Subgroups — C₈×D₄×D₃₇ #18003265

Order 4736.

616 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	C₂²×C₈×D₃₇	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₃₇	C₂	1	1	yes	maximal, characteristic
3	C₈×D₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×#64436	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557598	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₃₇	C₂	1	2	yes	maximal, complemented
7	#14557596	C₂	1	2	yes	maximal, complemented
8	C₄⋊C₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊C₈×D₃₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₃₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
14	C₂²×#64436	C₂²	1	2	yes	complemented
15	C₂×C₈×D₃₇	—	2	8	no	—
16	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₃₇	C₂²	1	2	yes	—
18	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₃₇	C₂²	1	2	yes	complemented
20	C₂×C₈×D₃₇	C₂²	1	2	yes	complemented
21	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
22	D₄×Dic₃₇	C₄	1	1	yes	characteristic
23	#11924498	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₃₇	C₂²	1	2	yes	—
25	#11924496	C₄	1	2	yes	—
26	C₄⋊C₄×D₃₇	C₄	1	1	yes	characteristic
27	C₂²⋊C₄×D₃₇	C₄	1	2	yes	—
28	C₄²×D₃₇	C₂²	1	1	yes	characteristic
29	C₄⋊#64436	C₂²	1	1	yes	characteristic, complemented
30	#11924465	C₂²	1	2	yes	complemented
31	C₂²⋊#64436	C₂²	1	2	yes	complemented
32	#11924448	C₂²	1	2	yes	complemented
33	#11924448	C₂²	1	2	yes	complemented
34	C₄×#64436	C₂²	1	1	yes	characteristic, complemented
35	C₈×Dic₃₇	C₂²	1	1	yes	characteristic, complemented
36	C₈×Dic₃₇	C₂²	1	1	yes	characteristic, complemented
37	C₂²×C₈×C₃₇	C₂²	1	2	yes	complemented
38	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊C₈×C₃₇	C₂²	1	2	yes	complemented
41	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₃₇	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
46	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
47	D₄×D₃₇	C₈	1	8	yes	direct product factor
48	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₃₇	—	2	4	no	—
50	C₂×C₄×D₃₇	—	2	4	no	—
51	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₃₇	C₂³	1	2	yes	—
56	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₃₇	—	2	4	no	—
58	C₂×C₈×C₃₇	C₂³	1	2	yes	—
59	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
63	C₄²×C₃₇	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
65	#10596938	C₂×C₄	1	2	yes	—
66	#10596938	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
68	#10596936	C₂×C₄	1	2	yes	—
69	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
72	C₂×#64436	—	2	4	no	—
