Subgroups — #472×D₃₇ #18004299

Order 4736.

488 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	AGL(1,ℤ/8)×D₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557804	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557803	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557802	C₂	1	2	yes	maximal, complemented
7	#14557801	C₂	1	2	yes	maximal, complemented
8	#14557800	C₂	1	1	yes	maximal, characteristic, complemented
9	#108×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#100×D₃₇	C₂	1	2	yes	maximal, complemented
11	C₃₇×#472	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₃₇	C₂²	1	1	yes	characteristic
13	C₂×D₄×D₃₇	C₂²	1	2	yes	—
14	#11924564	C₂²	1	2	yes	complemented
15	#11924563	—	2	2	no	—
16	#11924562	C₂²	1	2	yes	complemented
17	#11924561	C₂²	1	2	yes	complemented
18	#11924560	—	2	2	no	—
19	SD₁₆×D₃₇	—	2	2	no	—
20	C₂×D₂₉₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×#10596939	C₂²	1	2	yes	complemented
22	D₈×D₃₇	—	2	2	no	—
23	D₈×D₃₇	—	2	1	no	—
24	D₈×D₃₇	—	2	1	no	—
25	D₈×D₃₇	—	2	1	no	—
26	D₈×D₃₇	—	2	1	no	—
27	#11924544	—	2	1	no	—
28	#11924544	C₂²	1	1	yes	characteristic, complemented
29	#11924543	C₂²	1	1	yes	characteristic, complemented
30	#11924542	—	2	1	no	—
31	#11924542	C₂²	1	1	yes	characteristic, complemented
32	M₁₆×D₃₇	—	2	1	no	—
33	M₁₆×D₃₇	C₂²	1	1	yes	characteristic
34	M₁₆×D₃₇	C₂²	1	2	yes	—
35	C₂×C₈×D₃₇	—	2	1	no	—
36	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
37	#11924472	C₂²	1	1	yes	characteristic, complemented
38	#11924471	C₂²	1	2	yes	complemented
39	#11924453	C₂²	1	2	yes	complemented
40	#11924452	C₂²	1	2	yes	complemented
41	#11924452	C₂²	1	2	yes	complemented
42	AGL(1,ℤ/8)×C₃₇	C₂²	1	2	yes	complemented
43	C₂×D₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
44	#131×C₃₇	C₂²	1	1	yes	characteristic, complemented
45	#108×C₃₇	C₂²	1	1	yes	characteristic, complemented
46	#100×C₃₇	C₂²	1	2	yes	complemented
47	C₂³×D₃₇	—	2	2	no	—
48	#41×C₃₇	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₃₇	C₂³	1	2	yes	—
50	C₂×C₂²⋊D₃₇	—	2	2	no	—
51	#10596966	D₄	1	1	yes	characteristic
52	Q₈×D₃₇	D₄	1	1	yes	characteristic
53	#10596964	—	2	1	no	—
54	#10596964	D₄	1	1	yes	characteristic
55	D₄×D₃₇	—	2	1	no	—
56	D₄×D₃₇	—	2	2	no	—
57	D₄×D₃₇	—	2	2	no	—
58	D₄×D₃₇	—	2	2	no	—
59	D₄×D₃₇	—	2	2	no	—
60	D₄×D₃₇	D₄	1	1	yes	characteristic
61	#10596962	—	2	1	no	—
62	#10596962	C₂³	1	1	yes	characteristic
63	C₂×D₁₄₈	C₂³	1	2	yes	—
64	C₂×C₄×D₃₇	—	2	1	no	—
65	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
66	SD₁₆×C₃₇	—	2	2	no	—
67	D₈×C₃₇	—	2	2	no	—
68	D₈×C₃₇	—	2	1	no	—
69	D₈×C₃₇	—	2	1	no	—
70	M₁₆×C₃₇	—	2	1	no	—
71	M₁₆×C₃₇	C₂³	1	1	yes	characteristic
72	M₁₆×C₃₇	C₂³	1	2	yes	—
73	C₂×C₈×C₃₇	—	2	1	no	—
74	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
75	#10596941	—	2	2	no	—
76	#10596940	—	2	2	no	—
77	#10596939	—	2	2	no	—
78	#10596939	—	2	2	no	—
