Subgroups — #18003468

Order 4736.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924557	C₂	1	1	yes	maximal, characteristic
2	#14558015	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558012	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557635	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557634	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557632	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#10596942	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557625	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10596939	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557613	C₂	1	1	yes	maximal, characteristic, complemented
11	#107×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#14557533	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557531	C₂	1	1	yes	maximal, characteristic, complemented
14	#14557490	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₇×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10596966	C₂²	1	1	yes	characteristic
17	C₂×#10596964	C₂²	1	1	yes	characteristic
18	#11924557	C₄	1	8	yes	complemented
19	C₂×#10596942	C₂²	1	1	yes	characteristic
20	C₂×#10596930	C₂²	1	1	yes	characteristic
21	C₂×#10596939	C₂²	1	1	yes	characteristic
22	C₂×#10596929	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
24	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
26	#11924501	C₂²	1	1	yes	characteristic
27	C₄×D₁₄₈	—	2	1	no	—
28	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
29	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
30	#11924498	—	2	1	no	—
31	#11924498	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₃₇	—	2	1	no	—
33	#11924496	—	2	1	no	—
34	#11924493	—	2	1	no	—
35	#11924493	C₂²	1	1	yes	characteristic
36	#11924492	—	2	1	no	—
37	C₂×#10596936	—	2	1	no	—
38	C₄⋊C₄×D₃₇	—	2	1	no	—
39	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
40	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
41	#11924470	C₂²	1	1	yes	characteristic
42	#11924469	C₂²	1	1	yes	characteristic
43	#11924469	C₂²	1	1	yes	characteristic
44	#11924461	C₂²	1	1	yes	characteristic
45	#11924460	C₂²	1	1	yes	characteristic
46	#11924459	C₂²	1	1	yes	characteristic
47	#11924458	C₂²	1	1	yes	characteristic
48	#11924457	C₂²	1	1	yes	characteristic
49	#11924456	C₂²	1	1	yes	characteristic
50	#11924446	C₂²	1	1	yes	characteristic
51	C₄×#64436	C₂²	1	1	yes	characteristic
52	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
56	#107×C₃₇	C₂²	1	1	yes	characteristic
57	#103×C₃₇	C₂²	1	1	yes	characteristic
58	#102×C₃₇	C₂²	1	1	yes	characteristic
59	#97×C₃₇	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	—	2	1	no	—
63	C₂×C₂²⋊D₃₇	—	2	1	no	—
64	C₂²×Dic₃₇	—	2	1	no	—
65	C₂²×Dic₃₇	—	2	1	no	—
66	#10596966	—	2	2	no	—
67	#10596966	C₂×C₄	1	4	yes	complemented
68	#10596964	—	2	2	no	—
69	#10596964	C₂×C₄	1	4	yes	complemented
70	C₂×D₁₄₈	—	2	1	no	—