73	C₂×#64436	C₂³	1	2	yes	—
74	C₂×#64436	C₂³	1	1	yes	characteristic
75	C₂×#64436	C₂³	1	1	yes	characteristic
76	C₈×D₃₇	—	2	4	no	—
77	C₈×D₃₇	—	2	2	no	—
78	C₈×D₃₇	—	2	4	no	—
79	C₈×D₃₇	—	2	4	no	—
80	C₈×D₃₇	D₄	1	4	yes	direct product factor
81	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
82	C₂²×D₃₇	—	2	4	no	—
83	C₂²×D₃₇	—	2	4	no	—
84	C₂²×D₃₇	C₂×C₈	1	8	yes	complemented
85	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
86	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
87	D₄×C₃₇	C₂×C₈	1	4	yes	complemented
88	C₂×C₄×C₃₇	—	2	2	no	—
89	C₂×C₄×C₃₇	—	2	2	no	—
90	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₃₇	C₂×C₈	1	8	yes	complemented
95	C₂×Dic₃₇	—	2	4	no	—
96	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
98	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
100	D₁₄₈	C₂×C₈	1	4	yes	complemented
101	C₄×D₃₇	—	2	4	no	—
102	C₄×D₃₇	—	2	2	no	—
103	C₄×D₃₇	—	2	2	no	—
104	C₄×D₃₇	—	2	2	no	—
105	C₄×D₃₇	—	2	4	no	—
106	C₄×D₃₇	C₂×C₈	1	4	yes	complemented
107	C₄×D₃₇	C₂×D₄	1	2	yes	—
108	C₄×D₃₇	#41	1	2	yes	—
109	C₈×C₃₇	—	2	2	no	—
110	C₈×C₃₇	—	2	1	no	—
111	C₈×C₃₇	C₂×D₄	1	2	yes	complemented
112	#64436	—	2	2	no	—
113	#64436	—	2	1	no	—
114	#64436	C₂×D₄	1	2	yes	complemented
115	C₂²×C₃₇	—	2	2	no	—
116	C₂²×C₃₇	—	2	2	no	—
117	C₂²×C₃₇	C₂²×C₈	1	4	yes	complemented
118	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₃₇	—	2	2	no	—
120	C₂×D₃₇	—	2	2	no	—
121	C₂×D₃₇	—	2	16	no	—
122	C₂×D₃₇	C₄×D₄	1	2	yes	—
123	C₂×D₃₇	C₂²×C₈	1	2	yes	—
124	C₂×D₃₇	C₂²×C₈	1	4	yes	complemented
125	C₂×D₃₇	#131	1	2	yes	—
126	C₄×C₃₇	—	2	2	no	—
127	C₄×C₃₇	—	2	1	no	—
128	C₄×C₃₇	C₂²×C₈	1	2	yes	complemented
129	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
131	Dic₃₇	—	2	2	no	—
132	Dic₃₇	—	2	1	no	—
133	Dic₃₇	C₄×D₄	1	2	yes	—
134	Dic₃₇	C₂²×C₈	1	2	yes	complemented
135	C₂×C₈×D₄	—	37	1	no	maximal, Sylow
136	C₂×C₃₇	—	2	4	no	—
137	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₃₇	C₂³×C₈	1	1	yes	characteristic
139	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
140	D₃₇	—	2	4	no	—
141	D₃₇	C₈×D₄	1	4	yes	direct product factor
142	C₂³×C₈	—	37	2	no	—
143	C₂×C₄×D₄	—	37	1	no	—
144	C₈×D₄	—	37	2	no	—
145	C₈×D₄	—	37	1	no	—
146	C₈×D₄	—	37	1	no	—
147	C₈×D₄	—	37	2	no	—
148	C₈×D₄	—	37	1	no	—
149	C₈×D₄	D₃₇	1	1	yes	characteristic, direct product factor
150	C₂×C₄⋊C₈	—	37	1	no	—
151	C₂×C₂²⋊C₈	—	37	2	no	—
152	C₂×C₄×C₈	—	37	1	no	—
153	C₃₇	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
154	C₂²×D₄	—	37	1	no	—
155	C₂³×C₄	—	37	2	no	—
156	C₂²×C₈	—	37	1	no	—
157	C₂²×C₈	—	37	2	no	—
158	C₂²×C₈	—	37	2	no	—
