Subgroups — #24917607

Order 10624.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128246	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128026	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128022	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127984	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127979	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127899	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127894	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127846	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127813	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127810	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127804	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127802	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127755	C₂	1	1	yes	maximal, characteristic, complemented
15	#449×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	—	2	1	no	—
17	C₂×#13050577	C₂²	1	1	yes	characteristic
18	C₂×#13050575	C₂²	1	1	yes	characteristic
19	#14581791	C₂²	1	1	yes	characteristic
20	#14581788	C₂²	1	1	yes	characteristic
21	#14581778	—	2	1	no	—
22	#14581766	C₂²	1	1	yes	characteristic
23	#14581764	—	2	1	no	—
24	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
25	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
26	D₄×Dic₈₃	—	2	1	no	—
27	D₄×Dic₈₃	—	2	1	no	—
28	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
30	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
31	#14581754	C₂²	1	1	yes	characteristic
32	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
33	#14581751	—	2	1	no	—
34	#14581751	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₈₃	—	2	1	no	—
36	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
37	#14581726	C₂²	1	1	yes	characteristic
38	#14581725	C₂²	1	1	yes	characteristic
39	#14581724	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	#14581716	C₂²	1	1	yes	characteristic
44	#14581714	C₂²	1	1	yes	characteristic
45	#14581714	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	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	#14581702	C₂²	1	1	yes	characteristic
52	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
55	#121×C₈₃	C₂²	1	1	yes	characteristic
56	#103×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	—	4	4	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₃	—	2	1	no	—
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂²×Dic₈₃	—	2	1	no	—
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂²×Dic₈₃	—	2	1	no	—
68	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
69	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₃	—	2	1	no	—
71	#13050577	—	2	2	no	—
72	#13050575	—	2	2	no	—
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	Dic₈₃⋊C₄	—	2	1	no	—
81	C₂²⋊Dic₈₃	—	2	1	no	—
82	C₂²⋊Dic₈₃	—	2	1	no	—
83	C₂²⋊Dic₈₃	—	2	1	no	—
84	C₂²⋊Dic₈₃	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	C₄×Dic₈₃	—	2	1	no	—
88	C₄×Dic₈₃	—	2	1	no	—
89	C₄×Dic₈₃	—	2	1	no	—
90	C₄×Dic₈₃	D₄	1	1	yes	characteristic
91	C₄×Dic₈₃	D₄	1	1	yes	characteristic
92	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
93	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
94	#41×C₈₃	—	2	2	no	—
95	#41×C₈₃	—	2	2	no	—
96	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	—	2	1	no	—
99	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₃	—	2	1	no	—
102	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
103	SD₁₆×C₈₃	—	2	2	no	—
104	D₈×C₈₃	—	2	2	no	—
105	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₃	—	2	1	no	—
109	C₂³×C₈₃	—	2	1	no	—