71	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₃₇	—	2	1	no	—
73	C₂×C₄×D₃₇	—	2	1	no	—
74	C₂×C₄×D₃₇	—	2	1	no	—
75	C₂×C₄×D₃₇	—	2	1	no	—
76	C₂×C₄×D₃₇	—	2	1	no	—
77	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
78	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
79	SD₁₆×C₃₇	C₂×C₄	1	4	yes	complemented
80	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₃₇	—	2	1	no	—
83	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₃₇	—	2	1	no	—
86	C₄²×C₃₇	—	2	1	no	—
87	C₄²×C₃₇	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₃₇	—	2	1	no	—
89	#10596942	C₂×C₄	1	4	yes	complemented
90	#10596939	C₂×C₄	1	4	yes	complemented
91	#10596938	—	2	1	no	—
92	#10596938	—	2	1	no	—
93	#10596938	D₄	1	1	yes	characteristic
94	#10596938	D₄	1	1	yes	characteristic
95	C₄⋊Dic₃₇	—	2	1	no	—
96	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
98	#10596936	—	2	1	no	—
99	#10596936	—	2	1	no	—
100	#10596936	—	2	1	no	—
101	#10596936	—	2	1	no	—
102	#10596936	D₄	1	1	yes	characteristic
103	#10596936	D₄	1	1	yes	characteristic
104	C₄×Dic₃₇	—	2	1	no	—
105	C₄×Dic₃₇	—	2	1	no	—
106	C₄×Dic₃₇	—	2	1	no	—
107	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
108	C₂×#64436	C₂³	1	1	yes	characteristic
109	C₂×#64436	C₂³	1	1	yes	characteristic
110	#10596930	C₂×C₄	1	4	yes	complemented
111	#10596929	—	2	2	no	—
112	C₈×D₃₇	C₂×C₄	1	4	yes	complemented
113	C₂³×C₃₇	—	2	1	no	—
114	C₂²×D₃₇	—	2	1	no	—
115	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₃₇	—	2	1	no	—
117	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
118	D₄×C₃₇	—	2	1	no	—
119	D₄×C₃₇	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₃₇	—	2	1	no	—
121	C₂×C₄×C₃₇	—	2	1	no	—
122	C₂×C₄×C₃₇	—	2	1	no	—
123	C₂×C₄×C₃₇	—	8	1	no	—
124	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₃₇	—	2	4	no	—
128	C₂×Dic₃₇	—	2	1	no	—
129	C₂×Dic₃₇	—	2	4	no	—
130	C₂×Dic₃₇	—	2	1	no	—
131	C₂×Dic₃₇	—	2	1	no	—
132	C₂×Dic₃₇	—	8	1	no	—
133	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
136	D₁₄₈	—	2	4	no	—
137	D₁₄₈	—	2	1	no	—
138	D₁₄₈	C₂²×C₄	1	2	yes	—
139	C₄×D₃₇	—	2	4	no	—
140	C₄×D₃₇	—	8	1	no	—
141	C₄×D₃₇	—	8	1	no	—
142	C₄×D₃₇	—	8	1	no	—
143	C₄×D₃₇	—	8	1	no	—
144	C₄×D₃₇	C₂²×C₄	1	2	yes	—
145	C₄×D₃₇	#41	1	2	yes	—
146	Dic₇₄	—	2	1	no	—
147	Dic₇₄	C₂²×C₄	1	2	yes	—
148	C₈×C₃₇	—	2	1	no	—
149	C₈×C₃₇	C₂²×C₄	1	2	yes	—
150	#64436	—	2	1	no	—
151	#64436	C₂²×C₄	1	2	yes	—
152	C₂²×C₃₇	—	2	2	no	—
153	C₂²×C₃₇	—	4	1	no	—
154	C₂²×C₃₇	—	4	1	no	—
155	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₃₇	—	2	1	no	—
157	C₂×D₃₇	—	2	1	no	—
158	C₂×D₃₇	—	2	2	no	—
159	C₂×D₃₇	—	4	1	no	—
160	C₂×D₃₇	—	4	1	no	—
161	C₂×D₃₇	C₄×D₄	1	2	yes	—
162	C₄×C₃₇	—	2	2	no	—
163	C₄×C₃₇	—	2	1	no	—
164	C₄×C₃₇	—	2	1	no	—
165	C₄×C₃₇	—	4	1	no	—
166	C₄×C₃₇	—	4	1	no	—
167	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
168	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
169	Dic₃₇	—	2	2	no	—
170	Dic₃₇	—	2	1	no	—
171	Dic₃₇	—	2	1	no	—
172	Dic₃₇	—	4	1	no	—
