Subgroups — #478×D₃₇ #18004349

Order 4736.

448 conjugacy classes of subgroups.

The subgroup lattice has 352 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, complemented
2	C₄×D₄×D₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557836	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557835	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557834	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557833	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557832	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557831	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557830	C₂	1	1	yes	maximal, characteristic, complemented
10	#107×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄⋊C₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#102×D₃₇	C₂	1	1	yes	maximal, characteristic
14	#102×D₃₇	C₂	1	1	yes	maximal, characteristic
15	C₃₇×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₃₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₃₇	—	2	1	no	—
18	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
20	#11924528	C₂²	1	1	yes	characteristic, complemented
21	#11924527	C₂²	1	1	yes	characteristic, complemented
22	#11924527	C₂²	1	1	yes	characteristic, complemented
23	#11924526	—	2	1	no	—
24	#11924526	—	2	1	no	—
25	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
26	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
27	#11924498	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊D₃₇	—	2	1	no	—
29	#11924496	—	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	C₂²⋊C₄×D₃₇	—	2	1	no	—
36	C₄²×D₃₇	C₂²	1	1	yes	characteristic
37	#11924470	C₂²	1	1	yes	characteristic, complemented
38	#11924469	C₂²	1	1	yes	characteristic, complemented
39	#11924469	C₂²	1	1	yes	characteristic, complemented
40	#11924468	C₂²	1	1	yes	characteristic, complemented
41	#11924467	C₂²	1	1	yes	characteristic, complemented
42	#11924467	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊#64436	C₂²	1	1	yes	characteristic, complemented
44	#11924465	C₂²	1	1	yes	characteristic, complemented
45	#11924465	C₂²	1	1	yes	characteristic, complemented
46	#11924458	C₂²	1	1	yes	characteristic
47	#11924458	C₂²	1	1	yes	characteristic
48	#11924457	C₂²	1	1	yes	characteristic
49	#11924457	C₂²	1	1	yes	characteristic
50	#11924456	C₂²	1	1	yes	characteristic
51	#11924456	C₂²	1	1	yes	characteristic
52	#125×C₃₇	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
54	#107×C₃₇	C₂²	1	1	yes	characteristic, complemented
55	#106×C₃₇	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
57	#102×C₃₇	C₂²	1	1	yes	characteristic
58	#102×C₃₇	C₂²	1	1	yes	characteristic
59	C₂³×D₃₇	—	2	1	no	—
60	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	—	2	1	no	—
62	C₂×C₂²⋊D₃₇	—	2	1	no	—
63	C₂²×Dic₃₇	—	2	1	no	—
64	D₄×D₃₇	—	2	2	no	—
65	D₄×D₃₇	Q₈	1	4	yes	—