79	#10596939	—	2	2	no	—
80	#10596939	—	2	2	no	—
81	#10596934	—	2	1	no	—
82	#10596934	C₂³	1	1	yes	characteristic
83	#10596934	C₂³	1	2	yes	—
84	C₂×#64436	—	2	1	no	—
85	C₂×#64436	C₂³	1	1	yes	characteristic
86	D₂₉₆	—	2	2	no	—
87	D₂₉₆	—	2	1	no	—
88	D₂₉₆	—	2	1	no	—
89	#10596930	—	2	2	no	—
90	#10596929	—	2	1	no	—
91	#10596929	—	2	2	no	—
92	#10596929	—	2	1	no	—
93	#10596929	—	2	1	no	—
94	C₈×D₃₇	—	2	1	no	—
95	C₈×D₃₇	—	2	2	no	—
96	C₈×D₃₇	—	2	1	no	—
97	C₈×D₃₇	—	2	1	no	—
98	C₈×D₃₇	D₄	1	1	yes	characteristic
99	C₈×D₃₇	D₄	1	1	yes	characteristic
100	C₈×D₃₇	D₄	1	1	yes	characteristic
101	C₈×D₃₇	D₄	1	1	yes	characteristic
102	C₂³×C₃₇	—	2	2	no	—
103	C₂²×D₃₇	—	2	1	no	—
104	C₂²×D₃₇	—	2	2	no	—
105	C₂²×D₃₇	—	4	2	no	—
106	C₂²×D₃₇	—	4	2	no	—
107	C₂²×D₃₇	—	8	4	no	—
108	C₂²×D₃₇	#41	1	1	yes	characteristic
109	Q₈×C₃₇	C₂×D₄	1	1	yes	characteristic
110	D₄×C₃₇	—	2	1	no	—
111	D₄×C₃₇	—	2	2	no	—
112	D₄×C₃₇	—	2	2	no	—
113	D₄×C₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₇	—	2	1	no	—
115	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₃₇	—	2	1	no	—
117	C₂²⋊D₃₇	—	2	1	no	—
118	C₂²⋊D₃₇	—	2	1	no	—
119	C₂²⋊D₃₇	—	4	2	no	—
120	C₂²⋊D₃₇	—	4	2	no	—
121	C₂×Dic₃₇	—	2	1	no	—
122	C₂×Dic₃₇	#41	1	1	yes	characteristic
123	D₁₄₈	—	2	1	no	—
124	D₁₄₈	—	2	2	no	—
125	D₁₄₈	—	2	2	no	—
126	D₁₄₈	C₂×D₄	1	1	yes	characteristic
127	C₄×D₃₇	—	2	1	no	—
128	C₄×D₃₇	—	2	1	no	—
129	C₄×D₃₇	—	2	1	no	—
130	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
131	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
132	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
133	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
134	Dic₇₄	—	2	1	no	—
135	Dic₇₄	C₂×D₄	1	1	yes	characteristic
136	C₈×C₃₇	—	2	1	no	—
137	C₈×C₃₇	—	2	2	no	—
138	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
139	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
140	#64436	—	2	1	no	—
141	#64436	—	2	2	no	—
142	#64436	C₂×D₄	1	1	yes	characteristic
143	#64436	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₃₇	—	2	1	no	—
145	C₂²×C₃₇	—	4	2	no	—
146	C₂²×C₃₇	—	8	2	no	—
147	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
148	C₂×D₃₇	—	2	2	no	—
149	C₂×D₃₇	—	2	1	no	—
150	C₂×D₃₇	—	4	2	no	—
151	C₂×D₃₇	—	4	2	no	—
152	C₂×D₃₇	—	8	4	no	—
153	C₂×D₃₇	—	8	4	no	—
154	C₂×D₃₇	—	8	2	no	—
155	C₂×D₃₇	#121	1	1	yes	characteristic
156	C₂×D₃₇	#121	1	1	yes	characteristic
157	C₄×C₃₇	—	2	1	no	—
158	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
160	Dic₃₇	—	2	1	no	—
161	Dic₃₇	#121	1	1	yes	characteristic
162	Dic₃₇	#121	1	1	yes	characteristic
163	C₂×#472	—	37	1	no	maximal, Sylow
164	C₂×C₃₇	—	2	1	no	—
165	C₂×C₃₇	—	4	1	no	—
166	C₂×C₃₇	—	8	2	no	—
167	C₂×C₃₇	C₂×#121	1	1	yes	characteristic
168	D₃₇	—	2	1	no	—
169	D₃₇	—	4	1	no	—
170	D₃₇	—	8	2	no	—
171	D₃₇	#472	1	2	yes	direct product factor
172	C₂×AGL(1,ℤ/8)	—	37	2	no	—
173	C₂²×D₈	—	37	1	no	—
174	C₂×#131	—	37	1	no	—
175	#472	—	37	2	no	—
