Subgroups — #24918417

Order 10624.

394 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128253	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128247	C₂	1	1	yes	maximal, characteristic
3	#19128079	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128072	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128050	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128046	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127926	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127925	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127922	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127921	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127865	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127863	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127861	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127751	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050580	C₂²	1	1	yes	characteristic
20	C₂×#13050579	C₂²	1	1	yes	characteristic
21	C₂×#13050578	C₂²	1	1	yes	characteristic
22	C₂×#13050577	C₂²	1	1	yes	characteristic
23	#14581791	C₂²	1	1	yes	characteristic
24	#14581790	C₂²	1	1	yes	characteristic
25	#14581790	C₂²	1	1	yes	characteristic
26	#14581788	C₂²	1	1	yes	characteristic
27	#14581782	—	2	1	no	—
28	#14581781	C₂²	1	1	yes	characteristic
29	#14581779	C₂²	1	1	yes	characteristic
30	#14581771	C₂²	1	1	yes	characteristic
31	#14581769	—	2	1	no	—
32	#14581764	—	2	1	no	—
33	#14581764	—	2	1	no	—
34	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
35	#14581757	—	2	1	no	—
36	#14581757	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₈₃	—	2	1	no	—
38	C₄×C₂²⋊D₈₃	—	2	1	no	—
39	#14581748	—	2	1	no	—
40	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581716	C₂²	1	1	yes	characteristic
45	#14581715	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	4	1	no	—
60	#13050590	—	4	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
69	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₃	—	2	1	no	—
71	Dic₃₃₂	—	4	1	no	—
72	#13050580	—	4	1	no	—
73	#13050580	—	4	1	no	—
74	#13050579	—	4	1	no	—
75	#13050578	—	4	1	no	—
76	#13050577	—	4	1	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	C₂³	1	1	yes	characteristic
81	Dic₈₃⋊C₄	—	2	1	no	—
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	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	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	—	2	1	no	—
95	C₄×Dic₈₃	D₄	1	1	yes	characteristic
96	C₄×Dic₈₃	D₄	1	1	yes	characteristic
97	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
98	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
99	#41×C₈₃	—	4	1	no	—
100	#41×C₈₃	—	4	1	no	—
101	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₈₃	—	2	1	no	—
104	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₈₃	—	2	1	no	—
106	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
107	Q₁₆×C₈₃	—	4	1	no	—
108	SD₁₆×C₈₃	—	4	1	no	—
109	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
110	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₈₃	—	2	1	no	—
112	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
113	C₂²⋊C₄×C₈₃	—	2	1	no	—
114	C₂³×C₈₃	—	2	1	no	—
115	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₈₃	—	2	1	no	—
117	Q₈×C₈₃	—	2	1	no	—
