Subgroups — #24920402

Order 10624.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128411	C₂	1	1	yes	maximal, characteristic
2	#19128111	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128107	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128100	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128008	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128008	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127998	C₂	1	1	yes	maximal, characteristic
8	#19127998	C₂	1	1	yes	maximal, characteristic
9	#19127988	C₂	1	1	yes	maximal, characteristic
10	#19127986	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127832	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127799	C₂	1	1	yes	maximal, characteristic
13	#19127799	C₂	1	1	yes	maximal, characteristic
14	C₄×#13050577	C₂	1	1	yes	maximal, characteristic
15	#557×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050577	C₂²	1	1	yes	characteristic
17	#14581792	C₂²	1	1	yes	characteristic
18	#14581784	C₂²	1	1	yes	characteristic
19	#14581784	C₂²	1	1	yes	characteristic
20	#14581770	—	2	1	no	—
21	#14581770	—	2	1	no	—
22	#14581768	—	2	1	no	—
23	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
24	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
25	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×D₈₃	—	2	1	no	—
27	#14581725	C₂²	1	1	yes	characteristic
28	#14581725	C₂²	1	1	yes	characteristic
29	#14581725	C₂²	1	1	yes	characteristic
30	#14581725	C₂²	1	1	yes	characteristic
31	#14581723	C₂²	1	1	yes	characteristic
32	#14581723	C₂²	1	1	yes	characteristic
33	#14581723	C₂²	1	1	yes	characteristic
34	#14581723	C₂²	1	1	yes	characteristic
35	#14581723	C₂²	1	1	yes	characteristic
36	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
37	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
39	#14581716	C₂²	1	1	yes	characteristic
40	#14581716	C₂²	1	1	yes	characteristic
41	#14581716	C₂²	1	1	yes	characteristic
42	#14581712	C₂²	1	1	yes	characteristic
43	#14581712	C₂²	1	1	yes	characteristic
44	#14581712	C₂²	1	1	yes	characteristic
45	#14581703	C₂²	1	1	yes	characteristic
46	#14581703	C₂²	1	1	yes	characteristic
47	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₃	—	2	1	no	—
49	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
50	#125×C₈₃	—	2	1	no	—
51	#125×C₈₃	—	2	1	no	—
52	#125×C₈₃	C₂²	1	1	yes	characteristic
53	#125×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₃	—	2	1	no	—
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₈₃	—	2	1	no	—
62	C₂×C₄×D₈₃	—	2	1	no	—
63	C₂×C₄×D₈₃	—	2	1	no	—
64	#13050577	Q₈	1	4	yes	complemented
65	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₈₃	—	2	1	no	—
70	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
73	Dic₈₃⋊C₄	—	2	1	no	—
74	Dic₈₃⋊C₄	—	2	1	no	—
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	(C₂×D₈₃)⋊C₄	—	2	1	no	—
78	(C₂×D₈₃)⋊C₄	—	2	1	no	—
79	(C₂×D₈₃)⋊C₄	—	2	1	no	—
80	C₂×Q₈×C₈₃	—	2	1	no	—