176	#472	—	37	1	no	—
177	#472	—	37	1	no	—
178	#472	—	37	1	no	—
179	#472	—	37	2	no	—
180	#472	D₃₇	1	1	yes	characteristic, direct product factor
181	C₂×#108	—	37	1	no	—
182	C₂×#100	—	37	2	no	—
183	C₃₇	C₂×#472	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	37	1	no	—
185	C₂²×D₄	—	37	2	no	—
186	AGL(1,ℤ/8)	—	37	2	no	—
187	AGL(1,ℤ/8)	—	37	2	no	—
188	AGL(1,ℤ/8)	—	37	2	no	—
189	AGL(1,ℤ/8)	—	74	2	no	—
190	AGL(1,ℤ/8)	—	74	2	no	—
191	AGL(1,ℤ/8)	C₂×D₃₇	1	2	yes	complemented
192	C₂×SD₁₆	—	74	2	no	—
193	C₂×D₈	—	37	2	no	—
194	C₂×D₈	—	37	1	no	—
195	C₂×D₈	—	74	1	no	—
196	C₂×D₈	—	74	1	no	—
197	C₂×D₈	—	74	1	no	—
198	C₂×D₈	—	74	1	no	—
199	C₂×D₈	—	74	2	no	—
200	C₂×D₈	C₂×D₃₇	1	1	yes	characteristic, complemented
201	#131	—	37	1	no	—
202	#131	—	37	1	no	—
203	#131	—	37	1	no	—
204	#131	—	74	1	no	—
205	#131	—	74	1	no	—
206	#131	C₂×D₃₇	1	1	yes	characteristic, complemented
207	C₂×M₁₆	—	37	1	no	—
208	C₂×M₁₆	—	37	2	no	—
209	C₂×M₁₆	—	74	1	no	—
210	C₂²×C₈	—	37	1	no	—
211	C₂²×C₈	—	74	1	no	—
212	#108	—	37	2	no	—
213	#108	—	37	1	no	—
214	#108	C₂×D₃₇	1	1	yes	characteristic, complemented
215	#100	—	37	2	no	—
216	#100	—	37	2	no	—
217	#100	—	37	2	no	—
218	#100	C₂×D₃₇	1	2	yes	complemented
219	C₂⁴	—	74	2	no	—
220	#41	—	37	1	no	—
221	#41	—	37	1	no	—
222	#41	—	37	1	no	—
223	#41	—	74	1	no	—
224	#41	—	74	1	no	—
225	#41	C₂²×D₃₇	1	1	yes	characteristic
226	C₂×Q₈	—	37	1	no	—
227	C₂×D₄	—	37	1	no	—
228	C₂×D₄	—	37	2	no	—
229	C₂×D₄	—	74	2	no	—
230	C₂×D₄	—	74	2	no	—
231	C₂×D₄	—	74	2	no	—
232	C₂×D₄	—	74	2	no	—
233	C₂×D₄	—	74	2	no	—
234	C₂×D₄	—	74	1	no	—
235	C₂×D₄	C₂²×D₃₇	1	2	yes	—
236	C₂²×C₄	—	37	1	no	—
237	C₂²×C₄	—	74	1	no	—
238	SD₁₆	—	2	2	no	—
239	SD₁₆	—	74	2	no	—
240	SD₁₆	—	74	2	no	—
241	SD₁₆	—	74	2	no	—
242	D₈	—	2	2	no	—
243	D₈	—	2	1	no	—
244	D₈	—	2	1	no	—
245	D₈	—	74	2	no	—
246	D₈	—	74	2	no	—
247	D₈	—	74	2	no	—
248	D₈	—	74	2	no	—
249	D₈	—	74	2	no	—
250	D₈	—	74	1	no	—
251	D₈	—	74	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	—	37	1	no	—
254	M₁₆	—	37	2	no	—
255	M₁₆	—	74	1	no	—
256	M₁₆	—	74	1	no	—
257	M₁₆	—	74	2	no	—
258	M₁₆	—	74	1	no	—
259	M₁₆	—	74	1	no	—
260	M₁₆	C₂²×D₃₇	1	1	yes	characteristic
261	M₁₆	C₂²×D₃₇	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	37	1	no	—
264	C₂×C₈	—	37	1	no	—
265	C₂×C₈	—	37	1	no	—
266	C₂×C₈	—	37	1	no	—
267	C₂×C₈	—	37	1	no	—
268	C₂×C₈	—	74	1	no	—
269	C₂×C₈	—	74	1	no	—
270	C₂×C₈	—	74	2	no	—
271	C₂×C₈	—	74	1	no	—
272	C₂×C₈	—	74	1	no	—
273	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
274	C₂³	—	2	2	no	—
275	C₂³	—	37	1	no	—
276	C₂³	—	296	4	no	—
277	C₂³	—	74	1	no	—
278	C₂³	—	74	2	no	—
279	C₂³	—	148	2	no	—
280	C₂³	—	148	2	no	—
281	Q₈	—	37	1	no	—
282	Q₈	—	74	1	no	—
283	Q₈	D₄×D₃₇	1	1	yes	characteristic
284	D₄	—	2	1	no	—
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	37	1	no	—
288	D₄	—	74	1	no	—
289	D₄	—	74	1	no	—
290	D₄	—	74	1	no	—
291	D₄	—	74	1	no	—
