Subgroups — #24918056

Order 10624.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128248	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128233	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128109	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128089	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128064	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128059	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127953	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127951	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127948	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127946	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13050579	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13050578	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13050575	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13050573	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×D₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	—	2	1	no	—
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×#13050573	C₂²	1	1	yes	characteristic
21	C₂×#13050573	C₂²	1	1	yes	characteristic
22	#14581791	C₂²	1	1	yes	characteristic
23	#14581788	C₂²	1	1	yes	characteristic
24	#14581785	—	2	1	no	—
25	#14581783	C₂²	1	1	yes	characteristic
26	#14581779	C₂²	1	1	yes	characteristic
27	#14581778	—	2	1	no	—
28	#14581774	—	2	1	no	—
29	#14581772	C₂²	1	1	yes	characteristic
30	#14581767	C₂²	1	1	yes	characteristic
31	#14581765	—	2	1	no	—
32	#14581763	C₂²	1	1	yes	characteristic
33	#14581763	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
35	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₈₃	—	2	1	no	—
37	C₄×C₂²⋊D₈₃	—	2	1	no	—
38	#14581751	—	2	1	no	—
39	C₄²×D₈₃	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581715	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	#14581702	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	#102×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	4	4	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₃₂	—	2	1	no	—
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
71	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
72	#13050579	—	2	2	no	—
73	#13050578	—	2	2	no	—
74	#13050575	—	2	2	no	—
75	#13050573	—	2	2	no	—
76	#13050573	—	2	2	no	—
77	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	—	2	1	no	—
80	C₄⋊Dic₈₃	—	2	1	no	—
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	D₄	1	1	yes	characteristic
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	—	2	1	no	—
100	C₂²×C₄×C₈₃	—	2	1	no	—
101	D₈×C₈₃	—	2	2	no	—
102	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₂²⋊C₄×C₈₃	—	2	1	no	—
108	C₄²×C₈₃	C₂³	1	1	yes	characteristic
109	C₂³×C₈₃	—	2	1	no	—
110	C₂³×C₈₃	—	2	1	no	—
111	C₂²×D₈₃	—	2	1	no	—
112	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
113	D₄×C₈₃	—	2	1	no	—
114	D₄×C₈₃	—	2	1	no	—
115	D₄×C₈₃	—	2	1	no	—
116	D₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	4	2	no	—
120	C₂×C₄×C₈₃	—	4	2	no	—
121	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₈₃	—	4	2	no	—
125	C₂²⋊D₈₃	—	4	2	no	—
126	C₂²⋊D₈₃	—	4	2	no	—
127	C₂²⋊D₈₃	—	4	2	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	—	4	2	no	—
