Subgroups — #24917673

Order 10624.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128309	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128180	C₂	1	1	yes	maximal, characteristic
3	#19128035	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128033	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128011	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127999	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127910	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127904	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127864	C₂	1	1	yes	maximal, characteristic
10	#19127860	C₂	1	1	yes	maximal, characteristic
11	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127801	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127760	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050578	C₂²	1	1	yes	characteristic
20	#14581790	C₂²	1	1	yes	characteristic
21	#14581787	—	2	1	no	—
22	#14581786	—	2	1	no	—
23	#14581784	—	2	1	no	—
24	#14581783	C₂²	1	1	yes	characteristic
25	#14581782	—	2	1	no	—
26	#14581772	C₂²	1	1	yes	characteristic
27	#14581768	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
30	#14581757	C₂²	1	1	yes	characteristic
31	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₈₃	—	2	1	no	—
33	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
34	#14581749	C₂²	1	1	yes	characteristic
35	#14581748	—	2	1	no	—
36	C₂×Dic₈₃⋊C₄	—	2	1	no	—
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581725	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	#14581717	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581715	C₂²	1	1	yes	characteristic
47	#14581715	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#14581703	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	#106×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	2	2	no	—
60	#13050589	—	2	2	no	—
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂²×Dic₈₃	—	2	1	no	—
65	C₂²×Dic₈₃	—	2	1	no	—
66	C₂×C₄×D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
68	Dic₃₃₂	—	2	2	no	—
69	#13050580	—	2	2	no	—
70	#13050578	—	2	2	no	—
71	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
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	—	2	1	no	—
85	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
86	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
91	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	—	2	1	no	—
99	SD₁₆×C₈₃	—	2	2	no	—
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	—	2	1	no	—
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	C₄²×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂³×C₈₃	—	2	1	no	—
109	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	D₄×C₈₃	—	2	1	no	—
113	D₄×C₈₃	#41	1	2	yes	—
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	—	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	—	2	4	no	—
122	C₂×Dic₈₃	—	2	4	no	—
123	C₂×Dic₈₃	—	2	1	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	C₄×D₈₃	—	2	1	no	—
132	C₄×D₈₃	—	2	1	no	—
133	C₄×D₈₃	—	8	1	no	—
134	Dic₁₆₆	—	2	1	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	2	1	no	—
137	Dic₁₆₆	#41	1	2	yes	—
138	C₈×C₈₃	—	2	1	no	—
139	C₈×C₈₃	—	2	1	no	—
140	C₈₃⋊C₈	—	2	1	no	—
141	C₈₃⋊C₈	—	2	1	no	—
142	C₂²×C₈₃	—	2	2	no	—
143	C₂²×C₈₃	—	4	1	no	—
144	C₂²×C₈₃	—	4	1	no	—
145	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₈₃	—	2	1	no	—
147	C₂×D₈₃	—	2	1	no	—
148	C₂×D₈₃	—	2	1	no	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	—	4	1	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
155	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
156	Dic₈₃	—	2	2	no	—
157	Dic₈₃	—	2	1	no	—
158	Dic₈₃	—	2	1	no	—
159	Dic₈₃	—	2	1	no	—
160	Dic₈₃	—	4	1	no	—
161	Dic₈₃	—	4	1	no	—
162	Dic₈₃	—	4	1	no	—
163	C₂×C₈₃	—	4	2	no	—
164	C₂×C₈₃	#525	1	1	yes	characteristic
165	C₂×C₈₃	#577	1	1	yes	characteristic
166	C₂×C₈₃	#578	1	1	yes	characteristic
167	D₈₃	—	4	1	no	—
168	#3120	—	83	1	no	maximal, Sylow
169	C₈₃	#3120	1	1	yes	characteristic, complemented, Sylow
170	#541	—	83	1	no	—
171	#520	—	83	1	no	—
172	#484	—	83	1	no	—
173	#483	—	83	1	no	—
174	#479	—	83	1	no	—
175	#478	—	83	1	no	—
176	#462	—	83	1	no	—
177	#461	—	83	1	no	—
178	#452	—	83	1	no	—
179	#452	—	83	1	no	—
180	#441	—	83	1	no	—
181	#440	—	83	1	no	—
182	C₄×Q₁₆	—	83	1	no	—
183	C₄×SD₁₆	D₈₃	1	1	yes	characteristic, complemented
184	#433	—	83	1	no	—
