Subgroups — C₂×C₃₇×#1589 #22639136

Order 9472.

822 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#436	C₂	1	2	yes	maximal, complemented
3	C₃₇×#1589	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₇×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₈²×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₃₇	C₄	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₇	—	2	2	no	—
9	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
10	C₄×D₈×C₃₇	C₄	1	4	yes	—
11	C₄×D₈×C₃₇	C₂²	1	4	yes	—
12	C₃₇×#436	—	2	4	no	—
13	C₃₇×#436	C₂²	1	8	yes	complemented
14	C₂×#107×C₃₇	C₄	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
16	C₂×#102×C₃₇	C₄	1	2	yes	—
17	C₂×C₂²⋊C₈×C₃₇	—	2	2	no	—
18	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#335	C₂²	1	4	yes	complemented
22	C₃₇×D₄⋊C₈	C₂²	1	8	yes	complemented
23	C₈²×C₃₇	C₂²	1	4	yes	complemented
24	C₂²×D₄×C₃₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₃₇	—	2	2	no	—
26	C₂×D₈×C₃₇	—	2	4	no	—
27	C₂×D₈×C₃₇	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₃₇	—	2	8	no	—
29	C₂×M₁₆×C₃₇	—	4	8	no	—
30	C₂²×C₈×C₃₇	—	2	1	no	—
31	C₂²×C₈×C₃₇	—	2	2	no	—
32	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
36	C₄×D₄×C₃₇	—	2	2	no	—
37	C₄×D₄×C₃₇	—	2	2	no	—
38	C₄×D₄×C₃₇	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₃₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
42	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
43	#107×C₃₇	C₂×C₄	1	4	yes	—
44	C₄⋊C₈×C₃₇	—	2	2	no	—
45	C₄⋊C₈×C₃₇	C₂³	1	4	yes	—
46	#102×C₃₇	C₂×C₄	1	8	yes	—
47	C₂²⋊C₈×C₃₇	—	2	8	no	—
48	C₄×C₈×C₃₇	—	2	1	no	—
49	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₃₇	D₄	1	2	yes	—
51	C₄×C₈×C₃₇	C₂³	1	2	yes	—
52	C₄×C₈×C₃₇	C₂³	1	2	yes	—
53	C₄×C₈×C₃₇	C₂³	1	2	yes	—
54	C₂⁴×C₃₇	—	2	2	no	—
55	C₂×D₄×C₃₇	—	2	2	no	—
56	C₂×D₄×C₃₇	—	2	4	no	—
57	C₂×D₄×C₃₇	—	2	4	no	—
58	C₂×D₄×C₃₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₃₇	—	2	2	no	—
60	C₂²×C₄×C₃₇	—	2	4	no	—
61	C₂²×C₄×C₃₇	—	2	4	no	—
62	C₂²×C₄×C₃₇	—	4	4	no	—
63	C₂²×C₄×C₃₇	—	4	4	no	—
64	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
67	D₈×C₃₇	—	2	8	no	—
68	M₁₆×C₃₇	—	4	16	no	—
69	C₂×C₈×C₃₇	—	2	4	no	—
70	C₂×C₈×C₃₇	—	2	2	no	—
71	C₂×C₈×C₃₇	—	2	2	no	—
72	C₂×C₈×C₃₇	—	2	2	no	—
73	C₂×C₈×C₃₇	—	2	4	no	—
74	C₂×C₈×C₃₇	—	4	4	no	—
75	C₂×C₈×C₃₇	M₁₆	1	4	yes	complemented
76	C₂×C₈×C₃₇	D₈	1	4	yes	complemented
77	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
81	C₄⋊C₄×C₃₇	—	2	2	no	—
82	C₄⋊C₄×C₃₇	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₃₇	—	2	4	no	—
84	C₂²⋊C₄×C₃₇	—	2	4	no	—
85	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₃₇	#41	1	1	yes	characteristic
88	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
89	C₂³×C₃₇	—	2	4	no	—
90	C₂³×C₃₇	—	4	4	no	—
91	C₂³×C₃₇	—	4	4	no	—
92	C₂³×C₃₇	—	4	4	no	—
93	C₂³×C₃₇	C₄×D₄	1	1	yes	characteristic
94	D₄×C₃₇	—	2	4	no	—
95	D₄×C₃₇	—	2	4	no	—
96	D₄×C₃₇	—	4	4	no	—
97	C₂×C₄×C₃₇	—	2	2	no	—
98	C₂×C₄×C₃₇	—	2	4	no	—
99	C₂×C₄×C₃₇	—	4	4	no	—
100	C₂×C₄×C₃₇	—	4	4	no	—
101	C₂×C₄×C₃₇	—	4	4	no	—