81	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
82	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₈₃	—	2	1	no	—
84	C₄⋊C₄×C₈₃	—	2	1	no	—
85	C₄⋊C₄×C₈₃	—	2	1	no	—
86	C₄⋊C₄×C₈₃	—	2	1	no	—
87	C₄⋊C₄×C₈₃	—	2	1	no	—
88	C₄⋊C₄×C₈₃	—	2	1	no	—
89	C₄⋊C₄×C₈₃	—	2	1	no	—
90	C₄⋊C₄×C₈₃	—	2	1	no	—
91	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₄²×C₈₃	—	2	1	no	—
98	C₄²×C₈₃	—	2	1	no	—
99	C₄²×C₈₃	—	2	1	no	—
100	C₄²×C₈₃	C₂³	1	1	yes	characteristic
101	C₄²×C₈₃	C₂³	1	1	yes	characteristic
102	C₄²×C₈₃	C₂³	1	1	yes	characteristic
103	C₂²×D₈₃	—	2	1	no	—
104	Q₈×C₈₃	—	2	1	no	—
105	Q₈×C₈₃	—	4	1	no	—
106	Q₈×C₈₃	—	4	2	no	—
107	Q₈×C₈₃	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	D₃₃₂	—	2	1	no	—
123	D₃₃₂	C₂×Q₈	1	2	yes	—
124	C₄×D₈₃	—	4	2	no	—
125	C₄×D₈₃	—	8	1	no	—
126	C₄×D₈₃	—	8	1	no	—
127	Dic₁₆₆	—	8	1	no	—
128	C₈₃⋊C₈	—	2	1	no	—
129	C₈₃⋊C₈	—	2	1	no	—
130	C₈₃⋊C₈	—	2	1	no	—
131	C₈₃⋊C₈	C₂×Q₈	1	2	yes	—
132	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₈₃	—	2	2	no	—
134	C₂×D₈₃	—	4	1	no	—
135	C₂×D₈₃	—	4	1	no	—
136	C₄×C₈₃	—	2	2	no	—
137	C₄×C₈₃	—	2	1	no	—
138	C₄×C₈₃	—	2	1	no	—
139	C₄×C₈₃	—	2	1	no	—
140	C₄×C₈₃	—	2	1	no	—
141	C₄×C₈₃	—	2	1	no	—
142	C₄×C₈₃	—	2	1	no	—
143	C₄×C₈₃	—	4	1	no	—
144	C₄×C₈₃	—	4	1	no	—
145	C₄×C₈₃	—	4	1	no	—
146	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₈₃	—	4	1	no	—
149	Dic₈₃	—	4	1	no	—
150	Dic₈₃	—	4	1	no	—
151	C₂×C₈₃	D₄×Q₈	1	1	yes	characteristic
152	C₂×C₈₃	#575	1	1	yes	characteristic
153	C₂×C₈₃	#577	1	1	yes	characteristic
154	D₈₃	—	4	2	no	—
155	#3388	—	83	1	no	maximal, Sylow
156	C₈₃	#3388	1	1	yes	characteristic, complemented, Sylow
157	#557	D₈₃	1	1	yes	characteristic, complemented
158	#554	—	83	1	no	—
159	#501	—	83	1	no	—
160	#501	—	83	1	no	—
161	#499	—	83	1	no	—
162	#479	—	83	1	no	—
163	#479	—	83	1	no	—
164	#478	—	83	1	no	—
165	#478	—	83	1	no	—
166	#476	—	83	1	no	—
167	#475	—	83	1	no	—
168	#446	—	83	1	no	—
169	#440	—	83	1	no	—
170	#440	—	83	1	no	—
171	C₄×SD₁₆	—	83	1	no	—
172	C₂×SD₁₆	—	83	1	no	—
173	C₄⋊Q₈	—	2	1	no	—
174	C₄⋊Q₈	—	83	1	no	—
175	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
176	#125	—	2	1	no	—
177	#125	—	2	1	no	—
178	#125	—	83	1	no	—
179	#125	—	83	1	no	—
180	#125	C₂×D₈₃	1	1	yes	characteristic
181	#125	C₂×D₈₃	1	1	yes	characteristic
182	C₂²⋊Q₈	—	166	1	no	—
183	C₂²⋊Q₈	—	166	1	no	—
184	C₂²⋊Q₈	—	166	1	no	—
185	C₄×Q₈	—	2	1	no	—
186	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
187	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
188	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
189	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
190	C₄×D₄	—	83	1	no	—
191	C₄×D₄	—	83	1	no	—
192	C₄×D₄	—	83	1	no	—
193	C₂×C₄⋊C₄	—	166	1	no	—
194	#107	—	83	1	no	—
195	#107	—	83	1	no	—
196	#107	—	83	1	no	—
197	#107	—	83	1	no	—
198	#106	—	83	1	no	—
199	#106	—	83	1	no	—
200	#106	—	83	1	no	—
201	#106	—	83	1	no	—
202	#106	—	83	1	no	—
203	C₄⋊C₈	—	83	1	no	—
204	C₄⋊C₈	—	83	1	no	—