159	C₂²×C₈	—	37	2	no	—
160	C₂²×C₈	—	37	1	no	—
161	C₂²×C₈	—	37	2	no	—
162	C₂²×C₈	—	74	8	no	—
163	C₂²×C₈	C₂×D₃₇	1	2	yes	complemented
164	C₄×D₄	—	37	1	no	—
165	C₄×D₄	—	37	2	no	—
166	C₄×D₄	—	37	1	no	—
167	C₄×D₄	—	37	2	no	—
168	C₄×D₄	—	37	1	no	—
169	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	37	1	no	—
171	C₂×C₂²⋊C₄	—	37	2	no	—
172	C₂×C₄²	—	37	1	no	—
173	C₄⋊C₈	—	37	2	no	—
174	C₄⋊C₈	—	37	1	no	—
175	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
176	C₂²⋊C₈	—	37	2	no	—
177	C₂²⋊C₈	—	37	2	no	—
178	C₂²⋊C₈	—	37	2	no	—
179	C₂²⋊C₈	C₂×D₃₇	1	2	yes	complemented
180	C₄×C₈	—	37	1	no	—
181	C₄×C₈	—	37	1	no	—
182	C₄×C₈	—	37	1	no	—
183	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
184	C₂⁴	—	37	2	no	—
185	C₂×D₄	—	37	2	no	—
186	C₂×D₄	—	37	8	no	—
187	C₂×D₄	—	37	1	no	—
188	C₂×D₄	C₄×D₃₇	1	1	yes	characteristic
189	C₂²×C₄	—	37	1	no	—
190	C₂²×C₄	—	37	1	no	—
191	C₂²×C₄	—	37	2	no	—
192	C₂²×C₄	—	37	1	no	—
193	C₂²×C₄	—	37	2	no	—
194	C₂²×C₄	—	37	2	no	—
195	C₂²×C₄	—	37	2	no	—
196	C₂²×C₄	—	74	4	no	—
197	C₂²×C₄	—	74	4	no	—
198	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	37	1	no	—
201	C₂×C₈	—	37	4	no	—
202	C₂×C₈	—	37	1	no	—
203	C₂×C₈	—	37	2	no	—
204	C₂×C₈	—	74	4	no	—
205	C₂×C₈	—	74	2	no	—
206	C₂×C₈	—	74	4	no	—
207	C₂×C₈	—	74	4	no	—
208	C₂×C₈	—	74	4	no	—
209	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₃₇	1	2	yes	—
211	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
212	C₄⋊C₄	—	37	2	no	—
213	C₄⋊C₄	—	37	1	no	—
214	C₄⋊C₄	C₄×D₃₇	1	1	yes	characteristic
215	C₂²⋊C₄	—	37	2	no	—
216	C₂²⋊C₄	—	37	2	no	—
217	C₂²⋊C₄	—	37	2	no	—
218	C₂²⋊C₄	C₄×D₃₇	1	2	yes	—
219	C₄²	—	37	1	no	—
220	C₄²	—	37	1	no	—
221	C₄²	—	37	1	no	—
222	C₄²	C₂²×D₃₇	1	1	yes	characteristic
223	C₂³	—	37	1	no	—
224	C₂³	—	37	8	no	—
225	C₂³	—	37	2	no	—
226	C₂³	—	74	4	no	—
227	C₂³	—	74	4	no	—
228	C₂³	C₂×C₄×D₃₇	1	2	yes	—
229	D₄	—	37	8	no	—
230	D₄	—	37	4	no	—
231	D₄	C₈×D₃₇	1	4	yes	direct product factor
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	37	1	no	—
235	C₂×C₄	—	37	1	no	—
236	C₂×C₄	—	37	4	no	—
237	C₂×C₄	—	37	2	no	—
238	C₂×C₄	—	37	2	no	—
239	C₂×C₄	—	37	1	no	—
240	C₂×C₄	—	37	2	no	—
241	C₂×C₄	—	74	4	no	—
242	C₂×C₄	—	74	4	no	—
243	C₂×C₄	—	74	2	no	—
244	C₂×C₄	—	74	4	no	—
245	C₂×C₄	—	74	2	no	—
246	C₂×C₄	—	74	2	no	—
247	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	37	2	no	—
254	C₈	—	74	2	no	—
255	C₈	—	74	1	no	—
256	C₈	D₄×D₃₇	1	2	yes	direct product factor
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	37	2	no	—
260	C₂²	—	37	2	no	—
261	C₂²	—	37	2	no	—
262	C₂²	—	37	4	no	—
263	C₂²	—	74	16	no	—
