Subgroups — #18005065

Order 4736.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924565	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11924563	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924557	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#14558108	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558096	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557925	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557924	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557920	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557919	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557917	C₂	1	1	yes	maximal, characteristic, complemented
12	#14557914	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557904	C₂	1	1	yes	maximal, characteristic, complemented
14	#14557487	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10596966	C₂²	1	1	yes	characteristic
17	C₂×#10596964	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₃₇	C₂²	1	1	yes	characteristic
20	#11924565	—	2	4	no	—
21	#11924563	—	2	4	no	—
22	#11924557	—	2	4	no	—
23	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×#10596942	C₂²	1	1	yes	characteristic, complemented
25	C₂×#10596942	C₂²	1	1	yes	characteristic, complemented
26	C₂×#10596930	C₂²	1	1	yes	characteristic
27	C₂×#10596930	C₂²	1	1	yes	characteristic, complemented
28	C₂×#10596940	C₂²	1	1	yes	characteristic
29	C₂×#10596940	C₂²	1	1	yes	characteristic, complemented
30	C₂×#10596941	C₂²	1	1	yes	characteristic
31	C₂×#10596941	C₂²	1	1	yes	characteristic
32	SD₁₆×D₃₇	—	2	4	no	—
33	C₂×D₂₉₆	C₂²	1	1	yes	characteristic
34	C₂×#10596939	C₂²	1	1	yes	characteristic
35	C₂×#10596939	C₂²	1	1	yes	characteristic, complemented
36	C₂×#10596929	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
38	#11924534	C₂²	1	1	yes	characteristic, complemented
39	#11924532	C₂²	1	1	yes	characteristic
40	#11924525	C₂²	1	1	yes	characteristic, complemented
41	#11924524	C₂²	1	1	yes	characteristic
42	#11924524	C₂²	1	1	yes	characteristic, complemented
43	#11924523	C₂²	1	1	yes	characteristic
44	#11924521	—	2	2	no	—
45	#11924514	—	2	2	no	—
46	#11924508	—	2	2	no	—
47	#11924504	—	2	2	no	—
48	#11924493	C₂²	1	1	yes	characteristic
49	#11924447	C₂²	1	1	yes	characteristic, complemented
50	#11924446	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₃₇	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic, complemented
55	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
57	#127×C₃₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₃₇	—	2	1	no	—
60	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₃₇	—	2	2	no	—
63	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₃₇	—	2	1	no	—
66	C₂×C₂²⋊D₃₇	—	2	1	no	—
67	C₂×C₂²⋊D₃₇	—	2	2	no	—
68	C₂²×Dic₃₇	—	2	1	no	—
69	#10596966	—	2	2	no	—
70	#10596966	—	2	2	no	—
71	Q₈×D₃₇	—	2	2	no	—
72	Q₈×D₃₇	—	2	2	no	—
73	#10596964	—	2	2	no	—
74	#10596964	—	2	2	no	—
75	D₄×D₃₇	—	2	2	no	—
76	D₄×D₃₇	—	2	2	no	—
77	C₂×D₁₄₈	—	2	1	no	—
78	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
79	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₃₇	—	2	1	no	—
