Subgroups — #24919035

Order 10624.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14581824	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Q₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₃₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128363	C₂	1	1	yes	maximal, characteristic
5	#19128095	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127960	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127931	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127923	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127911	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127866	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127858	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127858	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127841	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127774	C₂	1	1	yes	maximal, characteristic, complemented
15	#441×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581824	—	2	4	no	—
21	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
22	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
23	C₂×#13050580	C₂²	1	1	yes	characteristic
24	C₂×#13050580	C₂²	1	1	yes	characteristic
25	C₂×#13050580	C₂²	1	1	yes	characteristic
26	Q₁₆×D₈₃	—	2	4	no	—
27	C₂×#13050579	C₂²	1	1	yes	characteristic
28	C₂×#13050579	C₂²	1	1	yes	characteristic
29	C₂×#13050577	C₂²	1	1	yes	characteristic
30	C₂×#13050573	C₂²	1	1	yes	characteristic
31	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
32	#14581792	—	2	1	no	—
33	#14581792	C₂²	1	1	yes	characteristic
34	#14581780	—	2	1	no	—
35	#14581780	C₂²	1	1	yes	characteristic
36	#14581772	—	2	1	no	—
37	#14581772	—	2	1	no	—
38	#14581772	—	2	1	no	—
39	#14581772	C₂²	1	1	yes	characteristic
40	#14581772	C₂²	1	1	yes	characteristic
41	#14581772	C₂²	1	1	yes	characteristic
42	#14581767	—	2	1	no	—
43	#14581767	C₂²	1	1	yes	characteristic
44	C₄×D₃₃₂	—	2	1	no	—
45	C₄×D₃₃₂	—	2	1	no	—
46	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
47	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
48	#14581715	C₂²	1	1	yes	characteristic
49	#14581715	C₂²	1	1	yes	characteristic
50	#14581715	C₂²	1	1	yes	characteristic
51	#14581712	C₂²	1	1	yes	characteristic
52	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
53	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
57	#106×C₈₃	C₂²	1	1	yes	characteristic
58	#103×C₈₃	C₂²	1	1	yes	characteristic
59	#103×C₈₃	C₂²	1	1	yes	characteristic
60	#97×C₈₃	C₂²	1	1	yes	characteristic
61	#13050591	—	2	2	no	—
62	#13050591	—	4	1	no	—
63	C₂×Dic₁₆₆	—	2	1	no	—
64	C₂×Dic₁₆₆	—	2	1	no	—
65	C₂×Dic₁₆₆	—	2	1	no	—
66	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
69	Q₈×D₈₃	—	2	2	no	—
70	Q₈×D₈₃	—	2	2	no	—
71	Q₈×D₈₃	—	2	2	no	—
72	Q₈×D₈₃	—	4	1	no	—
73	Q₈×D₈₃	—	4	1	no	—
74	Q₈×D₈₃	—	4	1	no	—
75	C₂×D₃₃₂	—	2	1	no	—
76	C₂×D₃₃₂	—	2	1	no	—
77	C₂×D₃₃₂	D₄	1	1	yes	characteristic
78	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₈₃	—	2	1	no	—
80	C₂×C₄×D₈₃	—	2	1	no	—
81	C₂×C₄×D₈₃	—	2	1	no	—
82	C₂×C₄×D₈₃	—	2	1	no	—
83	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
