Subgroups — #24920548

Order 10624.

380 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	#19128437	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128433	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128089	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128082	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127921	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127920	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127918	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127914	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127894	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127893	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127829	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127803	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127802	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#496×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050579	C₂²	1	1	yes	characteristic
18	C₂×#13050578	C₂²	1	1	yes	characteristic
19	C₂×#13050575	C₂²	1	1	yes	characteristic
20	C₂×#13050575	C₂²	1	1	yes	characteristic
21	#14581788	C₂²	1	1	yes	characteristic
22	#14581781	C₂²	1	1	yes	characteristic
23	#14581779	—	2	1	no	—
24	#14581779	C₂²	1	1	yes	characteristic
25	#14581779	C₂²	1	1	yes	characteristic
26	#14581778	—	2	1	no	—
27	#14581776	—	2	1	no	—
28	#14581774	—	2	1	no	—
29	#14581773	—	2	1	no	—
30	#14581773	—	2	1	no	—
31	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
34	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
35	D₄×Dic₈₃	—	2	1	no	—
36	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
37	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
38	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
39	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
40	#14581726	C₂²	1	1	yes	characteristic
41	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	#14581713	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
56	#127×C₈₃	C₂²	1	1	yes	characteristic
57	#124×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₃	—	2	1	no	—
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂²×Dic₈₃	—	2	1	no	—
65	C₂²×Dic₈₃	—	2	1	no	—
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	#13050580	—	4	1	no	—
69	#13050579	—	2	2	no	—
70	#13050578	—	4	1	no	—
71	#13050575	—	4	1	no	—
72	#13050575	—	4	1	no	—
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
81	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₈₃	—	2	1	no	—
83	C₂²⋊Dic₈₃	—	2	1	no	—
84	C₂²⋊Dic₈₃	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	—	2	1	no	—
99	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
102	SD₁₆×C₈₃	—	2	2	no	—
103	SD₁₆×C₈₃	—	4	1	no	—
104	D₈×C₈₃	—	2	2	no	—
105	D₈×C₈₃	—	4	1	no	—
106	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₃	—	2	1	no	—
109	C₂²⋊C₄×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂³×C₈₃	—	2	1	no	—
112	C₂³×C₈₃	—	2	1	no	—
113	C₂³×C₈₃	—	2	1	no	—
114	C₂²×D₈₃	—	2	1	no	—
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	D₄×C₈₃	—	2	1	no	—
118	D₄×C₈₃	—	2	1	no	—
119	D₄×C₈₃	—	2	1	no	—
120	D₄×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	2	1	no	—
122	D₄×C₈₃	—	2	1	no	—
