Subgroups — #24920539

Order 10624.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128425	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128420	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128094	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128082	C₂	1	1	yes	maximal, characteristic
5	#19127925	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127921	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127918	C₂	1	1	yes	maximal, characteristic
8	#19127913	C₂	1	1	yes	maximal, characteristic
9	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127908	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	Q₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#497×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050579	C₂²	1	1	yes	characteristic
19	C₂×#13050577	C₂²	1	1	yes	characteristic
20	C₂×#13050575	C₂²	1	1	yes	characteristic
21	#14581790	C₂²	1	1	yes	characteristic
22	#14581787	—	2	1	no	—
23	#14581787	—	2	1	no	—
24	#14581785	—	2	1	no	—
25	#14581785	—	2	1	no	—
26	#14581781	C₂²	1	1	yes	characteristic
27	#14581781	C₂²	1	1	yes	characteristic
28	#14581779	C₂²	1	1	yes	characteristic
29	#14581772	—	2	1	no	—
30	#14581769	—	2	1	no	—
31	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
32	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
33	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
36	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
37	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
38	#14581750	—	2	1	no	—
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	#14581716	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581714	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×Q₁₆×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	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
57	#124×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
62	C₂²×Dic₈₃	—	2	1	no	—
63	C₂×C₄×D₈₃	—	2	1	no	—
64	#13050580	—	4	1	no	—
65	#13050580	—	4	1	no	—
66	#13050579	—	2	2	no	—
67	#13050577	—	4	1	no	—
68	#13050575	—	4	1	no	—
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	—	2	1	no	—
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
81	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₈₃	—	2	1	no	—
83	C₂²⋊Dic₈₃	—	2	1	no	—
84	C₂²⋊Dic₈₃	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	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	C₄×Dic₈₃	—	2	1	no	—
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
100	Q₁₆×C₈₃	—	2	2	no	—
101	Q₁₆×C₈₃	—	4	1	no	—
102	SD₁₆×C₈₃	—	2	2	no	—
103	SD₁₆×C₈₃	—	4	1	no	—
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₃	—	2	1	no	—
107	C₄⋊C₄×C₈₃	—	2	1	no	—
108	C₂²⋊C₄×C₈₃	—	2	1	no	—
109	C₂²⋊C₄×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂³×C₈₃	—	2	1	no	—
112	C₂²×D₈₃	—	2	1	no	—
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	—	2	1	no	—
119	D₄×C₈₃	—	2	1	no	—
120	D₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	4	2	no	—
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	—	2	1	no	—
137	C₄×D₈₃	—	4	2	no	—
138	Dic₁₆₆	—	2	1	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	—	4	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈×C₈₃	#41	1	2	yes	—
143	C₈₃⋊C₈	—	2	1	no	—
144	C₈₃⋊C₈	—	2	1	no	—
145	C₂²×C₈₃	—	4	1	no	—
146	C₂²×C₈₃	—	4	1	no	—
147	C₂²×C₈₃	—	4	1	no	—
148	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₈₃	—	4	1	no	—
150	C₂×D₈₃	—	4	1	no	—
151	C₂×D₈₃	—	4	1	no	—
152	C₄×C₈₃	—	2	1	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	4	1	no	—
155	C₄×C₈₃	—	4	1	no	—
156	C₄×C₈₃	—	4	1	no	—
157	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
158	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
159	Dic₈₃	—	2	1	no	—
160	Dic₈₃	—	2	1	no	—
161	Dic₈₃	—	2	1	no	—
162	Dic₈₃	—	2	1	no	—
163	Dic₈₃	—	4	1	no	—
164	Dic₈₃	—	4	1	no	—
165	Dic₈₃	—	4	1	no	—
166	C₂×C₈₃	—	8	1	no	—
167	C₂×C₈₃	#550	1	1	yes	characteristic
168	C₂×C₈₃	#575	1	1	yes	characteristic
169	C₂×C₈₃	#578	1	1	yes	characteristic
170	D₈₃	—	8	1	no	—
171	#3402	—	83	1	no	maximal, Sylow
172	C₈₃	#3402	1	1	yes	characteristic, complemented, Sylow
173	#556	—	83	1	no	—
174	#556	—	83	1	no	—
175	#497	—	83	1	no	—
176	#497	D₈₃	1	1	yes	characteristic, complemented
177	#495	—	83	1	no	—
178	#464	—	83	1	no	—
179	#464	—	83	1	no	—
180	#463	—	83	1	no	—
181	#463	—	83	1	no	—
182	#462	—	83	1	no	—
183	#462	—	83	1	no	—
184	#446	—	83	1	no	—
185	#440	—	83	1	no	—
186	#440	—	83	1	no	—
