Subgroups — #17997478

Order 4736.

504 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924492	C₂	1	1	yes	maximal, characteristic
2	#14557430	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557428	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557427	C₂	1	1	yes	maximal, characteristic
5	#14557426	C₂	1	1	yes	maximal, characteristic
6	#14557421	C₂	1	1	yes	maximal, characteristic
7	#14557398	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557397	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557391	C₂	1	1	yes	maximal, characteristic
10	C₄×#10596936	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557384	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²⋊C₄×Dic₃₇	C₂	1	1	yes	maximal, characteristic
13	#14557379	C₂	1	1	yes	maximal, characteristic, complemented
14	#14557378	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₇×#388	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₃₇	C₂²	1	1	yes	characteristic
17	#11924492	C₄	1	8	yes	complemented
18	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
19	C₂×#10596936	C₂²	1	1	yes	characteristic
20	C₂×#10596936	C₂²	1	1	yes	characteristic
21	C₂×#10596936	C₂²	1	1	yes	characteristic
22	C₂×#10596936	C₂²	1	1	yes	characteristic
23	C₂×#10596936	C₂²	1	1	yes	characteristic
24	C₂×#10596936	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₃₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
35	#11924443	C₂²	1	1	yes	characteristic
36	#11924443	C₂²	1	1	yes	characteristic
37	#11924443	C₂²	1	1	yes	characteristic
38	#11924443	C₂²	1	1	yes	characteristic
39	#11924443	C₂²	1	1	yes	characteristic
40	#11924443	C₂²	1	1	yes	characteristic
41	#11924443	C₂²	1	1	yes	characteristic
42	#11924443	C₂²	1	1	yes	characteristic
43	#11924443	C₂²	1	1	yes	characteristic
44	#11924443	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
49	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
50	#95×C₃₇	C₂²	1	1	yes	characteristic
51	#95×C₃₇	C₂²	1	1	yes	characteristic
52	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
59	C₂²×Dic₃₇	—	2	2	no	—
60	C₂²×Dic₃₇	—	2	2	no	—
61	C₂²×Dic₃₇	C₂×C₄	1	4	yes	—
62	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
63	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
64	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
65	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
66	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
67	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
68	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
69	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₃₇	—	4	1	no	—
71	C₂²⋊C₄×C₃₇	—	2	2	no	—
72	C₂²⋊C₄×C₃₇	—	4	1	no	—
73	C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
74	C₄²×C₃₇	—	2	2	no	—
75	C₂²⋊Dic₃₇	—	2	2	no	—
76	C₂²⋊Dic₃₇	—	4	1	no	—
77	C₂²⋊Dic₃₇	C₂×C₄	1	4	yes	—
78	C₄⋊Dic₃₇	—	4	1	no	—
79	#10596936	—	2	2	no	—
80	#10596936	—	2	2	no	—
81	#10596936	—	4	1	no	—
82	#10596936	—	4	1	no	—
83	#10596936	C₂×C₄	1	4	yes	—
84	#10596936	C₂×C₄	1	4	yes	—
85	C₄×Dic₃₇	—	2	2	no	—
86	C₄×Dic₃₇	—	2	2	no	—
87	C₄×Dic₃₇	—	2	2	no	—
88	C₄×Dic₃₇	—	2	2	no	—
89	C₄×Dic₃₇	—	2	2	no	—
90	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
91	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
92	C₂³×C₃₇	—	2	1	no	—
93	C₂³×C₃₇	—	2	1	no	—
94	C₂³×C₃₇	—	2	1	no	—
95	C₂³×C₃₇	—	2	1	no	—
96	C₂³×C₃₇	—	2	1	no	—
97	C₂³×C₃₇	—	2	1	no	—
98	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