123	D₄×C₈₃	—	4	2	no	—
124	D₄×C₈₃	—	8	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	C₂²⋊D₈₃	—	8	1	no	—
131	C₂×Dic₈₃	—	2	1	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	8	1	no	—
135	C₂×Dic₈₃	—	8	1	no	—
136	C₂×Dic₈₃	—	8	1	no	—
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	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
141	D₃₃₂	—	2	1	no	—
142	D₃₃₂	—	2	1	no	—
143	C₄×D₈₃	—	8	1	no	—
144	Dic₁₆₆	—	2	1	no	—
145	Dic₁₆₆	—	2	1	no	—
146	Dic₁₆₆	—	4	1	no	—
147	C₈×C₈₃	—	2	1	no	—
148	C₈×C₈₃	#41	1	2	yes	—
149	C₈₃⋊C₈	—	2	1	no	—
150	C₈₃⋊C₈	—	2	1	no	—
151	C₂²×C₈₃	—	4	1	no	—
152	C₂²×C₈₃	—	4	1	no	—
153	C₂²×C₈₃	—	4	1	no	—
154	C₂²×C₈₃	—	4	1	no	—
155	C₂²×C₈₃	—	4	1	no	—
156	C₂²×C₈₃	—	4	1	no	—
157	C₂²×C₈₃	—	4	1	no	—
158	C₂²×C₈₃	—	4	1	no	—
159	C₂²×C₈₃	—	4	1	no	—
160	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
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	C₂×#41	1	1	yes	characteristic
168	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
169	Dic₈₃	—	2	1	no	—
170	Dic₈₃	—	2	1	no	—
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	4	1	no	—
174	Dic₈₃	—	4	1	no	—
175	Dic₈₃	—	4	1	no	—
176	C₂×C₈₃	—	8	1	no	—
177	C₂×C₈₃	—	8	1	no	—
178	C₂×C₈₃	—	8	1	no	—
179	C₂×C₈₃	#550	1	1	yes	characteristic
180	C₂×C₈₃	#575	1	1	yes	characteristic
181	C₂×C₈₃	#576	1	1	yes	characteristic
182	D₈₃	—	8	1	no	—
183	#3403	—	83	1	no	maximal, Sylow
184	C₈₃	#3403	1	1	yes	characteristic, complemented, Sylow
185	#559	—	83	1	no	—
186	#559	—	83	1	no	—
187	#496	—	83	1	no	—
188	#496	D₈₃	1	1	yes	characteristic, complemented
189	#495	—	83	1	no	—
190	#464	—	83	1	no	—
191	#464	—	83	1	no	—
192	#463	—	83	1	no	—
193	#463	—	83	1	no	—
194	#459	—	83	1	no	—
195	#459	—	83	1	no	—
196	#446	—	83	1	no	—
197	#440	—	83	1	no	—
198	#440	—	83	1	no	—
199	C₄×D₈	—	83	1	no	—
200	C₂×Q₁₆	—	83	1	no	—
201	C₂×SD₁₆	—	83	1	no	—
202	C₂×SD₁₆	—	83	1	no	—
203	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
204	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
205	C₂×D₈	—	83	1	no	—
206	C₂×D₈	—	83	1	no	—
207	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
208	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
209	#127	—	83	1	no	—
210	#127	C₂×D₈₃	1	1	yes	characteristic
211	#124	—	83	1	no	—
212	#124	—	83	1	no	—
213	#124	—	83	1	no	—
214	#124	—	166	1	no	—
215	#124	—	166	1	no	—
216	#124	C₂×D₈₃	1	1	yes	characteristic
217	#123	—	166	1	no	—
218	#123	—	166	1	no	—
219	#123	—	166	1	no	—
220	#123	—	166	1	no	—
221	C₄×Q₈	—	83	1	no	—
222	C₄×Q₈	—	83	1	no	—
223	C₄×Q₈	—	83	1	no	—
224	C₄×D₄	—	83	1	no	—
225	C₄×D₄	—	83	1	no	—
226	C₄×D₄	—	83	1	no	—
227	C₄×D₄	—	83	1	no	—
228	C₄×D₄	—	166	1	no	—
229	C₄×D₄	—	166	1	no	—
230	#107	—	83	1	no	—
231	C₄⋊C₈	—	83	1	no	—
232	C₄⋊C₈	—	83	1	no	—
233	C₄⋊C₈	—	83	1	no	—
234	#103	—	83	1	no	—
235	#103	—	83	1	no	—
236	#102	—	83	1	no	—
237	#102	—	83	1	no	—
238	#102	—	83	1	no	—
239	#102	—	83	1	no	—
240	#97	—	83	1	no	—
241	#97	C₂×D₈₃	1	1	yes	characteristic
242	C₄×C₈	—	83	1	no	—
243	C₂×Q₈	—	83	1	no	—
244	C₂×Q₈	—	83	1	no	—
245	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	83	1	no	—
249	C₂×D₄	—	166	1	no	—
250	C₂×D₄	—	166	1	no	—
251	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
