Subgroups — #24919321

Order 10624.

420 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	Q₈×C₂²⋊D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128351	C₂	1	1	yes	maximal, characteristic
3	#19128050	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128032	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128030	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127984	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127928	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127927	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127904	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127838	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₈₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14581716	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#448×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	—	2	1	no	—
17	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×#13050578	C₂²	1	1	yes	characteristic
19	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581790	—	2	1	no	—
23	#14581790	C₂²	1	1	yes	characteristic
24	#14581774	—	2	1	no	—
25	#14581773	—	2	1	no	—
26	#14581771	—	2	1	no	—
27	#14581771	C₂²	1	1	yes	characteristic
28	#14581769	—	2	1	no	—
29	#14581768	—	2	1	no	—
30	#14581768	C₂²	1	1	yes	characteristic
31	#14581763	C₂²	1	1	yes	characteristic
32	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
33	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₈₃	—	2	1	no	—
35	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
38	#14581724	C₂²	1	1	yes	characteristic
39	#14581724	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	Dic₈₃⋊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	#14581716	C₂²	1	1	yes	characteristic
46	#14581716	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581715	C₂²	1	1	yes	characteristic
49	#14581713	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
55	#121×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	#13050589	—	4	4	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	Q₈×D₈₃	—	2	2	no	—
64	Q₈×D₈₃	—	4	1	no	—
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
67	C₂²×Dic₈₃	—	2	1	no	—
68	C₂²×Dic₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
72	#13050578	—	4	1	no	—
73	C₂×C₈₃⋊C₈	—	4	1	no	—
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×D₈₃	—	4	1	no	—
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	—	2	1	no	—
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	2	1	no	—
91	C₂²⋊Dic₈₃	—	2	1	no	—
92	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
93	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	—	2	1	no	—
98	C₂×Q₈×C₈₃	—	2	2	no	—
99	C₂×Q₈×C₈₃	—	4	1	no	—
100	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₈₃	—	2	1	no	—
103	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₈₃	—	2	1	no	—
105	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
106	SD₁₆×C₈₃	—	2	2	no	—
107	SD₁₆×C₈₃	—	4	1	no	—
108	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
109	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
111	C₂²⋊C₄×C₈₃	—	2	1	no	—
112	C₂³×C₈₃	—	2	1	no	—
113	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	Q₈×C₈₃	—	2	1	no	—
118	Q₈×C₈₃	—	4	2	no	—
119	Q₈×C₈₃	#41	1	2	yes	complemented
120	D₄×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	2	1	no	—
122	D₄×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	—	4	2	no	—
129	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₈₃	—	4	2	no	—
132	C₂²⋊D₈₃	SD₁₆	1	4	yes	complemented
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	2	1	no	—
135	C₂×Dic₈₃	—	2	1	no	—
136	C₂×Dic₈₃	—	2	1	no	—
137	C₂×Dic₈₃	—	4	2	no	—
138	C₂×Dic₈₃	—	4	2	no	—
139	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
142	C₄×D₈₃	—	2	1	no	—
143	C₄×D₈₃	—	2	1	no	—
144	C₄×D₈₃	—	4	2	no	—
145	C₄×D₈₃	—	4	2	no	—
146	Dic₁₆₆	—	2	1	no	—
147	Dic₁₆₆	—	2	1	no	—
148	Dic₁₆₆	—	4	2	no	—
149	Dic₁₆₆	—	4	2	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	—	2	1	no	—
155	C₂²×C₈₃	—	2	1	no	—
156	C₂²×C₈₃	—	4	1	no	—
157	C₂²×C₈₃	—	4	1	no	—
158	C₂²×C₈₃	—	4	1	no	—
159	C₂²×C₈₃	C₂×SD₁₆	1	2	yes	—
160	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₈₃	—	2	1	no	—
162	C₂×D₈₃	—	2	1	no	—
163	C₂×D₈₃	C₂×SD₁₆	1	2	yes	—
164	C₄×C₈₃	—	2	2	no	—
165	C₄×C₈₃	—	2	1	no	—
166	C₄×C₈₃	—	4	1	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
169	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
170	Dic₈₃	—	2	1	no	—
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	4	1	no	—
