Subgroups — #24920171

Order 10624.

396 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	#19128410	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128372	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128066	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128064	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128046	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128001	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128000	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127896	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127894	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127894	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127829	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127810	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127781	C₂	1	1	yes	maximal, characteristic, complemented
15	#442×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	—	2	1	no	—
17	C₂×#13050575	C₂²	1	1	yes	characteristic
18	C₂×#13050575	C₂²	1	1	yes	characteristic
19	#14581791	C₂²	1	1	yes	characteristic
20	#14581789	C₂²	1	1	yes	characteristic
21	#14581788	C₂²	1	1	yes	characteristic
22	#14581788	C₂²	1	1	yes	characteristic
23	#14581784	C₂²	1	1	yes	characteristic
24	#14581783	C₂²	1	1	yes	characteristic
25	#14581771	—	2	1	no	—
26	#14581769	—	2	1	no	—
27	#14581769	—	2	1	no	—
28	#14581766	—	2	1	no	—
29	#14581765	—	2	1	no	—
30	#14581765	—	2	1	no	—
31	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
32	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
33	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
34	#14581754	C₂²	1	1	yes	characteristic
35	#14581754	C₂²	1	1	yes	characteristic
36	C₂×Dic₈₃⋊C₄	—	2	1	no	—
37	#14581725	C₂²	1	1	yes	characteristic
38	#14581723	C₂²	1	1	yes	characteristic
39	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
40	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
42	#14581714	C₂²	1	1	yes	characteristic
43	#14581714	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	#14581712	C₂²	1	1	yes	characteristic
49	#14581712	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	#106×C₈₃	C₂²	1	1	yes	characteristic
56	#102×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	4	4	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
62	C₂×D₃₃₂	—	2	1	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₂²⋊D₈₃	—	2	1	no	—
69	C₂×C₂²⋊D₈₃	—	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	—	2	1	no	—
74	#13050575	—	2	2	no	—
75	#13050575	—	2	2	no	—
76	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	—	2	1	no	—
79	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	Dic₈₃⋊C₄	—	2	1	no	—
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	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
93	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₈₃	—	2	1	no	—
97	C₂²×C₄×C₈₃	—	2	1	no	—
98	D₈×C₈₃	—	2	2	no	—
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₃	—	2	1	no	—
104	C₂²⋊C₄×C₈₃	—	2	1	no	—
105	C₄²×C₈₃	C₂³	1	1	yes	characteristic
106	C₂³×C₈₃	—	2	1	no	—
107	C₂³×C₈₃	—	2	1	no	—
108	C₂²×D₈₃	—	2	1	no	—
109	C₂²×D₈₃	—	2	1	no	—
110	D₄×C₈₃	—	2	1	no	—
111	D₄×C₈₃	—	2	1	no	—
112	D₄×C₈₃	—	2	1	no	—
113	D₄×C₈₃	#41	1	2	yes	complemented
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	4	2	no	—
