Subgroups — #24918542

Order 10624.

364 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128311	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128275	C₂	1	1	yes	maximal, characteristic
3	#19128094	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128079	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128068	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128057	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127926	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127925	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127910	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127865	C₂	1	1	yes	maximal, characteristic
12	#19127861	C₂	1	1	yes	maximal, characteristic
13	#19127857	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127759	C₂	1	1	yes	maximal, characteristic, complemented
15	#451×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050580	C₂²	1	1	yes	characteristic
20	C₂×#13050580	C₂²	1	1	yes	characteristic
21	C₂×#13050579	C₂²	1	1	yes	characteristic
22	C₂×#13050577	C₂²	1	1	yes	characteristic
23	#14581790	C₂²	1	1	yes	characteristic
24	#14581790	C₂²	1	1	yes	characteristic
25	#14581790	C₂²	1	1	yes	characteristic
26	#14581785	—	2	1	no	—
27	#14581785	—	2	1	no	—
28	#14581783	—	2	1	no	—
29	#14581783	C₂²	1	1	yes	characteristic
30	#14581782	—	2	1	no	—
31	#14581781	C₂²	1	1	yes	characteristic
32	#14581781	C₂²	1	1	yes	characteristic
33	#14581775	—	2	1	no	—
34	#14581775	—	2	1	no	—
35	#14581771	C₂²	1	1	yes	characteristic
36	#14581768	—	2	1	no	—
37	#14581768	—	2	1	no	—
38	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
39	#14581757	C₂²	1	1	yes	characteristic
40	#14581751	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	#14581716	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581715	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	4	1	no	—
60	#13050590	—	4	1	no	—
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	—	2	1	no	—
66	Dic₃₃₂	—	4	1	no	—
67	#13050580	—	4	1	no	—
68	#13050580	—	4	1	no	—
69	#13050580	—	4	1	no	—
70	#13050579	—	4	1	no	—
71	#13050577	—	4	1	no	—
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
77	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
90	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	(C₂×D₈₃)⋊C₄	—	2	1	no	—
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	—	2	1	no	—
97	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
98	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₈₃	—	2	1	no	—
100	C₂×Q₈×C₈₃	—	4	1	no	—
101	C₂×Q₈×C₈₃	—	4	1	no	—
102	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₈₃	—	2	1	no	—
106	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
107	Q₁₆×C₈₃	—	4	1	no	—
108	Q₁₆×C₈₃	—	4	1	no	—
109	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
110	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₈₃	—	2	1	no	—
112	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
113	C₂²⋊C₄×C₈₃	—	2	1	no	—
114	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	Q₈×C₈₃	—	2	1	no	—
118	Q₈×C₈₃	—	2	1	no	—
119	Q₈×C₈₃	—	2	1	no	—
120	Q₈×C₈₃	—	2	1	no	—
121	Q₈×C₈₃	—	2	1	no	—
122	Q₈×C₈₃	—	4	2	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	—	2	1	no	—
127	C₂×C₄×C₈₃	—	2	1	no	—
128	C₂×C₄×C₈₃	—	2	1	no	—
129	C₂×C₄×C₈₃	—	4	2	no	—
130	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₈₃	—	4	2	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	2	1	no	—
135	C₂×Dic₈₃	—	2	1	no	—
136	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
140	D₃₃₂	—	2	1	no	—
141	D₃₃₂	—	2	1	no	—
142	C₄×D₈₃	—	4	2	no	—
143	Dic₁₆₆	—	2	1	no	—
144	Dic₁₆₆	—	2	1	no	—
145	Dic₁₆₆	—	2	1	no	—
146	Dic₁₆₆	—	2	1	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	1	no	—
153	C₂²×C₈₃	—	2	1	no	—
154	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₈₃	—	4	1	no	—
156	C₂×D₈₃	—	4	1	no	—
157	C₂×D₈₃	—	4	1	no	—
158	C₄×C₈₃	—	2	1	no	—
159	C₄×C₈₃	—	4	1	no	—
160	C₄×C₈₃	—	4	1	no	—
161	C₄×C₈₃	—	4	1	no	—
162	C₄×C₈₃	—	4	1	no	—
163	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₈₃	—	2	1	no	—
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	—	4	1	no	—
172	C₂×C₈₃	—	4	1	no	—
173	C₂×C₈₃	#535	1	1	yes	characteristic
174	C₂×C₈₃	#578	1	1	yes	characteristic
175	C₂×C₈₃	#578	1	1	yes	characteristic
176	D₈₃	—	8	1	no	—
177	#3223	—	83	1	no	maximal, Sylow
178	C₈₃	#3223	1	1	yes	characteristic, complemented, Sylow
179	#541	—	83	1	no	—
180	#536	—	83	1	no	—
181	#497	—	83	1	no	—
182	#494	—	83	1	no	—
183	#491	—	83	1	no	—
184	#488	—	83	1	no	—
185	#464	—	83	1	no	—
186	#464	—	83	1	no	—
187	#462	—	83	1	no	—