173	Dic₈₃	—	4	1	no	—
174	Dic₈₃	—	4	1	no	—
175	Dic₈₃	—	4	1	no	—
176	Dic₈₃	C₂×SD₁₆	1	2	yes	—
177	C₂×C₈₃	—	2	2	no	—
178	C₂×C₈₃	—	8	1	no	—
179	C₂×C₈₃	#546	1	1	yes	characteristic
180	C₂×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
181	C₂×C₈₃	#578	1	1	yes	characteristic
182	D₈₃	—	2	2	no	—
183	#3298	—	83	1	no	maximal, Sylow
184	C₈₃	#3298	1	1	yes	characteristic, complemented, Sylow
185	D₄×Q₈	—	83	1	no	—
186	#547	—	83	1	no	—
187	#487	—	83	1	no	—
188	#483	—	83	1	no	—
189	#483	—	83	1	no	—
190	#475	—	83	1	no	—
191	#465	—	83	1	no	—
192	#465	—	83	1	no	—
193	#461	—	83	1	no	—
194	#448	—	83	1	no	—
195	#448	D₈₃	1	1	yes	characteristic, complemented
196	C₄×SD₁₆	—	83	1	no	—
197	C₈×D₄	—	83	1	no	—
198	C₂×#103	—	83	1	no	—
199	C₂×#103	—	83	1	no	—
200	C₂×#41	—	166	1	no	—
201	C₂²×Q₈	—	83	1	no	—
202	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
203	C₂×SD₁₆	—	83	1	no	—
204	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
205	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
206	C₂²×C₈	—	83	1	no	—
207	C₂²×C₈	—	83	1	no	—
208	C₄⋊Q₈	—	83	1	no	—
209	C₄⋊Q₈	—	83	1	no	—
210	C₄⋊Q₈	—	166	1	no	—
211	#123	—	166	1	no	—
212	#123	—	166	1	no	—
213	C₂²⋊Q₈	—	83	1	no	—
214	C₂²⋊Q₈	—	83	1	no	—
215	C₂²⋊Q₈	—	166	1	no	—
216	C₂²⋊Q₈	—	166	1	no	—
217	C₂²⋊Q₈	—	166	1	no	—
218	#121	—	83	1	no	—
219	#121	C₂×D₈₃	1	1	yes	characteristic
220	C₄×Q₈	—	83	1	no	—
221	C₄×D₄	—	83	1	no	—
222	C₄×D₄	—	83	1	no	—
223	C₄×D₄	—	166	1	no	—
224	C₂×C₄⋊C₄	—	83	1	no	—
225	C₂×C₄⋊C₄	—	83	1	no	—
226	#106	—	83	1	no	—
227	#106	—	83	1	no	—
228	#106	—	83	1	no	—
229	C₄⋊C₈	—	83	1	no	—
230	#103	—	83	1	no	—
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	—	83	1	no	—
237	#103	C₂×D₈₃	1	1	yes	characteristic
238	#103	C₂×D₈₃	1	1	yes	characteristic
239	#102	—	83	1	no	—
240	C₂²⋊C₈	—	83	1	no	—
241	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
242	C₄×C₈	—	83	1	no	—
243	#41	—	332	4	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	332	1	no	—
248	C₂×Q₈	—	83	1	no	—
249	C₂×Q₈	—	166	1	no	—
250	C₂×Q₈	—	166	1	no	—
251	C₂×Q₈	—	166	2	no	—
252	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
253	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	83	1	no	—
256	C₂×D₄	—	166	1	no	—
257	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	83	1	no	—
260	C₂²×C₄	—	166	1	no	—
261	C₂²×C₄	—	166	1	no	—
262	C₂²×C₄	—	166	1	no	—
263	C₂²×C₄	—	166	1	no	—
264	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
265	SD₁₆	—	2	2	no	—
266	SD₁₆	—	4	1	no	—
267	SD₁₆	—	332	1	no	—
268	C₂×C₈	—	332	1	no	—
269	C₂×C₈	—	332	1	no	—
270	C₂×C₈	—	83	1	no	—
271	C₂×C₈	—	83	1	no	—
272	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
274	C₄⋊C₄	—	83	1	no	—
275	C₄⋊C₄	—	83	1	no	—
276	C₄⋊C₄	—	83	1	no	—
277	C₄⋊C₄	—	83	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	—	166	1	no	—
284	C₄⋊C₄	—	166	1	no	—
285	C₄⋊C₄	—	166	1	no	—
286	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	83	1	no	—
289	C₂²⋊C₄	—	83	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	C₄²	—	83	1	no	—
295	C₄²	—	166	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	83	1	no	—
298	C₂³	D₄×D₈₃	1	1	yes	characteristic
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	332	2	no	—
304	Q₈	—	332	2	no	—
305	Q₈	—	166	1	no	—
306	Q₈	—	166	1	no	—
307	Q₈	#13050589	1	2	yes	complemented
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	332	2	no	—
312	D₄	—	83	4	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	332	2	no	—
320	C₂×C₄	—	332	2	no	—
321	C₂×C₄	—	332	2	no	—
322	C₂×C₄	—	332	2	no	—
323	C₂×C₄	—	83	1	no	—
324	C₂×C₄	—	83	1	no	—
325	C₂×C₄	—	83	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	—	166	1	no	—
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	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	83	2	no	—
344	C₂²	—	166	1	no	—
345	C₂²	—	166	1	no	—
346	C₂²	SD₁₆×D₈₃	1	2	yes	—
347	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	332	1	no	—
353	C₄	—	332	1	no	—
354	C₄	—	332	1	no	—
355	C₄	—	332	1	no	—
356	C₄	—	83	2	no	—
357	C₄	—	166	1	no	—
358	C₄	—	166	1	no	—
359	C₄	C₂×#13050589	1	1	yes	characteristic
360	C₄	#14581837	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	8	1	no	—
363	C₂	—	166	2	no	—
364	C₂	#19128338	1	1	yes	characteristic
365	C₂	C₂×SD₁₆×D₈₃	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	#448×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128351	C₂
hypercenter	#448	D₈₃
hypocenter	C₈₃	#3298
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #448

