Subgroups — #24917990

Order 10624.

340 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128331	C₂	1	1	yes	maximal, characteristic
2	#19128328	C₂	1	1	yes	maximal, characteristic
3	#19128110	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128109	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128108	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128102	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128100	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128098	C₂	1	1	yes	maximal, characteristic, complemented
9	#19128097	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127742	C₂	1	1	yes	maximal, characteristic
11	#19127740	C₂	1	1	yes	maximal, characteristic
12	#19127738	C₂	1	1	yes	maximal, characteristic
13	#106×D₈₃	C₂	1	1	yes	maximal, characteristic
14	#19127658	C₂	1	1	yes	maximal, characteristic
15	#501×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050573	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
18	#14581792	C₂²	1	1	yes	characteristic
19	#14581791	—	2	1	no	—
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581790	—	2	1	no	—
23	#14581784	C₂²	1	1	yes	characteristic
24	#14581783	C₂²	1	1	yes	characteristic
25	#14581783	C₂²	1	1	yes	characteristic
26	#14581782	—	2	1	no	—
27	#14581782	—	2	1	no	—
28	#14581772	—	2	1	no	—
29	#14581770	—	2	1	no	—
30	#14581770	—	2	1	no	—
31	#14581768	—	2	1	no	—
32	#14581750	C₂²	1	1	yes	characteristic
33	#14581750	C₂²	1	1	yes	characteristic
34	#14581749	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
37	#14581726	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581724	C₂²	1	1	yes	characteristic
43	#14581723	C₂²	1	1	yes	characteristic
44	#14581723	C₂²	1	1	yes	characteristic
45	#14581723	C₂²	1	1	yes	characteristic
46	#14581723	C₂²	1	1	yes	characteristic
47	#14581723	C₂²	1	1	yes	characteristic
48	#14581723	C₂²	1	1	yes	characteristic
49	#14581702	C₂²	1	1	yes	characteristic
50	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	#125×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	#106×C₈₃	C₂²	1	1	yes	characteristic
57	#106×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×C₄×D₈₃	—	2	1	no	—
63	C₂×C₄×D₈₃	—	2	1	no	—
64	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
65	#13050573	Q₈	1	4	yes	complemented
66	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₈×D₈₃	Q₈	1	4	yes	complemented
69	C₄⋊Dic₈₃	—	2	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	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
85	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
89	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
90	C₄×Dic₈₃	—	2	1	no	—
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₈₃	—	2	1	no	—
94	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₃	—	2	1	no	—
97	C₄⋊C₄×C₈₃	—	2	1	no	—
98	C₄⋊C₄×C₈₃	—	2	1	no	—
