Subgroups — #24920218

Order 10624.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128436	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128411	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128066	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128061	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128056	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128009	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128002	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127982	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127917	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127917	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127897	C₂	1	1	yes	maximal, characteristic
12	#19127814	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127803	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×D₆₆₄	C₂	1	1	yes	maximal, characteristic, complemented
15	#446×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050579	C₂²	1	1	yes	characteristic
17	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
18	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
19	#14581792	C₂²	1	1	yes	characteristic
20	#14581789	C₂²	1	1	yes	characteristic
21	#14581784	C₂²	1	1	yes	characteristic
22	#14581784	C₂²	1	1	yes	characteristic
23	#14581781	—	2	1	no	—
24	#14581780	C₂²	1	1	yes	characteristic
25	#14581780	C₂²	1	1	yes	characteristic
26	#14581777	—	2	1	no	—
27	#14581777	—	2	1	no	—
28	#14581770	—	2	1	no	—
29	#14581770	—	2	1	no	—
30	#14581768	—	2	1	no	—
31	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
32	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
33	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
34	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
35	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
36	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
37	#14581754	—	2	1	no	—
38	C₄⋊C₄×D₈₃	—	2	1	no	—
39	#14581726	C₂²	1	1	yes	characteristic
40	#14581726	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581714	C₂²	1	1	yes	characteristic
49	#14581714	C₂²	1	1	yes	characteristic
50	#14581714	C₂²	1	1	yes	characteristic
51	#14581714	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
56	#97×C₈₃	C₂²	1	1	yes	characteristic
57	#97×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×D₃₃₂	—	2	1	no	—
62	C₂×D₃₃₂	—	2	1	no	—
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	#13050579	—	2	2	no	—
72	D₆₆₄	—	2	2	no	—
73	D₆₆₄	—	2	2	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	(C₂×D₈₃)⋊C₄	—	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	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
102	C₄²×C₈₃	C₂³	1	1	yes	characteristic
103	C₄²×C₈₃	C₂³	1	1	yes	characteristic
104	C₄²×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²×D₈₃	—	2	1	no	—
106	C₂²×D₈₃	—	2	1	no	—
107	C₂²×D₈₃	—	2	1	no	—
108	Q₈×C₈₃	—	4	1	no	—
109	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₈₃	—	2	1	no	—
117	C₂×Dic₈₃	—	2	1	no	—
118	C₂×Dic₈₃	—	2	1	no	—
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	D₃₃₂	—	2	1	no	—
122	D₃₃₂	—	2	1	no	—
