Subgroups — #24919337

Order 10624.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128406	C₂	1	1	yes	maximal, characteristic
2	#19128364	C₂	1	1	yes	maximal, characteristic
3	#19128050	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128038	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127995	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127943	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127940	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127910	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127859	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127858	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127795	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₈₃)⋊C₈	C₂	1	1	yes	maximal, characteristic
13	C₂×#14581715	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#484×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
20	C₂×#13050580	C₂²	1	1	yes	characteristic
21	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
23	#14581791	—	2	1	no	—
24	#14581791	C₂²	1	1	yes	characteristic
25	#14581790	C₂²	1	1	yes	characteristic
26	#14581782	—	2	1	no	—
27	#14581772	C₂²	1	1	yes	characteristic
28	#14581772	C₂²	1	1	yes	characteristic
29	#14581771	—	2	1	no	—
30	#14581771	C₂²	1	1	yes	characteristic
31	#14581770	—	2	1	no	—
32	#14581770	—	2	1	no	—
33	#14581769	—	2	1	no	—
34	#14581757	C₂²	1	1	yes	characteristic
35	#14581757	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₈₃	—	2	1	no	—
37	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
38	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
39	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
40	#14581725	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	#14581715	C₂²	1	1	yes	characteristic
46	#14581715	C₂²	1	1	yes	characteristic
47	#14581715	C₂²	1	1	yes	characteristic
48	#14581715	C₂²	1	1	yes	characteristic
49	#14581715	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×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#107×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	2	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×Dic₁₆₆	—	4	1	no	—
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
67	Q₈×D₈₃	—	2	2	no	—
68	Q₈×D₈₃	—	4	1	no	—
69	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
70	C₂²×Dic₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
74	Dic₃₃₂	—	2	2	no	—
75	Dic₃₃₂	—	4	1	no	—
76	#13050580	—	4	1	no	—
77	C₂×C₈₃⋊C₈	—	4	1	no	—
78	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₈×D₈₃	—	4	1	no	—
81	C₄⋊Dic₈₃	—	2	1	no	—
82	C₄⋊Dic₈₃	—	2	1	no	—
83	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
89	Dic₈₃⋊C₄	—	2	1	no	—
90	Dic₈₃⋊C₄	—	2	1	no	—
91	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
92	C₂²⋊Dic₈₃	—	2	1	no	—
93	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
94	(C₂×D₈₃)⋊C₄	—	2	1	no	—
95	(C₂×D₈₃)⋊C₄	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
97	C₄×Dic₈₃	—	2	1	no	—
98	C₄×Dic₈₃	—	2	1	no	—
99	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₃	—	2	1	no	—
102	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	—	2	1	no	—