99	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄²×C₈₃	C₂³	1	1	yes	characteristic
104	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
105	Q₈×C₈₃	—	8	1	no	—
106	C₂×C₄×C₈₃	—	2	1	no	—
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₈₃	—	2	1	no	—
114	C₂×Dic₈₃	—	2	1	no	—
115	C₂×Dic₈₃	—	2	1	no	—
116	C₂×Dic₈₃	—	2	1	no	—
117	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
120	C₄×D₈₃	—	8	1	no	—
121	C₄×D₈₃	—	8	1	no	—
122	C₄×D₈₃	C₂×Q₈	1	2	yes	—
123	C₄×D₈₃	C₂×Q₈	1	2	yes	—
124	Dic₁₆₆	—	8	1	no	—
125	Dic₁₆₆	—	8	1	no	—
126	Dic₁₆₆	—	8	1	no	—
127	C₈×C₈₃	C₂×Q₈	1	2	yes	—
128	C₈×C₈₃	C₂×Q₈	1	2	yes	—
129	C₈₃⋊C₈	C₂×Q₈	1	2	yes	—
130	C₈₃⋊C₈	C₂×Q₈	1	2	yes	—
131	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₈₃	—	2	1	no	—
133	C₂×D₈₃	—	2	1	no	—
134	C₂×D₈₃	C₄⋊Q₈	1	2	yes	—
135	C₄×C₈₃	—	2	1	no	—
136	C₄×C₈₃	—	2	1	no	—
137	C₄×C₈₃	—	4	1	no	—
138	C₄×C₈₃	—	4	1	no	—
139	C₄×C₈₃	—	4	1	no	—
140	C₄×C₈₃	—	4	1	no	—
141	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
142	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
143	Dic₈₃	—	2	1	no	—
144	Dic₈₃	—	2	1	no	—
145	Dic₈₃	—	4	1	no	—
146	Dic₈₃	—	4	1	no	—
147	Dic₈₃	—	4	1	no	—
148	Dic₈₃	—	4	1	no	—
149	Dic₈₃	C₄⋊Q₈	1	2	yes	—
150	C₂×C₈₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₈₃	#577	1	1	yes	characteristic
152	C₂×C₈₃	#577	1	1	yes	characteristic
153	D₈₃	—	2	2	no	—
154	#3163	—	83	1	no	maximal, Sylow
155	C₈₃	#3163	1	1	yes	characteristic, complemented, Sylow
156	#544	—	83	1	no	—
157	#544	—	83	1	no	—
158	#501	—	83	1	no	—
159	#501	—	83	1	no	—
160	#501	—	83	1	no	—
161	#501	D₈₃	1	1	yes	characteristic, complemented
162	#499	—	83	1	no	—
163	#499	—	83	1	no	—
164	#498	—	83	1	no	—
165	#498	—	83	1	no	—
166	#428	—	83	1	no	—
167	#428	—	83	1	no	—
168	#427	—	83	1	no	—
169	C₂×#106	—	83	1	no	—
170	#405	—	83	1	no	—
171	C₂×M₁₆	—	83	1	no	—
172	C₂²×C₈	—	83	1	no	—
173	C₄⋊Q₈	—	83	1	no	—
174	C₄⋊Q₈	—	83	1	no	—
175	C₄⋊Q₈	—	83	1	no	—
176	C₄⋊Q₈	—	166	1	no	—
177	C₄⋊Q₈	—	166	1	no	—
178	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
179	#125	—	83	1	no	—
180	#125	—	83	1	no	—
181	#125	—	83	1	no	—
182	#125	—	166	1	no	—
183	#125	—	166	1	no	—
184	#125	C₂×D₈₃	1	1	yes	characteristic
185	C₂²⋊Q₈	—	166	1	no	—
186	C₂²⋊Q₈	—	166	1	no	—
187	C₂²⋊Q₈	—	166	1	no	—
188	C₂²⋊Q₈	—	166	1	no	—
189	#117	—	83	1	no	—
190	#117	—	83	1	no	—
191	#117	—	83	1	no	—
192	C₂×C₄⋊C₄	—	83	1	no	—
193	C₂×C₄⋊C₄	—	83	1	no	—
194	#107	—	83	1	no	—
195	#107	—	83	1	no	—
196	#107	—	83	1	no	—
197	#107	—	83	1	no	—
198	#107	C₂×D₈₃	1	1	yes	characteristic
199	#107	C₂×D₈₃	1	1	yes	characteristic
200	#106	—	83	1	no	—
201	#106	—	83	1	no	—
202	#106	—	83	1	no	—
203	#106	—	83	1	no	—
204	#106	—	83	1	no	—
205	#106	—	83	1	no	—
206	#106	—	83	1	no	—
207	#106	—	83	1	no	—
208	#106	C₂×D₈₃	1	1	yes	characteristic
209	#106	C₂×D₈₃	1	1	yes	characteristic
210	#97	—	83	1	no	—
211	#97	C₂×D₈₃	1	1	yes	characteristic
