Subgroups — #24917521

Order 10624.

434 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128248	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13050591	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128027	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128023	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127994	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127899	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127895	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127849	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127845	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13050577	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×D₆₆₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13050575	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127752	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×#13050590	—	2	1	no	—
18	C₂×#13050577	C₂²	1	1	yes	characteristic
19	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
20	C₂×#13050575	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581788	C₂²	1	1	yes	characteristic
23	#14581778	—	2	1	no	—
24	#14581767	C₂²	1	1	yes	characteristic
25	#14581765	—	2	1	no	—
26	#14581763	C₂²	1	1	yes	characteristic
27	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
28	C₄×D₃₃₂	—	2	1	no	—
29	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
30	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
31	#14581754	—	2	1	no	—
32	#14581754	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₃	—	2	1	no	—
34	#14581750	—	2	1	no	—
35	#14581750	C₂²	1	1	yes	characteristic
36	C₄²×D₈₃	—	2	1	no	—
37	C₄²×D₈₃	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	#14581716	C₂²	1	1	yes	characteristic
44	#14581714	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	#106×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050591	—	2	2	no	—
60	#13050591	—	2	2	no	—
61	#13050590	—	4	4	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	—	2	1	no	—
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₂²⋊D₈₃	—	2	1	no	—
69	C₂×C₂²⋊D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	C₂×C₄×D₈₃	—	2	1	no	—
74	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
75	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
76	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
77	#13050577	—	2	2	no	—
78	D₆₆₄	—	2	2	no	—
79	#13050575	—	2	2	no	—
80	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₃	—	2	1	no	—
83	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
85	Dic₈₃⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	D₄	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	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	—	2	1	no	—
100	SD₁₆×C₈₃	—	2	2	no	—
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	—	2	1	no	—
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	—	2	1	no	—
108	C₄²×C₈₃	C₂³	1	1	yes	characteristic
109	C₂³×C₈₃	—	2	1	no	—
110	C₂²×D₈₃	—	2	1	no	—
111	C₂²×D₈₃	—	2	1	no	—
112	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	#41	1	2	yes	complemented
115	D₄×C₈₃	—	2	1	no	—
116	D₄×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	—	4	2	no	—
121	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₈₃	—	4	2	no	—
125	C₂²⋊D₈₃	—	4	2	no	—
