Subgroups — #35303634

Order 21248.

486 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908387	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
4	C₂×#19127447	C₂	1	2	yes	maximal
5	C₂×#500×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#14581716	—	2	4	no	—
9	C₂×C₈×Dic₈₃	—	2	1	no	—
10	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
11	#19127455	—	2	1	no	—
12	#19127454	—	2	2	no	—
13	#19127447	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
15	#500×C₈₃	—	2	2	no	—
16	#500×C₈₃	C₄	1	4	yes	complemented
17	C₂×#107×C₈₃	—	2	1	no	—
18	C₂×#107×C₈₃	—	2	1	no	—
19	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂²×C₈₃⋊C₈	—	2	1	no	—
21	C₂²×C₈₃⋊C₈	—	2	2	no	—
22	C₂×C₄⋊Dic₈₃	D₄	1	2	yes	—
23	C₂×C₄×Dic₈₃	—	2	1	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
26	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
27	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
28	#14581716	—	4	8	no	—
29	C₈×Dic₈₃	—	4	2	no	—
30	C₈₃⋊#95	—	2	1	no	—
31	C₂²×Q₈×C₈₃	—	2	2	no	—
32	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₈₃	—	2	2	no	—
35	C₄⋊Q₈×C₈₃	—	2	2	no	—
36	C₄⋊Q₈×C₈₃	—	2	2	no	—
37	C₄⋊Q₈×C₈₃	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
40	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	#107×C₈₃	—	2	4	no	—
42	#107×C₈₃	—	2	4	no	—
43	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₈₃	D₄	1	2	yes	—
45	C₂²×Dic₈₃	—	2	2	no	—
46	C₂×C₈₃⋊C₈	—	2	2	no	—
47	C₂×C₈₃⋊C₈	—	4	1	no	—
48	C₂×C₈₃⋊C₈	—	4	1	no	—
49	C₂×C₈₃⋊C₈	—	4	2	no	—
50	C₂×C₈₃⋊C₈	—	4	4	no	—
51	C₄⋊Dic₈₃	—	2	2	no	—
52	C₄⋊Dic₈₃	D₈	1	4	yes	—
53	C₄×Dic₈₃	—	4	1	no	—
54	C₄×Dic₈₃	—	4	1	no	—
55	C₂×Q₈×C₈₃	—	2	4	no	—
56	C₂×Q₈×C₈₃	—	4	2	no	—
57	C₂×Q₈×C₈₃	—	4	4	no	—
58	C₂×Q₈×C₈₃	—	4	4	no	—
59	C₂²×C₄×C₈₃	—	4	2	no	—
60	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
63	C₂×C₈×C₈₃	—	2	1	no	—
64	C₂×C₈×C₈₃	—	2	1	no	—
65	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
68	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
69	C₄⋊C₄×C₈₃	—	2	4	no	—
70	C₄⋊C₄×C₈₃	—	2	4	no	—
71	C₄⋊C₄×C₈₃	—	4	2	no	—
72	C₄⋊C₄×C₈₃	—	4	2	no	—
73	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₈₃	#41	1	1	yes	characteristic
77	C₂³×C₈₃	#121	1	1	yes	characteristic
78	Q₈×C₈₃	—	4	8	no	—
79	Q₈×C₈₃	—	8	2	no	—
80	Q₈×C₈₃	—	8	2	no	—
81	C₂×C₄×C₈₃	—	2	2	no	—
82	C₂×C₄×C₈₃	—	4	4	no	—
83	C₂×C₄×C₈₃	—	8	4	no	—
84	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
88	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
89	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
93	C₂×C₄×C₈₃	#135	1	2	yes	—
94	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
96	C₂×Dic₈₃	—	2	4	no	—
97	C₂×Dic₈₃	—	4	2	no	—
98	C₂×Dic₈₃	—	4	2	no	—
99	C₈×C₈₃	—	2	2	no	—
100	C₈×C₈₃	#103	1	4	yes	complemented
101	C₈₃⋊C₈	—	4	2	no	—
102	C₈₃⋊C₈	—	8	2	no	—
103	C₂²×C₈₃	#390	1	1	yes	characteristic
104	C₂²×C₈₃	#461	1	1	yes	characteristic
105	C₂²×C₈₃	#462	1	1	yes	characteristic
106	C₂²×C₈₃	#466	1	2	yes	—
107	C₂²×C₈₃	#473	1	2	yes	—
108	C₄×C₈₃	—	2	1	no	—
109	C₄×C₈₃	—	2	1	no	—
110	C₄×C₈₃	—	8	4	no	—
111	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
112	C₄×C₈₃	#419	1	1	yes	characteristic
113	C₄×C₈₃	C₄×D₈	1	2	yes	—
114	C₄×C₈₃	#461	1	1	yes	characteristic
115	C₄×C₈₃	#462	1	1	yes	characteristic
116	C₄×C₈₃	#493	1	1	yes	characteristic
117	C₄×C₈₃	#495	1	1	yes	characteristic
118	Dic₈₃	—	4	4	no	—
119	#9731	—	83	1	no	maximal, Sylow
120	C₂×C₈₃	#1673	1	2	yes	—
121	C₂×C₈₃	#1674	1	2	yes	—
122	C₂×C₈₃	#1970	1	1	yes	characteristic
123	C₂×C₈₃	#1979	1	1	yes	characteristic
124	C₂×C₈₃	#1981	1	1	yes	characteristic
125	C₂×#500	D₈₃	1	1	yes	characteristic
126	#1981	—	83	2	no	—
127	C₈×C₄⋊C₄	—	83	1	no	—
128	C₂×#335	—	83	1	no	—
129	C₂×#332	—	83	2	no	—
130	C₈₃	#9731	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
132	#500	—	2	2	no	—
133	#500	Dic₈₃	1	4	yes	complemented
134	C₂×#107	—	2	1	no	—
