Subgroups — #485×D₃₇ #18004484

Order 4736.

416 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
2	#117×D₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557877	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557876	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557875	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557874	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557873	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557872	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557871	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	#103×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
15	C₃₇×#485	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
20	#11924516	C₂²	1	1	yes	characteristic
21	#11924515	C₂²	1	1	yes	characteristic
22	#11924514	—	2	1	no	—
23	#11924513	—	2	1	no	—
24	#11924512	C₂²	1	1	yes	characteristic
25	#11924495	C₂²	1	1	yes	characteristic
26	#11924494	C₂²	1	1	yes	characteristic
27	#11924494	C₂²	1	1	yes	characteristic
28	#11924493	—	2	1	no	—
29	#11924492	—	2	1	no	—
30	C₄⋊C₄×D₃₇	—	2	1	no	—
31	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×D₃₇	—	2	1	no	—
35	C₂²⋊C₄×D₃₇	—	2	1	no	—
36	C₄²×D₃₇	—	2	1	no	—
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	#11924461	C₂²	1	1	yes	characteristic, complemented
44	#11924461	C₂²	1	1	yes	characteristic, complemented
45	#11924460	C₂²	1	1	yes	characteristic, complemented
46	#11924460	C₂²	1	1	yes	characteristic, complemented
47	#11924459	C₂²	1	1	yes	characteristic, complemented
48	#11924459	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊#64436	C₂²	1	1	yes	characteristic
50	#11924448	C₂²	1	1	yes	characteristic
51	#11924448	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
53	#117×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	#103×C₃₇	C₂²	1	1	yes	characteristic, complemented
57	#103×C₃₇	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
59	C₂³×D₃₇	D₄	1	1	yes	characteristic
60	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
63	Q₈×D₃₇	—	2	2	no	—
64	Q₈×D₃₇	—	2	2	no	—
65	C₂×C₄×D₃₇	—	2	2	no	—
66	C₂×C₄×D₃₇	—	2	2	no	—
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	D₄	1	1	yes	characteristic
72	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
73	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
75	C₂×Dic₇₄	—	2	1	no	—
76	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₃₇	—	2	1	no	—
85	C₄²×C₃₇	—	2	1	no	—
86	C₂²⋊Dic₃₇	—	2	1	no	—
87	C₂²⋊Dic₃₇	—	2	1	no	—
88	#10596938	—	2	1	no	—
89	#10596938	—	2	1	no	—
90	#10596938	—	2	1	no	—
91	#10596938	—	2	1	no	—
92	C₄⋊Dic₃₇	—	2	1	no	—
93	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
96	#10596936	—	2	1	no	—
97	#10596936	—	2	1	no	—
98	#10596936	—	2	1	no	—
99	#10596936	—	2	1	no	—
100	#10596936	—	2	1	no	—
101	C₄×Dic₃₇	—	2	1	no	—
102	C₄×Dic₃₇	—	2	1	no	—
103	C₄×Dic₃₇	—	2	1	no	—
104	C₂×#64436	C₂³	1	1	yes	characteristic
105	C₂×#64436	C₂³	1	1	yes	characteristic
106	C₈×D₃₇	—	2	2	no	—
107	C₈×D₃₇	—	2	2	no	—
108	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₃₇	—	2	2	no	—
110	C₂²×D₃₇	—	2	2	no	—
111	C₂²×D₃₇	—	2	2	no	—
112	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₃₇	—	2	1	no	—
115	Q₈×C₃₇	—	2	1	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	—	2	1	no	—
118	C₂×C₄×C₃₇	—	2	1	no	—
119	C₂×C₄×C₃₇	—	2	1	no	—
120	C₂×C₄×C₃₇	—	2	1	no	—
121	C₂×C₄×C₃₇	—	2	1	no	—