117	C₂×C₄×C₈₃	—	8	1	no	—
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₈₃	—	4	2	no	—
122	C₂²⋊D₈₃	—	4	2	no	—
123	C₂²⋊D₈₃	—	8	1	no	—
124	C₂²⋊D₈₃	—	8	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	4	2	no	—
128	C₂×Dic₈₃	—	8	1	no	—
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	D₃₃₂	—	2	1	no	—
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	—	2	1	no	—
137	D₃₃₂	—	4	2	no	—
138	D₃₃₂	—	8	1	no	—
139	C₄×D₈₃	—	4	2	no	—
140	C₄×D₈₃	—	4	2	no	—
141	Dic₁₆₆	—	2	2	no	—
142	Dic₁₆₆	—	8	1	no	—
143	C₈×C₈₃	—	2	1	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈₃⋊C₈	—	2	1	no	—
146	C₈₃⋊C₈	—	2	1	no	—
147	C₂²×C₈₃	—	2	2	no	—
148	C₂²×C₈₃	—	4	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	—	4	1	no	—
151	C₂²×C₈₃	—	4	1	no	—
152	C₂²×C₈₃	—	4	1	no	—
153	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₈₃	—	4	1	no	—
155	C₂×D₈₃	—	4	1	no	—
156	C₂×D₈₃	—	4	1	no	—
157	C₂×D₈₃	—	4	1	no	—
158	C₂×D₈₃	—	4	1	no	—
159	C₂×D₈₃	—	4	1	no	—
160	C₄×C₈₃	—	2	1	no	—
161	C₄×C₈₃	—	2	1	no	—
162	C₄×C₈₃	—	4	1	no	—
163	C₄×C₈₃	—	4	1	no	—
164	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
165	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
166	Dic₈₃	—	2	1	no	—
167	Dic₈₃	—	2	1	no	—
168	Dic₈₃	—	2	1	no	—
169	Dic₈₃	—	4	1	no	—
170	Dic₈₃	—	4	1	no	—
171	Dic₈₃	AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₈₃	—	4	2	no	—
173	C₂×C₈₃	—	8	1	no	—
174	C₂×C₈₃	#548	1	1	yes	characteristic
175	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
176	C₂×C₈₃	#576	1	1	yes	characteristic
177	D₈₃	—	8	1	no	—
178	D₈₃	—	8	1	no	—
179	#3366	—	83	1	no	maximal, Sylow
180	C₈₃	#3366	1	1	yes	characteristic, complemented, Sylow
181	#554	—	83	1	no	—
182	#550	—	83	1	no	—
183	#490	—	83	1	no	—
184	#490	—	83	1	no	—
185	#486	—	83	1	no	—
186	#478	—	83	1	no	—
187	#478	—	83	1	no	—
188	#476	—	83	1	no	—
189	#459	—	83	1	no	—
190	#459	—	83	1	no	—
191	#459	—	83	1	no	—
192	#446	—	83	1	no	—
193	#442	—	83	1	no	—
194	#442	D₈₃	1	1	yes	characteristic, complemented
195	C₄×D₈	—	83	1	no	—
196	C₂×#41	—	166	1	no	—
197	C₂×D₈	—	83	1	no	—
198	C₂×D₈	—	83	1	no	—
199	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	83	1	no	—
201	#127	—	83	1	no	—
202	#127	—	83	1	no	—
203	#127	—	83	1	no	—
204	#125	—	83	1	no	—
205	#125	—	83	1	no	—
206	C₂²⋊Q₈	—	166	1	no	—
207	C₂²⋊Q₈	—	166	1	no	—
208	C₂²⋊Q₈	—	166	1	no	—
209	#121	—	166	1	no	—
210	#121	—	166	1	no	—
211	#121	—	166	1	no	—
212	C₄×Q₈	—	83	1	no	—
213	C₄×D₄	—	83	1	no	—
214	C₄×D₄	—	83	1	no	—
215	C₄×D₄	—	83	1	no	—
216	C₄×D₄	—	83	1	no	—
217	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
218	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	166	1	no	—
220	#107	—	83	1	no	—
221	#106	—	83	1	no	—
222	#106	C₂×D₈₃	1	1	yes	characteristic
223	C₄⋊C₈	—	83	1	no	—
224	C₄⋊C₈	—	83	1	no	—
225	C₄⋊C₈	—	83	1	no	—
226	#102	—	83	1	no	—
227	#102	—	83	1	no	—
228	#102	—	83	1	no	—
229	#102	—	83	1	no	—
230	#102	—	83	1	no	—
231	#102	—	83	1	no	—
232	#102	—	83	1	no	—
233	#102	—	83	1	no	—
234	#102	C₂×D₈₃	1	1	yes	characteristic
235	#102	C₂×D₈₃	1	1	yes	characteristic
236	#97	—	83	1	no	—
237	#97	C₂×D₈₃	1	1	yes	characteristic
238	C₄×C₈	—	83	1	no	—
239	#41	—	332	4	no	—
240	C₂×Q₈	—	83	1	no	—
241	C₂×Q₈	—	166	1	no	—
242	C₂×D₄	—	83	1	no	—
243	C₂×D₄	—	83	1	no	—
244	C₂×D₄	—	166	1	no	—
245	C₂×D₄	—	166	1	no	—
246	C₂×D₄	—	166	1	no	—
247	C₂×D₄	—	166	1	no	—