185	C₂×#41	—	83	1	no	—
186	C₂×Q₁₆	—	83	1	no	—
187	C₂×Q₁₆	—	83	1	no	—
188	C₂×SD₁₆	—	83	1	no	—
189	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	83	1	no	—
191	#126	—	166	1	no	—
192	#126	—	166	1	no	—
193	#125	—	83	1	no	—
194	#125	—	166	1	no	—
195	#125	—	166	1	no	—
196	C₂²⋊Q₈	—	83	1	no	—
197	C₂²⋊Q₈	—	83	1	no	—
198	C₄×Q₈	—	83	1	no	—
199	C₄×Q₈	—	83	1	no	—
200	C₄×Q₈	—	83	1	no	—
201	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
202	C₄×D₄	—	83	1	no	—
203	C₄×D₄	—	166	1	no	—
204	C₄×D₄	—	166	1	no	—
205	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
206	#117	—	83	1	no	—
207	#117	—	166	1	no	—
208	C₂×C₄⋊C₄	—	83	1	no	—
209	C₂×C₄⋊C₄	—	166	1	no	—
210	#107	—	83	1	no	—
211	#107	—	83	1	no	—
212	#106	—	83	1	no	—
213	#106	C₂×D₈₃	1	1	yes	characteristic
214	C₄⋊C₈	—	83	1	no	—
215	C₄⋊C₈	—	83	1	no	—
216	#103	—	83	1	no	—
217	#103	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#103	—	83	1	no	—
221	#103	C₂×D₈₃	1	1	yes	characteristic
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	#97	—	83	1	no	—
227	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
228	#41	—	166	2	no	—
229	#41	—	166	2	no	—
230	C₂×Q₈	—	83	1	no	—
231	C₂×Q₈	—	83	1	no	—
232	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
233	C₂×D₄	—	166	1	no	—
234	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	83	1	no	—
237	C₂²×C₄	—	166	1	no	—
238	C₂²×C₄	—	166	1	no	—
239	C₂²×C₄	—	166	1	no	—
240	Q₁₆	—	166	2	no	—
241	Q₁₆	—	166	2	no	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	166	2	no	—
244	C₂×C₈	—	83	1	no	—
245	C₂×C₈	—	83	1	no	—
246	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	83	1	no	—
251	C₄⋊C₄	—	83	1	no	—
252	C₄⋊C₄	—	83	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	—	166	1	no	—
255	C₄⋊C₄	—	166	1	no	—
256	C₄⋊C₄	—	166	1	no	—
257	C₄⋊C₄	—	166	1	no	—
258	C₄⋊C₄	—	166	1	no	—
259	C₄⋊C₄	—	166	1	no	—
260	C₄⋊C₄	—	166	1	no	—
261	C₄⋊C₄	—	166	1	no	—
262	C₄⋊C₄	—	166	1	no	—
263	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	83	1	no	—
267	C₂²⋊C₄	—	83	1	no	—
268	C₂²⋊C₄	—	166	1	no	—
269	C₂²⋊C₄	—	166	1	no	—
270	C₂²⋊C₄	—	166	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	83	1	no	—
273	C₄²	—	166	1	no	—
274	C₄²	—	166	1	no	—
275	C₄²	—	166	1	no	—
276	C₄²	C₂²×D₈₃	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	—	83	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	83	2	no	—
282	Q₈	—	166	1	no	—
283	Q₈	—	166	1	no	—
284	Q₈	—	166	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	166	4	no	—
287	D₄	#13050590	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	83	1	no	—
293	C₂×C₄	—	83	1	no	—
294	C₂×C₄	—	83	1	no	—
295	C₂×C₄	—	664	1	no	—
296	C₂×C₄	—	664	1	no	—
297	C₂×C₄	—	166	1	no	—
298	C₂×C₄	—	166	1	no	—
299	C₂×C₄	—	166	1	no	—
300	C₂×C₄	—	166	4	no	—
301	C₂×C₄	—	166	1	no	—
302	C₂×C₄	—	166	1	no	—
303	C₂×C₄	—	166	1	no	—
304	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
305	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	166	1	no	—
310	C₈	—	166	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	166	1	no	—
315	C₂²	—	166	1	no	—
316	C₂²	—	166	1	no	—
317	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	332	1	no	—
324	C₄	—	332	1	no	—
325	C₄	—	332	1	no	—
326	C₄	—	166	1	no	—
327	C₄	—	166	2	no	—
328	C₄	—	166	1	no	—
329	C₄	—	166	1	no	—
330	C₄	#41×D₈₃	1	1	yes	characteristic
331	C₄	C₂×#13050590	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	332	1	no	—
334	C₂	#19128210	1	1	yes	characteristic
335	C₂	#19128507	1	1	yes	characteristic
336	C₂	#19128515	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₄×SD₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#13050589	C₂²
hypercenter	C₄×SD₁₆	D₈₃
hypocenter	C₈₃	#3120
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #19128309	C₂	semidirect
3. #19128035	C₂	semidirect
4. #19128033	C₂	semidirect
5. #19128011	C₂	semidirect
6. #19127999	C₂	semidirect
7. #19127910	C₂	semidirect
8. #19127904	C₂	semidirect
11. #19127806	C₂	semidirect
12. #19127801	C₂	semidirect
13. C₄×Dic₃₃₂	C₂	semidirect
14. #19127760	C₂	semidirect
15. C₄×SD₁₆×C₈₃	C₂	semidirect
169. C₈₃	#3120	semidirect
183. C₄×SD₁₆	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128309	2
#19128180	2
#19128035	2
#19128033	2
#19128011	2
#19127999	2
#19127910	2
#19127904	2
#19127864	2
#19127860	2
#19127806	2
#19127801	2
C₄×Dic₃₃₂	2
#19127760	2
C₄×SD₁₆×C₈₃	2
#3120	83