Subgroups — #24918577

Order 10624.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128323	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128271	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128094	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128082	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128059	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128050	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127913	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127911	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127907	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127860	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127855	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127847	C₂	1	1	yes	maximal, characteristic
13	#19127841	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127761	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050580	C₂²	1	1	yes	characteristic
20	C₂×#13050579	C₂²	1	1	yes	characteristic
21	C₂×#13050579	C₂²	1	1	yes	characteristic
22	C₂×#13050577	C₂²	1	1	yes	characteristic
23	C₂×#13050575	C₂²	1	1	yes	characteristic
24	#14581792	C₂²	1	1	yes	characteristic
25	#14581791	C₂²	1	1	yes	characteristic
26	#14581790	C₂²	1	1	yes	characteristic
27	#14581786	—	2	1	no	—
28	#14581781	C₂²	1	1	yes	characteristic
29	#14581780	—	2	1	no	—
30	#14581779	C₂²	1	1	yes	characteristic
31	#14581777	—	2	1	no	—
32	#14581774	—	2	1	no	—
33	#14581774	—	2	1	no	—
34	#14581770	—	2	1	no	—
35	#14581769	—	2	1	no	—
36	#14581769	—	2	1	no	—
37	#14581768	C₂²	1	1	yes	characteristic
38	#14581767	C₂²	1	1	yes	characteristic
39	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
40	#14581749	C₂²	1	1	yes	characteristic
41	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581716	C₂²	1	1	yes	characteristic
45	#14581715	C₂²	1	1	yes	characteristic
46	#14581715	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	#124×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	4	1	no	—
60	#13050589	—	4	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
65	Q₈×D₈₃	—	4	1	no	—
66	Q₈×D₈₃	—	4	1	no	—
67	C₂×D₃₃₂	—	2	1	no	—
68	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₈₃	—	2	1	no	—
70	C₂²×Dic₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
74	#13050580	—	4	1	no	—
75	#13050580	—	4	1	no	—
76	#13050579	—	4	1	no	—
77	#13050579	—	4	1	no	—
78	#13050577	—	4	1	no	—
79	#13050575	—	4	1	no	—
80	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₃	—	2	1	no	—
83	C₄⋊Dic₈₃	—	2	1	no	—
84	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
85	Dic₈₃⋊C₄	—	2	1	no	—
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	—	2	1	no	—
88	Dic₈₃⋊C₄	—	2	1	no	—
89	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
90	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
91	C₂²⋊Dic₈₃	—	2	1	no	—
92	C₂²⋊Dic₈₃	—	2	1	no	—
93	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	(C₂×D₈₃)⋊C₄	—	2	1	no	—
95	(C₂×D₈₃)⋊C₄	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	—	2	1	no	—
97	(C₂×D₈₃)⋊C₄	—	2	1	no	—
98	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
99	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
104	Q₁₆×C₈₃	—	4	1	no	—
105	SD₁₆×C₈₃	—	4	1	no	—
106	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₃	—	2	1	no	—
109	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₃	—	2	1	no	—
111	C₂²⋊C₄×C₈₃	—	2	1	no	—
112	C₄²×C₈₃	—	2	1	no	—
