Subgroups — #24920480

Order 10624.

406 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128420	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128371	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128089	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128072	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127914	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127913	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127904	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127899	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127898	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127894	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127829	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127803	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127802	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#496×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	—	2	1	no	—
17	C₂×#13050579	C₂²	1	1	yes	characteristic
18	C₂×#13050578	C₂²	1	1	yes	characteristic
19	C₂×#13050578	C₂²	1	1	yes	characteristic
20	C₂×#13050577	C₂²	1	1	yes	characteristic
21	C₂×#13050575	C₂²	1	1	yes	characteristic
22	#14581790	C₂²	1	1	yes	characteristic
23	#14581788	C₂²	1	1	yes	characteristic
24	#14581788	C₂²	1	1	yes	characteristic
25	#14581785	—	2	1	no	—
26	#14581785	—	2	1	no	—
27	#14581779	C₂²	1	1	yes	characteristic
28	#14581769	—	2	1	no	—
29	#14581764	—	2	1	no	—
30	#14581764	—	2	1	no	—
31	#14581763	—	2	1	no	—
32	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
35	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
36	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
37	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
38	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
39	#14581748	—	2	1	no	—
40	#14581726	C₂²	1	1	yes	characteristic
41	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	#14581713	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
56	#127×C₈₃	C₂²	1	1	yes	characteristic
57	#124×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	4	4	no	—
60	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂²×Dic₈₃	—	2	1	no	—
68	C₂²×Dic₈₃	—	2	1	no	—
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	#13050579	—	2	2	no	—
72	#13050578	—	4	1	no	—
73	#13050578	—	4	1	no	—
74	#13050577	—	4	1	no	—
75	#13050575	—	4	1	no	—
76	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	—	2	1	no	—
79	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
86	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	2	1	no	—
91	C₂²⋊Dic₈₃	—	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	C₂³	1	1	yes	characteristic
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₈₃	—	2	1	no	—
99	C₂×D₄×C₈₃	—	2	1	no	—
100	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
103	SD₁₆×C₈₃	—	2	2	no	—
104	SD₁₆×C₈₃	—	4	1	no	—
105	D₈×C₈₃	—	2	2	no	—
106	D₈×C₈₃	—	4	1	no	—
107	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
108	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₃	—	2	1	no	—
110	C₂²⋊C₄×C₈₃	—	2	1	no	—
111	C₄²×C₈₃	C₂³	1	1	yes	characteristic
112	C₂³×C₈₃	—	2	1	no	—
113	C₂³×C₈₃	—	2	1	no	—
114	C₂³×C₈₃	—	2	1	no	—
115	C₂²×D₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	Q₈×C₈₃	—	2	1	no	—
