Subgroups — #35319690

Order 21248.

428 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916265	C₂	1	1	yes	maximal, characteristic
2	#24909200	C₂	1	1	yes	maximal, characteristic
3	#24909192	C₂	1	1	yes	maximal, characteristic
4	#24907347	C₂	1	1	yes	maximal, characteristic, complemented
5	#24905578	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905565	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#347×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581814	C₂²	1	1	yes	characteristic
9	C₄×#13050590	—	2	1	no	—
10	#19127814	C₄	1	2	yes	complemented
11	#19127809	C₄	1	2	yes	complemented
12	#19127803	—	2	2	no	—
13	C₂×#14581728	C₂²	1	1	yes	characteristic
14	#19127739	C₂²	1	1	yes	characteristic
15	#19127739	C₂²	1	1	yes	characteristic
16	#19127698	—	2	1	no	—
17	#19127656	—	2	1	no	—
18	#19127512	C₄	1	4	yes	complemented
19	#19127509	C₄	1	4	yes	complemented
20	#19127461	—	2	1	no	—
21	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
24	#347×C₈₃	—	2	2	no	—
25	C₂×#13050590	—	2	1	no	—
26	#14581814	—	2	1	no	—
27	#14581814	—	2	1	no	—
28	#14581814	C₂×C₄	1	2	yes	—
29	#14581814	C₂×C₄	1	2	yes	—
30	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#13050579	—	2	1	no	—
32	C₂×D₆₆₄	C₂×C₄	1	1	yes	characteristic
33	C₂×#13050574	—	2	1	no	—
34	C₄×Dic₁₆₆	—	2	2	no	—
35	C₄×D₃₃₂	—	2	2	no	—
36	C₄×C₂²⋊D₈₃	—	4	2	no	—
37	#14581751	—	2	1	no	—
38	#14581751	—	2	1	no	—
39	#14581749	—	4	1	no	—
40	C₄²×D₈₃	—	4	1	no	—
41	#14581728	—	2	1	no	—
42	#14581728	C₂×C₄	1	2	yes	complemented
43	#14581726	—	2	1	no	—
44	#14581726	—	2	1	no	—
45	#14581724	C₂×C₄	1	2	yes	—
46	#14581724	C₂×C₄	1	2	yes	complemented
47	#14581717	—	2	2	no	—
48	#14581714	—	2	2	no	—
49	#14581702	—	4	1	no	—
50	C₈×Dic₈₃	—	4	1	no	—
51	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
54	M₃₂×C₈₃	—	2	1	no	—
55	M₃₂×C₈₃	—	2	1	no	—
56	M₃₂×C₈₃	—	2	1	no	—
57	M₃₂×C₈₃	C₂×C₄	1	2	yes	complemented
58	C₂×C₁₆×C₈₃	—	2	1	no	—
59	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	complemented
60	#97×C₈₃	C₂×C₄	1	2	yes	—
61	#97×C₈₃	D₄	1	2	yes	—
62	#13050590	—	2	2	no	—
63	#13050590	—	2	2	no	—
64	#13050590	—	4	1	no	—
65	#13050590	—	4	1	no	—
66	C₂×Dic₁₆₆	—	2	1	no	—
67	C₂×D₃₃₂	—	2	1	no	—
68	C₂×C₂²⋊D₈₃	—	4	1	no	—
69	C₂×C₄×D₈₃	—	4	1	no	—
70	C₂×C₄×D₈₃	—	4	2	no	—
71	Dic₃₃₂	—	2	1	no	—
72	Dic₃₃₂	C₄²	1	2	yes	—
73	#13050579	—	2	2	no	—
74	#13050579	—	2	2	no	—
75	D₆₆₄	—	2	1	no	—
76	D₆₆₄	C₄²	1	2	yes	—
77	#13050574	—	2	2	no	—
78	#13050574	—	4	1	no	—
79	C₂×C₈₃⋊C₈	—	4	1	no	—
80	C₄⋊Dic₈₃	—	2	2	no	—
81	C₄⋊Dic₈₃	—	2	2	no	—
82	Dic₈₃⋊C₄	—	4	2	no	—
83	C₂²⋊Dic₈₃	—	4	1	no	—
84	C₂²⋊Dic₈₃	—	4	1	no	—
85	(C₂×D₈₃)⋊C₄	—	4	2	no	—
86	C₄×Dic₈₃	—	4	2	no	—
87	C₄×Dic₈₃	—	4	1	no	—
88	C₄×Dic₈₃	—	4	1	no	—
89	C₂²×C₄×C₈₃	—	2	1	no	—
90	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₈₃	—	2	1	no	—
92	C₂×C₈×C₈₃	—	2	1	no	—
93	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
94	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
100	C₄²×C₈₃	—	2	1	no	—
101	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
102	C₁₆×C₈₃	—	2	2	no	—
103	C₁₆×C₈₃	—	2	2	no	—
104	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂²×D₈₃	—	4	1	no	—
