Subgroups — #479×D₃₇ #18004369

Order 4736.

384 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×D₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557843	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557842	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557841	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557840	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557839	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557838	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557837	C₂	1	1	yes	maximal, characteristic, complemented
10	#107×D₃₇	C₂	1	1	yes	maximal, characteristic
11	#106×D₃₇	C₂	1	1	yes	maximal, characteristic
12	C₄⋊C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
13	#103×D₃₇	C₂	1	1	yes	maximal, characteristic
14	#103×D₃₇	C₂	1	1	yes	maximal, characteristic
15	C₃₇×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
19	#11924528	C₂²	1	1	yes	characteristic
20	#11924527	C₂²	1	1	yes	characteristic
21	#11924527	C₂²	1	1	yes	characteristic
22	#11924526	—	2	1	no	—
23	#11924526	—	2	1	no	—
24	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₇₄	—	2	1	no	—
26	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
27	#11924501	—	2	1	no	—
28	#11924501	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₃₇	—	2	1	no	—
30	C₄⋊C₄×D₃₇	—	2	1	no	—
31	C₄⋊C₄×D₃₇	—	2	1	no	—
32	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
35	C₄²×D₃₇	—	2	1	no	—
36	C₄²×D₃₇	C₂²	1	1	yes	characteristic
37	#11924470	C₂²	1	1	yes	characteristic
38	#11924469	C₂²	1	1	yes	characteristic
39	#11924469	C₂²	1	1	yes	characteristic
40	#11924468	C₂²	1	1	yes	characteristic
41	#11924467	C₂²	1	1	yes	characteristic
42	#11924467	C₂²	1	1	yes	characteristic
43	C₄⋊#64436	C₂²	1	1	yes	characteristic
44	#11924465	C₂²	1	1	yes	characteristic
45	#11924465	C₂²	1	1	yes	characteristic
46	#11924461	C₂²	1	1	yes	characteristic
47	#11924461	C₂²	1	1	yes	characteristic
48	#11924460	C₂²	1	1	yes	characteristic
49	#11924460	C₂²	1	1	yes	characteristic
50	#11924459	C₂²	1	1	yes	characteristic
51	#11924459	C₂²	1	1	yes	characteristic
52	#125×C₃₇	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
54	#107×C₃₇	C₂²	1	1	yes	characteristic
55	#106×C₃₇	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
57	#103×C₃₇	C₂²	1	1	yes	characteristic
58	#103×C₃₇	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
60	Q₈×D₃₇	—	2	2	no	—
61	Q₈×D₃₇	Q₈	1	4	yes	—
62	C₂×C₄×D₃₇	—	2	1	no	—
63	C₂×C₄×D₃₇	—	2	1	no	—
64	C₂×C₄×D₃₇	—	2	1	no	—
65	C₂×C₄×D₃₇	—	2	1	no	—
66	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
67	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
68	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
69	C₂×Dic₇₄	—	2	1	no	—
70	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₃₇	—	2	1	no	—
74	C₄⋊C₄×C₃₇	—	2	1	no	—
75	C₄⋊C₄×C₃₇	—	2	1	no	—
76	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
79	C₄²×C₃₇	—	2	1	no	—
80	C₄²×C₃₇	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₃₇	—	2	1	no	—
82	C₄⋊Dic₃₇	—	2	1	no	—
83	C₄⋊Dic₃₇	—	2	1	no	—
84	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
87	#10596936	—	2	1	no	—
88	#10596936	—	2	1	no	—
89	#10596936	—	2	1	no	—
90	#10596936	—	2	1	no	—
91	#10596936	—	2	1	no	—
92	#10596936	—	2	1	no	—