173	Dic₃₇	—	4	1	no	—
174	Dic₃₇	C₄×D₄	1	2	yes	—
175	#2951	—	37	1	no	maximal, Sylow
176	C₂×C₃₇	—	4	2	no	—
177	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×C₃₇	#576	1	1	yes	characteristic
179	C₂×C₃₇	#578	1	1	yes	characteristic
180	D₃₇	—	2	2	no	—
181	D₃₇	—	4	2	no	—
182	C₂×#135	—	37	1	no	—
183	#520	—	37	1	no	—
184	#520	—	37	1	no	—
185	#440	—	37	1	no	—
186	#440	—	37	1	no	—
187	#440	—	37	1	no	—
188	#440	D₃₇	1	1	yes	characteristic, complemented
189	C₄×Q₁₆	—	37	1	no	—
190	C₄×Q₁₆	—	37	1	no	—
191	C₄×D₈	—	37	1	no	—
192	C₄×D₈	—	37	1	no	—
193	C₂×#107	—	37	1	no	—
194	#417	—	37	1	no	—
195	#417	—	37	1	no	—
196	#405	—	37	1	no	—
197	C₃₇	#2951	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	37	1	no	—
199	C₂×#41	—	37	1	no	—
200	#135	—	37	8	no	—
201	C₂×Q₁₆	—	37	1	no	—
202	C₂×SD₁₆	—	37	1	no	—
203	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
204	C₂×D₈	—	37	1	no	—
205	C₂×M₁₆	—	37	1	no	—
206	C₂²×C₈	—	37	1	no	—
207	C₄×Q₈	—	37	1	no	—
208	C₄×Q₈	—	37	1	no	—
209	C₄×Q₈	—	37	1	no	—
210	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
211	C₄×D₄	—	37	1	no	—
212	C₄×D₄	—	37	1	no	—
213	C₄×D₄	—	37	1	no	—
214	C₄×D₄	—	74	1	no	—
215	C₄×D₄	—	74	1	no	—
216	C₄×D₄	—	74	1	no	—
217	C₄×D₄	—	74	1	no	—
218	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
219	#117	—	37	1	no	—
220	#117	—	74	1	no	—
221	#117	—	74	1	no	—
222	C₂×C₄⋊C₄	—	37	1	no	—
223	C₂×C₄⋊C₄	—	37	1	no	—
224	C₂×C₄⋊C₄	—	74	1	no	—
225	C₂×C₄⋊C₄	—	74	1	no	—
226	#107	—	37	1	no	—
227	#107	—	37	1	no	—
228	#107	—	37	1	no	—
229	#107	C₂×D₃₇	1	1	yes	characteristic
230	#103	—	37	1	no	—
231	#103	—	37	1	no	—
232	#103	—	37	1	no	—
233	#103	C₂×D₃₇	1	1	yes	characteristic
234	#102	—	37	1	no	—
235	#102	—	37	1	no	—
236	#102	—	37	1	no	—
237	#102	C₂×D₃₇	1	1	yes	characteristic
238	#97	—	37	1	no	—
239	#97	C₂×D₃₇	1	1	yes	characteristic
240	C₄×C₈	—	37	1	no	—
241	C₄×C₈	—	37	1	no	—
242	#41	—	37	4	no	—
243	#41	—	37	4	no	—
244	#41	—	74	2	no	—
245	#41	—	74	2	no	—
246	C₂×Q₈	—	37	1	no	—
247	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
248	C₂×D₄	—	37	1	no	—
249	C₂×D₄	—	74	1	no	—
250	C₂×D₄	—	74	1	no	—
251	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	37	1	no	—
254	C₂²×C₄	—	74	1	no	—
255	C₂²×C₄	—	74	1	no	—
256	C₂²×C₄	—	74	1	no	—
257	C₂²×C₄	—	74	1	no	—
258	C₂²×C₄	—	74	1	no	—
259	C₂²×C₄	—	74	1	no	—
260	C₂²×C₄	—	74	1	no	—
261	Q₁₆	—	37	4	no	—
262	SD₁₆	—	37	4	no	—
263	SD₁₆	C₄×D₃₇	1	4	yes	complemented
264	D₈	—	37	4	no	—
265	M₁₆	—	74	2	no	—
266	C₂×C₈	—	37	1	no	—
267	C₂×C₈	—	37	4	no	—
268	C₂×C₈	—	37	1	no	—
269	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	37	1	no	—
273	C₄⋊C₄	—	37	1	no	—
274	C₄⋊C₄	—	37	1	no	—
275	C₄⋊C₄	—	37	1	no	—
276	C₄⋊C₄	—	74	1	no	—
277	C₄⋊C₄	—	74	1	no	—
278	C₄⋊C₄	—	74	1	no	—
279	C₄⋊C₄	—	74	1	no	—
280	C₄⋊C₄	—	74	1	no	—
281	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	37	1	no	—
285	C₂²⋊C₄	—	37	1	no	—
286	C₂²⋊C₄	—	74	1	no	—