137	C₄×D₈₃	—	2	2	no	—
138	C₄×D₈₃	—	2	1	no	—
139	C₄×D₈₃	—	2	1	no	—
140	C₄×D₈₃	—	4	1	no	—
141	C₄×D₈₃	—	4	2	no	—
142	Dic₁₆₆	—	2	1	no	—
143	Dic₁₆₆	—	2	1	no	—
144	Dic₁₆₆	—	4	2	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈×C₈₃	#41	1	2	yes	complemented
147	C₈₃⋊C₈	—	2	1	no	—
148	C₈₃⋊C₈	#41	1	2	yes	complemented
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	—	4	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	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₈₃	—	2	1	no	—
157	C₂×D₈₃	—	2	1	no	—
158	C₂×D₈₃	—	2	1	no	—
159	C₂×D₈₃	—	4	1	no	—
160	C₂×D₈₃	—	4	1	no	—
161	C₂×D₈₃	—	4	1	no	—
162	C₄×C₈₃	—	2	1	no	—
163	C₄×C₈₃	—	4	1	no	—
164	C₄×C₈₃	—	4	1	no	—
165	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
166	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
167	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
168	Dic₈₃	—	2	1	no	—
169	Dic₈₃	—	2	1	no	—
170	Dic₈₃	—	2	1	no	—
171	Dic₈₃	—	4	1	no	—
172	Dic₈₃	—	4	1	no	—
173	Dic₈₃	—	4	1	no	—
174	C₂×C₈₃	—	8	1	no	—
175	C₂×C₈₃	—	8	1	no	—
176	C₂×C₈₃	#532	1	1	yes	characteristic
177	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂×C₈₃	#575	1	1	yes	characteristic
179	D₈₃	—	4	1	no	—
180	D₈₃	—	8	1	no	—
181	#3171	—	83	1	no	maximal, Sylow
182	C₈₃	#3171	1	1	yes	characteristic, complemented, Sylow
183	#532	—	83	1	no	—
184	#529	—	83	1	no	—
185	#501	—	83	1	no	—
186	#496	—	83	1	no	—
187	#490	—	83	1	no	—
188	#489	—	83	1	no	—
189	#469	—	83	1	no	—
190	#469	—	83	1	no	—
191	#468	—	83	1	no	—
192	#468	—	83	1	no	—
193	C₄×SD₁₆	—	83	1	no	—
194	C₄×SD₁₆	—	83	1	no	—
195	C₄×D₈	—	83	1	no	—
196	C₄×D₈	D₈₃	1	1	yes	characteristic, complemented
197	C₄×M₁₆	—	83	1	no	—
198	C₂×#41	—	166	1	no	—
199	C₂×SD₁₆	—	83	1	no	—
200	C₂×SD₁₆	—	83	1	no	—
201	C₂×D₈	—	83	1	no	—
202	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
203	C₂×M₁₆	—	83	1	no	—
204	C₂×M₁₆	—	83	1	no	—
205	C₄⋊Q₈	—	83	1	no	—
206	#127	—	83	1	no	—
207	#126	—	166	1	no	—
208	#125	—	83	1	no	—
209	#124	—	83	1	no	—
210	#124	—	166	1	no	—
211	#123	—	166	1	no	—
212	C₂²⋊Q₈	—	83	1	no	—
213	#121	—	83	1	no	—
214	#121	—	83	1	no	—
215	#121	—	83	1	no	—
216	#121	—	166	1	no	—
217	C₄×Q₈	—	83	1	no	—
218	C₄×D₄	—	83	1	no	—
219	C₄×D₄	—	166	1	no	—
220	C₄×D₄	—	166	1	no	—
221	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
222	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
223	#117	—	166	1	no	—
224	C₂×C₄²	—	83	1	no	—
225	#107	—	83	1	no	—
226	#107	C₂×D₈₃	1	1	yes	characteristic
227	#106	—	83	1	no	—
228	#106	—	83	1	no	—
229	#103	—	83	1	no	—
230	#103	—	83	1	no	—
231	#102	—	83	1	no	—
232	#102	—	83	1	no	—
233	#102	—	83	1	no	—
234	#102	—	83	1	no	—
235	#102	C₂×D₈₃	1	1	yes	characteristic
236	#102	C₂×D₈₃	1	1	yes	characteristic
237	#97	—	83	1	no	—
238	#97	—	83	1	no	—
239	C₄×C₈	—	83	1	no	—
240	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
241	#41	—	332	4	no	—
242	C₂×Q₈	—	83	1	no	—
243	C₂×Q₈	—	166	1	no	—
244	C₂×D₄	—	83	1	no	—
245	C₂×D₄	—	166	1	no	—
246	C₂×D₄	—	166	1	no	—
247	C₂×D₄	—	166	1	no	—
248	C₂×D₄	—	166	1	no	—
249	C₂×D₄	—	166	1	no	—
250	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
251	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	83	1	no	—