66	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₃₇	—	2	1	no	—
68	C₂×C₄×D₃₇	—	2	1	no	—
69	C₂×C₄×D₃₇	—	2	1	no	—
70	C₂×C₄×D₃₇	—	2	1	no	—
71	C₂×C₄×D₃₇	—	2	1	no	—
72	C₂×C₄×D₃₇	—	4	4	no	—
73	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
74	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
75	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₃₇	—	2	1	no	—
79	C₄⋊C₄×C₃₇	—	2	1	no	—
80	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₃₇	—	2	1	no	—
84	C₄²×C₃₇	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₃₇	—	2	1	no	—
86	#10596938	—	2	1	no	—
87	#10596938	—	2	1	no	—
88	C₄⋊Dic₃₇	—	2	1	no	—
89	C₄⋊Dic₃₇	—	2	1	no	—
90	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
93	#10596936	—	2	1	no	—
94	#10596936	—	2	1	no	—
95	#10596936	—	2	1	no	—
96	#10596936	—	2	1	no	—
97	#10596936	—	2	1	no	—
98	#10596936	—	2	1	no	—
99	#10596936	—	2	1	no	—
100	C₄×Dic₃₇	D₄	1	1	yes	characteristic
101	C₄×Dic₃₇	D₄	1	1	yes	characteristic
102	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
103	C₂×#64436	C₂³	1	1	yes	characteristic
104	C₂×#64436	C₂³	1	1	yes	characteristic
105	C₈×D₃₇	—	2	2	no	—
106	C₈×D₃₇	—	2	2	no	—
107	C₂³×C₃₇	—	2	1	no	—
108	C₂²×D₃₇	—	2	1	no	—
109	C₂²×D₃₇	—	2	4	no	—
110	C₂²×D₃₇	—	4	2	no	—
111	C₂²×D₃₇	—	4	2	no	—
112	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
113	D₄×C₃₇	—	2	1	no	—
114	D₄×C₃₇	C₂×Q₈	1	2	yes	—
115	C₂×C₄×C₃₇	—	2	1	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	—	2	1	no	—
118	C₂×C₄×C₃₇	—	4	2	no	—
119	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₃₇	—	2	4	no	—
123	C₂×Dic₃₇	—	2	1	no	—
124	C₂×Dic₃₇	—	2	1	no	—
125	C₂×Dic₃₇	—	2	1	no	—
126	C₂×Dic₃₇	—	4	2	no	—
127	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
130	D₁₄₈	—	2	1	no	—
131	D₁₄₈	C₂×Q₈	1	2	yes	—
132	C₄×D₃₇	—	2	2	no	—
133	C₄×D₃₇	—	2	2	no	—
134	C₄×D₃₇	—	4	2	no	—
135	C₄×D₃₇	—	4	2	no	—
136	C₄×D₃₇	—	4	2	no	—
137	C₄×D₃₇	—	4	2	no	—
138	C₄×D₃₇	—	4	2	no	—
139	C₄×D₃₇	C₂×Q₈	1	2	yes	—
140	C₄×D₃₇	#41	1	2	yes	—
141	C₈×C₃₇	—	2	1	no	—
142	C₈×C₃₇	—	2	1	no	—
143	#64436	—	2	1	no	—
144	#64436	—	2	1	no	—
145	C₂²×C₃₇	—	2	2	no	—
146	C₂²×C₃₇	—	4	1	no	—
147	C₂²×C₃₇	—	4	1	no	—
148	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₃₇	—	2	2	no	—
150	C₂×D₃₇	—	4	1	no	—
151	C₂×D₃₇	—	4	1	no	—
152	C₂×D₃₇	—	4	8	no	—
153	C₂×D₃₇	C₂²⋊Q₈	1	2	yes	—
154	C₂×D₃₇	#135	1	2	yes	—
155	C₂×D₃₇	AGL(1,ℤ/8)	1	2	yes	—
156	C₄×C₃₇	—	2	1	no	—
157	C₄×C₃₇	—	2	1	no	—
158	C₄×C₃₇	—	4	1	no	—
159	C₄×C₃₇	—	4	1	no	—
160	C₄×C₃₇	—	4	1	no	—
161	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
162	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
163	Dic₃₇	—	2	1	no	—
164	Dic₃₇	—	2	1	no	—
165	Dic₃₇	—	4	1	no	—
166	Dic₃₇	—	4	1	no	—
167	Dic₃₇	—	4	1	no	—
168	Dic₃₇	C₂²⋊Q₈	1	2	yes	—
169	C₂×#478	—	37	1	no	maximal, Sylow
170	C₂×C₃₇	—	4	2	no	—
171	C₂×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