106	C₄⋊C₄×C₈₃	—	2	1	no	—
107	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×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	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	8	1	no	—
122	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₈₃	Q₁₆	1	4	yes	complemented
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	4	2	no	—
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	C₄×D₈₃	—	2	1	no	—
134	C₄×D₈₃	—	2	1	no	—
135	C₄×D₈₃	—	4	2	no	—
136	C₄×D₈₃	—	8	1	no	—
137	Dic₁₆₆	—	2	1	no	—
138	Dic₁₆₆	—	2	1	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	2	1	no	—
142	Dic₁₆₆	—	4	2	no	—
143	Dic₁₆₆	—	4	2	no	—
144	Dic₁₆₆	—	8	1	no	—
145	Dic₁₆₆	#41	1	2	yes	—
146	C₈×C₈₃	—	2	1	no	—
147	C₈×C₈₃	—	2	1	no	—
148	C₈₃⋊C₈	—	2	1	no	—
149	C₈₃⋊C₈	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	C₂×Q₁₆	1	2	yes	—
153	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₈₃	—	2	1	no	—
155	C₂×D₈₃	—	2	1	no	—
156	C₂×D₈₃	C₂×Q₁₆	1	2	yes	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	—	4	1	no	—
160	C₄×C₈₃	—	4	1	no	—
161	C₄×C₈₃	—	4	1	no	—
162	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
163	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₈₃	—	2	2	no	—
165	Dic₈₃	—	2	1	no	—
166	Dic₈₃	—	2	1	no	—
167	Dic₈₃	—	4	1	no	—
168	Dic₈₃	—	4	1	no	—
169	Dic₈₃	—	4	1	no	—
170	Dic₈₃	C₂×Q₁₆	1	2	yes	—
171	C₂×C₈₃	—	2	2	no	—
172	C₂×C₈₃	#546	1	1	yes	characteristic
173	C₂×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
174	C₂×C₈₃	#577	1	1	yes	characteristic
175	D₈₃	—	2	2	no	—
176	#3299	—	83	1	no	maximal, Sylow
177	C₈₃	#3299	1	1	yes	characteristic, complemented, Sylow
178	#554	—	83	1	no	—
179	D₄×Q₈	—	83	1	no	—
180	#487	—	83	1	no	—
181	#484	—	83	1	no	—
182	#484	D₈₃	1	1	yes	characteristic, complemented
183	#477	—	83	1	no	—
184	#467	—	83	1	no	—
185	#467	—	83	1	no	—
186	#462	—	83	1	no	—
187	#451	—	83	1	no	—
188	#451	—	83	1	no	—
189	C₄×Q₁₆	—	83	1	no	—
190	C₈×D₄	—	83	1	no	—
191	C₂×#103	—	83	1	no	—
192	C₂×#103	—	83	1	no	—
193	C₂²×Q₈	—	83	1	no	—
194	C₂²×Q₈	—	83	1	no	—
195	C₂×Q₁₆	—	83	1	no	—
196	C₂×Q₁₆	—	83	1	no	—
197	C₂×Q₁₆	—	83	1	no	—
198	C₂²×C₈	—	83	1	no	—
199	C₂²×C₈	—	83	1	no	—
200	C₄⋊Q₈	—	83	1	no	—
201	C₄⋊Q₈	—	83	1	no	—
202	C₄⋊Q₈	—	166	1	no	—
203	#125	—	166	1	no	—
204	C₂²⋊Q₈	—	83	1	no	—
205	C₂²⋊Q₈	—	83	1	no	—
206	C₂²⋊Q₈	—	83	1	no	—
207	C₂²⋊Q₈	—	166	1	no	—
208	C₂²⋊Q₈	—	166	1	no	—
209	C₂²⋊Q₈	—	166	1	no	—
210	C₂²⋊Q₈	—	166	1	no	—
211	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
212	C₄×Q₈	—	83	1	no	—
213	C₄×Q₈	—	83	1	no	—
214	C₄×D₄	—	83	1	no	—
215	C₄×D₄	—	166	1	no	—
216	C₄×D₄	—	166	1	no	—
217	C₂×C₄⋊C₄	—	83	1	no	—
218	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
219	#107	—	83	1	no	—
220	#107	C₂×D₈₃	1	1	yes	characteristic
221	#107	C₂×D₈₃	1	1	yes	characteristic
222	C₄⋊C₈	—	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	#103	C₂×D₈₃	1	1	yes	characteristic
232	#103	C₂×D₈₃	1	1	yes	characteristic
233	C₂²⋊C₈	—	83	1	no	—
234	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
235	C₄×C₈	—	83	1	no	—
236	C₂×Q₈	—	332	1	no	—
237	C₂×Q₈	—	332	1	no	—
238	C₂×Q₈	—	83	1	no	—
