Subgroups — #24918887

Order 10624.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128323	C₂	1	1	yes	maximal, characteristic
2	#19128312	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128008	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128002	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127994	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127984	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127956	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127950	C₂	1	1	yes	maximal, characteristic
9	#19127939	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127933	C₂	1	1	yes	maximal, characteristic
11	#19127925	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127925	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127903	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127726	C₂	1	1	yes	maximal, characteristic, complemented
15	#462×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050577	C₂²	1	1	yes	characteristic
17	C₂×#13050577	C₂²	1	1	yes	characteristic
18	C₂×#13050573	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581787	—	2	1	no	—
22	#14581787	—	2	1	no	—
23	#14581786	—	2	1	no	—
24	#14581784	C₂²	1	1	yes	characteristic
25	#14581782	—	2	1	no	—
26	#14581782	—	2	1	no	—
27	#14581781	C₂²	1	1	yes	characteristic
28	#14581780	—	2	1	no	—
29	#14581777	—	2	1	no	—
30	#14581770	—	2	1	no	—
31	#14581768	C₂²	1	1	yes	characteristic
32	#14581768	C₂²	1	1	yes	characteristic
33	#14581768	C₂²	1	1	yes	characteristic
34	#14581767	C₂²	1	1	yes	characteristic
35	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
36	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
37	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
38	#14581749	C₂²	1	1	yes	characteristic
39	#14581726	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
46	#14581716	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581716	C₂²	1	1	yes	characteristic
49	#14581716	C₂²	1	1	yes	characteristic
50	#14581714	C₂²	1	1	yes	characteristic
51	#14581714	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#103×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×D₃₃₂	—	2	1	no	—
61	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₈₃	—	2	1	no	—
63	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
64	#13050577	—	4	1	no	—
65	#13050577	—	4	1	no	—
66	#13050573	—	4	1	no	—
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₈×D₈₃	—	4	1	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
82	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
83	(C₂×D₈₃)⋊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	D₄	1	1	yes	characteristic
91	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	Q₁₆×C₈₃	—	4	1	no	—
99	Q₁₆×C₈₃	—	4	1	no	—
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	—	2	1	no	—
103	C₄⋊C₄×C₈₃	—	2	1	no	—
104	C₄⋊C₄×C₈₃	—	2	1	no	—
105	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
106	C₄²×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂²×D₈₃	—	2	1	no	—
109	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	2	1	no	—
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	2	1	no	—
115	Q₈×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	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	D₃₃₂	—	2	1	no	—
130	D₃₃₂	—	2	1	no	—
131	D₃₃₂	—	8	1	no	—
132	C₄×D₈₃	—	2	1	no	—
133	C₄×D₈₃	—	2	1	no	—
134	C₄×D₈₃	—	8	1	no	—
135	Dic₁₆₆	—	8	1	no	—
136	C₈×C₈₃	—	2	1	no	—
137	C₈×C₈₃	—	2	1	no	—
138	C₈₃⋊C₈	—	2	1	no	—
139	C₈₃⋊C₈	—	2	1	no	—
140	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₈₃	—	2	1	no	—
142	C₂×D₈₃	—	2	1	no	—
143	C₂×D₈₃	—	2	1	no	—
144	C₂×D₈₃	—	4	1	no	—
145	C₂×D₈₃	—	4	1	no	—
146	C₂×D₈₃	—	4	1	no	—
147	C₄×C₈₃	—	2	1	no	—
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	4	1	no	—
150	C₄×C₈₃	—	4	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	—	4	1	no	—
153	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
154	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₈₃	—	2	1	no	—
156	Dic₈₃	—	2	1	no	—
157	Dic₈₃	—	2	1	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	—	4	1	no	—
161	C₂×C₈₃	#537	1	1	yes	characteristic
162	C₂×C₈₃	#577	1	1	yes	characteristic
163	C₂×C₈₃	#578	1	1	yes	characteristic
164	D₈₃	—	4	1	no	—
165	D₈₃	—	8	1	no	—
166	#3258	—	83	1	no	maximal, Sylow
167	C₈₃	#3258	1	1	yes	characteristic, complemented, Sylow