126	C₂²⋊D₈₃	—	4	2	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	D₃₃₂	—	2	4	no	—
133	D₃₃₂	—	2	1	no	—
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	—	4	2	no	—
137	D₃₃₂	#41	1	2	yes	complemented
138	C₄×D₈₃	—	2	4	no	—
139	C₄×D₈₃	—	2	2	no	—
140	C₄×D₈₃	—	2	1	no	—
141	C₄×D₈₃	—	2	1	no	—
142	C₄×D₈₃	—	2	2	no	—
143	C₄×D₈₃	—	4	2	no	—
144	C₄×D₈₃	—	4	2	no	—
145	C₄×D₈₃	—	4	2	no	—
146	Dic₁₆₆	—	4	2	no	—
147	C₈×C₈₃	—	2	1	no	—
148	C₈×C₈₃	—	2	1	no	—
149	C₈₃⋊C₈	—	2	1	no	—
150	C₈₃⋊C₈	—	2	1	no	—
151	C₂²×C₈₃	—	4	1	no	—
152	C₂²×C₈₃	—	4	1	no	—
153	C₂²×C₈₃	—	4	1	no	—
154	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₈₃	—	2	1	no	—
156	C₂×D₈₃	—	2	2	no	—
157	C₂×D₈₃	—	2	1	no	—
158	C₂×D₈₃	—	2	1	no	—
159	C₂×D₈₃	—	4	1	no	—
160	C₂×D₈₃	—	4	1	no	—
161	C₂×D₈₃	—	4	1	no	—
162	C₂×D₈₃	—	4	1	no	—
163	C₂×D₈₃	—	4	1	no	—
164	C₄×C₈₃	—	2	2	no	—
165	C₄×C₈₃	—	2	1	no	—
166	C₄×C₈₃	—	4	1	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
169	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
170	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	4	1	no	—
174	Dic₈₃	—	4	1	no	—
175	Dic₈₃	#135	1	2	yes	—
176	C₂×C₈₃	—	8	1	no	—
177	C₂×C₈₃	#525	1	1	yes	characteristic
178	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
179	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	D₈₃	—	4	1	no	—
181	D₈₃	—	4	2	no	—
182	D₈₃	—	8	1	no	—
183	#3106	—	83	1	no	maximal, Sylow
184	C₈₃	#3106	1	1	yes	characteristic, complemented, Sylow
185	#532	—	83	1	no	—
186	C₄×#41	—	83	1	no	—
187	#482	—	83	1	no	—
188	#482	—	83	1	no	—
189	#477	—	83	1	no	—
190	#476	—	83	1	no	—
191	#460	—	83	1	no	—
192	#459	—	83	1	no	—
193	#449	—	83	1	no	—
194	#449	—	83	1	no	—
195	C₄×SD₁₆	—	83	1	no	—
196	C₄×SD₁₆	D₈₃	1	1	yes	characteristic, complemented
197	C₄×D₈	—	83	1	no	—
198	C₄×D₈	—	83	1	no	—
199	#431	—	83	1	no	—
200	C₂×#41	—	83	1	no	—
201	C₂×#41	—	166	1	no	—
202	C₂×SD₁₆	—	83	1	no	—
203	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
204	C₂×D₈	—	83	1	no	—
205	C₂×D₈	—	83	1	no	—
206	C₄⋊Q₈	—	83	1	no	—
207	#127	—	83	1	no	—
208	#124	—	166	1	no	—
209	#121	—	83	1	no	—
210	#121	—	83	1	no	—
211	#121	—	166	1	no	—
212	C₄×Q₈	—	83	1	no	—
213	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
214	C₄×D₄	—	83	1	no	—
215	C₄×D₄	—	83	1	no	—
216	C₄×D₄	—	83	1	no	—
217	C₄×D₄	—	166	1	no	—
218	C₄×D₄	—	166	1	no	—
219	C₄×D₄	—	166	1	no	—
220	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
221	#117	—	83	1	no	—
222	#117	—	166	1	no	—
223	C₂×C₄²	—	83	1	no	—
224	C₂×C₄²	—	166	1	no	—
225	#107	—	83	1	no	—
226	#107	—	83	1	no	—
227	#106	—	83	1	no	—
228	#106	C₂×D₈₃	1	1	yes	characteristic
229	C₄⋊C₈	—	83	1	no	—
230	C₄⋊C₈	—	83	1	no	—
231	#103	—	83	1	no	—
232	#103	C₂×D₈₃	1	1	yes	characteristic
233	#102	—	83	1	no	—
234	#102	—	83	1	no	—
235	#102	—	83	1	no	—
236	#102	—	83	1	no	—
237	#102	—	83	1	no	—
238	#102	C₂×D₈₃	1	1	yes	characteristic
239	C₂²⋊C₈	—	83	1	no	—
240	C₂²⋊C₈	—	83	1	no	—
241	C₄×C₈	—	83	1	no	—
242	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
243	#41	—	332	4	no	—
244	#41	—	166	2	no	—
245	#41	—	166	2	no	—
246	C₂×Q₈	—	166	1	no	—
247	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
248	C₂×D₄	—	83	1	no	—
249	C₂×D₄	—	83	1	no	—
250	C₂×D₄	—	166	1	no	—
251	C₂×D₄	—	166	1	no	—
252	C₂×D₄	—	166	1	no	—
253	C₂×D₄	—	166	1	no	—
254	C₂×D₄	—	166	1	no	—
255	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	83	1	no	—
