Subgroups — #24917508

Order 10624.

454 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128233	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128038	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128018	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128004	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128003	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127915	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127914	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127853	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127833	C₂	1	1	yes	maximal, characteristic
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	#19127752	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×D₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₈₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₈₃	—	2	1	no	—
18	C₂×#13050579	C₂²	1	1	yes	characteristic
19	C₂×#13050578	C₂²	1	1	yes	characteristic
20	C₂×#13050575	C₂²	1	1	yes	characteristic
21	#14581785	—	2	1	no	—
22	#14581783	C₂²	1	1	yes	characteristic, complemented
23	#14581779	C₂²	1	1	yes	characteristic, complemented
24	#14581774	—	2	1	no	—
25	#14581772	C₂²	1	1	yes	characteristic
26	#14581763	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic, complemented
28	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
30	#14581754	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₈₃	—	2	1	no	—
32	C₄×C₂²⋊D₈₃	—	2	1	no	—
33	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
34	#14581751	—	2	1	no	—
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₂²⋊C₄×D₈₃	—	2	1	no	—
37	C₄²×D₈₃	C₂²	1	1	yes	characteristic
38	#14581725	C₂²	1	1	yes	characteristic, complemented
39	#14581724	C₂²	1	1	yes	characteristic, complemented
40	#14581723	C₂²	1	1	yes	characteristic, complemented
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#14581717	C₂²	1	1	yes	characteristic, complemented
44	#14581715	C₂²	1	1	yes	characteristic, complemented
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic, complemented
48	#14581712	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
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, complemented
55	#107×C₈₃	C₂²	1	1	yes	characteristic, complemented
56	#102×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic, complemented
58	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₃	—	2	1	no	—
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	D₄×D₈₃	—	2	2	no	—
65	D₄×D₈₃	—	2	2	no	—
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	4	4	no	—
71	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
72	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
73	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
74	#13050579	—	2	2	no	—
75	#13050578	—	2	2	no	—
76	#13050575	—	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	4	no	—
112	C₂²×D₈₃	—	2	1	no	—
113	C₂²×D₈₃	—	4	2	no	—
114	C₂²×D₈₃	—	4	2	no	—
115	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
116	D₄×C₈₃	—	2	1	no	—
117	D₄×C₈₃	—	2	1	no	—
118	D₄×C₈₃	—	2	1	no	—
119	D₄×C₈₃	#41	1	2	yes	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	4	2	no	—
123	C₂×C₄×C₈₃	—	4	2	no	—
124	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₈₃	—	2	4	no	—
128	C₂²⋊D₈₃	—	4	2	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	—	2	1	no	—