123	D₃₃₂	—	2	1	no	—
124	D₃₃₂	—	2	1	no	—
125	D₃₃₂	—	2	1	no	—
126	D₃₃₂	—	4	2	no	—
127	D₃₃₂	—	8	1	no	—
128	D₃₃₂	#41	1	2	yes	—
129	C₄×D₈₃	—	4	2	no	—
130	C₄×D₈₃	—	8	1	no	—
131	C₄×D₈₃	—	8	1	no	—
132	C₄×D₈₃	—	8	1	no	—
133	C₄×D₈₃	—	8	1	no	—
134	Dic₁₆₆	—	2	1	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	8	1	no	—
137	C₈×C₈₃	—	2	1	no	—
138	C₈×C₈₃	—	2	1	no	—
139	C₈×C₈₃	—	2	1	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₈₃	—	2	2	no	—
143	C₂×D₈₃	—	4	1	no	—
144	C₂×D₈₃	—	4	1	no	—
145	C₂×D₈₃	—	4	1	no	—
146	C₂×D₈₃	—	4	1	no	—
147	C₂×D₈₃	—	4	1	no	—
148	C₂×D₈₃	—	4	1	no	—
149	C₂×D₈₃	—	4	1	no	—
150	C₂×D₈₃	—	4	1	no	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	—	2	1	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
158	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	—	4	1	no	—
161	Dic₈₃	—	4	1	no	—
162	Dic₈₃	—	4	1	no	—
163	Dic₈₃	—	4	1	no	—
164	C₂×C₈₃	#548	1	1	yes	characteristic
165	C₂×C₈₃	#575	1	1	yes	characteristic
166	C₂×C₈₃	#576	1	1	yes	characteristic
167	D₈₃	—	4	2	no	—
168	D₈₃	—	8	1	no	—
169	D₈₃	—	8	1	no	—
170	#3370	—	83	1	no	maximal, Sylow
171	C₈₃	#3370	1	1	yes	characteristic, complemented, Sylow
172	#559	—	83	1	no	—
173	#554	—	83	1	no	—
174	#490	—	83	1	no	—
175	#489	—	83	1	no	—
176	#488	—	83	1	no	—
177	#479	—	83	1	no	—
178	#478	—	83	1	no	—
179	#474	—	83	1	no	—
180	#463	—	83	1	no	—
181	#463	—	83	1	no	—
182	#459	—	83	1	no	—
183	#446	D₈₃	1	1	yes	characteristic, complemented
184	#442	—	83	1	no	—
185	#440	—	83	1	no	—
186	C₄×D₈	—	83	1	no	—
187	C₂×SD₁₆	—	83	1	no	—
188	C₂×D₈	—	83	1	no	—
189	C₂×D₈	—	83	1	no	—
190	C₄⋊Q₈	—	83	1	no	—
191	#127	—	83	1	no	—
192	#125	—	83	1	no	—
193	#125	—	83	1	no	—
194	#124	—	83	1	no	—
195	#124	—	83	1	no	—
196	#124	—	166	1	no	—
197	#123	—	166	1	no	—
198	#123	—	166	1	no	—
199	C₂²⋊Q₈	—	166	1	no	—
200	C₂²⋊Q₈	—	166	1	no	—
201	C₂²⋊Q₈	—	166	1	no	—
202	C₄×Q₈	—	83	1	no	—
203	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
204	C₄×D₄	—	83	1	no	—
205	C₄×D₄	—	83	1	no	—
206	C₄×D₄	—	83	1	no	—
207	C₄×D₄	—	83	1	no	—
208	C₄×D₄	—	83	1	no	—
209	C₄×D₄	—	166	1	no	—
210	C₂×C₄⋊C₄	—	166	1	no	—
211	#107	—	83	1	no	—
212	#107	—	83	1	no	—
213	#106	—	83	1	no	—
214	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
216	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
217	#103	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#102	—	83	1	no	—
221	#102	—	83	1	no	—
222	#102	—	83	1	no	—
223	#102	—	83	1	no	—
224	#102	—	83	1	no	—
225	#102	—	83	1	no	—
226	#102	—	83	1	no	—
227	#97	C₂×D₈₃	1	1	yes	characteristic
228	#97	C₂×D₈₃	1	1	yes	characteristic
229	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
230	C₂×Q₈	—	83	1	no	—
231	C₂×Q₈	—	166	1	no	—
232	C₂×Q₈	D₃₃₂	1	1	yes	characteristic
233	C₂×D₄	—	83	1	no	—
234	C₂×D₄	—	83	1	no	—
235	C₂×D₄	—	83	1	no	—
236	C₂×D₄	—	166	1	no	—
237	C₂×D₄	—	166	1	no	—
238	C₂²×C₄	—	166	1	no	—
239	C₂²×C₄	—	166	1	no	—
240	C₂²×C₄	—	166	1	no	—
241	C₂²×C₄	—	166	1	no	—
242	C₂²×C₄	—	166	1	no	—
243	SD₁₆	—	166	2	no	—
244	D₈	—	166	2	no	—
245	D₈	—	166	2	no	—
246	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
