Subgroups — #24918131

Order 10624.

350 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	#19128322	C₂	1	1	yes	maximal, characteristic
2	#19128313	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128110	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128095	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128056	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128052	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127959	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127947	C₂	1	1	yes	maximal, characteristic
9	#19127942	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127930	C₂	1	1	yes	maximal, characteristic
11	#19127809	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127803	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13050579	C₂	1	1	yes	maximal, characteristic, complemented
15	#405×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
18	C₂×#13050579	C₂²	1	1	yes	characteristic
19	C₂×#13050579	C₂²	1	1	yes	characteristic
20	#14581792	C₂²	1	1	yes	characteristic
21	#14581792	C₂²	1	1	yes	characteristic
22	#14581787	—	2	1	no	—
23	#14581785	—	2	1	no	—
24	#14581785	—	2	1	no	—
25	#14581784	C₂²	1	1	yes	characteristic
26	#14581782	—	2	1	no	—
27	#14581782	—	2	1	no	—
28	#14581780	C₂²	1	1	yes	characteristic
29	#14581778	—	2	1	no	—
30	#14581774	—	2	1	no	—
31	#14581771	C₂²	1	1	yes	characteristic
32	#14581771	C₂²	1	1	yes	characteristic
33	#14581771	C₂²	1	1	yes	characteristic
34	#14581770	—	2	1	no	—
35	#14581766	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
38	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
39	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
40	#14581726	C₂²	1	1	yes	characteristic
41	#14581726	C₂²	1	1	yes	characteristic
42	#14581724	C₂²	1	1	yes	characteristic
43	#14581724	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581717	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581717	C₂²	1	1	yes	characteristic
48	#14581717	C₂²	1	1	yes	characteristic
49	#14581717	C₂²	1	1	yes	characteristic
50	#14581714	C₂²	1	1	yes	characteristic
51	#14581714	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
54	#117×C₈₃	C₂²	1	1	yes	characteristic
55	#97×C₈₃	C₂²	1	1	yes	characteristic
56	#97×C₈₃	C₂²	1	1	yes	characteristic
57	C₄×C₈×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×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	—	2	1	no	—
66	Dic₃₃₂	—	2	2	no	—
67	Dic₃₃₂	—	2	2	no	—
68	#13050579	—	2	2	no	—
69	#13050579	—	2	2	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	—	2	1	no	—
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	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	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊Dic₈₃	—	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	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
96	M₁₆×C₈₃	—	2	2	no	—
97	C₂×C₈×C₈₃	—	2	2	no	—
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
106	C₄²×C₈₃	C₂³	1	1	yes	characteristic
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₃	—	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	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₈₃	—	8	1	no	—
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	D₃₃₂	—	2	1	no	—
127	D₃₃₂	—	2	1	no	—
128	C₄×D₈₃	—	8	1	no	—
129	Dic₁₆₆	—	2	1	no	—
130	Dic₁₆₆	—	2	1	no	—
131	Dic₁₆₆	—	2	1	no	—
132	Dic₁₆₆	—	2	1	no	—
133	Dic₁₆₆	—	2	1	no	—
134	Dic₁₆₆	—	2	1	no	—
135	Dic₁₆₆	—	8	1	no	—
136	C₈×C₈₃	—	2	1	no	—
137	C₈×C₈₃	—	2	1	no	—
138	C₈×C₈₃	#41	1	2	yes	—
139	C₈×C₈₃	#41	1	2	yes	—
140	C₂²×C₈₃	—	2	1	no	—
141	C₂²×C₈₃	—	2	1	no	—
142	C₂²×C₈₃	—	2	1	no	—
143	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
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	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
153	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
154	Dic₈₃	—	4	1	no	—
155	Dic₈₃	—	4	1	no	—
156	Dic₈₃	—	4	1	no	—
157	Dic₈₃	—	4	1	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	—	4	1	no	—
161	C₂×C₈₃	—	4	1	no	—
162	C₂×C₈₃	#532	1	1	yes	characteristic
163	C₂×C₈₃	#577	1	1	yes	characteristic
164	C₂×C₈₃	#578	1	1	yes	characteristic
165	D₈₃	—	8	1	no	—
166	#3178	—	83	1	no	maximal, Sylow
167	C₈₃	#3178	1	1	yes	characteristic, complemented, Sylow
168	#542	—	83	1	no	—