122	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₃₇	—	2	2	no	—
125	C₂×Dic₃₇	—	2	1	no	—
126	C₂×Dic₃₇	—	2	1	no	—
127	C₂×Dic₃₇	—	2	1	no	—
128	C₂×Dic₃₇	—	2	1	no	—
129	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
131	C₄×D₃₇	—	2	2	no	—
132	C₄×D₃₇	—	2	1	no	—
133	C₄×D₃₇	—	2	1	no	—
134	C₄×D₃₇	—	2	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	#41	1	2	yes	—
140	C₄×D₃₇	#41	1	2	yes	—
141	Dic₇₄	—	2	1	no	—
142	Dic₇₄	—	2	1	no	—
143	Dic₇₄	—	4	2	no	—
144	C₈×C₃₇	—	2	1	no	—
145	C₈×C₃₇	—	2	1	no	—
146	#64436	—	2	1	no	—
147	#64436	—	2	1	no	—
148	C₂²×C₃₇	—	2	1	no	—
149	C₂²×C₃₇	—	2	1	no	—
150	C₂²×C₃₇	—	2	1	no	—
151	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₃₇	—	2	1	no	—
153	C₂×D₃₇	—	2	1	no	—
154	C₂×D₃₇	—	2	1	no	—
155	C₂×D₃₇	—	4	4	no	—
156	C₂×D₃₇	#123	1	2	yes	—
157	C₂×D₃₇	#135	1	2	yes	—
158	C₂×D₃₇	#137	1	2	yes	—
159	C₄×C₃₇	—	2	1	no	—
160	C₄×C₃₇	—	4	1	no	—
161	C₄×C₃₇	—	4	1	no	—
162	C₄×C₃₇	—	4	1	no	—
163	C₄×C₃₇	—	4	1	no	—
164	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
165	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
166	Dic₃₇	—	2	1	no	—
167	Dic₃₇	—	4	1	no	—
168	Dic₃₇	—	4	1	no	—
169	Dic₃₇	—	4	1	no	—
170	Dic₃₇	—	4	1	no	—
171	Dic₃₇	#123	1	2	yes	—
172	C₂×#485	—	37	1	no	maximal, Sylow
173	C₂×C₃₇	—	4	1	no	—
174	C₂×C₃₇	C₂×#123	1	1	yes	characteristic
175	C₂×C₃₇	C₂×#135	1	1	yes	characteristic
176	C₂×C₃₇	C₂×#137	1	1	yes	characteristic
177	D₃₇	—	4	1	no	—
178	D₃₇	#485	1	4	yes	direct product factor
179	C₂×C₂²⋊Q₈	—	37	1	no	—
180	C₂×#117	—	37	1	no	—
181	#485	—	37	1	no	—
182	#485	—	37	1	no	—
183	#485	—	37	1	no	—
184	#485	—	37	1	no	—
185	#485	—	37	1	no	—
186	#485	—	37	1	no	—
187	#485	—	37	1	no	—
188	#485	D₃₇	1	1	yes	characteristic, direct product factor
189	C₂×#107	—	37	1	no	—
190	C₂×#106	—	37	1	no	—
191	C₂×#103	—	37	1	no	—
192	C₂×#103	—	37	1	no	—
193	C₂×C₂²⋊C₈	—	37	1	no	—
194	C₃₇	C₂×#485	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	37	1	no	—
196	C₂³×C₄	—	37	1	no	—
197	C₂²×C₈	—	37	1	no	—
198	C₂²×C₈	—	37	1	no	—
199	C₂²⋊Q₈	—	37	1	no	—
200	C₂²⋊Q₈	—	37	1	no	—
201	C₂²⋊Q₈	—	37	1	no	—
202	C₂²⋊Q₈	—	74	1	no	—
203	C₂²⋊Q₈	—	74	1	no	—
204	C₂²⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
205	#117	—	37	1	no	—
206	#117	—	37	1	no	—
207	#117	—	37	1	no	—
208	#117	—	74	1	no	—
209	#117	—	74	1	no	—
210	#117	C₂×D₃₇	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	37	1	no	—
212	C₂×C₄⋊C₄	—	37	1	no	—
213	C₂×C₄⋊C₄	—	37	1	no	—
214	C₂×C₄⋊C₄	—	74	1	no	—
215	C₂×C₂²⋊C₄	—	74	1	no	—
216	C₂×C₂²⋊C₄	—	74	1	no	—
217	C₂×C₄²	—	74	1	no	—
218	#107	—	37	1	no	—
219	#107	—	37	1	no	—
220	#107	—	37	1	no	—
221	#107	C₂×D₃₇	1	1	yes	characteristic, complemented
222	#106	—	37	1	no	—
223	#106	—	37	1	no	—
224	#106	—	37	1	no	—
225	#106	C₂×D₃₇	1	1	yes	characteristic, complemented
226	#103	—	37	1	no	—
227	#103	—	37	1	no	—
228	#103	—	37	1	no	—
229	#103	—	37	1	no	—
230	#103	—	37	1	no	—
231	#103	—	37	1	no	—
232	#103	C₂×D₃₇	1	1	yes	characteristic, complemented
233	#103	C₂×D₃₇	1	1	yes	characteristic, complemented
234	C₂²⋊C₈	—	37	1	no	—
235	C₂²⋊C₈	—	37	1	no	—
236	C₂²⋊C₈	—	37	1	no	—
237	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
238	C₂⁴	—	37	1	no	—
239	C₂×Q₈	—	37	1	no	—
240	C₂×Q₈	—	74	1	no	—
241	C₂×Q₈	—	74	2	no	—
242	C₂×Q₈	—	74	2	no	—
243	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
244	C₂²×C₄	—	37	1	no	—
245	C₂²×C₄	—	37	1	no	—
246	C₂²×C₄	—	37	1	no	—
247	C₂²×C₄	—	37	1	no	—
248	C₂²×C₄	—	37	1	no	—
249	C₂²×C₄	—	74	2	no	—