84	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
85	Dic₃₃₂	—	2	2	no	—
86	Dic₃₃₂	—	4	1	no	—
87	#13050580	—	2	2	no	—
88	#13050580	—	2	2	no	—
89	#13050580	—	2	2	no	—
90	#13050579	—	2	2	no	—
91	#13050579	—	4	1	no	—
92	#13050577	—	2	2	no	—
93	#13050573	—	2	2	no	—
94	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
95	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
96	C₈×D₈₃	—	2	2	no	—
97	C₄⋊Dic₈₃	—	2	1	no	—
98	C₄⋊Dic₈₃	—	2	1	no	—
99	C₄⋊Dic₈₃	—	2	1	no	—
100	C₄⋊Dic₈₃	—	2	1	no	—
101	C₄⋊Dic₈₃	—	2	1	no	—
102	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
103	(C₂×D₈₃)⋊C₄	—	2	1	no	—
104	(C₂×D₈₃)⋊C₄	—	2	1	no	—
105	(C₂×D₈₃)⋊C₄	—	2	1	no	—
106	(C₂×D₈₃)⋊C₄	—	2	1	no	—
107	(C₂×D₈₃)⋊C₄	—	2	1	no	—
108	(C₂×D₈₃)⋊C₄	—	2	1	no	—
109	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
110	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
111	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
112	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
113	Q₁₆×C₈₃	D₄	1	4	yes	complemented
114	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
115	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
116	C₄⋊C₄×C₈₃	—	2	1	no	—
117	C₄⋊C₄×C₈₃	—	2	1	no	—
118	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
119	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
120	C₄²×C₈₃	—	2	1	no	—
121	C₄²×C₈₃	—	2	1	no	—
122	C₄²×C₈₃	C₂³	1	1	yes	characteristic
123	C₂²×D₈₃	—	2	1	no	—
124	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
126	Q₈×C₈₃	—	2	1	no	—
127	Q₈×C₈₃	—	2	1	no	—
128	Q₈×C₈₃	C₂×D₄	1	2	yes	—
129	Q₈×C₈₃	C₂×D₄	1	2	yes	—
130	C₂×C₄×C₈₃	—	2	1	no	—
131	C₂×C₄×C₈₃	—	2	1	no	—
132	C₂×C₄×C₈₃	—	2	1	no	—
133	C₂×C₄×C₈₃	—	2	1	no	—
134	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
137	C₂×Dic₈₃	—	2	1	no	—
138	C₂×Dic₈₃	—	2	1	no	—
139	C₂×Dic₈₃	—	2	1	no	—
140	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
142	D₃₃₂	—	2	2	no	—
143	D₃₃₂	—	2	1	no	—
144	D₃₃₂	—	2	1	no	—
145	D₃₃₂	—	4	2	no	—
146	D₃₃₂	—	4	2	no	—
147	C₄×D₈₃	—	2	1	no	—
148	C₄×D₈₃	—	2	1	no	—
149	C₄×D₈₃	—	2	1	no	—
150	C₄×D₈₃	—	2	1	no	—
151	C₄×D₈₃	—	4	2	no	—
152	C₄×D₈₃	—	4	2	no	—
153	C₄×D₈₃	—	4	2	no	—
154	C₄×D₈₃	—	4	2	no	—
155	Dic₁₆₆	—	2	1	no	—
156	Dic₁₆₆	—	2	1	no	—
157	Dic₁₆₆	—	2	1	no	—
158	Dic₁₆₆	—	2	1	no	—
159	Dic₁₆₆	—	2	1	no	—
160	Dic₁₆₆	—	2	1	no	—
161	Dic₁₆₆	—	4	2	no	—
162	Dic₁₆₆	—	4	2	no	—
163	Dic₁₆₆	—	4	2	no	—
164	C₈×C₈₃	—	2	1	no	—
165	C₈×C₈₃	C₂×D₄	1	2	yes	—
166	C₈₃⋊C₈	—	2	1	no	—
167	C₈₃⋊C₈	—	2	1	no	—
168	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×D₈₃	—	2	1	no	—
170	C₂×D₈₃	—	2	1	no	—
171	C₂×D₈₃	—	2	1	no	—
172	C₂×D₈₃	—	2	1	no	—
173	C₂×D₈₃	—	2	1	no	—
174	C₂×D₈₃	—	4	1	no	—
175	C₂×D₈₃	—	4	1	no	—
176	C₂×D₈₃	—	4	1	no	—
177	C₂×D₈₃	#137	1	2	yes	complemented
178	C₄×C₈₃	—	2	2	no	—
179	C₄×C₈₃	—	2	2	no	—
180	C₄×C₈₃	—	2	1	no	—
181	C₄×C₈₃	—	2	1	no	—
182	C₄×C₈₃	—	4	1	no	—
183	C₄×C₈₃	—	4	1	no	—
184	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
185	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
186	Dic₈₃	—	2	1	no	—
187	Dic₈₃	—	2	1	no	—
188	Dic₈₃	—	4	1	no	—
189	Dic₈₃	—	4	1	no	—
