C₈	C₂³×C₃₇	1	4	yes	—
216	C₂⁴	C₄⋊C₄×C₃₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
223	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
224	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
225	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
226	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
233	M₁₆	—	2	16	no	—
234	M₁₆	—	4	8	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	4	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	M₁₆×C₃₇	1	8	yes	complemented
243	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
244	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
245	C₄²	C₄⋊C₄×C₃₇	1	2	yes	—
246	C₄²	C₄⋊C₄×C₃₇	1	2	yes	—
247	C₄²	C₄⋊C₄×C₃₇	1	2	yes	—
248	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
249	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
250	C₄²	C₂×D₄×C₃₇	1	2	yes	—
251	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
252	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
253	C₄²	C₂×Q₈×C₃₇	1	1	yes	characteristic
254	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
259	C₂³	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
277	C₂×C₄	C₂×M₁₆×C₃₇	1	4	yes	—
278	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	2	yes	—
279	C₂×C₄	#137×C₃₇	1	2	yes	—
280	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
282	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	4	4	no	—
285	C₈	—	4	2	no	—
286	C₈	—	4	2	no	—
287	C₈	C₂×M₁₆×C₃₇	1	8	yes	complemented
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	C₃₇×#423	1	2	yes	—
293	C₂²	C₃₇×#428	1	2	yes	—
294	C₂²	C₃₇×#430	1	2	yes	—
295	C₂²	C₂²×C₄⋊C₄×C₃₇	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	C₃₇×#423	1	2	yes	—
302	C₄	C₃₇×#428	1	2	yes	—
303	C₄	C₂²×M₁₆×C₃₇	1	2	yes	—
304	C₄	C₂×AGL(1,ℤ/8)×C₃₇	1	1	yes	characteristic
305	C₄	C₂×#137×C₃₇	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	C₃₇×#1569	1	4	yes	direct product factor
308	C₂	C₂×C₃₇×#423	1	1	yes	characteristic
309	C₂	C₂×C₃₇×#428	1	1	yes	characteristic
310	C₂	C₂×C₃₇×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₄²	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₂²×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₇×#1569	C₂	direct (N × H)
5. C₂×C₃₇×#335	C₂	semidirect
6. C₂×C₃₇×#334	C₂	semidirect
25. C₃₇×#335	C₂²	semidirect
26. C₃₇×#334	C₂²	semidirect
86. C₂×C₈×C₃₇	M₁₆	semidirect
131. C₈×C₃₇	C₂×M₁₆	semidirect
132. C₂×#1569	C₃₇	direct (N × H)
154. #1569	C₂×C₃₇	direct (N × H)
155. C₂×#335	C₂×C₃₇	semidirect
156. C₂×#334	C₂×C₃₇	semidirect
158. C₂×C₃₇	#1569	direct (N × H)
180. #335	C₂²×C₃₇	semidirect
181. #334	C₂²×C₃₇	semidirect
182. C₃₇	C₂×#1569	direct (N × H)
242. C₂×C₈	M₁₆×C₃₇	semidirect
287. C₈	C₂×M₁₆×C₃₇	semidirect
307. C₂	C₃₇×#1569	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×#432	2
C₂×C₃₇×#423	2
C₂×C₄×M₁₆×C₃₇	2
C₃₇×#1569	2
C₂×C₃₇×#335	2
C₂×C₃₇×#334	2
C₂×#1569	37

Subgroups — C2×C37×#1569 #22639047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#1569 #22639047