Subgroups — #24919750

Order 10624.

386 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128415	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₂²⋊D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128050	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128043	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128040	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127994	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127945	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127944	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	#19127776	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127710	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127708	C₂	1	1	yes	maximal, characteristic, complemented
15	#448×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050578	C₂²	1	1	yes	characteristic
18	C₂×#13050574	C₂²	1	1	yes	characteristic
19	C₂×#13050573	C₂²	1	1	yes	characteristic
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581790	—	2	1	no	—
22	#14581790	C₂²	1	1	yes	characteristic
23	#14581779	—	2	1	no	—
24	#14581778	—	2	1	no	—
25	#14581771	—	2	1	no	—
26	#14581771	C₂²	1	1	yes	characteristic
27	#14581768	—	2	1	no	—
28	#14581768	C₂²	1	1	yes	characteristic
29	#14581764	—	2	1	no	—
30	#14581763	C₂²	1	1	yes	characteristic
31	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
32	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₃	—	2	1	no	—
34	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
35	#14581751	C₂²	1	1	yes	characteristic
36	#14581750	C₂²	1	1	yes	characteristic
37	#14581748	—	2	1	no	—
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581724	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	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	Q₈×D₈₃	—	2	2	no	—
63	Q₈×D₈₃	—	4	1	no	—
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
69	#13050578	—	4	1	no	—
70	#13050574	—	4	1	no	—
71	#13050573	—	4	1	no	—
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₃	—	2	1	no	—
97	C₂×Q₈×C₈₃	—	2	2	no	—
98	C₂×Q₈×C₈₃	—	4	1	no	—
99	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₈₃	—	2	1	no	—
102	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₈₃	—	2	1	no	—
104	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
105	SD₁₆×C₈₃	—	2	2	no	—
106	SD₁₆×C₈₃	—	4	1	no	—
107	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
108	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₃	—	2	1	no	—
111	C₂³×C₈₃	—	2	1	no	—
112	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₈₃	—	2	1	no	—
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	Q₈×C₈₃	—	4	2	no	—
118	Q₈×C₈₃	#41	1	2	yes	—
119	D₄×C₈₃	—	2	1	no	—
120	D₄×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	8	1	no	—
122	C₂×C₄×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	—	2	1	no	—
127	C₂×C₄×C₈₃	—	4	2	no	—
128	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₃	—	2	2	no	—
131	C₂²⋊D₈₃	—	8	1	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	2	1	no	—
135	C₂×Dic₈₃	—	2	1	no	—
136	C₂×Dic₈₃	—	4	2	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	C₄×D₈₃	—	2	1	no	—
141	C₄×D₈₃	—	2	1	no	—
142	C₄×D₈₃	—	4	2	no	—
143	Dic₁₆₆	—	2	1	no	—
144	Dic₁₆₆	—	2	1	no	—
145	Dic₁₆₆	—	4	2	no	—
146	Dic₁₆₆	—	8	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	1	no	—
152	C₂²×C₈₃	—	2	1	no	—
153	C₂²×C₈₃	—	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	C₄×C₈₃	—	2	2	no	—
162	C₄×C₈₃	—	2	1	no	—
163	C₄×C₈₃	—	4	1	no	—
164	C₄×C₈₃	—	4	1	no	—
165	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
166	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
167	Dic₈₃	—	2	1	no	—
168	Dic₈₃	—	2	1	no	—
169	Dic₈₃	—	4	1	no	—
170	Dic₈₃	—	4	1	no	—
171	Dic₈₃	—	4	1	no	—
172	Dic₈₃	—	4	1	no	—
173	Dic₈₃	#137	1	2	yes	—
174	C₂×C₈₃	—	4	1	no	—
175	C₂×C₈₃	—	8	1	no	—
176	C₂×C₈₃	#546	1	1	yes	characteristic
177	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
178	C₂×C₈₃	#577	1	1	yes	characteristic
179	D₈₃	—	4	1	no	—
180	#3329	—	83	1	no	maximal, Sylow
181	C₈₃	#3329	1	1	yes	characteristic, complemented, Sylow
182	#555	—	83	1	no	—
183	D₄×Q₈	—	83	1	no	—
184	#487	—	83	1	no	—
185	#485	—	83	1	no	—
186	#485	—	83	1	no	—
187	#477	—	83	1	no	—
188	#468	—	83	1	no	—
189	#468	—	83	1	no	—
190	#461	—	83	1	no	—
191	#448	—	83	1	no	—
