Subgroups — #24919871

Order 10624.

352 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128422	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128415	C₂	1	1	yes	maximal, characteristic
3	#19128098	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128041	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128039	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128023	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128022	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127994	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127979	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127927	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×C₂²⋊D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127739	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127736	C₂	1	1	yes	maximal, characteristic, complemented
15	#465×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
17	#14581791	C₂²	1	1	yes	characteristic
18	#14581791	C₂²	1	1	yes	characteristic
19	#14581787	—	2	1	no	—
20	#14581785	—	2	1	no	—
21	#14581779	—	2	1	no	—
22	#14581778	—	2	1	no	—
23	#14581778	—	2	1	no	—
24	#14581768	C₂²	1	1	yes	characteristic
25	#14581764	—	2	1	no	—
26	#14581763	C₂²	1	1	yes	characteristic
27	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
28	#14581757	—	2	1	no	—
29	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
31	#14581751	C₂²	1	1	yes	characteristic
32	#14581751	C₂²	1	1	yes	characteristic
33	#14581750	C₂²	1	1	yes	characteristic
34	#14581750	C₂²	1	1	yes	characteristic
35	#14581748	—	2	1	no	—
36	#14581726	C₂²	1	1	yes	characteristic
37	#14581726	C₂²	1	1	yes	characteristic
38	#14581725	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581724	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	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581716	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	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	#121×C₈₃	C₂²	1	1	yes	characteristic
56	#106×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×C₂²⋊D₈₃	—	2	1	no	—
62	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
63	C₂²×Dic₈₃	—	2	1	no	—
64	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
65	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₈×D₈₃	—	2	2	no	—
68	C₄⋊Dic₈₃	—	2	1	no	—
69	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
73	Dic₈₃⋊C₄	—	2	1	no	—
74	Dic₈₃⋊C₄	—	2	1	no	—
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
77	C₂²⋊Dic₈₃	—	2	1	no	—
78	C₂²⋊Dic₈₃	—	2	1	no	—
79	C₂²⋊Dic₈₃	—	2	1	no	—
80	C₂²⋊Dic₈₃	—	2	1	no	—
81	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
82	(C₂×D₈₃)⋊C₄	—	2	1	no	—
83	(C₂×D₈₃)⋊C₄	—	2	1	no	—
84	(C₂×D₈₃)⋊C₄	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
86	C₄×Dic₈₃	—	2	1	no	—
87	C₄×Dic₈₃	—	2	1	no	—
88	C₄×Dic₈₃	—	2	1	no	—
89	C₄×Dic₈₃	—	2	1	no	—
90	C₄×Dic₈₃	—	2	1	no	—
91	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₈₃	—	2	1	no	—
94	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
96	SD₁₆×C₈₃	—	2	2	no	—
97	C₂×C₈×C₈₃	—	2	2	no	—
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	—	2	1	no	—
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	C₂³×C₈₃	—	2	1	no	—
106	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₈₃	—	2	1	no	—
109	Q₈×C₈₃	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₈₃	—	2	2	no	—
121	C₂²⋊D₈₃	—	8	1	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	4	2	no	—
127	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	C₄×D₈₃	—	2	1	no	—
131	C₄×D₈₃	—	2	1	no	—
132	Dic₁₆₆	—	4	2	no	—
133	Dic₁₆₆	—	8	1	no	—
134	C₈×C₈₃	—	2	1	no	—
135	C₈×C₈₃	#41	1	2	yes	—
136	C₈₃⋊C₈	—	2	1	no	—
137	C₈₃⋊C₈	—	2	1	no	—
138	C₂²×C₈₃	—	2	1	no	—
139	C₂²×C₈₃	—	2	1	no	—
140	C₂²×C₈₃	—	2	1	no	—
141	C₂²×C₈₃	—	4	1	no	—
142	C₂²×C₈₃	—	4	1	no	—
143	C₂²×C₈₃	—	4	1	no	—
144	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₈₃	—	2	1	no	—
146	C₂×D₈₃	—	2	1	no	—
147	C₂×D₈₃	—	2	1	no	—
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	4	1	no	—
150	C₄×C₈₃	—	4	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
153	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₈₃	—	2	1	no	—
155	Dic₈₃	—	2	1	no	—
156	Dic₈₃	—	4	1	no	—
157	Dic₈₃	—	4	1	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	#135	1	2	yes	—
161	C₂×C₈₃	—	4	1	no	—
162	C₂×C₈₃	—	8	1	no	—
163	C₂×C₈₃	#547	1	1	yes	characteristic
164	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
165	C₂×C₈₃	#577	1	1	yes	characteristic
166	D₈₃	—	4	1	no	—
167	#3340	—	83	1	no	maximal, Sylow
168	C₈₃	#3340	1	1	yes	characteristic, complemented, Sylow
169	#556	—	83	1	no	—
170	#555	—	83	1	no	—