292	D₄	—	74	2	no	—
293	D₄	—	74	2	no	—
294	D₄	—	148	2	no	—
295	D₄	—	148	2	no	—
296	D₄	D₄×D₃₇	1	1	yes	characteristic
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	37	1	no	—
299	C₂×C₄	—	37	1	no	—
300	C₂×C₄	—	37	1	no	—
301	C₂×C₄	—	37	1	no	—
302	C₂×C₄	—	37	1	no	—
303	C₂×C₄	—	74	1	no	—
304	C₂×C₄	—	74	1	no	—
305	C₂×C₄	—	74	1	no	—
306	C₂×C₄	—	74	1	no	—
307	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	—	37	1	no	—
311	C₈	—	37	1	no	—
312	C₈	—	74	1	no	—
313	C₈	—	74	2	no	—
314	C₈	D₄×D₃₇	1	1	yes	characteristic
315	C₈	D₄×D₃₇	1	1	yes	characteristic
316	C₂²	—	2	1	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	37	1	no	—
320	C₂²	—	37	1	no	—
321	C₂²	—	296	2	no	—
322	C₂²	—	296	4	no	—
323	C₂²	—	296	4	no	—
324	C₂²	—	74	2	no	—
325	C₂²	—	74	1	no	—
326	C₂²	—	148	2	no	—
327	C₂²	—	148	2	no	—
328	C₂²	#41×D₃₇	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	37	1	no	—
331	C₄	—	37	1	no	—
332	C₄	—	74	1	no	—
333	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
334	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	4	1	no	—
337	C₂	—	8	2	no	—
338	C₂	—	37	2	no	—
339	C₂	—	296	2	no	—
340	C₂	—	74	1	no	—
341	C₂	—	148	1	no	—
342	C₂	#121×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#121×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#472	C₂
socle	C₂×C₃₇	C₂×#121
hypercenter	#472	D₃₇
hypocenter	C₃₇	C₂×#472
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #472

Complements

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

N	H ≅ G/N	type
1. AGL(1,ℤ/8)×D₃₇	C₂	semidirect
2. C₂×D₈×D₃₇	C₂	semidirect
3. #131×D₃₇	C₂	semidirect
4. #14557804	C₂	semidirect
5. #14557803	C₂	semidirect
6. #14557802	C₂	semidirect
7. #14557801	C₂	semidirect
8. #14557800	C₂	semidirect
9. #108×D₃₇	C₂	semidirect
10. #100×D₃₇	C₂	semidirect
11. C₃₇×#472	C₂	semidirect
14. #11924564	C₂²	semidirect
16. #11924562	C₂²	semidirect
17. #11924561	C₂²	semidirect
20. C₂×D₂₉₆	C₂²	semidirect
21. C₂×#10596939	C₂²	semidirect
28. #11924544	C₂²	semidirect
29. #11924543	C₂²	semidirect
31. #11924542	C₂²	semidirect
37. #11924472	C₂²	semidirect
38. #11924471	C₂²	semidirect
39. #11924453	C₂²	semidirect
40. #11924452	C₂²	semidirect
41. #11924452	C₂²	semidirect
42. AGL(1,ℤ/8)×C₃₇	C₂²	semidirect
43. C₂×D₈×C₃₇	C₂²	semidirect
44. #131×C₃₇	C₂²	semidirect
45. #108×C₃₇	C₂²	semidirect
46. #100×C₃₇	C₂²	semidirect
171. D₃₇	#472	direct (N × H)
180. #472	D₃₇	direct (N × H)
183. C₃₇	C₂×#472	semidirect
191. AGL(1,ℤ/8)	C₂×D₃₇	semidirect
200. C₂×D₈	C₂×D₃₇	semidirect
206. #131	C₂×D₃₇	semidirect
214. #108	C₂×D₃₇	semidirect
218. #100	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
AGL(1,ℤ/8)×D₃₇	2
C₂×D₈×D₃₇	2
#131×D₃₇	2
#14557804	2
#14557803	2
#14557802	2
#14557801	2
#14557800	2
#108×D₃₇	2
#100×D₃₇	2
C₃₇×#472	2
C₂×#472	37

Subgroups — #472×D37 #18004299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #472×D₃₇ #18004299