190	Dic₈₃	—	4	1	no	—
191	C₂×C₈₃	D₄²	1	1	yes	characteristic
192	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
193	C₂×C₈₃	#578	1	1	yes	characteristic
194	D₈₃	—	2	2	no	—
195	D₈₃	—	4	1	no	—
196	D₈₃	—	8	1	no	—
197	#3270	—	83	1	no	maximal, Sylow
198	C₈₃	#3270	1	1	yes	characteristic, complemented, Sylow
199	C₂×#137	—	83	1	no	—
200	C₂²×Q₁₆	—	83	1	no	—
201	D₄×Q₈	—	83	1	no	—
202	#548	—	83	1	no	—
203	#497	—	83	1	no	—
204	#470	—	83	1	no	—
205	#465	—	83	1	no	—
206	#464	—	83	1	no	—
207	#462	—	83	1	no	—
208	#452	—	83	1	no	—
209	#451	—	83	1	no	—
210	#451	—	83	1	no	—
211	#448	—	83	1	no	—
212	#441	D₈₃	1	1	yes	characteristic, complemented
213	#435	—	83	1	no	—
214	C₂×#41	—	83	1	no	—
215	C₂²×Q₈	—	83	1	no	—
216	C₂²×Q₈	—	83	1	no	—
217	C₂²×Q₈	—	83	1	no	—
218	#137	—	166	4	no	—
219	C₂×Q₁₆	—	83	1	no	—
220	C₂×Q₁₆	—	83	1	no	—
221	C₂×Q₁₆	—	83	1	no	—
222	C₂×Q₁₆	—	83	1	no	—
223	C₂×Q₁₆	—	83	1	no	—
224	C₂×Q₁₆	—	166	4	no	—
225	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
226	C₂×SD₁₆	—	83	1	no	—
227	C₂×SD₁₆	—	83	1	no	—
228	C₂×SD₁₆	—	83	1	no	—
229	C₂×M₁₆	—	83	1	no	—
230	C₂²×C₈	—	83	1	no	—
231	C₄⋊Q₈	—	83	1	no	—
232	C₄⋊Q₈	—	166	1	no	—
233	#124	—	83	1	no	—
234	#124	—	166	1	no	—
235	C₂²⋊Q₈	—	83	1	no	—
236	C₂²⋊Q₈	—	83	1	no	—
237	C₂²⋊Q₈	—	83	1	no	—
238	C₂²⋊Q₈	—	166	1	no	—
239	C₂²⋊Q₈	—	166	1	no	—
240	C₂²⋊Q₈	—	166	1	no	—
241	#121	—	83	1	no	—
242	#121	—	166	1	no	—
243	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
244	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
245	C₄×D₄	—	83	1	no	—
246	C₄×D₄	—	166	1	no	—
247	C₄×D₄	—	166	1	no	—
248	#106	C₂×D₈₃	1	1	yes	characteristic
249	C₄⋊C₈	—	83	1	no	—
250	#103	—	83	1	no	—
251	#103	—	83	1	no	—
252	#103	—	83	1	no	—
253	#103	C₂×D₈₃	1	1	yes	characteristic
254	#103	C₂×D₈₃	1	1	yes	characteristic
255	#102	—	83	1	no	—
256	C₂²⋊C₈	—	83	1	no	—
257	C₂²⋊C₈	—	83	1	no	—
258	#97	C₂×D₈₃	1	1	yes	characteristic
259	#41	—	332	1	no	—
260	#41	—	166	2	no	—
261	C₂×Q₈	—	332	1	no	—
262	C₂×Q₈	—	332	1	no	—
263	C₂×Q₈	—	332	1	no	—
264	C₂×Q₈	—	83	1	no	—
265	C₂×Q₈	—	83	1	no	—
266	C₂×Q₈	—	83	1	no	—
267	C₂×Q₈	—	166	1	no	—
268	C₂×Q₈	—	166	1	no	—
269	C₂×Q₈	—	166	1	no	—
270	C₂×Q₈	—	166	2	no	—
271	C₂×Q₈	—	166	2	no	—
272	C₂×Q₈	—	166	2	no	—
273	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
274	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
275	C₂×D₄	—	83	1	no	—
276	C₂×D₄	—	83	1	no	—
277	C₂×D₄	—	166	1	no	—
278	C₂×D₄	—	166	1	no	—
279	C₂²×C₄	—	83	1	no	—
280	C₂²×C₄	—	83	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	Q₁₆	—	332	1	no	—
286	Q₁₆	—	166	2	no	—
287	Q₁₆	—	166	2	no	—
288	Q₁₆	—	166	2	no	—
289	Q₁₆	—	166	2	no	—
290	Q₁₆	D₃₃₂	1	4	yes	complemented
291	SD₁₆	—	332	1	no	—
292	SD₁₆	—	166	2	no	—
293	SD₁₆	—	166	2	no	—
294	M₁₆	—	166	2	no	—
295	C₂×C₈	—	83	1	no	—
296	C₂×C₈	—	83	1	no	—
297	C₂×C₈	—	166	2	no	—
298	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
299	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	83	1	no	—
303	C₄⋊C₄	—	166	1	no	—
304	C₄⋊C₄	—	166	1	no	—
305	C₄⋊C₄	—	166	1	no	—
306	C₄⋊C₄	—	166	1	no	—
307	C₄⋊C₄	—	166	1	no	—