102	C₂×C₄×C₃₇	—	4	8	no	—
103	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₇	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₃₇	#131	1	2	yes	—
108	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₃₇	#135	1	2	yes	—
110	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
113	C₈×C₃₇	—	2	4	no	—
114	C₈×C₃₇	—	2	2	no	—
115	C₈×C₃₇	—	2	2	no	—
116	C₈×C₃₇	—	4	4	no	—
117	C₈×C₃₇	C₂×M₁₆	1	4	yes	complemented
118	C₈×C₃₇	C₂×D₈	1	4	yes	complemented
119	C₂×#1589	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₃₇	—	4	4	no	—
121	C₂²×C₃₇	—	4	4	no	—
122	C₂²×C₃₇	—	4	4	no	—
123	C₂²×C₃₇	—	8	8	no	—
124	C₂²×C₃₇	#436	1	2	yes	—
125	C₂²×C₃₇	C₄×D₈	1	2	yes	—
126	C₂²×C₃₇	#443	1	2	yes	—
127	C₂²×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₃₇	—	2	1	no	—
129	C₄×C₃₇	—	2	1	no	—
130	C₄×C₃₇	—	4	4	no	—
131	C₄×C₃₇	#436	1	2	yes	—
132	C₄×C₃₇	C₄×D₈	1	2	yes	—
133	C₄×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
134	C₄×C₃₇	C₂×#131	1	1	yes	characteristic
135	C₄×C₃₇	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₃₇	C₂×#135	1	1	yes	characteristic
137	C₂×C₄×D₈	C₂×C₃₇	1	1	yes	characteristic
138	C₂×#436	C₂×C₃₇	1	2	yes	complemented
139	#1589	C₂×C₃₇	1	8	yes	direct product factor
140	C₂×#335	C₂×C₃₇	1	1	yes	characteristic, complemented
141	C₂×D₄⋊C₈	C₂×C₃₇	1	2	yes	complemented
142	C₂×C₈²	C₂×C₃₇	1	1	yes	characteristic, complemented
143	C₂×C₃₇	—	8	4	no	—
144	C₂×C₃₇	#1589	1	4	yes	direct product factor
145	C₂×C₃₇	C₂×#436	1	1	yes	characteristic
146	C₂×C₃₇	C₂×C₄×D₈	1	1	yes	characteristic
147	C₂×C₃₇	C₂×#443	1	1	yes	characteristic
148	C₂²×D₈	C₄×C₃₇	1	1	yes	characteristic
149	C₂²×M₁₆	—	2	2	no	—
150	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
151	C₄×D₈	C₄×C₃₇	1	4	yes	—
152	C₄×D₈	C₂²×C₃₇	1	4	yes	—
153	#436	—	2	4	no	—
154	#436	C₂²×C₃₇	1	8	yes	complemented
155	C₂×#107	C₄×C₃₇	1	1	yes	characteristic
156	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
157	C₂×#102	C₄×C₃₇	1	2	yes	—
158	C₂×C₂²⋊C₈	—	2	2	no	—
159	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
162	#335	C₂²×C₃₇	1	4	yes	complemented
163	D₄⋊C₈	C₂²×C₃₇	1	8	yes	complemented
164	C₈²	C₂²×C₃₇	1	4	yes	complemented
165	C₃₇	C₂×#1589	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×D₄	C₂×C₄×C₃₇	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂×D₈	—	2	4	no	—
169	C₂×D₈	C₂×C₄×C₃₇	1	4	yes	—
170	C₂×M₁₆	—	2	8	no	—
171	C₂×M₁₆	—	4	8	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	C₂×C₄×C₃₇	1	4	yes	—
181	C₄×D₄	C₂³×C₃₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
185	#107	C₂×C₄×C₃₇	1	4	yes	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	C₂³×C₃₇	1	4	yes	—
188	#102	C₂×C₄×C₃₇	1	8	yes	—
189	C₂²⋊C₈	—	2	8	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
192	C₄×C₈	D₄×C₃₇	1	2	yes	—
193	C₄×C₈	C₂³×C₃₇	1	2	yes	—
194	C₄×C₈	C₂³×C₃₇	1	2	yes	—
195	C₄×C₈	C₂³×C₃₇	1	2	yes	—
196	C₂⁴	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	C₂²×C₄×C₃₇	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