287	C₂²⋊C₄	—	74	1	no	—
288	C₂²⋊C₄	—	74	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	37	1	no	—
291	C₄²	—	74	1	no	—
292	C₄²	—	74	1	no	—
293	C₄²	—	74	1	no	—
294	C₄²	C₂²×D₃₇	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	37	1	no	—
297	C₂³	—	74	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	37	2	no	—
300	Q₈	—	74	1	no	—
301	Q₈	C₂×C₄×D₃₇	1	2	yes	—
302	D₄	—	2	1	no	—
303	D₄	—	37	2	no	—
304	D₄	—	74	4	no	—
305	D₄	—	74	4	no	—
306	D₄	—	74	1	no	—
307	D₄	C₂×C₄×D₃₇	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	37	1	no	—
313	C₂×C₄	—	37	2	no	—
314	C₂×C₄	—	37	2	no	—
315	C₂×C₄	—	37	1	no	—
316	C₂×C₄	—	37	1	no	—
317	C₂×C₄	—	296	1	no	—
318	C₂×C₄	—	296	1	no	—
319	C₂×C₄	—	296	1	no	—
320	C₂×C₄	—	296	1	no	—
321	C₂×C₄	—	296	1	no	—
322	C₂×C₄	—	74	4	no	—
323	C₂×C₄	—	74	4	no	—
324	C₂×C₄	—	74	1	no	—
325	C₂×C₄	—	74	1	no	—
326	C₂×C₄	—	74	1	no	—
327	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
328	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
329	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	37	2	no	—
332	C₈	—	74	1	no	—
333	C₈	C₂×C₄×D₃₇	1	2	yes	—
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	37	2	no	—
338	C₂²	—	74	1	no	—
339	C₂²	—	74	1	no	—
340	C₂²	—	74	2	no	—
341	C₂²	—	148	1	no	—
342	C₂²	—	148	1	no	—
343	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	37	2	no	—
350	C₄	—	74	2	no	—
351	C₄	—	74	1	no	—
352	C₄	—	74	1	no	—
353	C₄	—	148	1	no	—
354	C₄	—	148	1	no	—
355	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
356	C₄	#41×D₃₇	1	1	yes	characteristic
357	C₂	—	4	2	no	—
358	C₂	—	74	2	no	—
359	C₂	—	148	2	no	—
360	C₂	C₄×D₄×D₃₇	1	1	yes	characteristic
361	C₂	#14558337	1	1	yes	characteristic
362	C₂	#14558349	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#440	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#11924557	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#440	D₃₇
hypocenter	C₃₇	#2951
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #14558015	C₂	semidirect
3. #14558012	C₂	semidirect
4. #14557635	C₂	semidirect
5. #14557634	C₂	semidirect
6. #14557632	C₂	semidirect
7. C₄×#10596942	C₂	semidirect
8. #14557625	C₂	semidirect
9. C₄×#10596939	C₂	semidirect
10. #14557613	C₂	semidirect
11. #107×D₃₇	C₂	semidirect
12. #14557533	C₂	semidirect
13. #14557531	C₂	semidirect
14. #14557490	C₂	semidirect
15. C₃₇×#440	C₂	semidirect
18. #11924557	C₄	semidirect
67. #10596966	C₂×C₄	semidirect
69. #10596964	C₂×C₄	semidirect
79. SD₁₆×C₃₇	C₂×C₄	semidirect
89. #10596942	C₂×C₄	semidirect
90. #10596939	C₂×C₄	semidirect
110. #10596930	C₂×C₄	semidirect
112. C₈×D₃₇	C₂×C₄	semidirect
188. #440	D₃₇	semidirect
197. C₃₇	#2951	semidirect
263. SD₁₆	C₄×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924557	2
#14558015	2
#14558012	2
#14557635	2
#14557634	2
#14557632	2
C₄×#10596942	2
#14557625	2
C₄×#10596939	2
#14557613	2
#107×D₃₇	2
#14557533	2
#14557531	2
#14557490	2
C₃₇×#440	2
#2951	37