81	C₂×C₄×D₃₇	—	2	1	no	—
82	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
83	C₂×Dic₇₄	—	2	1	no	—
84	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
85	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
86	SD₁₆×C₃₇	—	2	2	no	—
87	SD₁₆×C₃₇	—	2	2	no	—
88	SD₁₆×C₃₇	—	2	2	no	—
89	SD₁₆×C₃₇	—	2	2	no	—
90	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₃₇	—	2	2	no	—
93	C₄²×C₃₇	C₂³	1	1	yes	characteristic
94	C₂²⋊Dic₃₇	—	2	2	no	—
95	#10596942	—	2	2	no	—
96	#10596942	—	2	2	no	—
97	#10596941	—	2	2	no	—
98	#10596941	—	2	2	no	—
99	#10596940	—	2	2	no	—
100	#10596940	—	2	2	no	—
101	#10596939	—	2	2	no	—
102	#10596939	—	2	2	no	—
103	#10596938	—	2	2	no	—
104	#10596938	—	2	2	no	—
105	#10596938	—	2	2	no	—
106	#10596938	D₄	1	2	yes	—
107	#10596936	—	2	2	no	—
108	#10596936	D₄	1	2	yes	—
109	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
110	C₂×#64436	C₂³	1	1	yes	characteristic
111	C₂×#64436	C₂³	1	1	yes	characteristic
112	Dic₁₄₈	—	2	2	no	—
113	D₂₉₆	—	2	2	no	—
114	#10596930	—	2	2	no	—
115	#10596930	—	2	2	no	—
116	#10596929	D₄	1	4	yes	complemented
117	C₈×D₃₇	D₄	1	4	yes	complemented
118	C₂³×C₃₇	—	2	1	no	—
119	C₂³×C₃₇	—	2	1	no	—
120	C₂²×D₃₇	—	2	1	no	—
121	C₂²×D₃₇	—	2	1	no	—
122	C₂²×D₃₇	—	4	2	no	—
123	C₂²×D₃₇	—	8	2	no	—
124	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₃₇	—	2	1	no	—
126	Q₈×C₃₇	—	2	1	no	—
127	Q₈×C₃₇	—	2	1	no	—
128	Q₈×C₃₇	—	2	1	no	—
129	D₄×C₃₇	—	2	1	no	—
130	D₄×C₃₇	—	2	1	no	—
131	D₄×C₃₇	—	2	1	no	—
132	D₄×C₃₇	—	2	1	no	—
133	D₄×C₃₇	—	8	2	no	—
134	C₂×C₄×C₃₇	—	2	1	no	—
135	C₂×C₄×C₃₇	—	2	1	no	—
136	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
137	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
138	C₂²⋊D₃₇	—	4	2	no	—
139	C₂²⋊D₃₇	—	4	2	no	—
140	C₂²⋊D₃₇	—	8	2	no	—
141	C₂×Dic₃₇	—	2	1	no	—
142	C₂×Dic₃₇	—	2	1	no	—
143	C₂×Dic₃₇	—	4	2	no	—
144	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
145	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
146	D₁₄₈	—	2	1	no	—
147	D₁₄₈	—	2	1	no	—
148	D₁₄₈	—	2	1	no	—
149	D₁₄₈	—	2	1	no	—
150	D₁₄₈	—	4	2	no	—
151	C₄×D₃₇	—	4	2	no	—
152	C₄×D₃₇	—	4	2	no	—
153	C₄×D₃₇	C₂×D₄	1	2	yes	—
154	C₄×D₃₇	C₂×D₄	1	2	yes	—
155	Dic₇₄	—	2	1	no	—
156	Dic₇₄	—	2	1	no	—
157	Dic₇₄	—	2	1	no	—
158	Dic₇₄	—	2	1	no	—
159	Dic₇₄	—	4	2	no	—
160	C₈×C₃₇	C₂×D₄	1	2	yes	—
161	C₈×C₃₇	C₂×D₄	1	2	yes	—
162	#64436	C₂×D₄	1	2	yes	—
163	#64436	C₂×D₄	1	2	yes	—
164	C₂²×C₃₇	—	4	1	no	—
165	C₂²×C₃₇	—	4	1	no	—
166	C₂²×C₃₇	—	4	2	no	—
167	C₂²×C₃₇	—	4	2	no	—
168	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×D₃₇	—	2	2	no	—
170	C₂×D₃₇	—	4	1	no	—
171	C₂×D₃₇	—	4	1	no	—
172	C₂×D₃₇	—	4	2	no	—
173	C₂×D₃₇	—	4	2	no	—
174	C₂×D₃₇	—	8	4	no	—
175	C₂×D₃₇	#127	1	2	yes	—
176	C₄×C₃₇	—	2	2	no	—
177	C₄×C₃₇	—	4	1	no	—
178	C₄×C₃₇	—	4	1	no	—
179	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
180	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
181	Dic₃₇	—	2	2	no	—
182	Dic₃₇	—	4	1	no	—
183	Dic₃₇	—	4	1	no	—
184	Dic₃₇	#127	1	2	yes	—
185	#3154	—	37	1	no	maximal, Sylow
186	C₂×C₃₇	—	8	1	no	—
187	C₂×C₃₇	—	8	1	no	—
188	C₂×C₃₇	C₂×#127	1	1	yes	characteristic
189	C₂×C₃₇	#577	1	2	yes	—
190	D₃₇	—	2	2	no	—