99	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₃₇	—	2	1	no	—
101	C₂×C₄×C₃₇	—	2	1	no	—
102	C₂×C₄×C₃₇	—	2	1	no	—
103	C₂×C₄×C₃₇	—	2	1	no	—
104	C₂×C₄×C₃₇	—	2	1	no	—
105	C₂×C₄×C₃₇	—	2	1	no	—
106	C₂×C₄×C₃₇	—	2	1	no	—
107	C₂×C₄×C₃₇	—	2	1	no	—
108	C₂×C₄×C₃₇	—	2	1	no	—
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	—	2	1	no	—
114	C₂×C₄×C₃₇	—	2	1	no	—
115	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₃₇	#41	1	2	yes	—
118	C₂×Dic₃₇	—	2	1	no	—
119	C₂×Dic₃₇	—	2	1	no	—
120	C₂×Dic₃₇	—	2	1	no	—
121	C₂×Dic₃₇	—	2	1	no	—
122	C₂×Dic₃₇	—	2	1	no	—
123	C₂×Dic₃₇	—	2	1	no	—
124	C₂×Dic₃₇	—	2	1	no	—
125	C₂×Dic₃₇	—	2	1	no	—
126	C₂×Dic₃₇	—	2	8	no	—
127	C₂×Dic₃₇	—	2	1	no	—
128	C₂×Dic₃₇	—	2	1	no	—
129	C₂×Dic₃₇	—	2	1	no	—
130	C₂×Dic₃₇	—	2	1	no	—
131	C₂×Dic₃₇	—	2	1	no	—
132	C₂×Dic₃₇	—	2	1	no	—
133	C₂×Dic₃₇	—	2	1	no	—
134	C₂×Dic₃₇	—	2	1	no	—
135	C₂×Dic₃₇	—	2	1	no	—
136	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
137	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
138	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
139	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
140	C₂×Dic₃₇	#41	1	2	yes	—
141	C₂×Dic₃₇	#41	1	2	yes	—
142	C₂×Dic₃₇	#41	1	2	yes	—
143	C₂²×C₃₇	—	2	2	no	—
144	C₂²×C₃₇	—	2	2	no	—
145	C₂²×C₃₇	—	2	2	no	—
146	C₂²×C₃₇	—	4	1	no	—
147	C₂²×C₃₇	—	4	1	no	—
148	C₂²×C₃₇	—	4	1	no	—
149	C₂²×C₃₇	—	4	1	no	—
150	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
151	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
152	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
153	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
154	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
155	C₂²×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
157	C₄×C₃₇	—	2	2	no	—
158	C₄×C₃₇	—	2	2	no	—
159	C₄×C₃₇	—	2	2	no	—
160	C₄×C₃₇	—	4	1	no	—
161	C₄×C₃₇	—	4	1	no	—
162	C₄×C₃₇	—	4	1	no	—
163	Dic₃₇	—	2	2	no	—
164	Dic₃₇	—	2	2	no	—
165	Dic₃₇	—	2	2	no	—
166	Dic₃₇	—	2	2	no	—
167	Dic₃₇	—	4	1	no	—
168	Dic₃₇	—	4	1	no	—
169	Dic₃₇	—	4	1	no	—
170	Dic₃₇	#117	1	4	yes	—
171	#2346	—	37	1	no	maximal, Sylow
172	C₂×C₃₇	—	4	2	no	—
173	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
174	C₂×C₃₇	C₄×#41	1	1	yes	characteristic
175	C₂×C₃₇	#520	1	1	yes	characteristic
176	C₂×C₃₇	#552	1	1	yes	characteristic
177	C₂×C₃₇	#552	1	1	yes	characteristic
178	C₂×C₃₇	#555	1	1	yes	characteristic
179	C₂×C₃₇	#555	1	1	yes	characteristic
180	C₂×#117	—	37	1	no	—
181	#389	—	37	1	no	—
182	#389	—	37	1	no	—
183	#388	—	37	1	no	—
184	#388	—	37	1	no	—
185	#388	—	37	1	no	—
186	#388	D₃₇	1	1	yes	characteristic
187	#382	—	37	1	no	—
188	#382	—	37	1	no	—
189	#380	—	37	1	no	—
190	C₄×C₄⋊C₄	—	37	1	no	—
191	C₄×C₄⋊C₄	—	37	1	no	—
192	C₄×C₂²⋊C₄	—	37	1	no	—
193	#376	—	37	1	no	—
194	#376	—	37	1	no	—
195	C₃₇	#2346	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	—	37	1	no	—
197	#117	—	37	8	no	—
198	C₂×C₄⋊C₄	—	37	1	no	—
199	C₂×C₄⋊C₄	—	37	1	no	—
200	C₂×C₄⋊C₄	—	37	1	no	—
201	C₂×C₄⋊C₄	—	37	1	no	—
202	C₂×C₄⋊C₄	—	37	1	no	—
203	C₂×C₄⋊C₄	—	37	1	no	—
204	C₂×C₄⋊C₄	—	37	1	no	—
205	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	37	1	no	—
207	C₂×C₂²⋊C₄	—	37	1	no	—
208	C₂×C₂²⋊C₄	—	37	1	no	—
209	C₂×C₂²⋊C₄	C₂×D₃₇	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	C₂×D₃₇	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	C₂×D₃₇	1	1	yes	characteristic
212	C₂×C₄²	—	37	1	no	—
213	C₂×C₄²	—	37	1	no	—
214	C₂×C₄²	—	37	1	no	—
215	C₂×C₄²	—	37	1	no	—
216	C₂×C₄²	—	37	1	no	—
217	C₂×C₄²	—	37	1	no	—
