Subgroups — #24919428

Order 10624.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128351	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128345	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128059	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128032	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128018	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127989	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127951	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127928	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127914	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127843	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127813	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127769	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127699	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#14581713	C₂	1	1	yes	maximal, characteristic, complemented
15	#447×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	—	2	1	no	—
17	C₂×#13050589	C₂²	1	1	yes	characteristic
18	C₂×#13050578	C₂²	1	1	yes	characteristic, complemented
19	C₂×#13050573	C₂²	1	1	yes	characteristic, complemented
20	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic, complemented
22	#14581779	C₂²	1	1	yes	characteristic
23	#14581774	—	2	1	no	—
24	#14581773	—	2	1	no	—
25	#14581773	—	2	1	no	—
26	#14581771	C₂²	1	1	yes	characteristic
27	#14581769	—	2	1	no	—
28	#14581764	—	2	1	no	—
29	#14581763	C₂²	1	1	yes	characteristic
30	#14581763	C₂²	1	1	yes	characteristic, complemented
31	#14581761	—	2	1	no	—
32	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
33	D₄×Dic₈₃	C₂²	1	1	yes	characteristic, complemented
34	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊Dic₈₃	—	2	1	no	—
38	#14581724	C₂²	1	1	yes	characteristic, complemented
39	#14581724	C₂²	1	1	yes	characteristic, complemented
40	#14581723	C₂²	1	1	yes	characteristic, complemented
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	#14581717	C₂²	1	1	yes	characteristic, complemented
43	#14581717	C₂²	1	1	yes	characteristic, complemented
44	#14581715	C₂²	1	1	yes	characteristic, complemented
45	#14581713	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	#14581713	C₂²	1	1	yes	characteristic, complemented
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	#14581702	C₂²	1	1	yes	characteristic, complemented
52	C₂²×D₄×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
55	#121×C₈₃	C₂²	1	1	yes	characteristic
56	#102×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	2	2	no	—
60	#13050589	—	4	1	no	—
61	#13050589	—	4	4	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
68	C₂²×Dic₈₃	—	2	1	no	—
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂²×Dic₈₃	—	2	1	no	—
71	C₂²×Dic₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
74	#13050578	—	4	1	no	—
75	#13050573	—	4	1	no	—
76	C₂×C₈₃⋊C₈	—	4	1	no	—
77	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	—	2	1	no	—
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	2	1	no	—
91	C₂²⋊Dic₈₃	—	2	1	no	—
92	C₂²⋊Dic₈₃	—	2	1	no	—
93	C₂²⋊Dic₈₃	—	2	1	no	—
94	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
95	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂⁴×C₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	—	2	2	no	—
99	C₂×D₄×C₈₃	—	2	1	no	—
100	C₂×D₄×C₈₃	—	2	1	no	—
101	C₂×D₄×C₈₃	—	4	1	no	—
102	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
106	D₈×C₈₃	—	2	2	no	—
107	D₈×C₈₃	—	4	1	no	—
108	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
109	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
111	C₂²⋊C₄×C₈₃	—	2	1	no	—
