Subgroups — #24919514

Order 10624.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128391	C₂	1	1	yes	maximal, characteristic
2	#19128069	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128039	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128029	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128008	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127957	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127940	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127908	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127857	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127857	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127773	C₂	1	1	yes	maximal, characteristic
13	#19127710	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#14581716	C₂	1	1	yes	maximal, characteristic, complemented
15	D₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050574	C₂²	1	1	yes	characteristic
20	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
21	#14581785	—	2	1	no	—
22	#14581784	C₂²	1	1	yes	characteristic
23	#14581782	—	2	1	no	—
24	#14581773	—	2	1	no	—
25	#14581771	C₂²	1	1	yes	characteristic
26	#14581771	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
28	#14581751	C₂²	1	1	yes	characteristic
29	#14581748	—	2	1	no	—
30	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
31	#14581725	C₂²	1	1	yes	characteristic
32	#14581723	C₂²	1	1	yes	characteristic
33	#14581723	C₂²	1	1	yes	characteristic
34	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
35	#14581716	C₂²	1	1	yes	characteristic
36	#14581716	C₂²	1	1	yes	characteristic
37	#14581716	C₂²	1	1	yes	characteristic
38	#14581716	C₂²	1	1	yes	characteristic
39	#14581716	C₂²	1	1	yes	characteristic
40	#14581716	C₂²	1	1	yes	characteristic
41	#14581716	C₂²	1	1	yes	characteristic
42	#14581716	C₂²	1	1	yes	characteristic
43	#14581716	C₂²	1	1	yes	characteristic
44	#14581715	C₂²	1	1	yes	characteristic
45	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
46	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
47	#14581703	C₂²	1	1	yes	characteristic
48	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
49	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
50	C₄⋊Q₈×C₈₃	—	2	1	no	—
51	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₈₃	—	2	1	no	—
53	C₂²⋊Q₈×C₈₃	—	2	1	no	—
54	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₈₃	—	2	1	no	—
58	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂²×Dic₈₃	—	2	1	no	—
61	#13050580	—	2	2	no	—
62	#13050580	—	4	1	no	—
63	#13050580	—	4	1	no	—
64	#13050574	—	4	1	no	—
65	C₂×C₈₃⋊C₈	—	4	1	no	—
66	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	Dic₈₃⋊C₄	—	2	1	no	—
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	C₂²⋊Dic₈₃	—	2	1	no	—
80	C₂²⋊Dic₈₃	—	2	1	no	—
81	C₂²⋊Dic₈₃	—	2	1	no	—
82	C₄×Dic₈₃	—	2	1	no	—
83	C₄×Dic₈₃	—	2	1	no	—
84	C₂×Q₈×C₈₃	—	2	1	no	—
85	C₂×Q₈×C₈₃	—	2	1	no	—
86	C₂×Q₈×C₈₃	—	2	2	no	—
87	C₂×Q₈×C₈₃	—	2	2	no	—
88	C₂×Q₈×C₈₃	—	4	1	no	—
89	C₂×Q₈×C₈₃	—	4	1	no	—
90	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₃	—	2	1	no	—
95	C₂²×C₄×C₈₃	—	2	1	no	—
96	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₃	—	2	1	no	—
99	C₄⋊C₄×C₈₃	—	2	1	no	—
100	C₄⋊C₄×C₈₃	—	2	1	no	—
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	—	2	1	no	—
103	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
109	C₄²×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	2	1	no	—
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	—	4	2	no	—
120	Q₈×C₈₃	#41	1	2	yes	—
121	D₄×C₈₃	—	2	2	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	—	2	1	no	—
128	C₂×C₄×C₈₃	—	2	1	no	—
129	C₂×C₄×C₈₃	—	2	1	no	—
130	C₂×C₄×C₈₃	—	4	2	no	—
131	C₂×C₄×C₈₃	—	4	2	no	—
132	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
141	C₂×Dic₈₃	—	4	2	no	—
142	Dic₁₆₆	—	2	1	no	—
143	Dic₁₆₆	—	2	1	no	—
144	C₈₃⋊C₈	—	2	1	no	—
145	C₈₃⋊C₈	—	2	1	no	—
146	C₈₃⋊C₈	—	2	1	no	—
147	C₈₃⋊C₈	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	—	2	1	no	—
153	C₂²×C₈₃	#137	1	2	yes	—
154	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	2	2	no	—
158	C₄×C₈₃	—	2	1	no	—
159	C₄×C₈₃	—	2	1	no	—
160	C₄×C₈₃	—	4	1	no	—
161	C₄×C₈₃	—	4	1	no	—
162	C₄×C₈₃	—	4	1	no	—
163	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
164	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₈₃	—	4	1	no	—
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	—	4	1	no	—
168	Dic₈₃	—	4	1	no	—
169	C₂×C₈₃	—	2	2	no	—
