Subgroups — #24919926

Order 10624.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128398	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128352	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128108	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128026	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128021	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128013	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128000	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127989	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127952	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127936	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127810	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127775	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127741	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#481×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	—	2	1	no	—
17	C₂×#13050575	C₂²	1	1	yes	characteristic
18	C₂×#13050574	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581785	—	2	1	no	—
22	#14581783	C₂²	1	1	yes	characteristic
23	#14581777	—	2	1	no	—
24	#14581775	—	2	1	no	—
25	#14581773	—	2	1	no	—
26	#14581770	—	2	1	no	—
27	#14581767	C₂²	1	1	yes	characteristic
28	#14581766	C₂²	1	1	yes	characteristic
29	#14581763	—	2	1	no	—
30	#14581763	C₂²	1	1	yes	characteristic
31	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
32	#14581754	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
34	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
35	#14581751	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	#14581723	C₂²	1	1	yes	characteristic
44	#14581723	C₂²	1	1	yes	characteristic
45	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	#14581712	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#14581702	C₂²	1	1	yes	characteristic
52	#121×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
54	#106×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	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	4	4	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×D₃₃₂	—	2	1	no	—
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
71	#13050575	—	2	2	no	—
72	#13050574	—	2	2	no	—
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₈×D₈₃	—	2	2	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	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₈₃	—	2	1	no	—
96	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₈₃	—	2	1	no	—
98	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂³×C₈₃	—	2	1	no	—
107	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₈₃	—	2	1	no	—
109	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
110	D₄×C₈₃	—	2	1	no	—
111	D₄×C₈₃	—	2	1	no	—
112	D₄×C₈₃	—	8	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	4	2	no	—
119	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₈₃	—	2	2	no	—
122	C₂²⋊D₈₃	—	4	2	no	—
123	C₂²⋊D₈₃	—	4	2	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	8	1	no	—
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	D₃₃₂	—	2	1	no	—
132	D₃₃₂	—	2	1	no	—
133	D₃₃₂	—	4	2	no	—
134	C₄×D₈₃	—	2	1	no	—
135	C₄×D₈₃	—	2	1	no	—
136	C₄×D₈₃	—	4	2	no	—
137	C₄×D₈₃	—	4	2	no	—
138	C₄×D₈₃	—	8	1	no	—
139	Dic₁₆₆	—	4	2	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈₃⋊C₈	—	2	1	no	—
143	C₈₃⋊C₈	#41	1	2	yes	complemented
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	4	1	no	—
148	C₂²×C₈₃	—	4	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₈₃	—	2	1	no	—
152	C₂×D₈₃	—	2	1	no	—
153	C₂×D₈₃	—	4	1	no	—
154	C₂×D₈₃	—	4	1	no	—
155	C₂×D₈₃	—	4	1	no	—
156	C₂×D₈₃	AGL(1,ℤ/8)	1	2	yes	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	—	4	1	no	—
160	C₄×C₈₃	—	4	1	no	—
161	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
162	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
163	Dic₈₃	—	2	1	no	—
164	Dic₈₃	—	2	1	no	—
165	Dic₈₃	—	2	1	no	—
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	—	4	1	no	—
168	Dic₈₃	—	4	1	no	—
169	C₂×C₈₃	—	4	1	no	—
170	C₂×C₈₃	—	8	1	no	—
171	C₂×C₈₃	#547	1	1	yes	characteristic
172	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂×C₈₃	#577	1	1	yes	characteristic
174	D₈₃	—	2	2	no	—
175	D₈₃	—	8	1	no	—
176	#3345	—	83	1	no	maximal, Sylow
177	C₈₃	#3345	1	1	yes	characteristic, complemented, Sylow
178	#553	—	83	1	no	—
179	#547	—	83	1	no	—
180	#501	—	83	1	no	—
181	#482	—	83	1	no	—
182	#481	—	83	1	no	—
183	#481	D₈₃	1	1	yes	characteristic, complemented
184	#480	—	83	1	no	—
185	#478	—	83	1	no	—
186	#476	—	83	1	no	—