110	C₂³×C₈₃	—	2	1	no	—
111	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₈₃	—	2	1	no	—
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	#41	1	2	yes	complemented
115	D₄×C₈₃	—	2	4	no	—
116	D₄×C₈₃	—	2	1	no	—
117	D₄×C₈₃	—	2	1	no	—
118	D₄×C₈₃	—	2	1	no	—
119	D₄×C₈₃	—	4	2	no	—
120	D₄×C₈₃	#41	1	2	yes	complemented
121	C₂×C₄×C₈₃	—	2	4	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	—	2	1	no	—
125	C₂×C₄×C₈₃	—	2	1	no	—
126	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₈₃	—	4	2	no	—
129	C₂²⋊D₈₃	—	4	2	no	—
130	C₂²⋊D₈₃	—	4	2	no	—
131	C₂×Dic₈₃	—	2	2	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	2	1	no	—
135	C₂×Dic₈₃	—	4	1	no	—
136	C₂×Dic₈₃	—	4	2	no	—
137	C₂×Dic₈₃	—	4	2	no	—
138	C₂×Dic₈₃	—	4	2	no	—
139	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
143	D₃₃₂	—	2	1	no	—
144	D₃₃₂	—	2	1	no	—
145	C₄×D₈₃	—	4	2	no	—
146	Dic₁₆₆	—	4	2	no	—
147	C₈×C₈₃	—	2	1	no	—
148	C₈×C₈₃	—	2	1	no	—
149	C₈₃⋊C₈	—	2	1	no	—
150	C₈₃⋊C₈	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	—	2	2	no	—
153	C₂²×C₈₃	—	2	1	no	—
154	C₂²×C₈₃	—	2	1	no	—
155	C₂²×C₈₃	—	4	1	no	—
156	C₂²×C₈₃	—	4	1	no	—
157	C₂²×C₈₃	—	4	1	no	—
158	C₂²×C₈₃	—	4	1	no	—
159	C₂²×C₈₃	—	4	1	no	—
160	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₈₃	—	4	1	no	—
162	C₂×D₈₃	—	4	1	no	—
163	C₂×D₈₃	—	4	1	no	—
164	C₄×C₈₃	—	2	1	no	—
165	C₄×C₈₃	—	2	2	no	—
166	C₄×C₈₃	—	4	1	no	—
167	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
168	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
169	Dic₈₃	—	2	1	no	—
170	Dic₈₃	—	2	1	no	—
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	4	1	no	—
173	Dic₈₃	—	4	1	no	—
174	Dic₈₃	—	4	1	no	—
175	Dic₈₃	AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₈₃	—	4	1	no	—
177	C₂×C₈₃	—	4	2	no	—
178	C₂×C₈₃	—	8	1	no	—
179	C₂×C₈₃	#525	1	1	yes	characteristic
180	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₈₃	#575	1	1	yes	characteristic
182	D₈₃	—	8	1	no	—
183	#3115	—	83	1	no	maximal, Sylow
184	C₈₃	#3115	1	1	yes	characteristic, complemented, Sylow
185	#532	—	83	1	no	—
186	C₄×#41	—	83	1	no	—
187	#482	—	83	1	no	—
188	#482	—	83	1	no	—
189	#475	—	83	1	no	—
190	#474	—	83	1	no	—
191	#460	—	83	1	no	—
192	#459	—	83	1	no	—
193	#449	—	83	1	no	—
194	#449	D₈₃	1	1	yes	characteristic, complemented
195	#442	—	83	1	no	—
196	#442	—	83	1	no	—
197	#440	—	83	1	no	—
198	#440	—	83	1	no	—
199	#432	—	83	1	no	—
200	C₂×#41	—	166	1	no	—
201	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
202	C₂×SD₁₆	—	83	1	no	—
203	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
204	C₂×D₈	—	83	1	no	—
205	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
206	C₄⋊Q₈	—	83	1	no	—
207	#127	—	83	1	no	—
208	#124	—	166	1	no	—
209	#121	—	83	1	no	—
210	#121	—	166	1	no	—
211	#121	C₂×D₈₃	1	1	yes	characteristic
212	C₄×Q₈	—	83	1	no	—
213	C₄×Q₈	—	83	1	no	—
214	C₄×D₄	—	83	1	no	—
215	C₄×D₄	—	83	1	no	—
216	C₄×D₄	—	83	1	no	—
217	C₄×D₄	—	83	1	no	—
218	C₄×D₄	—	166	1	no	—
219	C₄×D₄	—	166	1	no	—
220	C₄×D₄	—	166	1	no	—
221	#117	—	83	1	no	—
222	#117	—	166	1	no	—
223	C₂×C₄²	—	83	1	no	—
224	C₂×C₄²	—	166	1	no	—
225	#107	—	83	1	no	—
226	#107	—	83	1	no	—
227	#106	—	83	1	no	—
228	#106	—	83	1	no	—
229	C₄⋊C₈	—	83	1	no	—
230	C₄⋊C₈	—	83	1	no	—
231	#103	—	83	1	no	—
232	#103	C₂×D₈₃	1	1	yes	characteristic
233	#102	—	83	1	no	—
234	#102	—	83	1	no	—
235	#102	—	83	1	no	—
236	#102	—	83	1	no	—
237	#102	—	83	1	no	—
238	#102	C₂×D₈₃	1	1	yes	characteristic
239	C₂²⋊C₈	—	83	1	no	—
240	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
241	#97	—	83	1	no	—
242	#97	—	83	1	no	—
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	#41	—	332	4	no	—
