Subgroups — #24920292

Order 10624.

346 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128424	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128411	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128070	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	#19127987	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127917	C₂	1	1	yes	maximal, characteristic
10	#19127917	C₂	1	1	yes	maximal, characteristic
11	#19127907	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127814	C₂	1	1	yes	maximal, characteristic
13	#19127803	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13050579	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×#13050579	C₂²	1	1	yes	characteristic
18	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
19	#14581792	C₂²	1	1	yes	characteristic
20	#14581792	C₂²	1	1	yes	characteristic
21	#14581787	—	2	1	no	—
22	#14581787	—	2	1	no	—
23	#14581784	C₂²	1	1	yes	characteristic
24	#14581784	C₂²	1	1	yes	characteristic
25	#14581780	C₂²	1	1	yes	characteristic
26	#14581780	C₂²	1	1	yes	characteristic
27	#14581770	—	2	1	no	—
28	#14581770	—	2	1	no	—
29	#14581768	—	2	1	no	—
30	#14581768	—	2	1	no	—
31	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₆₆	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	#14581750	—	2	1	no	—
38	C₄⋊C₄×D₈₃	—	2	1	no	—
39	#14581726	C₂²	1	1	yes	characteristic
40	#14581724	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	#14581717	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581717	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×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	—	2	1	no	—
66	C₂×C₄×D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	#13050579	—	2	2	no	—
69	#13050579	—	2	2	no	—
70	D₆₆₄	—	2	2	no	—
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	—	2	1	no	—
73	C₄⋊Dic₈₃	—	2	1	no	—
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	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	(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	C₄×Dic₈₃	—	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	Q₈×C₈₃	—	4	1	no	—
108	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₈₃	—	2	1	no	—
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	#41	1	2	yes	—
125	C₄×D₈₃	—	4	2	no	—
126	C₄×D₈₃	—	4	2	no	—
127	C₄×D₈₃	—	8	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	—	8	1	no	—
134	C₈×C₈₃	—	2	1	no	—
135	C₈×C₈₃	—	2	1	no	—
136	C₈×C₈₃	—	2	1	no	—
137	C₈×C₈₃	—	2	1	no	—
138	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₈₃	—	2	2	no	—
140	C₂×D₈₃	—	4	1	no	—
141	C₂×D₈₃	—	4	1	no	—
142	C₂×D₈₃	—	4	1	no	—
143	C₂×D₈₃	—	4	1	no	—
144	C₂×D₈₃	—	4	1	no	—
145	C₄×C₈₃	—	2	1	no	—
146	C₄×C₈₃	—	2	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	C₂×#41	1	1	yes	characteristic
152	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₈₃	—	4	1	no	—
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	C₂×C₈₃	#548	1	1	yes	characteristic
160	C₂×C₈₃	#575	1	1	yes	characteristic
161	C₂×C₈₃	#577	1	1	yes	characteristic
162	D₈₃	—	4	2	no	—
163	D₈₃	—	8	1	no	—
164	#3376	—	83	1	no	maximal, Sylow
165	C₈₃	#3376	1	1	yes	characteristic, complemented, Sylow
166	#556	—	83	1	no	—
167	#554	—	83	1	no	—
168	#491	—	83	1	no	—
169	#489	—	83	1	no	—
170	#488	—	83	1	no	—
171	#479	—	83	1	no	—
172	#478	—	83	1	no	—
173	#475	—	83	1	no	—
174	#463	—	83	1	no	—
175	#463	—	83	1	no	—
176	#461	—	83	1	no	—
177	#446	D₈₃	1	1	yes	characteristic, complemented
178	#442	—	83	1	no	—
179	#440	—	83	1	no	—
180	C₄×SD₁₆	—	83	1	no	—
181	C₂×SD₁₆	—	83	1	no	—
182	C₂×SD₁₆	—	83	1	no	—
183	C₂×D₈	—	83	1	no	—
184	C₄⋊Q₈	—	83	1	no	—
185	C₄⋊Q₈	—	83	1	no	—
186	#126	—	166	1	no	—
187	#126	—	166	1	no	—
188	#125	—	83	1	no	—
189	#125	—	83	1	no	—
190	#124	—	83	1	no	—
191	#124	—	83	1	no	—
192	C₂²⋊Q₈	—	166	1	no	—
193	C₂²⋊Q₈	—	166	1	no	—
194	C₂²⋊Q₈	—	166	1	no	—
195	C₂²⋊Q₈	—	166	1	no	—
196	C₄×Q₈	—	83	1	no	—
197	C₄×Q₈	—	83	1	no	—
198	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