132	C₂×Dic₈₃	—	4	2	no	—
133	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
136	D₃₃₂	—	2	1	no	—
137	D₃₃₂	#41	1	2	yes	—
138	C₄×D₈₃	—	2	2	no	—
139	C₄×D₈₃	—	2	2	no	—
140	C₄×D₈₃	—	2	1	no	—
141	C₄×D₈₃	—	2	1	no	—
142	C₄×D₈₃	—	4	2	no	—
143	C₄×D₈₃	—	4	2	no	—
144	C₄×D₈₃	—	4	2	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	2	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	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₈₃	—	2	1	no	—
159	C₂×D₈₃	—	2	1	no	—
160	C₂×D₈₃	—	2	2	no	—
161	C₂×D₈₃	—	4	8	no	—
162	C₂×D₈₃	—	4	1	no	—
163	C₂×D₈₃	—	4	1	no	—
164	C₂×D₈₃	#135	1	2	yes	—
165	C₄×C₈₃	—	2	1	no	—
166	C₄×C₈₃	—	4	1	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
169	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
170	C₄×C₈₃	C₂×#41	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	—	4	2	no	—
178	C₂×C₈₃	—	8	1	no	—
179	C₂×C₈₃	#525	1	1	yes	characteristic
180	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
181	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	D₈₃	—	2	2	no	—
183	D₈₃	—	4	2	no	—
184	#3105	—	83	1	no	maximal, Sylow
185	C₈₃	#3105	1	1	yes	characteristic, complemented, Sylow
186	#529	—	83	1	no	—
187	C₂×C₄×D₄	—	83	1	no	—
188	#484	—	83	1	no	—
189	#481	—	83	1	no	—
190	#478	—	83	1	no	—
191	#478	—	83	1	no	—
192	#463	—	83	1	no	—
193	#463	—	83	1	no	—
194	#450	—	83	1	no	—
195	#447	—	83	1	no	—
196	C₄×SD₁₆	—	83	1	no	—
197	C₄×SD₁₆	—	83	1	no	—
198	C₄×D₈	—	83	1	no	—
199	C₄×D₈	D₈₃	1	1	yes	characteristic, complemented
200	#431	—	83	1	no	—
201	C₂²×D₄	—	83	1	no	—
202	C₂³×C₄	—	166	1	no	—
203	C₂×SD₁₆	—	83	1	no	—
204	C₂×SD₁₆	—	83	1	no	—
205	C₂×D₈	—	83	1	no	—
206	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
207	#126	—	166	1	no	—
208	#125	—	83	1	no	—
209	#124	—	83	1	no	—
210	#123	—	166	1	no	—
211	C₂²⋊Q₈	—	83	1	no	—
212	#121	—	83	1	no	—
213	C₄×Q₈	—	83	1	no	—
214	C₄×D₄	—	83	1	no	—
215	C₄×D₄	—	83	1	no	—
216	C₄×D₄	—	83	1	no	—
217	C₄×D₄	—	166	1	no	—
218	C₄×D₄	—	166	1	no	—
219	C₄×D₄	—	166	1	no	—
220	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
221	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic, complemented
222	#117	—	166	1	no	—
223	C₂×C₄⋊C₄	—	83	1	no	—
224	C₂×C₂²⋊C₄	—	166	1	no	—
225	C₂×C₄²	—	83	1	no	—
226	#107	—	83	1	no	—
227	#107	C₂×D₈₃	1	1	yes	characteristic, complemented
228	#106	—	83	1	no	—
229	#106	—	83	1	no	—
230	C₄⋊C₈	—	83	1	no	—
231	C₄⋊C₈	—	83	1	no	—
232	#103	—	83	1	no	—
233	#103	—	83	1	no	—
234	#102	—	83	1	no	—
235	#102	—	83	1	no	—
236	#102	—	83	1	no	—
237	#102	—	83	1	no	—
238	#102	C₂×D₈₃	1	1	yes	characteristic
239	#102	C₂×D₈₃	1	1	yes	characteristic, complemented
240	C₂²⋊C₈	—	83	1	no	—
241	C₂²⋊C₈	—	83	1	no	—
242	C₄×C₈	—	83	1	no	—
243	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic, complemented
244	C₂⁴	—	166	1	no	—
245	C₂×Q₈	—	83	1	no	—
246	C₂×D₄	—	83	1	no	—
247	C₂×D₄	—	166	1	no	—
248	C₂×D₄	—	166	2	no	—
249	C₂×D₄	—	166	1	no	—
250	C₂×D₄	—	166	2	no	—
251	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
252	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	332	4	no	—
256	C₂²×C₄	—	83	1	no	—
257	C₂²×C₄	—	83	1	no	—
258	C₂²×C₄	—	83	1	no	—
259	C₂²×C₄	—	166	1	no	—
260	C₂²×C₄	—	166	1	no	—
261	C₂²×C₄	—	166	1	no	—
262	C₂²×C₄	—	166	1	no	—
263	SD₁₆	—	166	2	no	—
264	SD₁₆	—	166	2	no	—
265	D₈	—	2	2	no	—