112	C₂³×C₈₃	—	2	1	no	—
113	C₂³×C₈₃	—	2	1	no	—
114	C₂³×C₈₃	—	2	1	no	—
115	C₂³×C₈₃	—	4	2	no	—
116	C₂³×C₈₃	—	4	2	no	—
117	C₂³×C₈₃	—	8	1	no	—
118	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
120	D₄×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	2	1	no	—
122	D₄×C₈₃	—	2	1	no	—
123	D₄×C₈₃	—	2	1	no	—
124	D₄×C₈₃	—	2	1	no	—
125	D₄×C₈₃	—	4	2	no	—
126	D₄×C₈₃	—	4	2	no	—
127	D₄×C₈₃	#41	1	2	yes	complemented
128	C₂×C₄×C₈₃	—	2	1	no	—
129	C₂×C₄×C₈₃	—	2	1	no	—
130	C₂×C₄×C₈₃	—	2	1	no	—
131	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
133	C₂²⋊D₈₃	—	2	2	no	—
134	C₂²⋊D₈₃	—	4	2	no	—
135	C₂²⋊D₈₃	—	4	2	no	—
136	C₂²⋊D₈₃	—	8	1	no	—
137	C₂×Dic₈₃	—	2	1	no	—
138	C₂×Dic₈₃	—	2	1	no	—
139	C₂×Dic₈₃	—	2	1	no	—
140	C₂×Dic₈₃	—	2	1	no	—
141	C₂×Dic₈₃	—	4	2	no	—
142	C₂×Dic₈₃	—	4	2	no	—
143	C₂×Dic₈₃	—	4	2	no	—
144	C₂×Dic₈₃	—	8	1	no	—
145	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
146	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
147	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
148	C₄×D₈₃	—	2	1	no	—
149	C₄×D₈₃	—	2	1	no	—
150	C₄×D₈₃	—	4	2	no	—
151	Dic₁₆₆	—	2	1	no	—
152	Dic₁₆₆	—	2	1	no	—
153	Dic₁₆₆	—	4	2	no	—
154	C₈×C₈₃	—	2	1	no	—
155	C₈×C₈₃	—	2	1	no	—
156	C₈₃⋊C₈	—	2	1	no	—
157	C₈₃⋊C₈	—	2	1	no	—
158	C₂²×C₈₃	—	2	1	no	—
159	C₂²×C₈₃	—	2	1	no	—
160	C₂²×C₈₃	—	2	2	no	—
161	C₂²×C₈₃	—	4	1	no	—
162	C₂²×C₈₃	—	4	1	no	—
163	C₂²×C₈₃	—	4	1	no	—
164	C₂²×C₈₃	—	4	1	no	—
165	C₂²×C₈₃	—	4	1	no	—
166	C₂²×C₈₃	—	4	1	no	—
167	C₂²×C₈₃	—	4	1	no	—
168	C₂²×C₈₃	—	4	1	no	—
169	C₂²×C₈₃	—	8	4	no	—
170	C₂²×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
171	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×D₈₃	—	2	1	no	—
173	C₂×D₈₃	—	2	1	no	—
174	C₂×D₈₃	—	2	1	no	—
175	C₄×C₈₃	—	2	1	no	—
176	C₄×C₈₃	—	4	1	no	—
177	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
178	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
179	Dic₈₃	—	2	1	no	—
180	Dic₈₃	—	2	1	no	—
181	Dic₈₃	—	2	1	no	—
182	Dic₈₃	—	4	1	no	—
183	Dic₈₃	—	4	1	no	—
184	Dic₈₃	—	4	1	no	—
185	Dic₈₃	—	4	1	no	—
186	C₂×C₈₃	—	2	2	no	—
187	C₂×C₈₃	—	4	2	no	—
188	C₂×C₈₃	—	8	1	no	—
189	C₂×C₈₃	—	8	1	no	—
190	C₂×C₈₃	#546	1	1	yes	characteristic
191	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₂×C₈₃	#577	1	1	yes	characteristic
193	D₈₃	—	4	1	no	—
194	#3306	—	83	1	no	maximal, Sylow
195	C₈₃	#3306	1	1	yes	characteristic, complemented, Sylow
196	#547	—	83	1	no	—
197	#546	—	83	1	no	—
198	#489	—	83	1	no	—
199	#483	—	83	1	no	—
200	#481	—	83	1	no	—
201	#476	—	83	1	no	—
202	#469	—	83	1	no	—
203	#465	—	83	1	no	—
204	#463	—	83	1	no	—
205	#449	—	83	1	no	—
206	#447	D₈₃	1	1	yes	characteristic, complemented
207	#442	—	83	1	no	—
208	#435	—	83	1	no	—
209	#417	—	83	1	no	—
210	C₂×#102	—	83	1	no	—
211	C₂×#41	—	83	1	no	—
212	C₂×#41	—	166	1	no	—
213	C₂²×D₄	C₂×D₈₃	1	1	yes	characteristic
214	C₂×SD₁₆	—	83	1	no	—
215	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
216	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
217	C₂×M₁₆	—	83	1	no	—
218	C₂²×C₈	—	83	1	no	—
219	C₄⋊Q₈	—	83	1	no	—
220	#124	—	83	1	no	—
221	#123	—	166	1	no	—
222	#123	—	166	1	no	—
223	#123	—	166	1	no	—
224	C₂²⋊Q₈	—	83	1	no	—
225	C₂²⋊Q₈	—	166	1	no	—
226	#121	—	83	1	no	—
227	#121	—	83	1	no	—
228	#121	—	166	1	no	—
229	#121	C₂×D₈₃	1	1	yes	characteristic
230	#120	—	166	1	no	—
231	C₄×D₄	—	83	1	no	—
232	C₄×D₄	—	83	1	no	—
233	C₄×D₄	—	83	1	no	—
234	C₂×C₄⋊C₄	—	83	1	no	—
235	C₂×C₄⋊C₄	—	83	1	no	—
236	C₂×C₂²⋊C₄	—	166	1	no	—
237	#106	—	83	1	no	—
238	#106	—	83	1	no	—
239	#106	—	83	1	no	—
240	C₄⋊C₈	—	83	1	no	—
241	#103	—	83	1	no	—
242	#103	—	83	1	no	—
243	#103	—	83	1	no	—
244	#102	—	83	1	no	—
245	#102	—	83	1	no	—