118	D₄×C₈₃	—	2	1	no	—
119	D₄×C₈₃	—	2	1	no	—
120	D₄×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	2	1	no	—
122	D₄×C₈₃	—	2	1	no	—
123	D₄×C₈₃	—	2	1	no	—
124	D₄×C₈₃	—	4	2	no	—
125	D₄×C₈₃	—	8	1	no	—
126	C₂×C₄×C₈₃	—	2	1	no	—
127	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₃	—	4	2	no	—
131	C₂²⋊D₈₃	—	4	2	no	—
132	C₂²⋊D₈₃	—	8	1	no	—
133	C₂²⋊D₈₃	—	8	1	no	—
134	C₂×Dic₈₃	—	2	1	no	—
135	C₂×Dic₈₃	—	2	1	no	—
136	C₂×Dic₈₃	—	2	1	no	—
137	C₂×Dic₈₃	—	4	2	no	—
138	C₂×Dic₈₃	—	4	2	no	—
139	C₂×Dic₈₃	—	4	2	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	2	no	—
148	Dic₁₆₆	—	2	1	no	—
149	Dic₁₆₆	—	2	1	no	—
150	C₈×C₈₃	—	2	1	no	—
151	C₈×C₈₃	#41	1	2	yes	complemented
152	C₈₃⋊C₈	—	2	1	no	—
153	C₈₃⋊C₈	—	2	1	no	—
154	C₂²×C₈₃	—	4	1	no	—
155	C₂²×C₈₃	—	4	1	no	—
156	C₂²×C₈₃	—	4	1	no	—
157	C₂²×C₈₃	—	4	1	no	—
158	C₂²×C₈₃	—	4	1	no	—
159	C₂²×C₈₃	—	4	1	no	—
160	C₂²×C₈₃	—	4	1	no	—
161	C₂²×C₈₃	—	4	1	no	—
162	C₂²×C₈₃	—	4	1	no	—
163	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₈₃	—	4	1	no	—
165	C₂×D₈₃	—	4	1	no	—
166	C₂×D₈₃	—	4	1	no	—
167	C₄×C₈₃	—	2	1	no	—
168	C₄×C₈₃	—	2	1	no	—
169	C₄×C₈₃	—	4	1	no	—
170	C₄×C₈₃	C₂×#41	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	—	2	1	no	—
175	Dic₈₃	—	4	1	no	—
176	Dic₈₃	—	4	1	no	—
177	Dic₈₃	—	4	1	no	—
178	Dic₈₃	AGL(1,ℤ/8)	1	2	yes	—
179	C₂×C₈₃	—	8	1	no	—
180	C₂×C₈₃	—	8	1	no	—
181	C₂×C₈₃	—	8	1	no	—
182	C₂×C₈₃	#550	1	1	yes	characteristic
183	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×C₈₃	#577	1	1	yes	characteristic
185	D₈₃	—	8	1	no	—
186	#3397	—	83	1	no	maximal, Sylow
187	C₈₃	#3397	1	1	yes	characteristic, complemented, Sylow
188	#556	—	83	1	no	—
189	#550	—	83	1	no	—
190	#496	—	83	1	no	—
191	#496	D₈₃	1	1	yes	characteristic, complemented
192	#492	—	83	1	no	—
193	#463	—	83	1	no	—
194	#463	—	83	1	no	—
195	#461	—	83	1	no	—
196	#460	—	83	1	no	—
197	#460	—	83	1	no	—
198	#459	—	83	1	no	—
199	#446	—	83	1	no	—
200	#440	—	83	1	no	—
201	#440	—	83	1	no	—
202	C₄×D₈	—	83	1	no	—
203	C₂×#41	—	166	1	no	—
204	C₂×SD₁₆	—	83	1	no	—
205	C₂×SD₁₆	—	83	1	no	—
206	C₂×SD₁₆	—	83	1	no	—
207	C₂×SD₁₆	—	83	1	no	—
208	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
209	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
210	C₂×D₈	—	83	1	no	—
211	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
212	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
213	C₄⋊Q₈	—	83	1	no	—
214	#127	—	83	1	no	—
215	#127	—	83	1	no	—
216	#127	C₂×D₈₃	1	1	yes	characteristic
217	#126	—	166	1	no	—
218	#126	—	166	1	no	—
219	#124	—	83	1	no	—
220	#124	C₂×D₈₃	1	1	yes	characteristic
221	C₂²⋊Q₈	—	166	1	no	—
222	#121	—	166	1	no	—
223	#121	—	166	1	no	—
224	#121	—	166	1	no	—
225	C₄×Q₈	—	83	1	no	—
226	C₄×Q₈	—	83	1	no	—
227	C₄×Q₈	—	83	1	no	—
228	C₄×D₄	—	83	1	no	—
229	C₄×D₄	—	83	1	no	—
230	C₄×D₄	—	83	1	no	—
231	C₄×D₄	—	83	1	no	—
232	#117	—	166	1	no	—
233	#107	—	83	1	no	—
234	C₄⋊C₈	—	83	1	no	—
235	C₄⋊C₈	—	83	1	no	—
236	C₄⋊C₈	—	83	1	no	—
237	#103	—	83	1	no	—
238	#103	—	83	1	no	—
239	#102	—	83	1	no	—
240	#102	—	83	1	no	—
241	#102	—	83	1	no	—
242	#102	—	83	1	no	—
243	#97	—	83	1	no	—
244	#97	C₂×D₈₃	1	1	yes	characteristic
245	C₄×C₈	—	83	1	no	—
246	#41	—	332	4	no	—
247	C₂×Q₈	—	83	1	no	—
248	C₂×Q₈	—	83	1	no	—
249	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	83	1	no	—