212	C₄×C₈	—	83	1	no	—
213	C₄×C₈	—	83	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	166	1	no	—
216	C₂×Q₈	—	166	1	no	—
217	C₂×Q₈	—	166	1	no	—
218	C₂²×C₄	—	83	1	no	—
219	C₂²×C₄	—	166	1	no	—
220	C₂²×C₄	—	166	1	no	—
221	M₁₆	—	83	4	no	—
222	C₂×C₈	—	83	1	no	—
223	C₂×C₈	—	83	1	no	—
224	C₂×C₈	—	83	4	no	—
225	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	83	1	no	—
231	C₄⋊C₄	—	83	1	no	—
232	C₄⋊C₄	—	83	1	no	—
233	C₄⋊C₄	—	83	1	no	—
234	C₄⋊C₄	—	83	1	no	—
235	C₄⋊C₄	—	83	1	no	—
236	C₄⋊C₄	—	166	1	no	—
237	C₄⋊C₄	—	166	1	no	—
238	C₄⋊C₄	—	166	1	no	—
239	C₄⋊C₄	—	166	1	no	—
240	C₄⋊C₄	—	166	1	no	—
241	C₄⋊C₄	—	166	1	no	—
242	C₄⋊C₄	—	166	1	no	—
243	C₄⋊C₄	—	166	1	no	—
244	C₄⋊C₄	—	166	1	no	—
245	C₄⋊C₄	—	166	1	no	—
246	C₄⋊C₄	—	166	1	no	—
247	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
251	C₂²⋊C₄	—	83	1	no	—
252	C₂²⋊C₄	—	83	1	no	—
253	C₂²⋊C₄	—	166	1	no	—
254	C₂²⋊C₄	—	166	1	no	—
255	C₄²	—	83	1	no	—
256	C₄²	—	166	1	no	—
257	C₄²	—	166	1	no	—
258	C₄²	C₂²×D₈₃	1	1	yes	characteristic
259	C₂³	—	83	1	no	—
260	Q₈	—	8	1	no	—
261	Q₈	—	664	1	no	—
262	Q₈	—	664	1	no	—
263	Q₈	—	664	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	83	1	no	—
269	C₂×C₄	—	83	2	no	—
270	C₂×C₄	—	83	2	no	—
271	C₂×C₄	—	83	1	no	—
272	C₂×C₄	—	83	1	no	—
273	C₂×C₄	—	664	1	no	—
274	C₂×C₄	—	664	1	no	—
275	C₂×C₄	—	166	1	no	—
276	C₂×C₄	—	166	1	no	—
277	C₂×C₄	—	166	1	no	—
278	C₂×C₄	—	166	1	no	—
279	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
280	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
282	C₈	—	83	2	no	—
283	C₈	—	83	2	no	—
284	C₈	Q₈×D₈₃	1	2	yes	—
285	C₈	Q₈×D₈₃	1	2	yes	—
286	C₂²	—	83	2	no	—
287	C₂²	—	166	1	no	—
288	C₂²	—	166	1	no	—
289	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	332	1	no	—
297	C₄	—	332	1	no	—
298	C₄	—	332	1	no	—
299	C₄	—	332	1	no	—
300	C₄	—	83	2	no	—
301	C₄	—	166	1	no	—
302	C₄	—	166	1	no	—
303	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
304	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
305	C₂	—	166	2	no	—
306	C₂	C₄⋊Q₈×D₈₃	1	1	yes	characteristic
307	C₂	#19128510	1	1	yes	characteristic
308	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	#501×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂²×D₈₃	C₂×D₄
hypercenter	#501	D₈₃
hypocenter	C₈₃	#3163
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
3. #19128110	C₂	semidirect
4. #19128109	C₂	semidirect
5. #19128108	C₂	semidirect
6. #19128102	C₂	semidirect
7. #19128100	C₂	semidirect
8. #19128098	C₂	semidirect
9. #19128097	C₂	semidirect
15. #501×C₈₃	C₂	semidirect
65. #13050573	Q₈	semidirect
68. C₈×D₈₃	Q₈	semidirect
155. C₈₃	#3163	semidirect
161. #501	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128331	2
#19128328	2
#19128110	2
#19128109	2
#19128108	2
#19128102	2
#19128100	2
#19128098	2
#19128097	2
#19127742	2
#19127740	2
#19127738	2
#106×D₈₃	2
#19127658	2
#501×C₈₃	2
#3163	83