246	#102	—	83	1	no	—
247	#102	—	83	1	no	—
248	#102	—	83	1	no	—
249	#102	C₂×D₈₃	1	1	yes	characteristic
250	#102	C₂×D₈₃	1	1	yes	characteristic
251	C₂²⋊C₈	—	83	1	no	—
252	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
253	#97	—	83	1	no	—
254	C₂⁴	—	2	1	no	—
255	#41	—	332	4	no	—
256	#41	—	332	1	no	—
257	#41	—	166	2	no	—
258	C₂×Q₈	—	83	1	no	—
259	C₂×Q₈	—	166	1	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	83	1	no	—
265	C₂×D₄	—	166	1	no	—
266	C₂×D₄	—	166	1	no	—
267	C₂×D₄	—	166	1	no	—
268	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
269	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
270	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
271	C₂²×C₄	—	83	1	no	—
272	C₂²×C₄	—	166	1	no	—
273	C₂²×C₄	—	166	1	no	—
274	C₂²×C₄	—	166	1	no	—
275	C₂²×C₄	—	166	1	no	—
276	C₂²×C₄	—	166	1	no	—
277	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
278	SD₁₆	—	332	1	no	—
279	D₈	—	2	2	no	—
280	D₈	—	4	1	no	—
281	M₁₆	—	332	1	no	—
282	C₂×C₈	—	332	1	no	—
283	C₂×C₈	—	83	1	no	—
284	C₂×C₈	—	83	1	no	—
285	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
286	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
287	C₄⋊C₄	—	83	1	no	—
288	C₄⋊C₄	—	83	1	no	—
289	C₄⋊C₄	—	83	1	no	—
290	C₄⋊C₄	—	83	1	no	—
291	C₄⋊C₄	—	166	1	no	—
292	C₄⋊C₄	—	166	1	no	—
293	C₄⋊C₄	—	166	1	no	—
294	C₄⋊C₄	—	166	1	no	—
295	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
296	C₂²⋊C₄	—	2	1	no	—
297	C₂²⋊C₄	—	83	1	no	—
298	C₂²⋊C₄	—	83	1	no	—
299	C₂²⋊C₄	—	166	1	no	—
300	C₂²⋊C₄	—	166	1	no	—
301	C₂²⋊C₄	—	166	1	no	—
302	C₂²⋊C₄	—	166	1	no	—
303	C₂²⋊C₄	—	166	1	no	—
304	C₂²⋊C₄	—	166	1	no	—
305	C₂²⋊C₄	—	166	1	no	—
306	C₄²	—	83	1	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	2	1	no	—
309	C₂³	—	2	1	no	—
310	C₂³	—	4	2	no	—
311	C₂³	—	4	2	no	—
312	C₂³	—	8	1	no	—
313	C₂³	—	83	1	no	—
314	C₂³	D₄×D₈₃	1	1	yes	characteristic
315	Q₈	—	332	2	no	—
316	Q₈	—	166	1	no	—
317	Q₈	—	166	1	no	—
318	D₄	—	2	1	no	—
319	D₄	—	2	1	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	2	1	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	332	2	no	—
326	D₄	—	332	2	no	—
327	D₄	—	664	1	no	—
328	D₄	—	166	2	no	—
329	D₄	#13050589	1	2	yes	complemented
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	332	2	no	—
334	C₂×C₄	—	332	2	no	—
335	C₂×C₄	—	332	2	no	—
336	C₂×C₄	—	332	2	no	—
337	C₂×C₄	—	83	1	no	—
338	C₂×C₄	—	83	1	no	—
339	C₂×C₄	—	83	1	no	—
340	C₂×C₄	—	664	1	no	—
341	C₂×C₄	—	166	1	no	—
342	C₂×C₄	—	166	1	no	—
343	C₂×C₄	—	166	1	no	—
344	C₂×C₄	—	166	1	no	—
345	C₂×C₄	—	166	1	no	—
346	C₂×C₄	—	166	1	no	—
347	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
348	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	166	1	no	—
352	C₈	—	166	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	2	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	—	4	1	no	—
364	C₂²	—	8	4	no	—
365	C₂²	—	166	1	no	—
366	C₂²	—	166	1	no	—
367	C₂²	—	166	1	no	—
368	C₂²	#14581816	1	2	yes	—
369	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	332	1	no	—
373	C₄	—	332	1	no	—
374	C₄	—	332	1	no	—
375	C₄	—	332	1	no	—
376	C₄	—	166	1	no	—
377	C₄	—	166	1	no	—
378	C₄	—	166	1	no	—
379	C₄	C₂×#13050589	1	1	yes	characteristic
380	C₄	#14581837	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	4	2	no	—
383	C₂	—	8	1	no	—
384	C₂	—	8	1	no	—
385	C₂	—	332	1	no	—
386	C₂	#19128338	1	1	yes	characteristic
387	C₂	C₂×#14581816	1	1	yes	characteristic
388	C₂	#19128511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#447×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#447	D₈₃
hypocenter	C₈₃	#3306
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #447