170	C₂×C₈₃	—	4	1	no	—
171	C₂×C₈₃	#546	1	1	yes	characteristic
172	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
173	C₂×C₈₃	#578	1	1	yes	characteristic
174	#3312	—	83	1	no	maximal, Sylow
175	C₈₃	#3312	1	1	yes	characteristic, complemented, Sylow
176	#552	—	83	1	no	—
177	D₄×Q₈	D₈₃	1	1	yes	characteristic
178	#491	—	83	1	no	—
179	#485	—	83	1	no	—
180	#483	—	83	1	no	—
181	#479	—	83	1	no	—
182	#470	—	83	1	no	—
183	#467	—	83	1	no	—
184	#462	—	83	1	no	—
185	#451	—	83	1	no	—
186	#451	—	83	1	no	—
187	#441	—	83	1	no	—
188	#435	—	83	1	no	—
189	#419	—	83	1	no	—
190	C₂×#103	—	83	1	no	—
191	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
192	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
193	C₂×Q₁₆	—	83	1	no	—
194	C₂×Q₁₆	—	83	1	no	—
195	C₂×Q₁₆	—	83	1	no	—
196	C₂×M₁₆	—	83	1	no	—
197	C₂²×C₈	—	83	1	no	—
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
200	#126	—	166	1	no	—
201	#125	—	83	1	no	—
202	#125	—	166	1	no	—
203	#123	—	166	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	C₂²⋊Q₈	—	83	1	no	—
207	C₂²⋊Q₈	—	83	1	no	—
208	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
209	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
210	C₄×Q₈	—	83	1	no	—
211	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
214	#117	—	83	1	no	—
215	#117	—	166	1	no	—
216	C₂×C₄⋊C₄	—	83	1	no	—
217	#107	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	—	83	1	no	—
220	C₄⋊C₈	—	83	1	no	—
221	#103	—	83	1	no	—
222	#103	—	83	1	no	—
223	#103	—	83	1	no	—
224	#103	—	83	1	no	—
225	#103	—	83	1	no	—
226	#103	—	83	1	no	—
227	#103	—	83	1	no	—
228	#103	—	83	1	no	—
229	#103	—	83	1	no	—
230	#103	—	83	1	no	—
231	C₂²⋊C₈	—	83	1	no	—
232	C₂²⋊C₈	—	83	1	no	—
233	#97	—	83	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	83	1	no	—
241	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
242	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₂×D₄	C₂²⋊D₈₃	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
250	Q₁₆	—	332	1	no	—
251	Q₁₆	—	332	1	no	—
252	Q₁₆	—	166	2	no	—
253	M₁₆	—	332	1	no	—
254	C₂×C₈	—	332	1	no	—
255	C₂×C₈	—	83	1	no	—
256	C₂×C₈	—	83	1	no	—
257	C₂×C₈	—	83	1	no	—
258	C₂×C₈	—	83	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	83	1	no	—
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	83	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	1	no	—
273	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	166	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
281	C₂²⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
282	C₄²	—	2	1	no	—
283	C₄²	—	166	1	no	—
284	C₄²	—	166	1	no	—
285	C₄²	C₂²×D₈₃	1	1	yes	characteristic
286	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
287	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	166	1	no	—
296	Q₈	—	166	1	no	—
297	Q₈	#13050589	1	2	yes	—
298	D₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	332	2	no	—
310	C₂×C₄	—	166	1	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	1	no	—
314	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
317	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
319	C₈	—	166	1	no	—
320	C₈	—	166	1	no	—
321	C₈	—	166	1	no	—
322	C₈	—	166	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	#14581823	1	2	yes	—
329	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	332	1	no	—
339	C₄	—	332	1	no	—
340	C₄	—	332	1	no	—
341	C₄	—	332	1	no	—
342	C₄	C₂×#13050589	1	1	yes	characteristic
343	C₄	#14581837	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	4	1	no	—
346	C₂	#19128345	1	1	yes	characteristic
347	C₂	C₂×#14581823	1	1	yes	characteristic
348	C₂	#19128516	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	D₄×Q₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₄×C₈₃	D₄
hypercenter	D₄×Q₈	D₈₃
hypocenter	C₈₃	#3312
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #19128069	C₂	semidirect
3. #19128039	C₂	semidirect
4. #19128029	C₂	semidirect
5. #19128008	C₂	semidirect
6. #19127957	C₂	semidirect
7. #19127940	C₂	semidirect
8. #19127908	C₂	semidirect
9. #19127857	C₂	semidirect
10. #19127857	C₂	semidirect
11. #19127806	C₂	semidirect
13. #19127710	C₂	semidirect
14. C₂×#14581716	C₂	semidirect
175. C₈₃	#3312	semidirect

Maximal subgroups

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

subgroup	index
#19128391	2
#19128069	2
#19128039	2
#19128029	2
#19128008	2
#19127957	2
#19127940	2
#19127908	2
#19127857	2
#19127806	2
#19127773	2
#19127710	2
C₂×#14581716	2
D₄×Q₈×C₈₃	2
#3312	83