255	C₂²×C₄	—	83	1	no	—
256	C₂²×C₄	—	83	1	no	—
257	C₂²×C₄	—	166	1	no	—
258	SD₁₆	—	166	2	no	—
259	SD₁₆	—	166	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	166	2	no	—
262	M₁₆	—	166	2	no	—
263	M₁₆	—	166	2	no	—
264	C₂×C₈	—	83	1	no	—
265	C₂×C₈	—	83	1	no	—
266	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
268	C₄⋊C₄	—	83	1	no	—
269	C₄⋊C₄	—	83	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	C₂²×D₈₃	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	166	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	C₄²	—	83	1	no	—
287	C₄²	—	83	1	no	—
288	C₄²	—	83	1	no	—
289	C₄²	—	166	1	no	—
290	C₄²	C₂²×D₈₃	1	1	yes	characteristic
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	83	1	no	—
294	C₂³	—	166	1	no	—
295	Q₈	—	332	2	no	—
296	Q₈	—	166	1	no	—
297	Q₈	—	166	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	332	2	no	—
303	D₄	—	332	2	no	—
304	D₄	—	332	2	no	—
305	D₄	—	332	2	no	—
306	D₄	—	332	2	no	—
307	D₄	—	166	1	no	—
308	D₄	—	166	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	332	1	no	—
314	C₂×C₄	—	332	2	no	—
315	C₂×C₄	—	83	1	no	—
316	C₂×C₄	—	83	1	no	—
317	C₂×C₄	—	83	1	no	—
318	C₂×C₄	—	166	1	no	—
319	C₂×C₄	—	166	1	no	—
320	C₂×C₄	—	166	1	no	—
321	C₂×C₄	—	166	2	no	—
322	C₂×C₄	—	166	1	no	—
323	C₂×C₄	—	166	1	no	—
324	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
325	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	83	2	no	—
329	C₈	—	166	1	no	—
330	C₈	#13050590	1	2	yes	complemented
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	332	1	no	—
338	C₂²	—	332	1	no	—
339	C₂²	—	332	1	no	—
340	C₂²	—	166	1	no	—
341	C₂²	—	166	1	no	—
342	C₂²	—	166	1	no	—
343	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	332	1	no	—
348	C₄	—	332	1	no	—
349	C₄	—	332	1	no	—
350	C₄	—	166	1	no	—
351	C₄	—	166	1	no	—
352	C₄	—	166	1	no	—
353	C₄	#14581816	1	2	yes	—
354	C₄	#41×D₈₃	1	1	yes	characteristic
355	C₄	C₂×#13050590	1	1	yes	characteristic
356	C₂	—	8	1	no	—
357	C₂	—	8	1	no	—
358	C₂	—	332	1	no	—
359	C₂	—	664	1	no	—
360	C₂	#19128248	1	1	yes	characteristic
361	C₂	C₂×#14581816	1	1	yes	characteristic
362	C₂	#19128498	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₄×D₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	C₄×D₈	D₈₃
hypocenter	C₈₃	#3171
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #19128248	C₂	semidirect
2. #19128233	C₂	semidirect
3. #19128109	C₂	semidirect
4. #19128089	C₂	semidirect
5. #19128064	C₂	semidirect
6. #19128059	C₂	semidirect
7. #19127953	C₂	semidirect
8. #19127951	C₂	semidirect
9. #19127948	C₂	semidirect
10. #19127946	C₂	semidirect
11. C₄×#13050579	C₂	semidirect
12. C₄×#13050578	C₂	semidirect
13. C₄×#13050575	C₂	semidirect
14. C₄×#13050573	C₂	semidirect
15. C₄×D₈×C₈₃	C₂	semidirect
146. C₈×C₈₃	#41	semidirect
148. C₈₃⋊C₈	#41	semidirect
182. C₈₃	#3171	semidirect
196. C₄×D₈	D₈₃	semidirect
330. C₈	#13050590	semidirect

Maximal subgroups

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

subgroup	index
#19128248	2
#19128233	2
#19128109	2
#19128089	2
#19128064	2
#19128059	2
#19127953	2
#19127951	2
#19127948	2
#19127946	2
C₄×#13050579	2
C₄×#13050578	2
C₄×#13050575	2
C₄×#13050573	2
C₄×D₈×C₈₃	2
#3171	83