264	C₂²	—	74	2	no	—
265	C₂²	—	74	2	no	—
266	C₂²	C₂×C₈×D₃₇	1	4	yes	complemented
267	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	37	2	no	—
271	C₄	—	37	2	no	—
272	C₄	—	74	2	no	—
273	C₄	—	74	1	no	—
274	C₄	C₂×C₈×D₃₇	1	2	yes	complemented
275	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
276	C₄	#41×D₃₇	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	37	4	no	—
279	C₂	—	74	4	no	—
280	C₂	C₄×D₄×D₃₇	1	1	yes	characteristic
281	C₂	C₂²×C₈×D₃₇	1	1	yes	characteristic
282	C₂	#131×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₃₇
commutator	C₂×C₃₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₈×D₄×C₃₇	C₂
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂×D₄×D₃₇	C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₈×D₄	D₃₇
hypocenter	C₃₇	C₂×C₈×D₄
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₃₇	C₂	semidirect
3. C₈×D₁₄₈	C₂	semidirect
4. D₄×#64436	C₂	semidirect
5. #14557598	C₂	semidirect
6. C₈×C₂²⋊D₃₇	C₂	semidirect
7. #14557596	C₂	semidirect
8. C₄⋊C₈×D₃₇	C₂	semidirect
9. C₂²⋊C₈×D₃₇	C₂	semidirect
10. C₄×C₈×D₃₇	C₂	semidirect
11. C₈×D₄×C₃₇	C₂	semidirect
14. C₂²×#64436	C₂²	semidirect
19. C₂×C₈×D₃₇	C₂²	semidirect
20. C₂×C₈×D₃₇	C₂²	semidirect
29. C₄⋊#64436	C₂²	semidirect
30. #11924465	C₂²	semidirect
31. C₂²⋊#64436	C₂²	semidirect
32. #11924448	C₂²	semidirect
33. #11924448	C₂²	semidirect
34. C₄×#64436	C₂²	semidirect
35. C₈×Dic₃₇	C₂²	semidirect
36. C₈×Dic₃₇	C₂²	semidirect
37. C₂²×C₈×C₃₇	C₂²	semidirect
39. C₄⋊C₈×C₃₇	C₂²	semidirect
40. C₂²⋊C₈×C₃₇	C₂²	semidirect
41. C₄×C₈×C₃₇	C₂²	semidirect
47. D₄×D₃₇	C₈	direct (N × H)
80. C₈×D₃₇	D₄	direct (N × H)
84. C₂²×D₃₇	C₂×C₈	semidirect
87. D₄×C₃₇	C₂×C₈	semidirect
94. C₂²⋊D₃₇	C₂×C₈	semidirect
100. D₁₄₈	C₂×C₈	semidirect
106. C₄×D₃₇	C₂×C₈	semidirect
111. C₈×C₃₇	C₂×D₄	semidirect
114. #64436	C₂×D₄	semidirect
117. C₂²×C₃₇	C₂²×C₈	semidirect
124. C₂×D₃₇	C₂²×C₈	semidirect
128. C₄×C₃₇	C₂²×C₈	semidirect
134. Dic₃₇	C₂²×C₈	semidirect
141. D₃₇	C₈×D₄	direct (N × H)
149. C₈×D₄	D₃₇	direct (N × H)
153. C₃₇	C₂×C₈×D₄	semidirect
163. C₂²×C₈	C₂×D₃₇	semidirect
175. C₄⋊C₈	C₂×D₃₇	semidirect
179. C₂²⋊C₈	C₂×D₃₇	semidirect
183. C₄×C₈	C₂×D₃₇	semidirect
231. D₄	C₈×D₃₇	direct (N × H)
256. C₈	D₄×D₃₇	direct (N × H)
266. C₂²	C₂×C₈×D₃₇	semidirect
274. C₄	C₂×C₈×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₃₇	2
C₄×D₄×D₃₇	2
C₈×D₁₄₈	2
D₄×#64436	2
#14557598	2
C₈×C₂²⋊D₃₇	2
#14557596	2
C₄⋊C₈×D₃₇	2
C₂²⋊C₈×D₃₇	2
C₄×C₈×D₃₇	2
C₈×D₄×C₃₇	2
C₂×C₈×D₄	37

Subgroups — C8×D4×D37 #18003265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×D₄×D₃₇ #18003265