248	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
250	C₄⋊C₄	—	83	1	no	—
251	C₄⋊C₄	—	83	1	no	—
252	C₄⋊C₄	—	83	1	no	—
253	C₄⋊C₄	—	83	1	no	—
254	C₄⋊C₄	—	166	1	no	—
255	C₄⋊C₄	—	166	1	no	—
256	C₄⋊C₄	—	166	1	no	—
257	C₄⋊C₄	—	166	1	no	—
258	C₄⋊C₄	—	166	1	no	—
259	C₄⋊C₄	—	166	1	no	—
260	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
261	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
262	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
263	C₂²⋊C₄	—	166	1	no	—
264	C₂²⋊C₄	—	166	1	no	—
265	C₂²⋊C₄	—	166	1	no	—
266	C₂²⋊C₄	—	166	1	no	—
267	C₂²⋊C₄	—	166	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	C₄²	—	166	1	no	—
273	C₄²	C₂²×D₈₃	1	1	yes	characteristic
274	C₄²	C₂²×D₈₃	1	1	yes	characteristic
275	C₄²	C₂²×D₈₃	1	1	yes	characteristic
276	C₂³	—	166	1	no	—
277	C₂³	—	166	1	no	—
278	C₂³	—	166	1	no	—
279	Q₈	—	4	1	no	—
280	Q₈	—	664	1	no	—
281	Q₈	—	166	1	no	—
282	Q₈	—	166	1	no	—
283	D₄	—	332	2	no	—
284	D₄	—	83	2	no	—
285	D₄	—	664	1	no	—
286	D₄	—	166	1	no	—
287	D₄	—	166	1	no	—
288	D₄	—	166	1	no	—
289	D₄	—	166	1	no	—
290	D₄	—	166	1	no	—
291	C₂×C₄	—	332	2	no	—
292	C₂×C₄	—	664	1	no	—
293	C₂×C₄	—	664	1	no	—
294	C₂×C₄	—	664	1	no	—
295	C₂×C₄	—	664	1	no	—
296	C₂×C₄	—	166	1	no	—
297	C₂×C₄	—	166	1	no	—
298	C₂×C₄	—	166	1	no	—
299	C₂×C₄	—	166	1	no	—
300	C₂×C₄	—	166	1	no	—
301	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
302	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
303	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
304	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
305	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
306	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₂²	—	332	1	no	—
313	C₂²	—	332	1	no	—
314	C₂²	—	332	1	no	—
315	C₂²	—	332	1	no	—
316	C₂²	—	332	1	no	—
317	C₂²	—	332	1	no	—
318	C₂²	—	332	1	no	—
319	C₂²	—	332	1	no	—
320	C₂²	—	166	2	no	—
321	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	332	1	no	—
329	C₄	—	332	1	no	—
330	C₄	—	332	1	no	—
331	C₄	—	332	1	no	—
332	C₄	—	332	1	no	—
333	C₄	#41×D₈₃	1	1	yes	characteristic
334	C₄	#14581840	1	1	yes	characteristic
335	C₂	—	332	2	no	—
336	C₂	—	664	1	no	—
337	C₂	—	664	1	no	—
338	C₂	#19128363	1	1	yes	characteristic
339	C₂	#19128501	1	1	yes	characteristic
340	C₂	#19128506	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#446×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127897	C₂
hypercenter	#446	D₈₃
hypocenter	C₈₃	#3370
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #19128436	C₂	semidirect
2. #19128411	C₂	semidirect
3. #19128066	C₂	semidirect
4. #19128061	C₂	semidirect
5. #19128056	C₂	semidirect
6. #19128009	C₂	semidirect
7. #19128002	C₂	semidirect
8. #19127982	C₂	semidirect
9. #19127917	C₂	semidirect
10. #19127917	C₂	semidirect
12. #19127814	C₂	semidirect
13. #19127803	C₂	semidirect
14. C₄×D₆₆₄	C₂	semidirect
15. #446×C₈₃	C₂	semidirect
171. C₈₃	#3370	semidirect
183. #446	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128436	2
#19128411	2
#19128066	2
#19128061	2
#19128056	2
#19128009	2
#19128002	2
#19127982	2
#19127917	2
#19127897	2
#19127814	2
#19127803	2
C₄×D₆₆₄	2
#446×C₈₃	2
#3370	83