106	C₂×C₄×C₈₃	—	2	2	no	—
107	C₂×C₄×C₈₃	—	8	1	no	—
108	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
114	C₂²⋊D₈₃	—	4	2	no	—
115	C₂²⋊D₈₃	—	4	2	no	—
116	C₂×Dic₈₃	—	4	1	no	—
117	C₂×Dic₈₃	—	4	2	no	—
118	C₂×Dic₈₃	—	4	2	no	—
119	D₃₃₂	—	2	2	no	—
120	D₃₃₂	—	4	1	no	—
121	C₄×D₈₃	—	4	2	no	—
122	C₄×D₈₃	—	4	2	no	—
123	C₄×D₈₃	—	8	4	no	—
124	Dic₁₆₆	—	2	2	no	—
125	Dic₁₆₆	—	4	1	no	—
126	C₈×C₈₃	—	2	2	no	—
127	C₈×C₈₃	C₂×C₄²	1	1	yes	characteristic
128	C₈×C₈₃	#117	1	1	yes	characteristic
129	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
130	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
131	C₈₃⋊C₈	—	4	2	no	—
132	C₂²×C₈₃	—	4	1	no	—
133	C₂²×C₈₃	#418	1	1	yes	characteristic
134	C₂²×C₈₃	#440	1	1	yes	characteristic
135	C₂²×C₈₃	#442	1	1	yes	characteristic
136	C₂×D₈₃	—	4	2	no	—
137	C₂×D₈₃	—	8	2	no	—
138	C₄×C₈₃	—	4	2	no	—
139	C₄×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₈₃	#417	1	1	yes	characteristic
141	C₄×C₈₃	#440	1	1	yes	characteristic
142	C₄×C₈₃	#442	1	1	yes	characteristic
143	Dic₈₃	—	4	2	no	—
144	Dic₈₃	—	8	2	no	—
145	Dic₈₃	—	8	2	no	—
146	#12649	—	83	1	no	maximal, Sylow
147	C₂×C₈₃	—	2	1	no	—
148	C₂×C₈₃	—	2	1	no	—
149	C₂×C₈₃	—	2	1	no	—
150	C₂×C₈₃	#1762	1	1	yes	characteristic
151	D₈₃	—	8	2	no	—
152	#2943	—	83	1	no	—
153	#2145	—	83	1	no	—
154	#2143	—	83	1	no	—
155	C₂×#347	D₈₃	1	1	yes	characteristic, complemented
156	#1773	—	83	1	no	—
157	#1389	—	83	1	no	—
158	#1384	—	83	1	no	—
159	C₈₃	#12649	1	1	yes	characteristic, complemented, Sylow
160	C₂×#135	—	83	1	no	—
161	C₄×#41	—	166	1	no	—
162	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
163	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
164	#442	—	83	2	no	—
165	#441	—	83	2	no	—
166	#440	—	166	2	no	—
167	C₂×#108	—	83	1	no	—
168	#427	—	83	1	no	—
169	#427	—	83	1	no	—
170	#416	—	166	1	no	—
171	#405	—	166	1	no	—
172	C₂×#97	C₂×D₈₃	1	1	yes	characteristic
173	#360	—	83	4	no	—
174	#359	—	83	4	no	—
175	#347	—	2	2	no	—
176	#339	—	166	1	no	—
177	C₂×#41	—	166	1	no	—
178	#135	—	83	2	no	—
179	#135	—	83	2	no	—
180	#135	—	166	1	no	—
181	#135	—	166	1	no	—
182	C₂×Q₁₆	—	83	1	no	—
183	C₂×SD₁₆	—	166	1	no	—
184	C₂×D₈	—	83	1	no	—
185	C₂×M₁₆	—	166	1	no	—
186	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
188	C₄×Q₈	—	166	2	no	—
189	C₄×D₄	—	332	2	no	—
190	C₄×D₄	—	166	2	no	—
191	#117	—	332	1	no	—
192	#117	—	166	1	no	—
193	#117	—	166	1	no	—
194	C₂×C₄²	—	332	1	no	—
195	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
196	M₃₂	—	2	1	no	—
197	M₃₂	—	2	1	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	C₄×D₈₃	1	2	yes	complemented
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	C₄×D₈₃	1	2	yes	complemented
202	#108	—	83	2	no	—
203	#108	—	166	1	no	—
204	#107	—	166	1	no	—
205	#107	—	166	1	no	—
206	#106	—	83	2	no	—
207	#106	—	83	2	no	—
208	#103	—	166	2	no	—
209	#102	—	166	2	no	—
210	#97	—	332	1	no	—
211	#97	C₄×D₈₃	1	2	yes	—
212	#97	C₂²⋊D₈₃	1	2	yes	—
213	C₄×C₈	—	332	1	no	—
214	#41	—	332	1	no	—
215	#41	—	332	1	no	—
216	#41	—	166	2	no	—
217	#41	—	166	2	no	—
218	C₂×Q₈	—	166	1	no	—
219	C₂×D₄	—	332	1	no	—
220	C₂×D₄	—	166	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	332	2	no	—