171	#498	—	83	1	no	—
172	#485	—	83	1	no	—
173	#485	—	83	1	no	—
174	#482	—	83	1	no	—
175	#482	—	83	1	no	—
176	#477	—	83	1	no	—
177	#474	—	83	1	no	—
178	#465	—	83	1	no	—
179	#465	D₈₃	1	1	yes	characteristic, complemented
180	C₄×SD₁₆	—	83	1	no	—
181	C₈×D₄	—	83	1	no	—
182	#427	—	83	1	no	—
183	#427	—	83	1	no	—
184	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
185	C₂²×C₈	—	83	1	no	—
186	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
187	C₄⋊Q₈	—	83	1	no	—
188	C₄⋊Q₈	—	83	1	no	—
189	#126	—	166	1	no	—
190	#126	—	166	1	no	—
191	#124	—	166	1	no	—
192	#124	—	166	1	no	—
193	#124	—	166	1	no	—
194	C₂²⋊Q₈	—	83	1	no	—
195	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
196	#121	—	83	1	no	—
197	#121	—	166	1	no	—
198	#121	C₂×D₈₃	1	1	yes	characteristic
199	C₄×Q₈	—	83	1	no	—
200	C₄×Q₈	—	166	1	no	—
201	C₄×D₄	—	83	1	no	—
202	C₄×D₄	—	83	1	no	—
203	#117	—	83	1	no	—
204	#117	—	83	1	no	—
205	#117	—	83	1	no	—
206	#117	—	83	1	no	—
207	#117	—	166	1	no	—
208	#107	—	83	1	no	—
209	#107	—	83	1	no	—
210	#107	—	83	1	no	—
211	#107	—	83	1	no	—
212	#106	—	83	1	no	—
213	#106	—	83	1	no	—
214	#106	—	83	1	no	—
215	#106	—	83	1	no	—
216	#106	C₂×D₈₃	1	1	yes	characteristic
217	C₄⋊C₈	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#103	C₂×D₈₃	1	1	yes	characteristic
221	#102	—	83	1	no	—
222	#102	—	83	1	no	—
223	#102	C₂×D₈₃	1	1	yes	characteristic
224	C₂²⋊C₈	—	83	1	no	—
225	C₂²⋊C₈	—	83	1	no	—
226	C₄×C₈	—	83	1	no	—
227	C₂×Q₈	—	166	1	no	—
228	C₂×Q₈	—	166	1	no	—
229	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	83	1	no	—
232	C₂×D₄	—	166	1	no	—
233	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
234	C₂²×C₄	—	83	1	no	—
235	C₂²×C₄	—	166	1	no	—
236	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
237	SD₁₆	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	83	1	no	—
240	C₂×C₈	—	83	1	no	—
241	C₂×C₈	—	166	2	no	—
242	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	83	1	no	—
246	C₄⋊C₄	—	83	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄⋊C₄	—	83	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	166	1	no	—
251	C₄⋊C₄	—	166	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	83	1	no	—
259	C₂²⋊C₄	—	83	1	no	—
260	C₂²⋊C₄	—	166	1	no	—
261	C₂²⋊C₄	—	166	1	no	—
262	C₂²⋊C₄	—	166	1	no	—
263	C₂²⋊C₄	—	166	1	no	—
264	C₂²⋊C₄	—	166	1	no	—
265	C₂²⋊C₄	—	166	1	no	—
266	C₂²⋊C₄	—	166	1	no	—
267	C₄²	—	83	1	no	—
268	C₄²	—	166	1	no	—
269	C₄²	—	166	1	no	—
270	C₄²	—	166	1	no	—
271	C₄²	—	166	1	no	—
272	C₄²	—	166	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	83	1	no	—
275	C₂³	D₄×D₈₃	1	1	yes	characteristic
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	332	2	no	—
279	Q₈	—	664	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	2	1	no	—
282	D₄	—	8	1	no	—
283	D₄	—	664	1	no	—
284	D₄	—	166	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	332	2	no	—
291	C₂×C₄	—	83	1	no	—
292	C₂×C₄	—	83	1	no	—
293	C₂×C₄	—	83	1	no	—
294	C₂×C₄	—	166	1	no	—
295	C₂×C₄	—	166	1	no	—
296	C₂×C₄	—	166	1	no	—
297	C₂×C₄	—	166	1	no	—
298	C₂×C₄	—	166	1	no	—
299	C₂×C₄	—	166	1	no	—
300	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	166	1	no	—
304	C₈	—	166	1	no	—
305	C₈	#13050589	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	166	1	no	—
313	C₂²	—	166	1	no	—
314	C₂²	—	166	1	no	—
315	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	332	1	no	—
321	C₄	—	332	1	no	—
322	C₄	—	332	1	no	—
323	C₄	—	332	1	no	—
324	C₄	—	83	2	no	—
325	C₄	—	166	1	no	—
326	C₄	—	166	1	no	—
327	C₄	C₂×#13050589	1	1	yes	characteristic
328	C₄	#14581839	1	1	yes	characteristic
329	C₂	—	4	1	no	—
330	C₂	—	8	1	no	—
331	C₂	—	332	1	no	—
332	C₂	#19128351	1	1	yes	characteristic
333	C₂	#135×D₈₃	1	1	yes	characteristic
334	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	#465×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128415	C₂
hypercenter	#465	D₈₃
hypocenter	C₈₃	#3340
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #19128422	C₂	semidirect
3. #19128098	C₂	semidirect
4. #19128041	C₂	semidirect
5. #19128039	C₂	semidirect
6. #19128023	C₂	semidirect
7. #19128022	C₂	semidirect
8. #19127994	C₂	semidirect
9. #19127979	C₂	semidirect
10. #19127927	C₂	semidirect
11. SD₁₆×Dic₈₃	C₂	semidirect
12. C₈×C₂²⋊D₈₃	C₂	semidirect
13. #19127739	C₂	semidirect
14. #19127736	C₂	semidirect
15. #465×C₈₃	C₂	semidirect
168. C₈₃	#3340	semidirect
179. #465	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128422	2
#19128415	2
#19128098	2
#19128041	2
#19128039	2
#19128023	2
#19128022	2
#19127994	2
#19127979	2
#19127927	2
SD₁₆×Dic₈₃	2
C₈×C₂²⋊D₈₃	2
#19127739	2
#19127736	2
#465×C₈₃	2
#3340	83