93	#10596936	—	2	1	no	—
94	#10596936	—	2	1	no	—
95	#10596936	—	2	1	no	—
96	C₄×Dic₃₇	—	2	1	no	—
97	C₄×Dic₃₇	—	2	1	no	—
98	C₄×Dic₃₇	—	2	1	no	—
99	C₄×Dic₃₇	D₄	1	1	yes	characteristic
100	C₄×Dic₃₇	D₄	1	1	yes	characteristic
101	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
102	C₂×#64436	C₂³	1	1	yes	characteristic
103	C₂×#64436	C₂³	1	1	yes	characteristic
104	C₈×D₃₇	—	2	2	no	—
105	C₈×D₃₇	—	2	2	no	—
106	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₃₇	—	2	1	no	—
108	Q₈×C₃₇	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₃₇	—	2	1	no	—
110	C₂×C₄×C₃₇	—	2	1	no	—
111	C₂×C₄×C₃₇	—	2	1	no	—
112	C₂×C₄×C₃₇	—	2	1	no	—
113	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₃₇	—	2	1	no	—
117	C₂×Dic₃₇	—	2	1	no	—
118	C₂×Dic₃₇	—	2	1	no	—
119	C₂×Dic₃₇	—	2	1	no	—
120	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
123	C₄×D₃₇	—	2	4	no	—
124	C₄×D₃₇	—	2	2	no	—
125	C₄×D₃₇	—	2	2	no	—
126	C₄×D₃₇	—	4	2	no	—
127	C₄×D₃₇	—	4	2	no	—
128	C₄×D₃₇	—	4	2	no	—
129	C₄×D₃₇	C₂×Q₈	1	2	yes	—
130	C₄×D₃₇	#41	1	2	yes	—
131	Dic₇₄	—	2	1	no	—
132	Dic₇₄	—	2	4	no	—
133	Dic₇₄	C₂×Q₈	1	2	yes	—
134	C₈×C₃₇	—	2	1	no	—
135	C₈×C₃₇	—	2	1	no	—
136	#64436	—	2	1	no	—
137	#64436	—	2	1	no	—
138	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₃₇	C₂²⋊Q₈	1	2	yes	—
140	C₂×D₃₇	#135	1	2	yes	—
141	C₂×D₃₇	#137	1	2	yes	—
142	C₄×C₃₇	—	2	2	no	—
143	C₄×C₃₇	—	2	1	no	—
144	C₄×C₃₇	—	2	1	no	—
145	C₄×C₃₇	—	4	1	no	—
146	C₄×C₃₇	—	4	1	no	—
147	C₄×C₃₇	—	4	1	no	—
148	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
150	Dic₃₇	—	2	2	no	—
151	Dic₃₇	—	2	1	no	—
152	Dic₃₇	—	2	1	no	—
153	Dic₃₇	—	4	1	no	—
154	Dic₃₇	—	4	1	no	—
155	Dic₃₇	—	4	1	no	—
156	Dic₃₇	C₂²⋊Q₈	1	2	yes	—
157	C₂×#479	—	37	1	no	maximal, Sylow
158	C₂×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
160	C₂×C₃₇	C₂×#137	1	1	yes	characteristic
161	D₃₇	#479	1	4	yes	direct product factor
162	C₂×#125	—	37	1	no	—
163	C₂×C₄×Q₈	—	37	1	no	—
164	#479	—	37	1	no	—
165	#479	—	37	1	no	—
166	#479	—	37	1	no	—
167	#479	—	37	1	no	—
168	#479	—	37	1	no	—
169	#479	—	37	1	no	—
170	#479	—	37	1	no	—
171	#479	D₃₇	1	1	yes	characteristic, direct product factor
172	C₂×#107	—	37	1	no	—
173	C₂×#106	—	37	1	no	—
174	C₂×C₄⋊C₈	—	37	1	no	—
175	C₂×#103	—	37	1	no	—
176	C₂×#103	—	37	1	no	—
177	C₃₇	C₂×#479	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	—	37	1	no	—
179	C₂²×C₈	—	37	1	no	—
180	C₂²×C₈	—	37	1	no	—
181	#125	—	37	1	no	—
182	#125	—	37	1	no	—
183	#125	—	37	1	no	—
184	#125	—	74	1	no	—
185	#125	—	74	1	no	—
186	#125	C₂×D₃₇	1	1	yes	characteristic
187	C₄×Q₈	—	37	1	no	—
188	C₄×Q₈	—	37	1	no	—
189	C₄×Q₈	—	37	1	no	—
190	C₄×Q₈	—	74	1	no	—
191	C₄×Q₈	—	74	1	no	—
192	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	37	1	no	—
194	C₂×C₄⋊C₄	—	37	1	no	—
195	C₂×C₄⋊C₄	—	37	1	no	—
196	C₂×C₄⋊C₄	—	74	1	no	—
197	C₂×C₄⋊C₄	—	74	1	no	—
198	C₂×C₄⋊C₄	—	74	1	no	—
199	C₂×C₄²	—	37	1	no	—
200	C₂×C₄²	—	74	1	no	—
201	#107	—	37	1	no	—
202	#107	—	37	1	no	—
203	#107	—	37	1	no	—
204	#107	C₂×D₃₇	1	1	yes	characteristic
205	#106	—	37	1	no	—
206	#106	—	37	1	no	—
207	#106	—	37	1	no	—
208	#106	C₂×D₃₇	1	1	yes	characteristic
209	C₄⋊C₈	—	37	1	no	—
210	C₄⋊C₈	—	37	1	no	—
211	C₄⋊C₈	—	37	1	no	—
212	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
213	#103	—	37	1	no	—