192	#448	D₈₃	1	1	yes	characteristic, complemented
193	C₄×SD₁₆	—	83	1	no	—
194	#436	—	83	1	no	—
195	#419	—	83	1	no	—
196	#419	—	83	1	no	—
197	C₂²×Q₈	—	83	1	no	—
198	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
199	C₂×SD₁₆	—	83	1	no	—
200	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
201	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
202	C₂×M₁₆	—	83	1	no	—
203	C₂×M₁₆	—	83	1	no	—
204	C₄⋊Q₈	—	83	1	no	—
205	C₄⋊Q₈	—	83	1	no	—
206	C₄⋊Q₈	—	166	1	no	—
207	#124	—	166	1	no	—
208	#124	—	166	1	no	—
209	C₂²⋊Q₈	—	83	1	no	—
210	C₂²⋊Q₈	—	83	1	no	—
211	C₂²⋊Q₈	—	166	1	no	—
212	C₂²⋊Q₈	—	166	1	no	—
213	#121	—	83	1	no	—
214	#121	—	166	1	no	—
215	#121	C₂×D₈₃	1	1	yes	characteristic
216	C₄×Q₈	—	83	1	no	—
217	C₄×D₄	—	83	1	no	—
218	C₄×D₄	—	83	1	no	—
219	C₄×D₄	—	166	1	no	—
220	#117	—	83	1	no	—
221	#117	—	83	1	no	—
222	#117	—	166	1	no	—
223	#107	—	83	1	no	—
224	#107	—	83	1	no	—
225	#106	—	83	1	no	—
226	C₄⋊C₈	—	83	1	no	—
227	#103	—	83	1	no	—
228	#103	—	83	1	no	—
229	#103	—	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	C₂×D₈₃	1	1	yes	characteristic
235	#103	C₂×D₈₃	1	1	yes	characteristic
236	#102	—	83	1	no	—
237	C₂²⋊C₈	—	83	1	no	—
238	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
239	C₄×C₈	—	83	1	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	332	1	no	—
244	C₂×Q₈	—	83	1	no	—
245	C₂×Q₈	—	166	1	no	—
246	C₂×Q₈	—	166	1	no	—
247	C₂×Q₈	—	166	2	no	—
248	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
249	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	83	1	no	—
252	C₂×D₄	—	166	1	no	—
253	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	83	1	no	—
256	C₂²×C₄	—	166	1	no	—
257	C₂²×C₄	—	166	1	no	—
258	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	4	1	no	—
261	SD₁₆	—	332	1	no	—
262	M₁₆	—	332	1	no	—
263	M₁₆	—	332	1	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	—	83	1	no	—
271	C₄⋊C₄	—	83	1	no	—
272	C₄⋊C₄	—	166	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	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	83	1	no	—
281	C₂²⋊C₄	—	83	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	C₄²	—	83	1	no	—
289	C₄²	—	166	1	no	—
290	C₄²	—	166	1	no	—
291	C₄²	—	166	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	83	1	no	—
294	C₂³	D₄×D₈₃	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	332	2	no	—
300	Q₈	—	664	1	no	—
301	Q₈	—	166	1	no	—
302	Q₈	—	166	1	no	—
303	Q₈	#13050589	1	2	yes	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	8	1	no	—
307	D₄	—	664	1	no	—
308	D₄	—	166	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	332	2	no	—
316	C₂×C₄	—	332	2	no	—
317	C₂×C₄	—	83	1	no	—
318	C₂×C₄	—	83	1	no	—
319	C₂×C₄	—	83	1	no	—
320	C₂×C₄	—	166	1	no	—
321	C₂×C₄	—	166	1	no	—
322	C₂×C₄	—	166	1	no	—
323	C₂×C₄	—	166	1	no	—
324	C₂×C₄	—	166	1	no	—
325	C₂×C₄	—	166	1	no	—
326	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	166	1	no	—
331	C₈	—	166	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	166	1	no	—
339	C₂²	—	166	1	no	—
340	C₂²	—	166	1	no	—
341	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	332	1	no	—
347	C₄	—	332	1	no	—
348	C₄	—	332	1	no	—
349	C₄	—	332	1	no	—
350	C₄	—	83	2	no	—
351	C₄	—	166	1	no	—
352	C₄	—	166	1	no	—
353	C₄	C₂×#13050589	1	1	yes	characteristic
354	C₄	#14581837	1	1	yes	characteristic
355	C₂	—	4	1	no	—
356	C₂	—	8	1	no	—
357	C₂	—	332	1	no	—
358	C₂	#19128338	1	1	yes	characteristic
359	C₂	#137×D₈₃	1	1	yes	characteristic
360	C₂	#19128507	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	#19128415	C₂
hypercenter	#448	D₈₃
hypocenter	C₈₃	#3329
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
2. Q₈×C₂²⋊D₈₃	C₂	semidirect
3. #19128050	C₂	semidirect
4. #19128043	C₂	semidirect
5. #19128040	C₂	semidirect
6. #19127994	C₂	semidirect
7. #19127945	C₂	semidirect
8. #19127944	C₂	semidirect
9. #19127904	C₂	semidirect
10. #19127838	C₂	semidirect
11. SD₁₆×Dic₈₃	C₂	semidirect
12. #19127776	C₂	semidirect
13. #19127710	C₂	semidirect
14. #19127708	C₂	semidirect
15. #448×C₈₃	C₂	semidirect
181. C₈₃	#3329	semidirect
192. #448	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128415	2
Q₈×C₂²⋊D₈₃	2
#19128050	2
#19128043	2
#19128040	2
#19127994	2
#19127945	2
#19127944	2
#19127904	2
#19127838	2
SD₁₆×Dic₈₃	2
#19127776	2
#19127710	2
#19127708	2
#448×C₈₃	2
#3329	83