199	C₄×D₄	—	83	1	no	—
200	C₄×D₄	—	83	1	no	—
201	C₄×D₄	—	83	1	no	—
202	C₄×D₄	—	83	1	no	—
203	#117	—	166	1	no	—
204	C₂×C₄⋊C₄	—	166	1	no	—
205	#107	—	83	1	no	—
206	#106	—	83	1	no	—
207	#106	—	83	1	no	—
208	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
209	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
210	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
211	#103	—	83	1	no	—
212	#103	—	83	1	no	—
213	#103	—	83	1	no	—
214	#103	—	83	1	no	—
215	#103	—	83	1	no	—
216	#103	—	83	1	no	—
217	#102	—	83	1	no	—
218	#102	—	83	1	no	—
219	#102	—	83	1	no	—
220	#102	—	83	1	no	—
221	#97	C₂×D₈₃	1	1	yes	characteristic
222	#97	C₂×D₈₃	1	1	yes	characteristic
223	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
224	C₂×Q₈	—	83	1	no	—
225	C₂×Q₈	—	83	1	no	—
226	C₂×Q₈	—	166	1	no	—
227	C₂×Q₈	D₃₃₂	1	1	yes	characteristic
228	C₂×D₄	—	83	1	no	—
229	C₂×D₄	—	83	1	no	—
230	C₂²×C₄	—	166	1	no	—
231	C₂²×C₄	—	166	1	no	—
232	C₂²×C₄	—	166	1	no	—
233	C₂²×C₄	—	166	1	no	—
234	SD₁₆	—	166	2	no	—
235	SD₁₆	—	166	2	no	—
236	D₈	—	166	2	no	—
237	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
239	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
241	C₄⋊C₄	—	83	1	no	—
242	C₄⋊C₄	—	83	1	no	—
243	C₄⋊C₄	—	83	1	no	—
244	C₄⋊C₄	—	83	1	no	—
245	C₄⋊C₄	—	166	1	no	—
246	C₄⋊C₄	—	166	1	no	—
247	C₄⋊C₄	—	166	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	D₃₃₂	1	1	yes	characteristic
255	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
256	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
257	C₂²⋊C₄	—	166	1	no	—
258	C₂²⋊C₄	—	166	1	no	—
259	C₂²⋊C₄	—	166	1	no	—
260	C₂²⋊C₄	—	166	1	no	—
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	C₄²	—	166	1	no	—
266	C₄²	—	166	1	no	—
267	C₄²	C₂²×D₈₃	1	1	yes	characteristic
268	C₄²	C₂²×D₈₃	1	1	yes	characteristic
269	C₄²	C₂²×D₈₃	1	1	yes	characteristic
270	C₂³	—	166	1	no	—
271	C₂³	—	166	1	no	—
272	Q₈	—	4	1	no	—
273	Q₈	—	664	1	no	—
274	Q₈	—	166	1	no	—
275	Q₈	—	166	1	no	—
276	Q₈	—	166	1	no	—
277	Q₈	—	166	1	no	—
278	D₄	—	83	2	no	—
279	D₄	—	166	1	no	—
280	D₄	—	166	1	no	—
281	D₄	—	166	1	no	—
282	C₂×C₄	—	332	2	no	—
283	C₂×C₄	—	332	2	no	—
284	C₂×C₄	—	664	1	no	—
285	C₂×C₄	—	664	1	no	—
286	C₂×C₄	—	166	1	no	—
287	C₂×C₄	—	166	1	no	—
288	C₂×C₄	—	166	1	no	—
289	C₂×C₄	—	166	1	no	—
290	C₂×C₄	—	166	1	no	—
291	C₂×C₄	—	166	1	no	—
292	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
293	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
294	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
295	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
296	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
297	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₂²	—	332	1	no	—
304	C₂²	—	332	1	no	—
305	C₂²	—	332	1	no	—
306	C₂²	—	332	1	no	—
307	C₂²	—	332	1	no	—
308	C₂²	—	166	2	no	—
309	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	332	1	no	—
317	C₄	—	332	1	no	—
318	C₄	—	332	1	no	—
319	C₄	—	332	1	no	—
320	C₄	—	332	1	no	—
321	C₄	—	332	1	no	—
322	C₄	#41×D₈₃	1	1	yes	characteristic
323	C₄	#14581840	1	1	yes	characteristic
324	C₂	—	332	2	no	—
325	C₂	—	664	1	no	—
326	C₂	#19128363	1	1	yes	characteristic
327	C₂	#19128501	1	1	yes	characteristic
328	C₂	#19128513	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	C₂×D₆₆₄	C₂²
hypercenter	#446	D₈₃
hypocenter	C₈₃	#3376
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. #19128424	C₂	semidirect
2. #19128411	C₂	semidirect
3. #19128070	C₂	semidirect
4. #19128061	C₂	semidirect
5. #19128056	C₂	semidirect
6. #19128009	C₂	semidirect
7. #19128002	C₂	semidirect
8. #19127987	C₂	semidirect
11. #19127907	C₂	semidirect
13. #19127803	C₂	semidirect
14. C₄×#13050579	C₂	semidirect
15. #446×C₈₃	C₂	semidirect
165. C₈₃	#3376	semidirect
177. #446	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128424	2
#19128411	2
#19128070	2
#19128061	2
#19128056	2
#19128009	2
#19128002	2
#19127987	2
#19127917	2
#19127907	2
#19127814	2
#19127803	2
C₄×#13050579	2
#446×C₈₃	2
#3376	83