205	C₄⋊C₈	—	83	1	no	—
206	#103	—	83	1	no	—
207	#103	—	83	1	no	—
208	#103	—	83	1	no	—
209	#102	—	83	1	no	—
210	#102	—	83	1	no	—
211	#102	—	83	1	no	—
212	#97	—	83	1	no	—
213	#97	—	83	1	no	—
214	C₄×C₈	—	83	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	166	1	no	—
217	C₂×Q₈	C₂²⋊D₈₃	1	1	yes	characteristic
218	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
219	C₂×D₄	—	83	1	no	—
220	C₂²×C₄	—	166	1	no	—
221	C₂²×C₄	—	166	1	no	—
222	C₂²×C₄	—	166	1	no	—
223	SD₁₆	—	83	4	no	—
224	C₂×C₈	—	83	1	no	—
225	C₂×C₈	—	83	1	no	—
226	C₂×C₈	—	83	1	no	—
227	C₂×C₈	—	83	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	83	1	no	—
237	C₄⋊C₄	—	83	1	no	—
238	C₄⋊C₄	—	83	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	C₂²⋊D₈₃	1	1	yes	characteristic
245	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
246	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
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	C₂²⋊C₄	—	166	1	no	—
251	C₂²⋊C₄	—	166	1	no	—
252	C₂²⋊C₄	—	166	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	C₂²×D₈₃	1	1	yes	characteristic
257	C₄²	C₂²×D₈₃	1	1	yes	characteristic
258	C₄²	C₂²×D₈₃	1	1	yes	characteristic
259	C₂³	—	166	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	4	1	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	664	1	no	—
264	Q₈	Q₈×D₈₃	1	2	yes	—
265	D₄	—	83	2	no	—
266	D₄	—	166	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	332	2	no	—
272	C₂×C₄	—	664	1	no	—
273	C₂×C₄	—	664	1	no	—
274	C₂×C₄	—	166	1	no	—
275	C₂×C₄	—	166	1	no	—
276	C₂×C₄	—	166	1	no	—
277	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
280	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
284	C₈	—	83	2	no	—
285	C₈	—	166	1	no	—
286	C₈	—	166	1	no	—
287	C₈	—	166	1	no	—
288	C₂²	—	332	1	no	—
289	C₂²	—	332	1	no	—
290	C₂²	—	166	2	no	—
291	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	332	1	no	—
303	C₄	—	332	1	no	—
304	C₄	—	332	1	no	—
305	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
306	C₄	#14581840	1	1	yes	characteristic
307	C₂	—	332	2	no	—
308	C₂	Q₈×C₂²⋊D₈₃	1	1	yes	characteristic
309	C₂	#19128499	1	1	yes	characteristic
310	C₂	#19128509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#557×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₃₃₂	C₂³
hypercenter	#557	D₈₃
hypocenter	C₈₃	#3388
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

Complements

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

N	H ≅ G/N	type
2. #19128111	C₂	semidirect
3. #19128107	C₂	semidirect
4. #19128100	C₂	semidirect
5. #19128008	C₂	semidirect
6. #19128008	C₂	semidirect
10. #19127986	C₂	semidirect
11. #19127832	C₂	semidirect
15. #557×C₈₃	C₂	semidirect
64. #13050577	Q₈	semidirect
156. C₈₃	#3388	semidirect
157. #557	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128411	2
#19128111	2
#19128107	2
#19128100	2
#19128008	2
#19127998	2
#19127988	2
#19127986	2
#19127832	2
#19127799	2
C₄×#13050577	2
#557×C₈₃	2
#3388	83