173	C₂×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	D₃₇	—	4	2	no	—
175	D₃₇	#478	1	4	yes	direct product factor
176	C₂×#125	—	37	1	no	—
177	C₂×C₄×D₄	—	37	1	no	—
178	#478	—	37	1	no	—
179	#478	—	37	1	no	—
180	#478	—	37	1	no	—
181	#478	—	37	1	no	—
182	#478	—	37	1	no	—
183	#478	—	37	1	no	—
184	#478	—	37	1	no	—
185	#478	D₃₇	1	1	yes	characteristic, direct product factor
186	C₂×#107	—	37	1	no	—
187	C₂×#106	—	37	1	no	—
188	C₂×C₄⋊C₈	—	37	1	no	—
189	C₂×#102	—	37	1	no	—
190	C₂×#102	—	37	1	no	—
191	C₃₇	C₂×#478	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	—	37	1	no	—
193	C₂³×C₄	—	74	1	no	—
194	C₂²×C₈	—	37	1	no	—
195	C₂²×C₈	—	37	1	no	—
196	#125	—	37	1	no	—
197	#125	—	37	1	no	—
198	#125	—	37	1	no	—
199	#125	—	74	1	no	—
200	#125	—	74	1	no	—
201	#125	C₂×D₃₇	1	1	yes	characteristic, complemented
202	C₄×D₄	—	37	1	no	—
203	C₄×D₄	—	37	1	no	—
204	C₄×D₄	—	37	1	no	—
205	C₄×D₄	—	74	1	no	—
206	C₄×D₄	—	74	1	no	—
207	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	37	1	no	—
209	C₂×C₄⋊C₄	—	37	1	no	—
210	C₂×C₄⋊C₄	—	37	1	no	—
211	C₂×C₄⋊C₄	—	74	1	no	—
212	C₂×C₄⋊C₄	—	74	1	no	—
213	C₂×C₂²⋊C₄	—	74	1	no	—
214	C₂×C₄²	—	37	1	no	—
215	#107	—	37	1	no	—
216	#107	—	37	1	no	—
217	#107	—	37	1	no	—
218	#107	C₂×D₃₇	1	1	yes	characteristic, complemented
219	#106	—	37	1	no	—
220	#106	—	37	1	no	—
221	#106	—	37	1	no	—
222	#106	C₂×D₃₇	1	1	yes	characteristic, complemented
223	C₄⋊C₈	—	37	1	no	—
224	C₄⋊C₈	—	37	1	no	—
225	C₄⋊C₈	—	37	1	no	—
226	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
227	#102	—	37	1	no	—
228	#102	—	37	1	no	—
229	#102	—	37	1	no	—
230	#102	—	37	1	no	—
231	#102	—	37	1	no	—
232	#102	—	37	1	no	—
233	#102	C₂×D₃₇	1	1	yes	characteristic
234	#102	C₂×D₃₇	1	1	yes	characteristic
235	C₂⁴	—	74	1	no	—
236	C₂×D₄	—	37	4	no	—
237	C₂×D₄	—	37	1	no	—
238	C₂×D₄	—	74	2	no	—
239	C₂×D₄	—	74	1	no	—
240	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	37	1	no	—
243	C₂²×C₄	—	37	1	no	—
244	C₂²×C₄	—	37	1	no	—
245	C₂²×C₄	—	74	1	no	—
246	C₂²×C₄	—	74	1	no	—
247	C₂²×C₄	—	74	1	no	—
248	C₂²×C₄	—	74	1	no	—
249	C₂²×C₄	—	74	1	no	—
250	C₂²×C₄	—	74	1	no	—
251	C₂²×C₄	—	148	4	no	—
252	C₂×C₈	—	37	1	no	—
253	C₂×C₈	—	37	1	no	—
254	C₂×C₈	—	74	2	no	—
255	C₂×C₈	—	74	2	no	—
256	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	37	1	no	—
261	C₄⋊C₄	—	37	1	no	—
262	C₄⋊C₄	—	37	1	no	—
263	C₄⋊C₄	—	74	1	no	—
264	C₄⋊C₄	—	74	1	no	—
265	C₄⋊C₄	—	74	1	no	—
266	C₄⋊C₄	—	74	1	no	—
267	C₄⋊C₄	—	74	1	no	—
268	C₄⋊C₄	—	74	1	no	—
269	C₄⋊C₄	—	74	1	no	—
270	C₄⋊C₄	—	74	1	no	—
271	C₄⋊C₄	—	74	1	no	—
272	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	74	1	no	—
277	C₂²⋊C₄	—	74	1	no	—
278	C₂²⋊C₄	—	74	1	no	—
279	C₄²	—	37	1	no	—
280	C₄²	—	37	1	no	—
281	C₄²	—	37	1	no	—
282	C₄²	C₂²×D₃₇	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	37	1	no	—
285	C₂³	—	74	4	no	—
286	C₂³	—	74	1	no	—