Complements

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

N	H ≅ G/N	type
1. #19128351	C₂	semidirect
2. #19128345	C₂	semidirect
3. #19128059	C₂	semidirect
4. #19128032	C₂	semidirect
5. #19128018	C₂	semidirect
6. #19127989	C₂	semidirect
7. #19127951	C₂	semidirect
8. #19127928	C₂	semidirect
9. #19127914	C₂	semidirect
10. #19127843	C₂	semidirect
11. #19127813	C₂	semidirect
12. #19127769	C₂	semidirect
13. #19127699	C₂	semidirect
14. C₂×#14581713	C₂	semidirect
15. #447×C₈₃	C₂	semidirect
18. C₂×#13050578	C₂²	semidirect
19. C₂×#13050573	C₂²	semidirect
21. #14581791	C₂²	semidirect
30. #14581763	C₂²	semidirect
33. D₄×Dic₈₃	C₂²	semidirect
36. C₄⋊C₄×D₈₃	C₂²	semidirect
38. #14581724	C₂²	semidirect
39. #14581724	C₂²	semidirect
40. #14581723	C₂²	semidirect
41. Dic₈₃⋊C₈	C₂²	semidirect
42. #14581717	C₂²	semidirect
43. #14581717	C₂²	semidirect
44. #14581715	C₂²	semidirect
49. #14581713	C₂²	semidirect
50. (C₂×D₈₃)⋊C₈	C₂²	semidirect
51. #14581702	C₂²	semidirect
127. D₄×C₈₃	#41	semidirect
195. C₈₃	#3306	semidirect
206. #447	D₈₃	semidirect
329. D₄	#13050589	semidirect

Maximal subgroups

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

subgroup	index
#19128351	2
#19128345	2
#19128059	2
#19128032	2
#19128018	2
#19127989	2
#19127951	2
#19127928	2
#19127914	2
#19127843	2
#19127813	2
#19127769	2
#19127699	2
C₂×#14581713	2
#447×C₈₃	2
#3306	83