246	C₂×Q₈	—	166	1	no	—
247	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	83	1	no	—
251	C₂×D₄	—	166	1	no	—
252	C₂×D₄	—	166	1	no	—
253	C₂×D₄	—	166	1	no	—
254	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
255	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	83	1	no	—
258	C₂²×C₄	—	83	1	no	—
259	C₂²×C₄	—	166	1	no	—
260	C₂²×C₄	—	166	1	no	—
261	C₂²×C₄	—	166	1	no	—
262	C₂²×C₄	—	166	1	no	—
263	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	166	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	166	2	no	—
268	C₂×C₈	—	83	1	no	—
269	C₂×C₈	—	83	1	no	—
270	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
272	C₄⋊C₄	—	83	1	no	—
273	C₄⋊C₄	—	83	1	no	—
274	C₄⋊C₄	—	83	1	no	—
275	C₄⋊C₄	—	166	1	no	—
276	C₄⋊C₄	—	166	1	no	—
277	C₄⋊C₄	—	166	1	no	—
278	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₂²⋊C₄	—	166	1	no	—
282	C₂²⋊C₄	—	166	1	no	—
283	C₂²⋊C₄	—	166	1	no	—
284	C₂²⋊C₄	—	166	1	no	—
285	C₂²⋊C₄	—	166	1	no	—
286	C₄²	—	83	1	no	—
287	C₄²	—	83	1	no	—
288	C₄²	—	83	1	no	—
289	C₄²	—	83	1	no	—
290	C₄²	—	166	1	no	—
291	C₄²	—	166	1	no	—
292	C₄²	—	166	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	166	1	no	—
296	C₂³	D₄×D₈₃	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	332	2	no	—
299	Q₈	#13050589	1	2	yes	complemented
300	D₄	—	2	4	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	332	2	no	—
306	D₄	—	332	2	no	—
307	D₄	—	332	2	no	—
308	D₄	—	166	1	no	—
309	D₄	—	166	1	no	—
310	D₄	#13050589	1	2	yes	complemented
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	332	2	no	—
317	C₂×C₄	—	332	2	no	—
318	C₂×C₄	—	332	2	no	—
319	C₂×C₄	—	332	2	no	—
320	C₂×C₄	—	332	1	no	—
321	C₂×C₄	—	83	1	no	—
322	C₂×C₄	—	83	1	no	—
323	C₂×C₄	—	83	1	no	—
324	C₂×C₄	—	83	1	no	—
325	C₂×C₄	—	166	1	no	—
326	C₂×C₄	—	166	1	no	—
327	C₂×C₄	—	166	1	no	—
328	C₂×C₄	—	166	2	no	—
329	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	166	1	no	—
334	C₈	—	166	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	332	1	no	—
345	C₂²	—	332	1	no	—
346	C₂²	—	332	1	no	—
347	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	332	1	no	—
352	C₄	—	332	1	no	—
353	C₄	—	332	1	no	—
354	C₄	—	83	2	no	—
355	C₄	—	166	1	no	—
356	C₄	—	166	1	no	—
357	C₄	—	166	1	no	—
358	C₄	C₂×#13050589	1	1	yes	characteristic
359	C₄	C₂×#13050589	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	4	2	no	—
362	C₂	—	8	1	no	—
363	C₂	—	664	1	no	—
364	C₂	#19128207	1	1	yes	characteristic
365	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
366	C₂	#19128498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#449×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#449	D₈₃
hypocenter	C₈₃	#3115
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #19128246	C₂	semidirect
2. #41×Dic₈₃	C₂	semidirect
3. #19128026	C₂	semidirect
4. #19128022	C₂	semidirect
5. #19127984	C₂	semidirect
6. #19127979	C₂	semidirect
7. #19127899	C₂	semidirect
8. #19127894	C₂	semidirect
9. #19127846	C₂	semidirect
10. #19127813	C₂	semidirect
11. #19127810	C₂	semidirect
12. #19127804	C₂	semidirect
13. #19127802	C₂	semidirect
14. #19127755	C₂	semidirect
15. #449×C₈₃	C₂	semidirect
114. Q₈×C₈₃	#41	semidirect
120. D₄×C₈₃	#41	semidirect
184. C₈₃	#3115	semidirect
194. #449	D₈₃	semidirect
299. Q₈	#13050589	semidirect
310. D₄	#13050589	semidirect

Maximal subgroups

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

subgroup	index
#19128246	2
#41×Dic₈₃	2
#19128026	2
#19128022	2
#19127984	2
#19127979	2
#19127899	2
#19127894	2
#19127846	2
#19127813	2
#19127810	2
#19127804	2
#19127802	2
#19127755	2
#449×C₈₃	2
#3115	83