287	C₂³	—	148	2	no	—
288	C₂³	—	148	2	no	—
289	D₄	—	2	1	no	—
290	D₄	—	37	2	no	—
291	D₄	—	74	4	no	—
292	D₄	—	74	1	no	—
293	D₄	Q₈×D₃₇	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	37	1	no	—
299	C₂×C₄	—	37	2	no	—
300	C₂×C₄	—	37	2	no	—
301	C₂×C₄	—	37	1	no	—
302	C₂×C₄	—	37	1	no	—
303	C₂×C₄	—	74	2	no	—
304	C₂×C₄	—	74	2	no	—
305	C₂×C₄	—	74	1	no	—
306	C₂×C₄	—	74	1	no	—
307	C₂×C₄	—	74	1	no	—
308	C₂×C₄	—	148	2	no	—
309	C₂×C₄	—	148	2	no	—
310	C₂×C₄	—	148	2	no	—
311	C₂×C₄	—	148	2	no	—
312	C₂×C₄	—	148	2	no	—
313	C₂×C₄	—	148	2	no	—
314	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
315	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	74	1	no	—
320	C₈	—	74	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	37	2	no	—
325	C₂²	—	37	2	no	—
326	C₂²	—	37	2	no	—
327	C₂²	—	74	2	no	—
328	C₂²	—	148	8	no	—
329	C₂²	—	148	1	no	—
330	C₂²	—	148	1	no	—
331	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	37	2	no	—
338	C₄	—	74	1	no	—
339	C₄	—	74	1	no	—
340	C₄	—	148	1	no	—
341	C₄	—	148	1	no	—
342	C₄	—	148	1	no	—
343	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
344	C₄	#41×D₃₇	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	37	4	no	—
347	C₂	—	148	2	no	—
348	C₂	C₂²⋊Q₈×D₃₇	1	1	yes	characteristic
349	C₂	#135×D₃₇	1	1	yes	characteristic
350	C₂	AGL(1,ℤ/8)×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×#478	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×D₄×D₃₇	C₂²
hypercenter	#478	D₃₇
hypocenter	C₃₇	C₂×#478
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #125×D₃₇	C₂	semidirect
3. #14557836	C₂	semidirect
4. #14557835	C₂	semidirect
5. #14557834	C₂	semidirect
6. #14557833	C₂	semidirect
7. #14557832	C₂	semidirect
8. #14557831	C₂	semidirect
9. #14557830	C₂	semidirect
10. #107×D₃₇	C₂	semidirect
11. #106×D₃₇	C₂	semidirect
12. C₄⋊C₈×D₃₇	C₂	semidirect
15. C₃₇×#478	C₂	semidirect
20. #11924528	C₂²	semidirect
21. #11924527	C₂²	semidirect
22. #11924527	C₂²	semidirect
37. #11924470	C₂²	semidirect
38. #11924469	C₂²	semidirect
39. #11924469	C₂²	semidirect
40. #11924468	C₂²	semidirect
41. #11924467	C₂²	semidirect
42. #11924467	C₂²	semidirect
43. C₄⋊#64436	C₂²	semidirect
44. #11924465	C₂²	semidirect
45. #11924465	C₂²	semidirect
52. #125×C₃₇	C₂²	semidirect
54. #107×C₃₇	C₂²	semidirect
55. #106×C₃₇	C₂²	semidirect
56. C₄⋊C₈×C₃₇	C₂²	semidirect
175. D₃₇	#478	direct (N × H)
185. #478	D₃₇	direct (N × H)
191. C₃₇	C₂×#478	semidirect
201. #125	C₂×D₃₇	semidirect
218. #107	C₂×D₃₇	semidirect
222. #106	C₂×D₃₇	semidirect
226. C₄⋊C₈	C₂×D₃₇	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×D₄×D₃₇	2
#14557836	2
#14557835	2
#14557834	2
#14557833	2
#14557832	2
#14557831	2
#14557830	2
#107×D₃₇	2
#106×D₃₇	2
C₄⋊C₈×D₃₇	2
#102×D₃₇	2
C₃₇×#478	2
C₂×#478	37

Subgroups — #478×D37 #18004349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #478×D₃₇ #18004349