113	C₄²×C₈₃	C₂³	1	1	yes	characteristic
114	C₂³×C₈₃	—	2	1	no	—
115	C₂²×D₈₃	—	2	1	no	—
116	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
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	D₄×C₈₃	—	2	1	no	—
122	D₄×C₈₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₈₃	—	4	2	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	—	2	1	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	4	2	no	—
134	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
137	D₃₃₂	—	2	1	no	—
138	D₃₃₂	—	2	1	no	—
139	D₃₃₂	—	8	1	no	—
140	C₄×D₈₃	—	2	1	no	—
141	C₄×D₈₃	—	2	1	no	—
142	C₄×D₈₃	—	4	2	no	—
143	C₄×D₈₃	—	8	1	no	—
144	Dic₁₆₆	—	2	1	no	—
145	Dic₁₆₆	—	2	1	no	—
146	Dic₁₆₆	—	2	1	no	—
147	Dic₁₆₆	—	2	1	no	—
148	Dic₁₆₆	—	4	2	no	—
149	Dic₁₆₆	—	8	1	no	—
150	C₈×C₈₃	—	2	1	no	—
151	C₈×C₈₃	—	2	1	no	—
152	C₈₃⋊C₈	—	2	1	no	—
153	C₈₃⋊C₈	—	2	1	no	—
154	C₂²×C₈₃	—	4	1	no	—
155	C₂²×C₈₃	—	4	1	no	—
156	C₂²×C₈₃	—	4	1	no	—
157	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₈₃	—	2	1	no	—
159	C₂×D₈₃	—	2	1	no	—
160	C₂×D₈₃	—	2	1	no	—
161	C₂×D₈₃	—	4	1	no	—
162	C₂×D₈₃	—	4	1	no	—
163	C₂×D₈₃	—	4	1	no	—
164	C₄×C₈₃	—	2	1	no	—
165	C₄×C₈₃	—	2	1	no	—
166	C₄×C₈₃	—	4	1	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	—	4	1	no	—
169	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
170	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	2	1	no	—
174	Dic₈₃	—	4	1	no	—
175	Dic₈₃	—	4	1	no	—
176	Dic₈₃	—	4	1	no	—
177	C₂×C₈₃	—	8	1	no	—
178	C₂×C₈₃	#535	1	1	yes	characteristic
179	C₂×C₈₃	#577	1	1	yes	characteristic
180	C₂×C₈₃	#578	1	1	yes	characteristic
181	D₈₃	—	4	1	no	—
182	D₈₃	—	8	1	no	—
183	#3225	—	83	1	no	maximal, Sylow
184	C₈₃	#3225	1	1	yes	characteristic, complemented, Sylow
185	#542	—	83	1	no	—
186	#536	—	83	1	no	—
187	#497	—	83	1	no	—
188	#495	—	83	1	no	—
189	#489	—	83	1	no	—
190	#487	—	83	1	no	—
191	#464	D₈₃	1	1	yes	characteristic, complemented
192	#463	—	83	1	no	—
193	#462	—	83	1	no	—
194	#461	—	83	1	no	—
195	#452	—	83	1	no	—
196	#451	—	83	1	no	—
197	#449	—	83	1	no	—
198	#448	—	83	1	no	—
199	#433	—	83	1	no	—
200	C₂×#41	—	83	1	no	—
201	C₂²×Q₈	—	83	1	no	—
202	C₂×Q₁₆	—	83	1	no	—
203	C₂×Q₁₆	—	83	1	no	—
204	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
205	C₂×SD₁₆	—	83	1	no	—
206	C₂×SD₁₆	—	83	1	no	—
207	C₂×SD₁₆	—	83	1	no	—
208	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
209	C₂×D₈	—	83	1	no	—
210	C₄⋊Q₈	—	83	1	no	—
211	C₄⋊Q₈	—	83	1	no	—
212	C₄⋊Q₈	—	83	1	no	—
213	#126	—	166	1	no	—
214	#124	—	83	1	no	—
215	#124	—	83	1	no	—
216	#124	—	166	1	no	—
217	#124	C₂×D₈₃	1	1	yes	characteristic
218	#123	—	166	1	no	—
219	#123	—	166	1	no	—
220	#123	—	166	1	no	—
221	C₂²⋊Q₈	—	83	1	no	—
222	C₂²⋊Q₈	—	166	1	no	—
223	C₂²⋊Q₈	—	166	1	no	—
224	C₂²⋊Q₈	—	166	1	no	—
225	#121	—	83	1	no	—
226	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
227	C₄×D₄	—	83	1	no	—
228	#117	—	83	1	no	—
229	C₄⋊C₈	—	83	1	no	—
230	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
231	#103	—	83	1	no	—
232	#103	—	83	1	no	—
233	#103	—	83	1	no	—
234	#103	—	83	1	no	—
235	#103	—	83	1	no	—
236	#103	C₂×D₈₃	1	1	yes	characteristic
237	#102	—	83	1	no	—
238	#102	C₂×D₈₃	1	1	yes	characteristic
239	C₂²⋊C₈	—	83	1	no	—
240	C₂²⋊C₈	—	83	1	no	—
241	#97	—	83	1	no	—
242	C₄×C₈	—	83	1	no	—
243	#41	—	332	1	no	—
244	#41	—	332	1	no	—
245	C₂×Q₈	—	332	1	no	—
246	C₂×Q₈	—	332	1	no	—
247	C₂×Q₈	—	83	1	no	—
248	C₂×Q₈	—	83	1	no	—
249	C₂×Q₈	—	166	1	no	—
250	C₂×Q₈	—	166	1	no	—