214	#103	—	37	1	no	—
215	#103	—	37	1	no	—
216	#103	—	37	1	no	—
217	#103	—	37	1	no	—
218	#103	—	37	1	no	—
219	#103	C₂×D₃₇	1	1	yes	characteristic
220	#103	C₂×D₃₇	1	1	yes	characteristic
221	C₂×Q₈	—	37	4	no	—
222	C₂×Q₈	—	37	1	no	—
223	C₂×Q₈	—	74	1	no	—
224	C₂×Q₈	—	74	2	no	—
225	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
226	C₂²×C₄	—	37	1	no	—
227	C₂²×C₄	—	37	1	no	—
228	C₂²×C₄	—	37	1	no	—
229	C₂²×C₄	—	74	1	no	—
230	C₂²×C₄	—	74	1	no	—
231	C₂²×C₄	—	74	1	no	—
232	C₂²×C₄	—	74	1	no	—
233	C₂×C₈	—	37	1	no	—
234	C₂×C₈	—	37	1	no	—
235	C₂×C₈	—	74	2	no	—
236	C₂×C₈	—	74	2	no	—
237	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	37	1	no	—
243	C₄⋊C₄	—	37	1	no	—
244	C₄⋊C₄	—	37	1	no	—
245	C₄⋊C₄	—	74	1	no	—
246	C₄⋊C₄	—	74	1	no	—
247	C₄⋊C₄	—	74	1	no	—
248	C₄⋊C₄	—	74	1	no	—
249	C₄⋊C₄	—	74	1	no	—
250	C₄⋊C₄	—	74	1	no	—
251	C₄⋊C₄	—	74	1	no	—
252	C₄⋊C₄	—	74	1	no	—
253	C₄⋊C₄	—	74	1	no	—
254	C₄⋊C₄	—	74	1	no	—
255	C₄⋊C₄	—	74	1	no	—
256	C₄⋊C₄	—	74	1	no	—
257	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
260	C₄²	—	2	1	no	—
261	C₄²	—	37	1	no	—
262	C₄²	—	37	1	no	—
263	C₄²	—	37	1	no	—
264	C₄²	—	74	1	no	—
265	C₄²	—	74	1	no	—
266	C₄²	—	74	1	no	—
267	C₄²	C₂²×D₃₇	1	1	yes	characteristic
268	C₂³	—	37	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	37	2	no	—
271	Q₈	—	74	4	no	—
272	Q₈	—	74	1	no	—
273	Q₈	Q₈×D₃₇	1	2	yes	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	37	1	no	—
279	C₂×C₄	—	37	2	no	—
280	C₂×C₄	—	37	2	no	—
281	C₂×C₄	—	37	1	no	—
282	C₂×C₄	—	37	1	no	—
283	C₂×C₄	—	74	2	no	—
284	C₂×C₄	—	74	2	no	—
285	C₂×C₄	—	74	4	no	—
286	C₂×C₄	—	74	1	no	—
287	C₂×C₄	—	74	1	no	—
288	C₂×C₄	—	74	1	no	—
289	C₂×C₄	—	74	1	no	—
290	C₂×C₄	—	148	2	no	—
291	C₂×C₄	—	148	2	no	—
292	C₂×C₄	—	148	2	no	—
293	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
294	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	74	1	no	—
299	C₈	—	74	1	no	—
300	C₂²	—	37	2	no	—
301	C₂²	—	37	2	no	—
302	C₂²	—	37	2	no	—
303	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	37	2	no	—
311	C₄	—	74	2	no	—
312	C₄	—	74	1	no	—
313	C₄	—	74	1	no	—
314	C₄	—	148	1	no	—
315	C₄	—	148	1	no	—
316	C₄	—	148	1	no	—
317	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
318	C₄	#41×D₃₇	1	1	yes	characteristic
319	C₂	—	37	4	no	—
320	C₂	C₂²⋊Q₈×D₃₇	1	1	yes	characteristic
321	C₂	#135×D₃₇	1	1	yes	characteristic
322	C₂	#137×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#479	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂²×D₃₇	C₂×D₄
hypercenter	#479	D₃₇
hypocenter	C₃₇	C₂×#479
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #14557843	C₂	semidirect
4. #14557842	C₂	semidirect
5. #14557841	C₂	semidirect
6. #14557840	C₂	semidirect
7. #14557839	C₂	semidirect
8. #14557838	C₂	semidirect
9. #14557837	C₂	semidirect
15. C₃₇×#479	C₂	semidirect
161. D₃₇	#479	direct (N × H)
171. #479	D₃₇	direct (N × H)
177. C₃₇	C₂×#479	semidirect

Maximal subgroups

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

subgroup	index
#125×D₃₇	2
C₄×Q₈×D₃₇	2
#14557843	2
#14557842	2
#14557841	2
#14557840	2
#14557839	2
#14557838	2
#14557837	2
#107×D₃₇	2
#106×D₃₇	2
C₄⋊C₈×D₃₇	2
#103×D₃₇	2
C₃₇×#479	2
C₂×#479	37

Subgroups — #479×D37 #18004369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #479×D₃₇ #18004369