266	D₈	—	166	2	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	—	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	C₂²×D₈₃	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	2	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	C₄²	—	83	1	no	—
290	C₄²	—	83	1	no	—
291	C₄²	—	83	1	no	—
292	C₄²	—	166	1	no	—
293	C₄²	C₂²×D₈₃	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	332	2	no	—
297	C₂³	—	332	2	no	—
298	C₂³	—	83	1	no	—
299	C₂³	—	166	4	no	—
300	C₂³	—	166	1	no	—
301	Q₈	—	166	1	no	—
302	Q₈	—	166	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	332	2	no	—
307	D₄	—	83	2	no	—
308	D₄	—	166	4	no	—
309	D₄	—	166	1	no	—
310	D₄	#13050590	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	332	2	no	—
316	C₂×C₄	—	332	2	no	—
317	C₂×C₄	—	332	2	no	—
318	C₂×C₄	—	332	2	no	—
319	C₂×C₄	—	83	1	no	—
320	C₂×C₄	—	83	1	no	—
321	C₂×C₄	—	83	1	no	—
322	C₂×C₄	—	166	2	no	—
323	C₂×C₄	—	166	1	no	—
324	C₂×C₄	—	166	1	no	—
325	C₂×C₄	—	166	1	no	—
326	C₂×C₄	—	166	2	no	—
327	C₂×C₄	—	166	1	no	—
328	C₂×C₄	—	166	1	no	—
329	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
330	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
331	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	166	1	no	—
335	C₈	—	166	1	no	—
336	C₂²	—	2	2	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	—	332	8	no	—
343	C₂²	—	332	1	no	—
344	C₂²	—	332	1	no	—
345	C₂²	—	83	2	no	—
346	C₂²	—	166	2	no	—
347	C₂²	—	166	1	no	—
348	C₂²	—	166	1	no	—
349	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
350	C₄	—	2	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	#14581816	1	2	yes	—
360	C₄	#41×D₈₃	1	1	yes	characteristic
361	C₄	C₂×#13050590	1	1	yes	characteristic
362	C₂	—	4	2	no	—
363	C₂	—	8	1	no	—
364	C₂	—	332	2	no	—
365	C₂	—	166	2	no	—
366	C₂	#19128210	1	1	yes	characteristic
367	C₂	#135×D₈₃	1	1	yes	characteristic
368	C₂	C₂×#14581816	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
prime radical	#19127833	C₂
hypercenter	C₄×D₈	D₈₃
hypocenter	C₈₃	#3105
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. #19128233	C₂	semidirect
2. C₄×D₄×D₈₃	C₂	semidirect
3. #19128038	C₂	semidirect
4. #19128018	C₂	semidirect
5. #19128004	C₂	semidirect
6. #19128003	C₂	semidirect
7. #19127915	C₂	semidirect
8. #19127914	C₂	semidirect
9. #19127853	C₂	semidirect
11. C₄×#13050579	C₂	semidirect
12. C₄×#13050578	C₂	semidirect
13. C₄×#13050575	C₂	semidirect
14. #19127752	C₂	semidirect
15. C₄×D₈×C₈₃	C₂	semidirect
22. #14581783	C₂²	semidirect
23. #14581779	C₂²	semidirect
27. C₄×Dic₁₆₆	C₂²	semidirect
38. #14581725	C₂²	semidirect
39. #14581724	C₂²	semidirect
40. #14581723	C₂²	semidirect
41. Dic₈₃⋊C₈	C₂²	semidirect
42. Dic₈₃⋊C₈	C₂²	semidirect
43. #14581717	C₂²	semidirect
44. #14581715	C₂²	semidirect
47. #14581713	C₂²	semidirect
51. C₄×C₈₃⋊C₈	C₂²	semidirect
54. C₄×D₄×C₈₃	C₂²	semidirect
55. #107×C₈₃	C₂²	semidirect
57. #102×C₈₃	C₂²	semidirect
58. C₄×C₈×C₈₃	C₂²	semidirect
185. C₈₃	#3105	semidirect
199. C₄×D₈	D₈₃	semidirect
221. C₄×D₄	C₂×D₈₃	semidirect
227. #107	C₂×D₈₃	semidirect
239. #102	C₂×D₈₃	semidirect
243. C₄×C₈	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128233	2
C₄×D₄×D₈₃	2
#19128038	2
#19128018	2
#19128004	2
#19128003	2
#19127915	2
#19127914	2
#19127853	2
#19127833	2
C₄×#13050579	2
C₄×#13050578	2
C₄×#13050575	2
#19127752	2
C₄×D₈×C₈₃	2
#3105	83