187	C₄×Q₁₆	—	83	1	no	—
188	C₂×Q₁₆	—	83	1	no	—
189	C₂×Q₁₆	—	83	1	no	—
190	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
191	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
192	C₂×SD₁₆	—	83	1	no	—
193	C₂×SD₁₆	—	83	1	no	—
194	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
195	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
196	C₂×D₈	—	83	1	no	—
197	C₄⋊Q₈	—	83	1	no	—
198	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
199	#126	—	166	1	no	—
200	#126	—	166	1	no	—
201	#126	—	166	1	no	—
202	#126	—	166	1	no	—
203	#124	—	83	1	no	—
204	#124	—	83	1	no	—
205	#124	—	83	1	no	—
206	#124	C₂×D₈₃	1	1	yes	characteristic
207	C₂²⋊Q₈	—	166	1	no	—
208	C₂²⋊Q₈	—	166	1	no	—
209	C₄×Q₈	—	83	1	no	—
210	C₄×Q₈	—	83	1	no	—
211	C₄×Q₈	—	83	1	no	—
212	C₄×Q₈	—	83	1	no	—
213	C₄×Q₈	—	83	1	no	—
214	C₄×D₄	—	83	1	no	—
215	C₄×D₄	—	83	1	no	—
216	#117	—	166	1	no	—
217	#117	—	166	1	no	—
218	#107	—	83	1	no	—
219	C₄⋊C₈	—	83	1	no	—
220	C₄⋊C₈	—	83	1	no	—
221	C₄⋊C₈	—	83	1	no	—
222	#103	—	83	1	no	—
223	#103	—	83	1	no	—
224	#103	—	83	1	no	—
225	#103	—	83	1	no	—
226	#102	—	83	1	no	—
227	#102	—	83	1	no	—
228	#97	—	83	1	no	—
229	#97	C₂×D₈₃	1	1	yes	characteristic
230	C₄×C₈	—	83	1	no	—
231	C₂×Q₈	—	83	1	no	—
232	C₂×Q₈	—	83	1	no	—
233	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
234	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
236	C₂×D₄	—	83	1	no	—
237	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
238	C₂²×C₄	—	166	1	no	—
239	C₂²×C₄	—	166	1	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	4	1	no	—
242	Q₁₆	—	332	1	no	—
243	Q₁₆	—	332	1	no	—
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	4	1	no	—
246	SD₁₆	—	332	1	no	—
247	SD₁₆	—	166	2	no	—
248	D₈	—	332	1	no	—
249	C₂×C₈	—	83	1	no	—
250	C₂×C₈	—	83	1	no	—
251	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	83	1	no	—
256	C₄⋊C₄	—	83	1	no	—
257	C₄⋊C₄	—	83	1	no	—
258	C₄⋊C₄	—	83	1	no	—
259	C₄⋊C₄	—	83	1	no	—
260	C₄⋊C₄	—	166	1	no	—
261	C₄⋊C₄	—	166	1	no	—
262	C₄⋊C₄	—	166	1	no	—
263	C₄⋊C₄	—	166	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	166	1	no	—
269	C₂²⋊C₄	—	166	1	no	—
270	C₂²⋊C₄	—	166	1	no	—
271	C₂²⋊C₄	—	166	1	no	—
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₂²⋊C₄	—	166	1	no	—
276	C₄²	—	83	1	no	—
277	C₄²	—	83	1	no	—
278	C₄²	—	166	1	no	—
279	C₄²	—	166	1	no	—
280	C₄²	—	166	1	no	—
281	C₄²	—	166	1	no	—
282	C₄²	C₂²×D₈₃	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	166	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	332	1	no	—
292	Q₈	—	166	1	no	—
293	Q₈	—	166	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	166	1	no	—
297	D₄	—	166	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	332	2	no	—
302	C₂×C₄	—	332	2	no	—
303	C₂×C₄	—	83	1	no	—
304	C₂×C₄	—	83	1	no	—
305	C₂×C₄	—	83	1	no	—
306	C₂×C₄	—	83	1	no	—
307	C₂×C₄	—	166	1	no	—
308	C₂×C₄	—	166	1	no	—
309	C₂×C₄	—	166	1	no	—
310	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
311	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	166	1	no	—
315	C₈	—	166	1	no	—
316	C₈	#13050589	1	2	yes	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	332	1	no	—
321	C₂²	—	332	1	no	—
322	C₂²	—	332	1	no	—
323	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	332	1	no	—
330	C₄	—	332	1	no	—
331	C₄	—	332	1	no	—
332	C₄	—	166	1	no	—
333	C₄	—	166	1	no	—
334	C₄	—	166	1	no	—
335	C₄	—	166	1	no	—
336	C₄	C₂×#13050589	1	1	yes	characteristic
337	C₄	#14581837	1	1	yes	characteristic
338	C₂	—	8	1	no	—
339	C₂	—	664	1	no	—
340	C₂	#19128371	1	1	yes	characteristic
341	C₂	#19128500	1	1	yes	characteristic
342	C₂	#19128517	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#497×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#13050575	C₂²
hypercenter	#497	D₈₃
hypocenter	C₈₃	#3402
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. #19128425	C₂	semidirect
2. #19128420	C₂	semidirect
3. #19128094	C₂	semidirect
5. #19127925	C₂	semidirect
6. #19127921	C₂	semidirect
9. #19127909	C₂	semidirect
10. #19127908	C₂	semidirect
11. #19127829	C₂	semidirect
12. #19127803	C₂	semidirect
13. #19127802	C₂	semidirect
14. Q₁₆×Dic₈₃	C₂	semidirect
15. #497×C₈₃	C₂	semidirect
172. C₈₃	#3402	semidirect
176. #497	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128425	2
#19128420	2
#19128094	2
#19128082	2
#19127925	2
#19127921	2
#19127918	2
#19127913	2
#19127909	2
#19127908	2
#19127829	2
#19127803	2
#19127802	2
Q₁₆×Dic₈₃	2
#497×C₈₃	2
#3402	83