208	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
209	D₈	—	2	8	no	—
210	M₁₆	—	4	16	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	M₁₆×C₃₇	1	4	yes	complemented
218	C₂×C₈	D₈×C₃₇	1	4	yes	complemented
219	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
220	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂²×C₄×C₃₇	1	4	yes	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
228	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
229	C₄²	#41×C₃₇	1	1	yes	characteristic
230	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
231	C₂³	—	2	4	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	C₄×D₄×C₃₇	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
246	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
248	C₂×C₄	C₂×M₁₆×C₃₇	1	2	yes	—
249	C₂×C₄	#131×C₃₇	1	2	yes	—
250	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
251	C₂×C₄	#135×C₃₇	1	2	yes	—
252	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	4	4	no	—
259	C₈	C₂×M₁₆×C₃₇	1	4	yes	complemented
260	C₈	C₂×D₈×C₃₇	1	4	yes	complemented
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	8	8	no	—
265	C₂²	C₃₇×#436	1	2	yes	—
266	C₂²	C₄×D₈×C₃₇	1	2	yes	—
267	C₂²	C₃₇×#443	1	2	yes	—
268	C₂²	C₂×C₄×D₄×C₃₇	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	C₃₇×#436	1	2	yes	—
273	C₄	C₄×D₈×C₃₇	1	2	yes	—
274	C₄	C₂²×M₁₆×C₃₇	1	1	yes	characteristic
275	C₄	C₂×#131×C₃₇	1	1	yes	characteristic
276	C₄	C₂²×D₈×C₃₇	1	1	yes	characteristic
277	C₄	C₂×#135×C₃₇	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	C₃₇×#1589	1	4	yes	direct product factor
280	C₂	C₂×C₃₇×#436	1	1	yes	characteristic
281	C₂	C₂×C₄×D₈×C₃₇	1	1	yes	characteristic
282	C₂	C₂×C₃₇×#443	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	C₄×D₄
prime radical	C₂²×D₈×C₃₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₂×C₄×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#436	C₂	semidirect
3. C₃₇×#1589	C₂	direct (N × H)
4. C₂×C₃₇×#335	C₂	semidirect
5. C₂×C₃₇×D₄⋊C₈	C₂	semidirect
6. C₂×C₈²×C₃₇	C₂	semidirect
13. C₃₇×#436	C₂²	semidirect
21. C₃₇×#335	C₂²	semidirect
22. C₃₇×D₄⋊C₈	C₂²	semidirect
23. C₈²×C₃₇	C₂²	semidirect
75. C₂×C₈×C₃₇	M₁₆	semidirect
76. C₂×C₈×C₃₇	D₈	semidirect
117. C₈×C₃₇	C₂×M₁₆	semidirect
118. C₈×C₃₇	C₂×D₈	semidirect
119. C₂×#1589	C₃₇	direct (N × H)
138. C₂×#436	C₂×C₃₇	semidirect
139. #1589	C₂×C₃₇	direct (N × H)
140. C₂×#335	C₂×C₃₇	semidirect
141. C₂×D₄⋊C₈	C₂×C₃₇	semidirect
142. C₂×C₈²	C₂×C₃₇	semidirect
144. C₂×C₃₇	#1589	direct (N × H)
154. #436	C₂²×C₃₇	semidirect
162. #335	C₂²×C₃₇	semidirect
163. D₄⋊C₈	C₂²×C₃₇	semidirect
164. C₈²	C₂²×C₃₇	semidirect
165. C₃₇	C₂×#1589	direct (N × H)
217. C₂×C₈	M₁₆×C₃₇	semidirect
218. C₂×C₈	D₈×C₃₇	semidirect
259. C₈	C₂×M₁₆×C₃₇	semidirect
260. C₈	C₂×D₈×C₃₇	semidirect
279. C₂	C₃₇×#1589	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×C₄×D₈×C₃₇	2
C₂×C₃₇×#436	2
C₃₇×#1589	2
C₂×C₃₇×#335	2
C₂×C₃₇×D₄⋊C₈	2
C₂×C₈²×C₃₇	2
C₂×#1589	37

Subgroups — C2×C37×#1589 #22639136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#1589 #22639136