Complements

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

N	H ≅ G/N	type
1. Q₈×C₂²⋊D₈₃	C₂	semidirect
3. #19128050	C₂	semidirect
4. #19128032	C₂	semidirect
5. #19128030	C₂	semidirect
6. #19127984	C₂	semidirect
7. #19127928	C₂	semidirect
8. #19127927	C₂	semidirect
9. #19127904	C₂	semidirect
10. #19127838	C₂	semidirect
11. SD₁₆×Dic₈₃	C₂	semidirect
12. (C₂²⋊D₈₃)⋊C₈	C₂	semidirect
13. C₂×#14581716	C₂	semidirect
14. #103×D₈₃	C₂	semidirect
15. #448×C₈₃	C₂	semidirect
119. Q₈×C₈₃	#41	semidirect
132. C₂²⋊D₈₃	SD₁₆	semidirect
184. C₈₃	#3298	semidirect
195. #448	D₈₃	semidirect
307. Q₈	#13050589	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₂²⋊D₈₃	2
#19128351	2
#19128050	2
#19128032	2
#19128030	2
#19127984	2
#19127928	2
#19127927	2
#19127904	2
#19127838	2
SD₁₆×Dic₈₃	2
(C₂²⋊D₈₃)⋊C₈	2
C₂×#14581716	2
#103×D₈₃	2
#448×C₈₃	2
#3298	83