135	C₂×#107	—	2	1	no	—
136	C₂×C₄⋊C₈	—	83	1	no	—
137	C₂×C₄⋊C₈	—	83	2	no	—
138	C₂×#103	—	166	4	no	—
139	C₂×C₄×C₈	—	166	1	no	—
140	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
141	C₄×C₄⋊C₄	—	83	1	no	—
142	#336	—	166	1	no	—
143	#335	—	166	2	no	—
144	#332	—	166	4	no	—
145	C₂²×Q₈	—	2	2	no	—
146	C₂²×C₈	—	166	2	no	—
147	C₂²×C₈	—	166	1	no	—
148	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	C₂×Dic₈₃	1	4	yes	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	83	2	no	—
157	C₂×C₄²	—	166	1	no	—
158	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
159	#107	—	2	4	no	—
160	#107	—	2	4	no	—
161	C₄⋊C₈	—	83	2	no	—
162	C₄⋊C₈	—	166	2	no	—
163	C₄⋊C₈	—	166	2	no	—
164	C₄⋊C₈	—	166	1	no	—
165	#103	—	332	8	no	—
166	C₄×C₈	—	332	2	no	—
167	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
168	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
169	#95	—	166	1	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	166	2	no	—
176	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
178	C₂²×C₄	#13050589	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	332	1	no	—
182	C₂×C₈	—	332	1	no	—
183	C₂×C₈	—	332	2	no	—
184	C₂×C₈	—	332	4	no	—
185	C₂×C₈	—	166	2	no	—
186	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
187	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
188	C₂×C₈	#13050577	1	2	yes	—
189	C₂×C₈	#13050580	1	2	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	83	4	no	—
195	C₄⋊C₄	—	166	2	no	—
196	C₄²	—	332	1	no	—
197	C₄²	—	332	1	no	—
198	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
199	C₄²	D₄×D₈₃	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
201	C₄²	#13050589	1	1	yes	characteristic
202	C₂³	#14581763	1	1	yes	characteristic
203	Q₈	—	4	8	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	332	2	no	—
210	C₂×C₄	—	332	2	no	—
211	C₂×C₄	—	166	4	no	—
212	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
213	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
214	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
215	C₂×C₄	#14581763	1	1	yes	characteristic
216	C₂×C₄	#14581779	1	1	yes	characteristic
217	C₂×C₄	#14581788	1	1	yes	characteristic
218	C₂×C₄	D₈×D₈₃	1	2	yes	—
219	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
220	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
221	C₂×C₄	#14581811	1	2	yes	—
222	C₂×C₄	#14581823	1	1	yes	characteristic
223	C₂×C₄	#14581823	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	332	2	no	—
226	C₈	—	664	2	no	—
227	C₈	#14581716	1	4	yes	complemented
228	C₂²	#19127604	1	1	yes	characteristic
229	C₂²	#19127903	1	1	yes	characteristic
230	C₂²	#19127908	1	1	yes	characteristic
231	C₂²	#19127934	1	2	yes	—
232	C₂²	#19127973	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	8	4	no	—
236	C₄	—	332	4	no	—
237	C₄	C₂×#14581716	1	1	yes	characteristic
238	C₄	#19127710	1	1	yes	characteristic
239	C₄	D₈×Dic₈₃	1	2	yes	—
240	C₄	#19127903	1	1	yes	characteristic
241	C₄	#19127908	1	1	yes	characteristic
242	C₄	#19128076	1	1	yes	characteristic
243	C₄	#19128084	1	1	yes	characteristic
244	C₂	#24906849	1	2	yes	—
245	C₂	#24906854	1	2	yes	—
246	C₂	#24908330	1	1	yes	characteristic
247	C₂	#24908379	1	1	yes	characteristic
248	C₂	#24908387	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581763
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#500×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#121
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#500	D₈₃
hypocenter	C₈₃	#9731
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
16. #500×C₈₃	C₄	semidirect
100. C₈×C₈₃	#103	semidirect
130. C₈₃	#9731	semidirect
133. #500	Dic₈₃	semidirect
227. C₈	#14581716	semidirect

Maximal subgroups

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

subgroup	index
#24908387	2
C₈×C₄⋊Dic₈₃	2
C₂×#19127454	2
C₂×#19127447	2
C₂×#500×C₈₃	2
#9731	83