218	C₂×C₄²	—	37	1	no	—
219	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
220	#95	—	37	1	no	—
221	#95	—	37	1	no	—
222	#95	—	37	1	no	—
223	#95	—	37	1	no	—
224	#95	—	37	1	no	—
225	#95	—	37	1	no	—
226	#95	—	37	1	no	—
227	#95	—	37	1	no	—
228	#95	—	37	1	no	—
229	#95	—	37	1	no	—
230	#95	C₂×D₃₇	1	1	yes	characteristic
231	#95	C₂×D₃₇	1	1	yes	characteristic
232	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
233	C₂²×C₄	—	37	1	no	—
234	C₂²×C₄	—	37	4	no	—
235	C₂²×C₄	—	37	1	no	—
236	C₂²×C₄	—	37	1	no	—
237	C₂²×C₄	—	37	1	no	—
238	C₂²×C₄	—	37	1	no	—
239	C₂²×C₄	—	37	1	no	—
240	C₂²×C₄	—	37	1	no	—
241	C₂²×C₄	—	37	1	no	—
242	C₂²×C₄	—	74	2	no	—
243	C₂²×C₄	—	74	2	no	—
244	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
246	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	37	4	no	—
252	C₄⋊C₄	—	37	4	no	—
253	C₄⋊C₄	—	74	2	no	—
254	C₄⋊C₄	—	74	2	no	—
255	C₄⋊C₄	—	148	1	no	—
256	C₄⋊C₄	—	148	1	no	—
257	C₄⋊C₄	—	148	1	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	37	4	no	—
261	C₂²⋊C₄	—	74	2	no	—
262	C₂²⋊C₄	—	148	1	no	—
263	C₂²⋊C₄	C₄×D₃₇	1	4	yes	—
264	C₄²	—	2	2	no	—
265	C₄²	—	37	4	no	—
266	C₄²	—	37	4	no	—
267	C₄²	—	74	2	no	—
268	C₄²	—	74	2	no	—
269	C₄²	—	74	2	no	—
270	C₄²	—	74	2	no	—
271	C₄²	—	74	2	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	C₂×C₄×D₃₇	1	2	yes	—
279	C₂³	C₂³×D₃₇	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	37	2	no	—
296	C₂×C₄	—	37	2	no	—
297	C₂×C₄	—	37	2	no	—
298	C₂×C₄	—	37	2	no	—
299	C₂×C₄	—	37	2	no	—
300	C₂×C₄	—	37	2	no	—
301	C₂×C₄	—	37	2	no	—
302	C₂×C₄	—	74	8	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	—	74	1	no	—
308	C₂×C₄	—	74	1	no	—
309	C₂×C₄	—	74	1	no	—
310	C₂×C₄	—	74	1	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	—	74	1	no	—
318	C₂×C₄	—	74	1	no	—
319	C₂×C₄	—	74	1	no	—
320	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
321	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
322	C₂×C₄	#10596964	1	2	yes	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
331	C₂²	#41×D₃₇	1	1	yes	characteristic
332	C₂²	#41×D₃₇	1	1	yes	characteristic
333	C₂²	#41×D₃₇	1	1	yes	characteristic
334	C₂²	C₂×#10596964	1	1	yes	characteristic
335	C₂²	#11924581	1	1	yes	characteristic
336	C₂²	#11924583	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	37	4	no	—
344	C₄	—	74	2	no	—
345	C₄	—	74	2	no	—
346	C₄	—	74	2	no	—
347	C₄	—	74	2	no	—
348	C₄	—	148	1	no	—
349	C₄	—	148	1	no	—
350	C₄	—	148	1	no	—
351	C₂	—	4	2	no	—
352	C₂	#117×D₃₇	1	1	yes	characteristic
353	C₂	C₄×#10596964	1	1	yes	characteristic
354	C₂	#14558012	1	1	yes	characteristic
355	C₂	#14558219	1	1	yes	characteristic
356	C₂	#14558220	1	1	yes	characteristic
357	C₂	#14558245	1	1	yes	characteristic
358	C₂	#14558247	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#388	C₂
socle, cosocle	C₂³×C₃₇	C₂⁴
prime radical	C₂⁴×C₃₇	C₂³
hypercenter	#388	D₃₇
hypocenter	C₃₇	#2346
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
2. #14557430	C₂	semidirect
3. #14557428	C₂	semidirect
7. #14557398	C₂	semidirect
8. #14557397	C₂	semidirect
10. C₄×#10596936	C₂	semidirect
11. #14557384	C₂	semidirect
13. #14557379	C₂	semidirect
14. #14557378	C₂	semidirect
17. #11924492	C₄	semidirect
195. C₃₇	#2346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924492	2
#14557430	2
#14557428	2
#14557427	2
#14557426	2
#14557421	2
#14557398	2
#14557397	2
#14557391	2
C₄×#10596936	2
#14557384	2
C₂²⋊C₄×Dic₃₇	2
#14557379	2
#14557378	2
C₃₇×#388	2
#2346	37