191	D₃₇	—	8	1	no	—
192	D₃₇	—	8	1	no	—
193	C₂×#137	—	37	1	no	—
194	C₂×AGL(1,ℤ/8)	—	37	1	no	—
195	C₂×#135	—	37	1	no	—
196	C₂²×SD₁₆	—	37	1	no	—
197	#536	—	37	1	no	—
198	#535	—	37	1	no	—
199	#497	—	37	1	no	—
200	#497	—	37	1	no	—
201	#496	—	37	1	no	—
202	#496	—	37	1	no	—
203	#495	—	37	1	no	—
204	#495	—	37	1	no	—
205	#492	—	37	1	no	—
206	#492	D₃₇	1	1	yes	characteristic, complemented
207	#405	—	37	1	no	—
208	C₃₇	#3154	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	37	1	no	—
210	C₂×#41	—	37	1	no	—
211	C₂²×Q₈	—	37	1	no	—
212	C₂²×D₄	—	37	1	no	—
213	#137	—	74	4	no	—
214	AGL(1,ℤ/8)	—	74	4	no	—
215	#135	—	74	4	no	—
216	C₂×Q₁₆	—	37	1	no	—
217	C₂×Q₁₆	—	37	1	no	—
218	C₂×Q₁₆	—	37	1	no	—
219	C₂×SD₁₆	—	37	1	no	—
220	C₂×SD₁₆	—	37	1	no	—
221	C₂×SD₁₆	—	37	1	no	—
222	C₂×SD₁₆	—	37	1	no	—
223	C₂×SD₁₆	—	37	1	no	—
224	C₂×SD₁₆	—	37	1	no	—
225	C₂×SD₁₆	—	74	4	no	—
226	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
227	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic, complemented
229	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic, complemented
230	C₂×D₈	—	37	1	no	—
231	C₂×D₈	—	37	1	no	—
232	C₂×D₈	—	37	1	no	—
233	C₂×M₁₆	—	37	1	no	—
234	C₂²×C₈	—	37	1	no	—
235	C₄⋊Q₈	—	37	1	no	—
236	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic, complemented
237	#127	—	37	1	no	—
238	#127	C₂×D₃₇	1	1	yes	characteristic
239	#124	—	37	1	no	—
240	#124	—	37	1	no	—
241	#124	—	37	1	no	—
242	#124	—	37	1	no	—
243	#123	—	74	2	no	—
244	C₂²⋊Q₈	—	74	2	no	—
245	#121	—	74	2	no	—
246	#120	—	74	2	no	—
247	#117	—	37	1	no	—
248	#97	—	37	1	no	—
249	#97	—	37	1	no	—
250	C₄×C₈	—	37	1	no	—
251	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
252	C₂⁴	—	74	1	no	—
253	#41	—	74	2	no	—
254	#41	—	74	2	no	—
255	#41	—	74	2	no	—
256	#41	—	74	2	no	—
257	C₂×Q₈	—	37	1	no	—
258	C₂×Q₈	—	37	1	no	—
259	C₂×Q₈	—	74	1	no	—
260	C₂×Q₈	—	74	2	no	—
261	C₂×Q₈	—	74	2	no	—
262	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
263	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	37	1	no	—
266	C₂×D₄	—	37	1	no	—
267	C₂×D₄	—	74	1	no	—
268	C₂×D₄	—	74	2	no	—
269	C₂×D₄	—	74	2	no	—
270	C₂×D₄	—	74	2	no	—
271	C₂×D₄	—	74	1	no	—
272	C₂×D₄	—	74	1	no	—
273	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
274	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
275	C₂²×C₄	—	37	1	no	—
276	C₂²×C₄	—	74	1	no	—
277	C₂²×C₄	—	74	1	no	—
278	C₂²×C₄	—	74	1	no	—
279	Q₁₆	—	74	2	no	—
280	Q₁₆	—	74	2	no	—
281	Q₁₆	—	74	2	no	—
282	SD₁₆	—	2	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	2	2	no	—
285	SD₁₆	—	2	2	no	—
286	SD₁₆	—	74	2	no	—
287	SD₁₆	—	74	2	no	—
288	SD₁₆	—	74	2	no	—
289	SD₁₆	—	74	2	no	—
290	SD₁₆	—	74	2	no	—
291	SD₁₆	—	74	2	no	—
292	D₈	—	74	2	no	—
293	D₈	—	74	2	no	—
294	D₈	—	74	2	no	—
295	M₁₆	—	37	4	no	—
296	C₂×C₈	—	37	1	no	—
297	C₂×C₈	—	37	1	no	—
298	C₂×C₈	—	37	4	no	—
299	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
300	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	37	2	no	—
303	C₄⋊C₄	—	74	2	no	—
304	C₂²⋊C₄	—	37	2	no	—
305	C₂²⋊C₄	—	74	2	no	—
306	C₂²⋊C₄	—	74	2	no	—
307	C₂²⋊C₄	—	74	2	no	—
308	C₂²⋊C₄	—	74	2	no	—
309	C₄²	—	37	1	no	—