118	Q₈×C₈₃	—	2	1	no	—
119	Q₈×C₈₃	—	2	1	no	—
120	Q₈×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	2	1	no	—
122	D₄×C₈₃	—	2	1	no	—
123	D₄×C₈₃	—	4	2	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	—	2	1	no	—
129	C₂×C₄×C₈₃	—	4	2	no	—
130	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₈₃	—	4	2	no	—
133	C₂²⋊D₈₃	—	4	2	no	—
134	C₂²⋊D₈₃	—	4	2	no	—
135	C₂×Dic₈₃	—	2	2	no	—
136	C₂×Dic₈₃	—	2	2	no	—
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	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
144	D₃₃₂	—	2	1	no	—
145	D₃₃₂	—	2	1	no	—
146	C₄×D₈₃	—	4	2	no	—
147	Dic₁₆₆	—	2	1	no	—
148	Dic₁₆₆	—	2	1	no	—
149	Dic₁₆₆	—	2	1	no	—
150	Dic₁₆₆	—	2	1	no	—
151	Dic₁₆₆	—	4	2	no	—
152	C₈×C₈₃	—	2	1	no	—
153	C₈×C₈₃	—	2	1	no	—
154	C₈₃⋊C₈	—	2	1	no	—
155	C₈₃⋊C₈	—	2	1	no	—
156	C₂²×C₈₃	—	2	1	no	—
157	C₂²×C₈₃	—	2	1	no	—
158	C₂²×C₈₃	—	2	1	no	—
159	C₂²×C₈₃	—	4	1	no	—
160	C₂²×C₈₃	—	4	1	no	—
161	C₂²×C₈₃	—	4	1	no	—
162	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₈₃	—	4	1	no	—
164	C₂×D₈₃	—	4	1	no	—
165	C₂×D₈₃	—	4	1	no	—
166	C₄×C₈₃	—	2	1	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	—	4	1	no	—
169	C₄×C₈₃	—	4	1	no	—
170	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
171	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
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	Dic₈₃	#135	1	2	yes	—
178	Dic₈₃	#137	1	2	yes	—
179	C₂×C₈₃	—	4	1	no	—
180	C₂×C₈₃	—	8	1	no	—
181	C₂×C₈₃	#535	1	1	yes	characteristic
182	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
183	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
184	D₈₃	—	8	1	no	—
185	#3210	—	83	1	no	maximal, Sylow
186	C₈₃	#3210	1	1	yes	characteristic, complemented, Sylow
187	#533	—	83	1	no	—
188	#532	—	83	1	no	—
189	#494	—	83	1	no	—
190	#492	—	83	1	no	—
191	#487	—	83	1	no	—
192	#486	—	83	1	no	—
193	#464	—	83	1	no	—
194	#464	—	83	1	no	—
195	#464	—	83	1	no	—
196	#464	—	83	1	no	—
197	#452	—	83	1	no	—
198	#452	—	83	1	no	—
199	#452	—	83	1	no	—
200	#452	D₈₃	1	1	yes	characteristic, complemented
201	#431	—	83	1	no	—
202	C₂×#41	—	83	1	no	—
203	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
204	C₂×Q₁₆	—	83	1	no	—
205	C₂×Q₁₆	—	83	1	no	—
206	C₂×Q₁₆	—	83	1	no	—
207	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
208	C₂×SD₁₆	—	83	1	no	—
209	C₂×SD₁₆	—	83	1	no	—
210	C₂×SD₁₆	—	83	1	no	—
211	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
212	C₄⋊Q₈	—	83	1	no	—
213	C₄⋊Q₈	—	83	1	no	—
214	C₄⋊Q₈	—	83	1	no	—
215	#127	—	83	1	no	—
216	#125	—	166	1	no	—
217	#124	—	83	1	no	—
218	#124	—	83	1	no	—
219	C₂²⋊Q₈	—	83	1	no	—
220	C₂²⋊Q₈	—	166	1	no	—
221	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
222	#121	—	166	1	no	—
223	#121	—	166	1	no	—
224	C₄×Q₈	—	83	1	no	—
225	C₄×Q₈	—	83	1	no	—
226	C₄×Q₈	—	166	1	no	—
227	C₄×D₄	—	166	1	no	—
228	C₄×D₄	—	166	1	no	—
229	#117	—	166	1	no	—
230	C₂×C₄²	—	83	1	no	—
231	C₄⋊C₈	—	83	1	no	—
232	C₄⋊C₈	—	83	1	no	—
233	#103	—	83	1	no	—
234	#103	—	83	1	no	—
235	#103	—	83	1	no	—
236	#103	C₂×D₈₃	1	1	yes	characteristic
237	#102	—	83	1	no	—
238	#102	—	83	1	no	—
239	#102	—	83	1	no	—
240	#102	C₂×D₈₃	1	1	yes	characteristic
241	C₂²⋊C₈	—	83	1	no	—
242	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
243	C₄×C₈	—	83	1	no	—
244	C₄×C₈	—	83	1	no	—
245	#41	—	4	1	no	—
246	#41	—	4	1	no	—