223	C₂²×C₄	—	332	1	no	—
224	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
225	Q₁₆	—	83	2	no	—
226	Q₁₆	—	166	1	no	—
227	SD₁₆	—	166	2	no	—
228	SD₁₆	—	166	2	no	—
229	D₈	—	83	2	no	—
230	D₈	—	166	1	no	—
231	M₁₆	—	332	1	no	—
232	M₁₆	—	166	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	332	1	no	—
236	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
241	C₂×C₈	#13050590	1	1	yes	characteristic
242	C₂×C₈	#13050590	1	1	yes	characteristic
243	C₄⋊C₄	—	332	2	no	—
244	C₄⋊C₄	—	166	2	no	—
245	C₄⋊C₄	—	166	2	no	—
246	C₂²⋊C₄	—	332	2	no	—
247	C₂²⋊C₄	—	332	1	no	—
248	C₂²⋊C₄	—	332	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	332	2	no	—
251	C₄²	—	332	1	no	—
252	C₄²	—	332	1	no	—
253	C₄²	(C₂×D₈₃)⋊C₄	1	2	yes	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	2	2	no	—
256	C₂³	—	332	1	no	—
257	C₂³	#14581820	1	1	yes	characteristic
258	Q₈	—	332	1	no	—
259	Q₈	—	166	2	no	—
260	D₄	—	332	2	no	—
261	D₄	—	332	2	no	—
262	D₄	—	332	1	no	—
263	D₄	—	166	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	—	332	1	no	—
267	C₂×C₄	—	332	2	no	—
268	C₂×C₄	—	332	2	no	—
269	C₂×C₄	—	332	2	no	—
270	C₂×C₄	—	332	2	no	—
271	C₂×C₄	—	664	4	no	—
272	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
274	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
275	C₂×C₄	#14581820	1	1	yes	characteristic
276	C₂×C₄	#14581820	1	1	yes	characteristic
277	C₂×C₄	#14581825	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	332	2	no	—
280	C₈	C₄²×D₈₃	1	1	yes	characteristic
281	C₈	#14581749	1	1	yes	characteristic
282	C₈	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
283	C₈	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
284	C₂²	—	4	1	no	—
285	C₂²	—	332	2	no	—
286	C₂²	—	664	2	no	—
287	C₂²	#19127707	1	1	yes	characteristic
288	C₂²	#19127803	1	1	yes	characteristic
289	C₂²	#19127814	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	332	2	no	—
292	C₄	—	664	2	no	—
293	C₄	—	664	2	no	—
294	C₄	C₄×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
295	C₄	#19127703	1	1	yes	characteristic
296	C₄	#19127803	1	1	yes	characteristic
297	C₄	#19127814	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	664	2	no	—
302	C₂	#24907289	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19127814
commutator	C₈×C₈₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#347×C₈₃	C₂
socle	C₂×C₈₃	#1762
prime radical	C₂×#14581814	C₂²
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#347	D₈₃
hypocenter	C₈₃	#12649
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #24907347	C₂	semidirect
5. #24905578	C₂	semidirect
6. #24905565	C₂	semidirect
7. C₂×#347×C₈₃	C₂	semidirect
10. #19127814	C₄	semidirect
11. #19127809	C₄	semidirect
18. #19127512	C₄	semidirect
19. #19127509	C₄	semidirect
42. #14581728	C₂×C₄	semidirect
46. #14581724	C₂×C₄	semidirect
57. M₃₂×C₈₃	C₂×C₄	semidirect
59. C₂×C₁₆×C₈₃	C₂×C₄	semidirect
155. C₂×#347	D₈₃	semidirect
159. C₈₃	#12649	semidirect
199. M₃₂	C₄×D₈₃	semidirect
201. C₂×C₁₆	C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24916265	2
#24909200	2
#24909192	2
#24907347	2
#24905578	2
#24905565	2
C₂×#347×C₈₃	2
#12649	83