251	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
252	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
253	C₂×D₄	—	83	1	no	—
254	C₂×D₄	—	166	1	no	—
255	C₂×D₄	—	166	1	no	—
256	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
257	C₂²×C₄	—	83	1	no	—
258	C₂²×C₄	—	166	1	no	—
259	C₂²×C₄	—	166	1	no	—
260	C₂²×C₄	—	166	1	no	—
261	Q₁₆	—	4	1	no	—
262	Q₁₆	—	332	1	no	—
263	Q₁₆	—	332	1	no	—
264	SD₁₆	—	4	1	no	—
265	SD₁₆	—	332	1	no	—
266	SD₁₆	—	332	1	no	—
267	SD₁₆	—	332	1	no	—
268	D₈	—	332	1	no	—
269	C₂×C₈	—	83	1	no	—
270	C₂×C₈	—	83	1	no	—
271	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	83	1	no	—
275	C₄⋊C₄	—	83	1	no	—
276	C₄⋊C₄	—	83	1	no	—
277	C₄⋊C₄	—	166	1	no	—
278	C₄⋊C₄	—	166	1	no	—
279	C₄⋊C₄	—	166	1	no	—
280	C₄⋊C₄	—	166	1	no	—
281	C₄⋊C₄	—	166	1	no	—
282	C₄⋊C₄	—	166	1	no	—
283	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	83	1	no	—
287	C₂²⋊C₄	—	83	1	no	—
288	C₂²⋊C₄	—	166	1	no	—
289	C₂²⋊C₄	—	166	1	no	—
290	C₂²⋊C₄	—	166	1	no	—
291	C₂²⋊C₄	—	166	1	no	—
292	C₂²⋊C₄	—	166	1	no	—
293	C₂²⋊C₄	—	166	1	no	—
294	C₂²⋊C₄	—	166	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	83	1	no	—
297	C₄²	C₂²×D₈₃	1	1	yes	characteristic
298	C₂³	—	2	1	no	—
299	C₂³	—	83	1	no	—
300	C₂³	—	166	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	332	2	no	—
306	Q₈	—	664	1	no	—
307	Q₈	—	166	1	no	—
308	Q₈	—	166	1	no	—
309	Q₈	—	166	1	no	—
310	Q₈	—	166	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	332	2	no	—
314	D₄	—	664	1	no	—
315	D₄	—	166	1	no	—
316	D₄	—	166	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	332	2	no	—
321	C₂×C₄	—	332	2	no	—
322	C₂×C₄	—	83	1	no	—
323	C₂×C₄	—	83	1	no	—
324	C₂×C₄	—	83	1	no	—
325	C₂×C₄	—	664	1	no	—
326	C₂×C₄	—	166	1	no	—
327	C₂×C₄	—	166	1	no	—
328	C₂×C₄	—	166	1	no	—
329	C₂×C₄	—	166	1	no	—
330	C₂×C₄	—	166	1	no	—
331	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
332	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	166	1	no	—
337	C₈	—	166	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	332	1	no	—
342	C₂²	—	332	1	no	—
343	C₂²	—	332	1	no	—
344	C₂²	—	166	1	no	—
345	C₂²	—	166	1	no	—
346	C₂²	—	166	1	no	—
347	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	332	1	no	—
354	C₄	—	332	1	no	—
355	C₄	—	332	1	no	—
356	C₄	—	166	1	no	—
357	C₄	—	166	1	no	—
358	C₄	—	166	1	no	—
359	C₄	#14581837	1	1	yes	characteristic
360	C₄	#14581838	1	1	yes	characteristic
361	C₂	—	8	1	no	—
362	C₂	—	332	1	no	—
363	C₂	—	664	1	no	—
364	C₂	#19128267	1	1	yes	characteristic
365	C₂	#19128508	1	1	yes	characteristic
366	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	#464×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127847	C₂
hypercenter	#464	D₈₃
hypocenter	C₈₃	#3225
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #19128323	C₂	semidirect
2. #19128271	C₂	semidirect
3. #19128094	C₂	semidirect
4. #19128082	C₂	semidirect
5. #19128059	C₂	semidirect
6. #19128050	C₂	semidirect
7. #19127913	C₂	semidirect
8. #19127911	C₂	semidirect
9. #19127907	C₂	semidirect
10. #19127860	C₂	semidirect
11. #19127855	C₂	semidirect
13. #19127841	C₂	semidirect
14. #19127761	C₂	semidirect
15. #464×C₈₃	C₂	semidirect
184. C₈₃	#3225	semidirect
191. #464	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128323	2
#19128271	2
#19128094	2
#19128082	2
#19128059	2
#19128050	2
#19127913	2
#19127911	2
#19127907	2
#19127860	2
#19127855	2
#19127847	2
#19127841	2
#19127761	2
#464×C₈₃	2
#3225	83