250	C₂²×C₄	—	74	2	no	—
251	C₂²×C₄	—	74	1	no	—
252	C₂²×C₄	—	74	1	no	—
253	C₂²×C₄	—	74	1	no	—
254	C₂²×C₄	—	74	1	no	—
255	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
256	C₂×C₈	—	37	1	no	—
257	C₂×C₈	—	37	1	no	—
258	C₂×C₈	—	74	2	no	—
259	C₂×C₈	—	74	2	no	—
260	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	37	1	no	—
264	C₄⋊C₄	—	37	1	no	—
265	C₄⋊C₄	—	37	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	—	2	1	no	—
277	C₂²⋊C₄	—	74	1	no	—
278	C₂²⋊C₄	—	74	1	no	—
279	C₂²⋊C₄	—	74	1	no	—
280	C₂²⋊C₄	—	74	1	no	—
281	C₂²⋊C₄	—	74	1	no	—
282	C₂²⋊C₄	—	74	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	74	1	no	—
285	C₄²	—	74	1	no	—
286	C₄²	—	74	1	no	—
287	C₂³	—	37	1	no	—
288	C₂³	—	37	1	no	—
289	C₂³	—	74	2	no	—
290	C₂³	—	74	2	no	—
291	C₂³	—	74	2	no	—
292	C₂³	D₄×D₃₇	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	74	1	no	—
296	Q₈	—	74	1	no	—
297	Q₈	—	148	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	37	1	no	—
305	C₂×C₄	—	37	2	no	—
306	C₂×C₄	—	37	2	no	—
307	C₂×C₄	—	37	1	no	—
308	C₂×C₄	—	74	2	no	—
309	C₂×C₄	—	74	2	no	—
310	C₂×C₄	—	74	2	no	—
311	C₂×C₄	—	74	1	no	—
312	C₂×C₄	—	74	1	no	—
313	C₂×C₄	—	74	1	no	—
314	C₂×C₄	—	74	1	no	—
315	C₂×C₄	—	74	1	no	—
316	C₂×C₄	—	74	1	no	—
317	C₂×C₄	—	148	2	no	—
318	C₂×C₄	—	148	2	no	—
319	C₂×C₄	—	148	2	no	—
320	C₂×C₄	—	148	2	no	—
321	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	74	1	no	—
326	C₈	—	74	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	37	2	no	—
331	C₂²	—	37	2	no	—
332	C₂²	—	37	2	no	—
333	C₂²	—	74	1	no	—
334	C₂²	—	74	1	no	—
335	C₂²	—	74	1	no	—
336	C₂²	—	148	4	no	—
337	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	37	2	no	—
344	C₄	—	74	1	no	—
345	C₄	—	148	1	no	—
346	C₄	—	148	1	no	—
347	C₄	—	148	1	no	—
348	C₄	—	148	1	no	—
349	C₄	#41×D₃₇	1	1	yes	characteristic
350	C₄	#41×D₃₇	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	37	4	no	—
353	C₂	—	148	1	no	—
354	C₂	#123×D₃₇	1	1	yes	characteristic
355	C₂	#135×D₃₇	1	1	yes	characteristic
356	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×#485	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂³×D₃₇	D₄
hypercenter	#485	D₃₇
hypocenter	C₃₇	C₂×#485
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #14557877	C₂	semidirect
4. #14557876	C₂	semidirect
5. #14557875	C₂	semidirect
6. #14557874	C₂	semidirect
7. #14557873	C₂	semidirect
8. #14557872	C₂	semidirect
9. #14557871	C₂	semidirect
10. #107×D₃₇	C₂	semidirect
11. #106×D₃₇	C₂	semidirect
12. #103×D₃₇	C₂	semidirect
13. #103×D₃₇	C₂	semidirect
15. C₃₇×#485	C₂	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. #11924461	C₂²	semidirect
44. #11924461	C₂²	semidirect
45. #11924460	C₂²	semidirect
46. #11924460	C₂²	semidirect
47. #11924459	C₂²	semidirect
48. #11924459	C₂²	semidirect
54. #107×C₃₇	C₂²	semidirect
55. #106×C₃₇	C₂²	semidirect
56. #103×C₃₇	C₂²	semidirect
57. #103×C₃₇	C₂²	semidirect
178. D₃₇	#485	direct (N × H)
188. #485	D₃₇	direct (N × H)
194. C₃₇	C₂×#485	semidirect
221. #107	C₂×D₃₇	semidirect
225. #106	C₂×D₃₇	semidirect
232. #103	C₂×D₃₇	semidirect
233. #103	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊Q₈×D₃₇	2
#117×D₃₇	2
#14557877	2
#14557876	2
#14557875	2
#14557874	2
#14557873	2
#14557872	2
#14557871	2
#107×D₃₇	2
#106×D₃₇	2
#103×D₃₇	2
C₂²⋊C₈×D₃₇	2
C₃₇×#485	2
C₂×#485	37

Subgroups — #485×D37 #18004484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #485×D₃₇ #18004484