252	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
253	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
254	C₂²×C₄	—	166	1	no	—
255	C₂²×C₄	—	166	1	no	—
256	C₂²×C₄	—	166	1	no	—
257	C₂²×C₄	—	166	1	no	—
258	Q₁₆	—	332	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	4	1	no	—
261	SD₁₆	—	332	1	no	—
262	SD₁₆	—	166	2	no	—
263	D₈	—	2	2	no	—
264	D₈	—	4	1	no	—
265	D₈	—	332	1	no	—
266	D₈	—	332	1	no	—
267	C₂×C₈	—	83	1	no	—
268	C₂×C₈	—	83	1	no	—
269	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
271	C₄⋊C₄	—	83	1	no	—
272	C₄⋊C₄	—	83	1	no	—
273	C₄⋊C₄	—	83	1	no	—
274	C₄⋊C₄	—	83	1	no	—
275	C₄⋊C₄	—	83	1	no	—
276	C₄⋊C₄	—	166	1	no	—
277	C₄⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₂²⋊C₄	—	166	1	no	—
282	C₂²⋊C₄	—	166	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	C₂²⋊C₄	—	166	1	no	—
289	C₂²⋊C₄	—	166	1	no	—
290	C₄²	—	83	1	no	—
291	C₄²	—	83	1	no	—
292	C₄²	—	166	1	no	—
293	C₄²	—	166	1	no	—
294	C₄²	C₂²×D₈₃	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	166	1	no	—
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	332	1	no	—
302	Q₈	—	166	1	no	—
303	Q₈	—	166	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	1	no	—
312	D₄	—	332	2	no	—
313	D₄	—	664	1	no	—
314	D₄	—	166	1	no	—
315	D₄	—	166	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	83	1	no	—
318	C₂×C₄	—	83	1	no	—
319	C₂×C₄	—	83	1	no	—
320	C₂×C₄	—	83	1	no	—
321	C₂×C₄	—	664	1	no	—
322	C₂×C₄	—	664	1	no	—
323	C₂×C₄	—	664	1	no	—
324	C₂×C₄	—	664	1	no	—
325	C₂×C₄	—	166	1	no	—
326	C₂×C₄	—	166	1	no	—
327	C₂×C₄	—	166	1	no	—
328	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
329	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	166	1	no	—
333	C₈	—	166	1	no	—
334	C₈	#13050589	1	2	yes	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	332	1	no	—
345	C₂²	—	332	1	no	—
346	C₂²	—	332	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	—	332	1	no	—
352	C₄	—	332	1	no	—
353	C₄	—	332	1	no	—
354	C₄	—	166	1	no	—
355	C₄	—	166	1	no	—
356	C₄	—	166	1	no	—
357	C₄	—	166	1	no	—
358	C₄	C₂×#13050589	1	1	yes	characteristic
359	C₄	#14581837	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	8	1	no	—
362	C₂	—	8	1	no	—
363	C₂	—	664	1	no	—
364	C₂	#19128371	1	1	yes	characteristic
365	C₂	#19128497	1	1	yes	characteristic
366	C₂	#19128505	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#496×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#496	D₈₃
hypocenter	C₈₃	#3403
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #19128437	C₂	semidirect
2. #19128433	C₂	semidirect
3. #19128089	C₂	semidirect
4. #19128082	C₂	semidirect
5. #19127921	C₂	semidirect
6. #19127920	C₂	semidirect
7. #19127918	C₂	semidirect
8. #19127914	C₂	semidirect
9. #19127894	C₂	semidirect
10. #19127893	C₂	semidirect
11. #19127829	C₂	semidirect
12. #19127803	C₂	semidirect
13. #19127802	C₂	semidirect
14. D₈×Dic₈₃	C₂	semidirect
15. #496×C₈₃	C₂	semidirect
184. C₈₃	#3403	semidirect
188. #496	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128437	2
#19128433	2
#19128089	2
#19128082	2
#19127921	2
#19127920	2
#19127918	2
#19127914	2
#19127894	2
#19127893	2
#19127829	2
#19127803	2
#19127802	2
D₈×Dic₈₃	2
#496×C₈₃	2
#3403	83