253	C₂×D₄	—	166	1	no	—
254	C₂×D₄	—	166	1	no	—
255	C₂×D₄	—	166	1	no	—
256	C₂×D₄	—	166	1	no	—
257	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
259	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
260	C₂²×C₄	—	166	1	no	—
261	C₂²×C₄	—	166	1	no	—
262	C₂²×C₄	—	166	1	no	—
263	C₂²×C₄	—	166	1	no	—
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	4	1	no	—
266	SD₁₆	—	332	1	no	—
267	SD₁₆	—	332	1	no	—
268	SD₁₆	—	332	1	no	—
269	SD₁₆	—	166	2	no	—
270	D₈	—	2	2	no	—
271	D₈	—	4	1	no	—
272	D₈	—	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	—	83	1	no	—
278	C₄⋊C₄	—	83	1	no	—
279	C₄⋊C₄	—	83	1	no	—
280	C₄⋊C₄	—	83	1	no	—
281	C₄⋊C₄	—	83	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	C₂²⋊C₄	—	2	1	no	—
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	166	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	—	166	1	no	—
297	C₄²	—	166	1	no	—
298	C₄²	C₂²×D₈₃	1	1	yes	characteristic
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	2	1	no	—
302	C₂³	—	166	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	166	2	no	—
306	Q₈	—	166	1	no	—
307	Q₈	—	166	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	1	no	—
316	D₄	—	332	2	no	—
317	D₄	—	332	2	no	—
318	D₄	—	664	1	no	—
319	D₄	—	664	1	no	—
320	D₄	—	166	1	no	—
321	D₄	—	166	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	332	2	no	—
324	C₂×C₄	—	332	2	no	—
325	C₂×C₄	—	332	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	D₄×D₈₃	1	1	yes	characteristic
335	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
336	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	166	1	no	—
339	C₈	—	166	1	no	—
340	C₈	#13050589	1	2	yes	complemented
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	332	1	no	—
351	C₂²	—	332	1	no	—
352	C₂²	—	332	1	no	—
353	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	332	1	no	—
358	C₄	—	332	1	no	—
359	C₄	—	332	1	no	—
360	C₄	—	83	2	no	—
361	C₄	—	166	1	no	—
362	C₄	—	166	1	no	—
363	C₄	—	166	1	no	—
364	C₄	C₂×#13050589	1	1	yes	characteristic
365	C₄	#14581837	1	1	yes	characteristic
366	C₂	—	8	1	no	—
367	C₂	—	8	1	no	—
368	C₂	—	8	1	no	—
369	C₂	—	664	1	no	—
370	C₂	#19128371	1	1	yes	characteristic
371	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
372	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	#496×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#496	D₈₃
hypocenter	C₈₃	#3397
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #19128420	C₂	semidirect
2. #19128371	C₂	semidirect
3. #19128089	C₂	semidirect
4. #19128072	C₂	semidirect
5. #19127914	C₂	semidirect
6. #19127913	C₂	semidirect
7. #19127904	C₂	semidirect
8. #19127899	C₂	semidirect
9. #19127898	C₂	semidirect
10. #19127894	C₂	semidirect
11. #19127829	C₂	semidirect
12. #19127803	C₂	semidirect
13. #19127802	C₂	semidirect
14. D₈×Dic₈₃	C₂	semidirect
15. #496×C₈₃	C₂	semidirect
151. C₈×C₈₃	#41	semidirect
187. C₈₃	#3397	semidirect
191. #496	D₈₃	semidirect
340. C₈	#13050589	semidirect

Maximal subgroups

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

subgroup	index
#19128420	2
#19128371	2
#19128089	2
#19128072	2
#19127914	2
#19127913	2
#19127904	2
#19127899	2
#19127898	2
#19127894	2
#19127829	2
#19127803	2
#19127802	2
D₈×Dic₈₃	2
#496×C₈₃	2
#3397	83