187	#469	—	83	1	no	—
188	#466	—	83	1	no	—
189	#442	—	83	1	no	—
190	#435	—	83	1	no	—
191	#428	—	83	1	no	—
192	C₂×#106	—	83	1	no	—
193	C₂×#41	—	166	1	no	—
194	C₂×D₈	—	83	1	no	—
195	C₂×M₁₆	—	83	1	no	—
196	C₂²×C₈	—	83	1	no	—
197	C₄⋊Q₈	—	83	1	no	—
198	#126	—	166	1	no	—
199	#125	—	83	1	no	—
200	#123	—	166	1	no	—
201	#123	—	166	1	no	—
202	#123	—	166	1	no	—
203	C₂²⋊Q₈	—	166	1	no	—
204	#121	—	83	1	no	—
205	#121	—	83	1	no	—
206	#121	—	83	1	no	—
207	#121	—	166	1	no	—
208	#121	C₂×D₈₃	1	1	yes	characteristic
209	C₄×D₄	—	83	1	no	—
210	C₄×D₄	—	83	1	no	—
211	C₄×D₄	—	83	1	no	—
212	C₄×D₄	—	166	1	no	—
213	#117	—	83	1	no	—
214	C₂×C₄⋊C₄	—	83	1	no	—
215	C₂×C₄⋊C₄	—	83	1	no	—
216	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
217	#107	—	83	1	no	—
218	#107	—	83	1	no	—
219	#107	—	83	1	no	—
220	#106	—	83	1	no	—
221	#106	—	83	1	no	—
222	#106	—	83	1	no	—
223	#106	—	83	1	no	—
224	#106	C₂×D₈₃	1	1	yes	characteristic
225	#106	C₂×D₈₃	1	1	yes	characteristic
226	C₄⋊C₈	—	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	C₂×D₈₃	1	1	yes	characteristic
232	#102	C₂×D₈₃	1	1	yes	characteristic
233	C₂²⋊C₈	—	83	1	no	—
234	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
235	#97	—	83	1	no	—
236	#41	—	332	4	no	—
237	C₂×Q₈	—	166	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	83	1	no	—
240	C₂×D₄	—	83	1	no	—
241	C₂×D₄	—	166	1	no	—
242	C₂×D₄	—	166	1	no	—
243	C₂×D₄	—	166	1	no	—
244	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	83	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	—	166	1	no	—
249	C₂²×C₄	—	166	1	no	—
250	C₂²×C₄	—	166	1	no	—
251	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
252	D₈	—	166	2	no	—
253	M₁₆	—	166	2	no	—
254	C₂×C₈	—	83	1	no	—
255	C₂×C₈	—	83	1	no	—
256	C₂×C₈	—	166	2	no	—
257	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	83	1	no	—
261	C₄⋊C₄	—	83	1	no	—
262	C₄⋊C₄	—	83	1	no	—
263	C₄⋊C₄	—	83	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	—	166	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	83	1	no	—
274	C₂²⋊C₄	—	83	1	no	—
275	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	166	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₄²	—	83	1	no	—
282	C₄²	—	166	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	83	1	no	—
285	C₂³	—	166	1	no	—
286	C₂³	D₄×D₈₃	1	1	yes	characteristic
287	Q₈	—	332	2	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	8	1	no	—
291	D₄	—	332	2	no	—
292	D₄	—	332	2	no	—
293	D₄	—	332	2	no	—
294	D₄	—	166	2	no	—
295	D₄	—	166	1	no	—
296	D₄	—	166	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	332	2	no	—
304	C₂×C₄	—	332	2	no	—
305	C₂×C₄	—	83	1	no	—
306	C₂×C₄	—	83	1	no	—
307	C₂×C₄	—	83	1	no	—
308	C₂×C₄	—	664	1	no	—
309	C₂×C₄	—	664	1	no	—
310	C₂×C₄	—	166	1	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	1	no	—
314	C₂×C₄	—	166	1	no	—
315	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	83	2	no	—
320	C₈	—	166	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	332	1	no	—
328	C₂²	—	332	1	no	—
329	C₂²	—	332	1	no	—
330	C₂²	—	83	2	no	—
331	C₂²	—	166	1	no	—
332	C₂²	—	166	1	no	—
333	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	332	1	no	—
339	C₄	—	332	1	no	—
340	C₄	—	332	1	no	—
341	C₄	—	166	1	no	—
342	C₄	—	166	1	no	—
343	C₄	—	166	1	no	—
344	C₄	#41×D₈₃	1	1	yes	characteristic
345	C₄	#14581839	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	8	1	no	—
348	C₂	—	664	1	no	—
349	C₂	—	166	2	no	—
350	C₂	#19128349	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
352	C₂	#19128507	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#481×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#481	D₈₃
hypocenter	C₈₃	#3345
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #481

Complements

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

N	H ≅ G/N	type
1. #19128398	C₂	semidirect
2. #19128352	C₂	semidirect
3. #19128108	C₂	semidirect
4. #19128026	C₂	semidirect
5. #19128021	C₂	semidirect
6. #19128013	C₂	semidirect
7. #19128000	C₂	semidirect
8. #19127989	C₂	semidirect
9. #19127952	C₂	semidirect
10. #19127936	C₂	semidirect
11. #19127810	C₂	semidirect
12. #19127775	C₂	semidirect
13. #19127741	C₂	semidirect
14. #106×D₈₃	C₂	semidirect
15. #481×C₈₃	C₂	semidirect
143. C₈₃⋊C₈	#41	semidirect
177. C₈₃	#3345	semidirect
183. #481	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128398	2
#19128352	2
#19128108	2
#19128026	2
#19128021	2
#19128013	2
#19128000	2
#19127989	2
#19127952	2
#19127936	2
#19127810	2
#19127775	2
#19127741	2
#106×D₈₃	2
#481×C₈₃	2
#3345	83