247	#41	—	332	1	no	—
248	#41	—	332	1	no	—
249	C₂×Q₈	—	83	1	no	—
250	C₂×Q₈	—	83	1	no	—
251	C₂×Q₈	—	166	1	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	—	166	1	no	—
258	C₂×D₄	—	166	1	no	—
259	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	83	1	no	—
262	C₂²×C₄	—	83	1	no	—
263	C₂²×C₄	—	166	1	no	—
264	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
265	Q₁₆	—	4	1	no	—
266	Q₁₆	—	332	1	no	—
267	Q₁₆	—	332	1	no	—
268	Q₁₆	—	332	1	no	—
269	SD₁₆	—	4	1	no	—
270	SD₁₆	—	332	1	no	—
271	SD₁₆	—	332	1	no	—
272	SD₁₆	—	332	1	no	—
273	C₂×C₈	—	83	1	no	—
274	C₂×C₈	—	83	1	no	—
275	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
276	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	83	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	—	166	1	no	—
287	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
288	C₂²⋊C₄	—	2	1	no	—
289	C₂²⋊C₄	—	166	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	—	83	1	no	—
296	C₄²	—	83	1	no	—
297	C₄²	—	83	1	no	—
298	C₄²	—	166	1	no	—
299	C₄²	—	166	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	166	1	no	—
302	C₂³	D₄×D₈₃	1	1	yes	characteristic
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	332	2	no	—
308	Q₈	—	166	1	no	—
309	Q₈	—	166	1	no	—
310	Q₈	—	166	1	no	—
311	Q₈	—	166	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	4	2	no	—
315	D₄	—	332	2	no	—
316	D₄	—	332	2	no	—
317	D₄	—	332	2	no	—
318	D₄	—	166	1	no	—
319	D₄	—	166	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	332	2	no	—
327	C₂×C₄	—	83	1	no	—
328	C₂×C₄	—	83	1	no	—
329	C₂×C₄	—	83	1	no	—
330	C₂×C₄	—	83	1	no	—
331	C₂×C₄	—	166	1	no	—
332	C₂×C₄	—	166	1	no	—
333	C₂×C₄	—	166	1	no	—
334	C₂×C₄	—	166	2	no	—
335	C₂×C₄	—	166	2	no	—
336	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
337	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	166	1	no	—
341	C₈	—	166	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	332	1	no	—
349	C₂²	—	332	1	no	—
350	C₂²	—	332	1	no	—
351	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	332	1	no	—
357	C₄	—	332	1	no	—
358	C₄	—	332	1	no	—
359	C₄	—	83	2	no	—
360	C₄	—	83	2	no	—
361	C₄	—	166	1	no	—
362	C₄	—	166	1	no	—
363	C₄	#14581837	1	1	yes	characteristic
364	C₄	#14581837	1	1	yes	characteristic
365	C₂	—	4	1	no	—
366	C₂	—	8	1	no	—
367	C₂	—	664	1	no	—
368	C₂	#19128266	1	1	yes	characteristic
369	C₂	#135×D₈₃	1	1	yes	characteristic
370	C₂	#137×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#452×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128247	C₂
hypercenter	#452	D₈₃
hypocenter	C₈₃	#3210
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #19128253	C₂	semidirect
3. #19128079	C₂	semidirect
4. #19128072	C₂	semidirect
5. #19128050	C₂	semidirect
6. #19128046	C₂	semidirect
7. #19127926	C₂	semidirect
8. #19127925	C₂	semidirect
9. #19127922	C₂	semidirect
10. #19127921	C₂	semidirect
11. #19127865	C₂	semidirect
12. #19127863	C₂	semidirect
13. #19127861	C₂	semidirect
14. #19127751	C₂	semidirect
15. #452×C₈₃	C₂	semidirect
186. C₈₃	#3210	semidirect
200. #452	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128253	2
#19128247	2
#19128079	2
#19128072	2
#19128050	2
#19128046	2
#19127926	2
#19127925	2
#19127922	2
#19127921	2
#19127865	2
#19127863	2
#19127861	2
#19127751	2
#452×C₈₃	2
#3210	83