310	C₄²	C₂²×D₃₇	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	37	1	no	—
314	C₂³	—	296	2	no	—
315	C₂³	—	74	1	no	—
316	C₂³	—	74	1	no	—
317	C₂³	—	148	2	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	74	1	no	—
323	Q₈	—	74	1	no	—
324	Q₈	—	74	1	no	—
325	Q₈	—	74	1	no	—
326	Q₈	—	148	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	2	1	no	—
331	D₄	—	8	2	no	—
332	D₄	—	296	2	no	—
333	D₄	—	74	1	no	—
334	D₄	—	74	1	no	—
335	D₄	—	74	1	no	—
336	D₄	—	74	1	no	—
337	D₄	—	148	2	no	—
338	D₄	—	148	2	no	—
339	D₄	—	148	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	37	1	no	—
343	C₂×C₄	—	37	2	no	—
344	C₂×C₄	—	37	2	no	—
345	C₂×C₄	—	37	2	no	—
346	C₂×C₄	—	74	1	no	—
347	C₂×C₄	—	74	1	no	—
348	C₂×C₄	—	148	2	no	—
349	C₂×C₄	—	148	2	no	—
350	C₂×C₄	—	148	2	no	—
351	C₂×C₄	D₄×D₃₇	1	2	yes	—
352	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
353	C₈	—	37	2	no	—
354	C₈	—	37	2	no	—
355	C₈	D₄×D₃₇	1	2	yes	—
356	C₈	D₄×D₃₇	1	2	yes	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	37	2	no	—
362	C₂²	—	296	4	no	—
363	C₂²	—	74	2	no	—
364	C₂²	—	148	1	no	—
365	C₂²	—	148	1	no	—
366	C₂²	—	148	2	no	—
367	C₂²	—	148	2	no	—
368	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	37	2	no	—
373	C₄	—	74	2	no	—
374	C₄	—	148	1	no	—
375	C₄	—	148	1	no	—
376	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
377	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
378	C₂	—	8	1	no	—
379	C₂	—	8	1	no	—
380	C₂	—	296	1	no	—
381	C₂	—	296	1	no	—
382	C₂	—	74	2	no	—
383	C₂	#127×D₃₇	1	1	yes	characteristic
384	C₂	#14558339	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#492	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	#492	D₃₇
hypocenter	C₃₇	#3154
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. C₂×#11924565	C₂	semidirect
2. C₂×#11924563	C₂	semidirect
3. C₂×#11924557	C₂	semidirect
4. C₂×SD₁₆×D₃₇	C₂	semidirect
5. #14558108	C₂	semidirect
6. #14558096	C₂	semidirect
7. #14557925	C₂	semidirect
8. #14557924	C₂	semidirect
9. #14557920	C₂	semidirect
10. #14557919	C₂	semidirect
11. #14557917	C₂	semidirect
12. #14557914	C₂	semidirect
13. #14557904	C₂	semidirect
14. #14557487	C₂	semidirect
15. C₃₇×#492	C₂	semidirect
23. C₂×Dic₁₄₈	C₂²	semidirect
24. C₂×#10596942	C₂²	semidirect
25. C₂×#10596942	C₂²	semidirect
27. C₂×#10596930	C₂²	semidirect
29. C₂×#10596940	C₂²	semidirect
35. C₂×#10596939	C₂²	semidirect
38. #11924534	C₂²	semidirect
40. #11924525	C₂²	semidirect
42. #11924524	C₂²	semidirect
49. #11924447	C₂²	semidirect
50. #11924446	C₂²	semidirect
51. C₈×Dic₃₇	C₂²	semidirect
54. C₂×SD₁₆×C₃₇	C₂²	semidirect
55. C₂×SD₁₆×C₃₇	C₂²	semidirect
56. C₄⋊Q₈×C₃₇	C₂²	semidirect
58. C₄×C₈×C₃₇	C₂²	semidirect
116. #10596929	D₄	semidirect
117. C₈×D₃₇	D₄	semidirect
206. #492	D₃₇	semidirect
208. C₃₇	#3154	semidirect
228. C₂×SD₁₆	C₂×D₃₇	semidirect
229. C₂×SD₁₆	C₂×D₃₇	semidirect
236. C₄⋊Q₈	C₂×D₃₇	semidirect
251. 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
C₂×#11924565	2
C₂×#11924563	2
C₂×#11924557	2
C₂×SD₁₆×D₃₇	2
#14558108	2
#14558096	2
#14557925	2
#14557924	2
#14557920	2
#14557919	2
#14557917	2
#14557914	2
#14557904	2
#14557487	2
C₃₇×#492	2
#3154	37