168	#542	—	83	1	no	—
169	#541	—	83	1	no	—
170	#479	—	83	1	no	—
171	#478	—	83	1	no	—
172	#477	—	83	1	no	—
173	#475	—	83	1	no	—
174	#470	—	83	1	no	—
175	#468	—	83	1	no	—
176	#467	—	83	1	no	—
177	#465	—	83	1	no	—
178	#464	—	83	1	no	—
179	#464	—	83	1	no	—
180	#462	D₈₃	1	1	yes	characteristic, complemented
181	#461	—	83	1	no	—
182	#424	—	83	1	no	—
183	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
184	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
185	C₂×SD₁₆	—	83	1	no	—
186	C₂×SD₁₆	—	83	1	no	—
187	C₂×M₁₆	—	83	1	no	—
188	C₂²×C₈	—	83	1	no	—
189	C₄⋊Q₈	—	83	1	no	—
190	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
191	#126	—	166	1	no	—
192	#126	—	166	1	no	—
193	#126	—	166	1	no	—
194	#125	—	83	1	no	—
195	#125	—	166	1	no	—
196	#125	—	166	1	no	—
197	#124	—	83	1	no	—
198	#124	—	166	1	no	—
199	#123	—	166	1	no	—
200	C₂²⋊Q₈	—	83	1	no	—
201	C₂²⋊Q₈	—	83	1	no	—
202	C₂²⋊Q₈	—	83	1	no	—
203	C₂²⋊Q₈	—	166	1	no	—
204	#121	—	83	1	no	—
205	C₄×Q₈	—	83	1	no	—
206	C₄×Q₈	—	83	1	no	—
207	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
208	C₄×D₄	—	83	1	no	—
209	#117	—	83	1	no	—
210	#107	—	83	1	no	—
211	#107	—	83	1	no	—
212	#106	—	83	1	no	—
213	#106	—	83	1	no	—
214	C₄⋊C₈	—	83	1	no	—
215	C₄⋊C₈	—	83	1	no	—
216	C₄⋊C₈	—	83	1	no	—
217	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#103	—	83	1	no	—
221	#103	—	83	1	no	—
222	#103	C₂×D₈₃	1	1	yes	characteristic
223	#103	C₂×D₈₃	1	1	yes	characteristic
224	#102	—	83	1	no	—
225	#102	—	83	1	no	—
226	C₂×Q₈	—	166	1	no	—
227	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
228	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
229	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
230	C₂×D₄	—	83	1	no	—
231	C₂×D₄	—	166	1	no	—
232	C₂²×C₄	—	83	1	no	—
233	C₂²×C₄	—	166	1	no	—
234	Q₁₆	—	4	1	no	—
235	Q₁₆	—	4	1	no	—
236	SD₁₆	—	332	1	no	—
237	SD₁₆	—	332	1	no	—
238	M₁₆	—	332	1	no	—
239	C₂×C₈	—	332	1	no	—
240	C₂×C₈	—	83	1	no	—
241	C₂×C₈	—	83	1	no	—
242	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄⋊C₄	—	83	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	83	1	no	—
251	C₄⋊C₄	—	83	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	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	C₂²×D₈₃	1	1	yes	characteristic
261	C₂²⋊C₄	—	83	1	no	—
262	C₂²⋊C₄	—	83	1	no	—
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	C₄²	—	2	1	no	—
271	C₄²	—	83	1	no	—
272	C₄²	—	166	1	no	—
273	C₄²	—	166	1	no	—
274	C₄²	C₂²×D₈₃	1	1	yes	characteristic
275	C₂³	—	83	1	no	—
276	C₂³	—	166	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	664	1	no	—
284	D₄	—	664	1	no	—
285	D₄	—	166	1	no	—
286	D₄	—	166	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	83	1	no	—
292	C₂×C₄	—	83	1	no	—
293	C₂×C₄	—	83	1	no	—
294	C₂×C₄	—	664	1	no	—
295	C₂×C₄	—	166	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	D₄×D₈₃	1	1	yes	characteristic
301	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	166	1	no	—
306	C₈	—	166	1	no	—
307	C₂²	—	332	1	no	—
308	C₂²	—	332	1	no	—
309	C₂²	—	332	1	no	—
310	C₂²	—	166	1	no	—
311	C₂²	—	166	1	no	—
312	C₂²	—	166	1	no	—
313	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	332	1	no	—
321	C₄	—	332	1	no	—
322	C₄	—	332	1	no	—
323	C₄	—	166	1	no	—
324	C₄	—	166	1	no	—
325	C₄	—	166	1	no	—
326	C₄	#14581837	1	1	yes	characteristic
327	C₄	#14581839	1	1	yes	characteristic
328	C₂	—	332	1	no	—
329	C₂	—	664	1	no	—
330	C₂	#19128280	1	1	yes	characteristic
331	C₂	#19128510	1	1	yes	characteristic
332	C₂	#19128514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#462×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#14581767	C₂²
hypercenter	#462	D₈₃
hypocenter	C₈₃	#3258
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
2. #19128312	C₂	semidirect
3. #19128008	C₂	semidirect
4. #19128002	C₂	semidirect
5. #19127994	C₂	semidirect
6. #19127984	C₂	semidirect
7. #19127956	C₂	semidirect
9. #19127939	C₂	semidirect
11. #19127925	C₂	semidirect
12. #19127925	C₂	semidirect
13. #19127903	C₂	semidirect
14. #19127726	C₂	semidirect
15. #462×C₈₃	C₂	semidirect
167. C₈₃	#3258	semidirect
180. #462	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128323	2
#19128312	2
#19128008	2
#19128002	2
#19127994	2
#19127984	2
#19127956	2
#19127950	2
#19127939	2
#19127933	2
#19127925	2
#19127903	2
#19127726	2
#462×C₈₃	2
#3258	83