258	C₂²×C₄	—	83	1	no	—
259	C₂²×C₄	—	83	1	no	—
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	—	166	2	no	—
266	D₈	—	166	2	no	—
267	D₈	—	166	2	no	—
268	C₂×C₈	—	83	1	no	—
269	C₂×C₈	—	83	1	no	—
270	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	83	1	no	—
274	C₄⋊C₄	—	83	1	no	—
275	C₄⋊C₄	—	166	1	no	—
276	C₄⋊C₄	—	166	1	no	—
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	—	166	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	C₂²⋊C₄	—	166	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	83	1	no	—
288	C₄²	—	83	1	no	—
289	C₄²	—	83	1	no	—
290	C₄²	—	166	1	no	—
291	C₄²	—	166	1	no	—
292	C₄²	C₂²×D₈₃	1	1	yes	characteristic
293	C₂³	—	2	1	no	—
294	C₂³	—	83	1	no	—
295	C₂³	—	166	1	no	—
296	C₂³	—	166	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	332	2	no	—
299	Q₈	#13050590	1	2	yes	complemented
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	332	2	no	—
303	D₄	—	332	2	no	—
304	D₄	—	332	2	no	—
305	D₄	—	332	2	no	—
306	D₄	—	83	2	no	—
307	D₄	—	166	4	no	—
308	D₄	—	166	1	no	—
309	D₄	—	166	1	no	—
310	D₄	—	166	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	332	2	no	—
316	C₂×C₄	—	332	2	no	—
317	C₂×C₄	—	332	2	no	—
318	C₂×C₄	—	83	1	no	—
319	C₂×C₄	—	83	1	no	—
320	C₂×C₄	—	83	1	no	—
321	C₂×C₄	—	166	2	no	—
322	C₂×C₄	—	166	4	no	—
323	C₂×C₄	—	166	1	no	—
324	C₂×C₄	—	166	1	no	—
325	C₂×C₄	—	166	2	no	—
326	C₂×C₄	—	166	1	no	—
327	C₂×C₄	—	166	1	no	—
328	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
329	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	166	1	no	—
334	C₈	—	166	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	332	1	no	—
339	C₂²	—	332	1	no	—
340	C₂²	—	332	1	no	—
341	C₂²	—	332	1	no	—
342	C₂²	—	332	1	no	—
343	C₂²	—	166	2	no	—
344	C₂²	—	166	1	no	—
345	C₂²	—	166	1	no	—
346	C₂²	—	166	1	no	—
347	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	332	1	no	—
353	C₄	—	332	1	no	—
354	C₄	—	83	2	no	—
355	C₄	—	166	1	no	—
356	C₄	—	166	1	no	—
357	C₄	#14581819	1	2	yes	—
358	C₄	#41×D₈₃	1	1	yes	characteristic
359	C₄	C₂×#13050590	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	332	2	no	—
362	C₂	—	332	1	no	—
363	C₂	—	664	1	no	—
364	C₂	#19128210	1	1	yes	characteristic
365	C₂	#135×D₈₃	1	1	yes	characteristic
366	C₂	C₂×#14581819	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₄×SD₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	C₄×SD₁₆	D₈₃
hypocenter	C₈₃	#3106
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #19128248	C₂	semidirect
2. C₄×#13050591	C₂	semidirect
3. #19128027	C₂	semidirect
4. #19128023	C₂	semidirect
5. #19127994	C₂	semidirect
6. #19127990	C₂	semidirect
7. #19127899	C₂	semidirect
8. #19127895	C₂	semidirect
9. #19127849	C₂	semidirect
10. #19127845	C₂	semidirect
11. C₄×#13050577	C₂	semidirect
12. C₄×D₆₆₄	C₂	semidirect
13. C₄×#13050575	C₂	semidirect
14. #19127752	C₂	semidirect
15. C₄×SD₁₆×C₈₃	C₂	semidirect
114. Q₈×C₈₃	#41	semidirect
137. D₃₃₂	#41	semidirect
184. C₈₃	#3106	semidirect
196. C₄×SD₁₆	D₈₃	semidirect
299. Q₈	#13050590	semidirect

Maximal subgroups

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

subgroup	index
#19128248	2
C₄×#13050591	2
#19128027	2
#19128023	2
#19127994	2
#19127990	2
#19127899	2
#19127895	2
#19127849	2
#19127845	2
C₄×#13050577	2
C₄×D₆₆₄	2
C₄×#13050575	2
#19127752	2
C₄×SD₁₆×C₈₃	2
#3106	83