169	#541	—	83	1	no	—
170	#501	—	83	1	no	—
171	#497	—	83	1	no	—
172	#488	—	83	1	no	—
173	#487	—	83	1	no	—
174	#470	—	83	1	no	—
175	#468	—	83	1	no	—
176	#467	—	83	1	no	—
177	#465	—	83	1	no	—
178	#441	—	83	1	no	—
179	#440	—	83	1	no	—
180	C₄×Q₁₆	—	83	1	no	—
181	C₄×SD₁₆	—	83	1	no	—
182	#405	D₈₃	1	1	yes	characteristic, complemented
183	C₂×Q₁₆	—	83	1	no	—
184	C₂×Q₁₆	—	83	1	no	—
185	C₂×SD₁₆	—	83	1	no	—
186	C₂×SD₁₆	—	83	1	no	—
187	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
188	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
189	C₄⋊Q₈	—	83	1	no	—
190	C₄⋊Q₈	—	83	1	no	—
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	—	83	1	no	—
208	C₄×D₄	—	83	1	no	—
209	#117	C₂×D₈₃	1	1	yes	characteristic
210	#107	—	83	1	no	—
211	#107	—	83	1	no	—
212	#106	—	83	1	no	—
213	#106	—	83	1	no	—
214	#103	—	83	1	no	—
215	#103	—	83	1	no	—
216	#103	—	83	1	no	—
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	#97	C₂×D₈₃	1	1	yes	characteristic
223	#97	C₂×D₈₃	1	1	yes	characteristic
224	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
225	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
226	C₂×Q₈	—	83	1	no	—
227	C₂×Q₈	—	83	1	no	—
228	C₂×Q₈	—	83	1	no	—
229	C₂×Q₈	—	166	1	no	—
230	C₂×D₄	—	83	1	no	—
231	C₂×D₄	—	166	1	no	—
232	C₂²×C₄	—	166	1	no	—
233	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
234	Q₁₆	—	166	2	no	—
235	Q₁₆	—	166	2	no	—
236	SD₁₆	—	166	2	no	—
237	SD₁₆	—	166	2	no	—
238	M₁₆	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₄⋊C₄	—	83	1	no	—
245	C₄⋊C₄	—	83	1	no	—
246	C₄⋊C₄	—	83	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄⋊C₄	—	166	1	no	—
249	C₄⋊C₄	—	166	1	no	—
250	C₄⋊C₄	—	166	1	no	—
251	C₄⋊C₄	—	166	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	D₃₃₂	1	1	yes	characteristic
260	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
261	C₂²⋊C₄	—	166	1	no	—
262	C₂²⋊C₄	—	166	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	D₃₃₂	1	1	yes	characteristic
269	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
270	C₄²	—	166	1	no	—
271	C₄²	—	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	C₂³	—	166	1	no	—
276	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
277	Q₈	—	664	1	no	—
278	Q₈	—	166	1	no	—
279	Q₈	—	166	1	no	—
280	Q₈	—	166	1	no	—
281	Q₈	—	166	1	no	—
282	Q₈	—	166	1	no	—
283	Q₈	—	166	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	—	664	1	no	—
290	C₂×C₄	—	166	1	no	—
291	C₂×C₄	—	166	1	no	—
292	C₂×C₄	—	166	1	no	—
293	C₂×C₄	—	166	1	no	—
294	C₂×C₄	—	166	1	no	—
295	C₂×C₄	—	166	1	no	—
296	C₂×C₄	—	166	1	no	—
297	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
298	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
299	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
300	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
301	C₂×C₄	C₂×D₃₃₂	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	#13050590	1	2	yes	—
306	C₈	#13050590	1	2	yes	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	332	1	no	—
311	C₂²	—	332	1	no	—
312	C₂²	—	332	1	no	—
313	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	332	1	no	—
320	C₄	—	332	1	no	—
321	C₄	—	332	1	no	—
322	C₄	—	332	1	no	—
323	C₄	—	332	1	no	—
324	C₄	—	332	1	no	—
325	C₄	—	332	1	no	—
326	C₄	C₂×#13050590	1	1	yes	characteristic
327	C₄	C₂×#13050590	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	664	1	no	—
330	C₂	#19128251	1	1	yes	characteristic
331	C₂	#19128513	1	1	yes	characteristic
332	C₂	#19128518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#405×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#14581766	C₂²
hypercenter	#405	D₈₃
hypocenter	C₈₃	#3178
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #19128313	C₂	semidirect
3. #19128110	C₂	semidirect
4. #19128095	C₂	semidirect
5. #19128056	C₂	semidirect
6. #19128052	C₂	semidirect
7. #19127959	C₂	semidirect
9. #19127942	C₂	semidirect
11. #19127809	C₂	semidirect
12. #19127803	C₂	semidirect
13. C₄×Dic₃₃₂	C₂	semidirect
14. C₄×#13050579	C₂	semidirect
15. #405×C₈₃	C₂	semidirect
167. C₈₃	#3178	semidirect
182. #405	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128322	2
#19128313	2
#19128110	2
#19128095	2
#19128056	2
#19128052	2
#19127959	2
#19127947	2
#19127942	2
#19127930	2
#19127809	2
#19127803	2
C₄×Dic₃₃₂	2
C₄×#13050579	2
#405×C₈₃	2
#3178	83