308	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
309	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
310	C₂²⋊C₄	—	83	1	no	—
311	C₂²⋊C₄	—	83	1	no	—
312	C₂²⋊C₄	—	166	1	no	—
313	C₂²⋊C₄	—	166	1	no	—
314	C₂²⋊C₄	—	166	1	no	—
315	C₂²⋊C₄	—	166	1	no	—
316	C₂²⋊C₄	—	166	1	no	—
317	C₂²⋊C₄	—	166	1	no	—
318	C₄²	—	2	1	no	—
319	C₄²	—	2	1	no	—
320	C₄²	C₂²×D₈₃	1	1	yes	characteristic
321	C₂³	—	83	1	no	—
322	C₂³	—	83	1	no	—
323	C₂³	—	166	1	no	—
324	Q₈	—	2	1	no	—
325	Q₈	—	2	1	no	—
326	Q₈	—	332	2	no	—
327	Q₈	—	332	2	no	—
328	Q₈	—	332	2	no	—
329	Q₈	—	166	1	no	—
330	Q₈	—	166	1	no	—
331	Q₈	—	166	1	no	—
332	Q₈	—	166	1	no	—
333	Q₈	—	166	1	no	—
334	Q₈	—	166	1	no	—
335	Q₈	C₂×D₃₃₂	1	2	yes	—
336	Q₈	C₂×D₃₃₂	1	2	yes	—
337	D₄	—	332	2	no	—
338	D₄	—	332	2	no	—
339	D₄	—	166	2	no	—
340	D₄	—	166	1	no	—
341	D₄	—	166	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	332	2	no	—
347	C₂×C₄	—	332	2	no	—
348	C₂×C₄	—	332	2	no	—
349	C₂×C₄	—	332	2	no	—
350	C₂×C₄	—	83	1	no	—
351	C₂×C₄	—	83	1	no	—
352	C₂×C₄	—	166	1	no	—
353	C₂×C₄	—	166	1	no	—
354	C₂×C₄	—	166	1	no	—
355	C₂×C₄	—	166	1	no	—
356	C₂×C₄	—	166	1	no	—
357	C₂×C₄	—	166	1	no	—
358	C₂×C₄	—	166	1	no	—
359	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
360	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
361	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
362	C₈	—	2	1	no	—
363	C₈	—	166	1	no	—
364	C₈	—	166	1	no	—
365	C₈	C₂×D₃₃₂	1	2	yes	—
366	C₂²	—	332	1	no	—
367	C₂²	—	332	1	no	—
368	C₂²	—	332	1	no	—
369	C₂²	—	83	2	no	—
370	C₂²	—	166	1	no	—
371	C₂²	—	166	1	no	—
372	C₂²	—	166	1	no	—
373	C₂²	—	166	1	no	—
374	C₂²	—	166	1	no	—
375	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
376	C₄	—	2	2	no	—
377	C₄	—	2	2	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	4	1	no	—
381	C₄	—	4	1	no	—
382	C₄	—	332	1	no	—
383	C₄	—	332	1	no	—
384	C₄	—	332	1	no	—
385	C₄	—	166	1	no	—
386	C₄	—	166	1	no	—
387	C₄	C₂²×D₃₃₂	1	1	yes	characteristic
388	C₄	#14581838	1	1	yes	characteristic
389	C₂	—	332	1	no	—
390	C₂	—	664	1	no	—
391	C₂	—	166	2	no	—
392	C₂	D₄×D₃₃₂	1	1	yes	characteristic
393	C₂	#137×D₈₃	1	1	yes	characteristic
394	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	#441×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128363	C₂
hypercenter	#441	D₈₃
hypocenter	C₈₃	#3270
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. C₂×#14581824	C₂	semidirect
2. C₂×Q₁₆×D₈₃	C₂	semidirect
3. Q₈×D₃₃₂	C₂	semidirect
5. #19128095	C₂	semidirect
6. #19127960	C₂	semidirect
7. #19127931	C₂	semidirect
8. #19127923	C₂	semidirect
9. #19127911	C₂	semidirect
10. #19127866	C₂	semidirect
11. #19127858	C₂	semidirect
12. #19127858	C₂	semidirect
13. #19127841	C₂	semidirect
14. #19127774	C₂	semidirect
15. #441×C₈₃	C₂	semidirect
113. Q₁₆×C₈₃	D₄	semidirect
177. C₂×D₈₃	#137	semidirect
198. C₈₃	#3270	semidirect
212. #441	D₈₃	semidirect
290. Q₁₆	D₃₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14581824	2
C₂×Q₁₆×D₈₃	2
Q₈×D₃₃₂	2
#19128363	2
#19128095	2
#19127960	2
#19127931	2
#19127923	2
#19127911	2
#19127866	2
#19127858	2
#19127841	2
#19127774	2
#441×C₈₃	2
#3270	83