239	C₂×Q₈	—	83	1	no	—
240	C₂×Q₈	—	83	1	no	—
241	C₂×Q₈	—	166	2	no	—
242	C₂×Q₈	—	166	1	no	—
243	C₂×Q₈	—	166	1	no	—
244	C₂×Q₈	—	166	2	no	—
245	C₂×Q₈	—	166	1	no	—
246	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
247	C₂×D₄	—	83	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	83	1	no	—
250	C₂²×C₄	—	166	1	no	—
251	C₂²×C₄	—	166	1	no	—
252	C₂²×C₄	—	166	1	no	—
253	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
254	Q₁₆	—	332	1	no	—
255	Q₁₆	—	332	1	no	—
256	Q₁₆	—	166	2	no	—
257	C₂×C₈	—	332	1	no	—
258	C₂×C₈	—	332	1	no	—
259	C₂×C₈	—	83	1	no	—
260	C₂×C₈	—	83	1	no	—
261	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	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	—	166	1	no	—
274	C₄⋊C₄	—	166	1	no	—
275	C₄⋊C₄	—	166	1	no	—
276	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
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	C₄²	—	83	1	no	—
286	C₄²	—	166	1	no	—
287	C₄²	—	166	1	no	—
288	C₂³	—	83	1	no	—
289	C₂³	D₄×D₈₃	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	332	2	no	—
293	Q₈	—	332	2	no	—
294	Q₈	—	83	2	no	—
295	Q₈	—	664	1	no	—
296	Q₈	—	166	1	no	—
297	Q₈	—	166	1	no	—
298	Q₈	—	166	1	no	—
299	Q₈	—	166	1	no	—
300	Q₈	—	166	1	no	—
301	D₄	—	83	4	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	—	2	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	332	2	no	—
310	C₂×C₄	—	332	2	no	—
311	C₂×C₄	—	83	1	no	—
312	C₂×C₄	—	83	1	no	—
313	C₂×C₄	—	83	1	no	—
314	C₂×C₄	—	664	1	no	—
315	C₂×C₄	—	166	1	no	—
316	C₂×C₄	—	166	1	no	—
317	C₂×C₄	—	166	1	no	—
318	C₂×C₄	—	166	1	no	—
319	C₂×C₄	—	166	1	no	—
320	C₂×C₄	—	166	1	no	—
321	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	166	1	no	—
326	C₈	—	166	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	83	2	no	—
330	C₂²	—	166	1	no	—
331	C₂²	—	166	1	no	—
332	C₂²	Q₁₆×D₈₃	1	2	yes	—
333	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	332	1	no	—
340	C₄	—	332	1	no	—
341	C₄	—	332	1	no	—
342	C₄	—	83	2	no	—
343	C₄	—	166	2	no	—
344	C₄	—	166	1	no	—
345	C₄	—	166	1	no	—
346	C₄	#41×D₈₃	1	1	yes	characteristic
347	C₄	#14581838	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	166	2	no	—
350	C₂	#19128343	1	1	yes	characteristic
351	C₂	C₂×Q₁₆×D₈₃	1	1	yes	characteristic
352	C₂	#19128511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#484×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×C₂²⋊D₈₃	D₄
hypercenter	#484	D₈₃
hypocenter	C₈₃	#3299
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
3. #19128050	C₂	semidirect
4. #19128038	C₂	semidirect
5. #19127995	C₂	semidirect
6. #19127943	C₂	semidirect
7. #19127940	C₂	semidirect
8. #19127910	C₂	semidirect
9. #19127859	C₂	semidirect
10. #19127858	C₂	semidirect
11. #19127795	C₂	semidirect
13. C₂×#14581715	C₂	semidirect
14. #103×D₈₃	C₂	semidirect
15. #484×C₈₃	C₂	semidirect
124. C₂²⋊D₈₃	Q₁₆	semidirect
177. C₈₃	#3299	semidirect
182. #484	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128406	2
#19128364	2
#19128050	2
#19128038	2
#19127995	2
#19127943	2
#19127940	2
#19127910	2
#19127859	2
#19127858	2
#19127795	2
(C₂²⋊D₈₃)⋊C₈	2
C₂×#14581715	2
#103×D₈₃	2
#484×C₈₃	2
#3299	83