188	#462	—	83	1	no	—
189	#452	—	83	1	no	—
190	#452	—	83	1	no	—
191	#451	—	83	1	no	—
192	#451	D₈₃	1	1	yes	characteristic, complemented
193	#433	—	83	1	no	—
194	C₂×#41	—	83	1	no	—
195	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
196	C₂×Q₁₆	—	83	1	no	—
197	C₂×Q₁₆	—	83	1	no	—
198	C₂×Q₁₆	—	83	1	no	—
199	C₂×Q₁₆	—	83	1	no	—
200	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
201	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
202	C₂×SD₁₆	—	83	1	no	—
203	C₂×SD₁₆	—	83	1	no	—
204	C₄⋊Q₈	—	83	1	no	—
205	C₄⋊Q₈	—	83	1	no	—
206	C₄⋊Q₈	—	83	1	no	—
207	#126	—	166	1	no	—
208	#126	—	166	1	no	—
209	#125	—	83	1	no	—
210	#125	—	166	1	no	—
211	#125	—	166	1	no	—
212	#124	—	83	1	no	—
213	#124	—	83	1	no	—
214	#123	—	166	1	no	—
215	#123	—	166	1	no	—
216	C₂²⋊Q₈	—	83	1	no	—
217	C₂²⋊Q₈	—	166	1	no	—
218	C₂²⋊Q₈	—	166	1	no	—
219	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
220	C₄×Q₈	—	83	1	no	—
221	C₄×Q₈	—	83	1	no	—
222	#117	—	83	1	no	—
223	C₄⋊C₈	—	83	1	no	—
224	C₄⋊C₈	—	83	1	no	—
225	#103	—	83	1	no	—
226	#103	—	83	1	no	—
227	#103	—	83	1	no	—
228	#103	—	83	1	no	—
229	#103	C₂×D₈₃	1	1	yes	characteristic
230	#103	C₂×D₈₃	1	1	yes	characteristic
231	#102	—	83	1	no	—
232	#102	—	83	1	no	—
233	C₂²⋊C₈	—	83	1	no	—
234	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
235	#97	—	83	1	no	—
236	C₄×C₈	—	83	1	no	—
237	#41	—	332	1	no	—
238	#41	—	332	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	83	1	no	—
243	C₂×Q₈	—	83	1	no	—
244	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
246	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
247	C₂×D₄	—	83	1	no	—
248	C₂×D₄	—	166	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	166	1	no	—
251	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
252	Q₁₆	—	4	1	no	—
253	Q₁₆	—	4	1	no	—
254	Q₁₆	—	332	1	no	—
255	Q₁₆	—	332	1	no	—
256	Q₁₆	—	332	1	no	—
257	Q₁₆	—	332	1	no	—
258	SD₁₆	—	332	1	no	—
259	SD₁₆	—	332	1	no	—
260	C₂×C₈	—	83	1	no	—
261	C₂×C₈	—	83	1	no	—
262	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	83	1	no	—
267	C₄⋊C₄	—	83	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	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	—	166	1	no	—
276	C₄⋊C₄	—	166	1	no	—
277	C₄⋊C₄	—	166	1	no	—
278	C₄⋊C₄	—	166	1	no	—
279	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	83	1	no	—
282	C₂²⋊C₄	—	83	1	no	—
283	C₂²⋊C₄	—	166	1	no	—
284	C₂²⋊C₄	—	166	1	no	—
285	C₂²⋊C₄	—	166	1	no	—
286	C₂²⋊C₄	—	166	1	no	—
287	C₂²⋊C₄	—	166	1	no	—
288	C₄²	—	83	1	no	—
289	C₄²	—	83	1	no	—
290	C₄²	—	166	1	no	—
291	C₄²	—	166	1	no	—
292	C₂³	—	166	1	no	—
293	C₂³	D₄×D₈₃	1	1	yes	characteristic
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	2	1	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	166	1	no	—
302	Q₈	—	166	1	no	—
303	Q₈	—	166	1	no	—
304	Q₈	—	166	1	no	—
305	D₄	—	332	2	no	—
306	D₄	—	166	1	no	—
307	D₄	—	166	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	332	2	no	—
316	C₂×C₄	—	83	1	no	—
317	C₂×C₄	—	83	1	no	—
318	C₂×C₄	—	83	1	no	—
319	C₂×C₄	—	83	1	no	—
320	C₂×C₄	—	166	1	no	—
321	C₂×C₄	—	166	1	no	—
322	C₂×C₄	—	166	1	no	—
323	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	166	1	no	—
328	C₈	—	166	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	332	1	no	—
333	C₂²	—	332	1	no	—
334	C₂²	—	332	1	no	—
335	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	332	1	no	—
342	C₄	—	332	1	no	—
343	C₄	—	332	1	no	—
344	C₄	—	166	1	no	—
345	C₄	—	166	1	no	—
346	C₄	—	166	1	no	—
347	C₄	—	166	1	no	—
348	C₄	#14581837	1	1	yes	characteristic
349	C₄	#14581837	1	1	yes	characteristic
350	C₂	—	4	1	no	—
351	C₂	—	664	1	no	—
352	C₂	#19128266	1	1	yes	characteristic
353	C₂	#19128514	1	1	yes	characteristic
354	C₂	#19128517	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#451×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#13050590	C₂²
hypercenter	#451	D₈₃
hypocenter	C₈₃	#3223
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
1. #19128311	C₂	semidirect
3. #19128094	C₂	semidirect
4. #19128079	C₂	semidirect
5. #19128068	C₂	semidirect
6. #19128057	C₂	semidirect
7. #19127926	C₂	semidirect
8. #19127925	C₂	semidirect
9. #19127910	C₂	semidirect
10. #19127909	C₂	semidirect
13. #19127857	C₂	semidirect
14. #19127759	C₂	semidirect
15. #451×C₈₃	C₂	semidirect
178. C₈₃	#3223	semidirect
192. #451	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128311	2
#19128275	2
#19128094	2
#19128079	2
#19128068	2
#19128057	2
#19127926	2
#19127925	2
#19127910	2
#19127909	2
#19127865	2
#19127861	2
#19127857	2
#19127759	2
#451×C₈₃	2
#3223	83