248	C₂×D₄	—	166	1	no	—
249	C₂×D₄	—	166	1	no	—
250	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
251	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	166	1	no	—
255	C₂²×C₄	—	166	1	no	—
256	C₂²×C₄	—	166	1	no	—
257	C₂²×C₄	—	166	1	no	—
258	D₈	—	2	2	no	—
259	D₈	—	166	2	no	—
260	D₈	—	166	2	no	—
261	C₂×C₈	—	83	1	no	—
262	C₂×C₈	—	83	1	no	—
263	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	83	1	no	—
267	C₄⋊C₄	—	83	1	no	—
268	C₄⋊C₄	—	83	1	no	—
269	C₄⋊C₄	—	83	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	1	no	—
273	C₄⋊C₄	—	166	1	no	—
274	C₄⋊C₄	—	166	1	no	—
275	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₂²⋊C₄	—	166	1	no	—
282	C₂²⋊C₄	—	166	1	no	—
283	C₄²	—	83	1	no	—
284	C₄²	—	83	1	no	—
285	C₄²	C₂²×D₈₃	1	1	yes	characteristic
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	166	1	no	—
289	C₂³	—	166	1	no	—
290	Q₈	—	664	1	no	—
291	Q₈	—	166	2	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	332	2	no	—
296	D₄	—	332	2	no	—
297	D₄	—	332	2	no	—
298	D₄	—	664	1	no	—
299	D₄	—	664	1	no	—
300	D₄	—	664	1	no	—
301	D₄	—	166	1	no	—
302	D₄	—	166	1	no	—
303	D₄	—	166	1	no	—
304	D₄	—	166	1	no	—
305	D₄	#13050590	1	2	yes	complemented
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	332	2	no	—
311	C₂×C₄	—	332	2	no	—
312	C₂×C₄	—	332	2	no	—
313	C₂×C₄	—	83	1	no	—
314	C₂×C₄	—	83	1	no	—
315	C₂×C₄	—	83	1	no	—
316	C₂×C₄	—	83	1	no	—
317	C₂×C₄	—	664	1	no	—
318	C₂×C₄	—	166	1	no	—
319	C₂×C₄	—	166	1	no	—
320	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
321	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	166	1	no	—
326	C₈	—	166	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	332	1	no	—
334	C₂²	—	332	1	no	—
335	C₂²	—	332	1	no	—
336	C₂²	—	332	1	no	—
337	C₂²	—	332	1	no	—
338	C₂²	—	332	1	no	—
339	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	332	1	no	—
345	C₄	—	332	1	no	—
346	C₄	—	83	2	no	—
347	C₄	—	166	1	no	—
348	C₄	—	166	1	no	—
349	C₄	—	166	1	no	—
350	C₄	C₂×#13050590	1	1	yes	characteristic
351	C₄	#14581839	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	8	1	no	—
354	C₂	—	664	1	no	—
355	C₂	—	664	1	no	—
356	C₂	#19128360	1	1	yes	characteristic
357	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
358	C₂	#19128505	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#442×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#442	D₈₃
hypocenter	C₈₃	#3366
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

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

N	H ≅ G/N	type
1. #19128410	C₂	semidirect
2. #19128372	C₂	semidirect
3. #19128066	C₂	semidirect
4. #19128064	C₂	semidirect
5. #19128046	C₂	semidirect
6. #19128001	C₂	semidirect
7. #19128000	C₂	semidirect
8. #19127990	C₂	semidirect
9. #19127896	C₂	semidirect
10. #19127894	C₂	semidirect
11. #19127894	C₂	semidirect
12. #19127829	C₂	semidirect
13. #19127810	C₂	semidirect
14. #19127781	C₂	semidirect
15. #442×C₈₃	C₂	semidirect
113. D₄×C₈₃	#41	semidirect
180. C₈₃	#3366	semidirect
194. #442	D₈₃	semidirect
305. D₄	#13050590	semidirect

Maximal subgroups

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

subgroup	index
#19128410	2
#19128372	2
#19128066	2
#19128064	2
#19128046	2
#19128001	2
#19128000	2
#19127990	2
#19127896	2
#19127894	2
#19127829	2
#19127810	2
#19127781	2
#442×C₈₃	2
#3366	83