Subgroups — #35320569

Order 21248.

438 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127527	C₂	1	1	yes	maximal, characteristic, complemented
3	#24907489	C₂	1	2	yes	maximal
4	#24905554	C₂	1	2	yes	maximal, complemented
5	C₂×#366×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₈₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₈₃	—	2	2	no	—
9	C₂×C₁₆×D₈₃	—	2	2	no	—
10	C₂×C₁₆×D₈₃	C₄	1	4	yes	complemented
11	C₂×#14581726	C₂²	1	2	yes	—
12	#19127558	—	2	2	no	—
13	#19127527	—	2	2	no	—
14	#19127463	—	2	2	no	—
15	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×#107×C₈₃	C₂²	1	2	yes	—
17	#366×C₈₃	—	2	2	no	—
18	C₂²×C₄×D₈₃	D₄	1	1	yes	characteristic
19	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
20	C₂×C₈×D₈₃	—	2	1	no	—
21	C₂×C₈×D₈₃	—	2	1	no	—
22	C₂×C₈×D₈₃	—	2	1	no	—
23	C₂×C₈×D₈₃	—	2	1	no	—
24	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₈₃	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₈₃	—	2	2	no	—
27	C₂×(C₂×D₈₃)⋊C₄	—	4	2	no	—
28	C₂×C₈₃⋊C₁₆	—	2	1	no	—
29	C₂×C₈₃⋊C₁₆	—	2	1	no	—
30	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
31	C₁₆×D₈₃	—	2	8	no	—
32	#14581726	—	2	2	no	—
33	#14581726	—	2	2	no	—
34	C₈₃⋊#95	—	4	2	no	—
35	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
37	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
39	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
40	#107×C₈₃	—	2	2	no	—
41	#107×C₈₃	—	2	2	no	—
42	C₂³×D₈₃	D₈	1	1	yes	characteristic
43	C₂²×Dic₈₃	—	4	2	no	—
44	C₂²×Dic₈₃	D₈	1	1	yes	characteristic
45	C₂×C₄×D₈₃	—	2	1	no	—
46	C₂×C₄×D₈₃	—	2	1	no	—
47	C₂×C₄×D₈₃	—	2	1	no	—
48	C₂×C₄×D₈₃	—	2	1	no	—
49	C₂×C₄×D₈₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×D₈₃	Q₁₆	1	2	yes	—
51	C₂×C₈₃⋊C₈	—	2	1	no	—
52	C₂×C₈₃⋊C₈	—	2	1	no	—
53	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₈×D₈₃	—	2	2	no	—
55	C₈×D₈₃	—	2	2	no	—
56	C₈×D₈₃	—	2	2	no	—
57	C₈×D₈₃	—	2	2	no	—
58	C₄⋊Dic₈₃	—	4	2	no	—
59	C₄⋊Dic₈₃	—	4	2	no	—
60	(C₂×D₈₃)⋊C₄	—	8	4	no	—
61	C₈₃⋊C₁₆	—	2	2	no	—
62	C₂²×C₄×C₈₃	—	4	2	no	—
63	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
69	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
70	C₄⋊C₄×C₈₃	—	4	2	no	—
71	C₄⋊C₄×C₈₃	—	4	2	no	—
72	C₁₆×C₈₃	C₂²⋊C₄	1	4	yes	complemented
73	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
74	C₂²×D₈₃	—	2	1	no	—
75	C₂²×D₈₃	—	2	1	no	—
76	C₂²×D₈₃	—	2	1	no	—
77	C₂²×D₈₃	—	2	1	no	—
78	C₂²×D₈₃	#107	1	2	yes	—
79	C₂²×D₈₃	D₁₆	1	2	yes	—
80	C₂²×D₈₃	Q₃₂	1	2	yes	—
81	C₂×C₄×C₈₃	—	8	2	no	—
82	C₂×C₄×C₈₃	—	8	2	no	—
83	C₂×C₄×C₈₃	—	8	2	no	—
84	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
89	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
90	C₂×Dic₈₃	—	2	1	no	—
91	C₂×Dic₈₃	—	2	1	no	—
92	C₂×Dic₈₃	—	8	2	no	—
93	C₂×Dic₈₃	—	8	2	no	—
94	C₂×Dic₈₃	—	8	2	no	—
95	C₂×Dic₈₃	#107	1	2	yes	—
96	C₄×D₈₃	—	2	2	no	—
97	C₄×D₈₃	—	2	2	no	—
98	C₄×D₈₃	—	2	2	no	—
99	C₄×D₈₃	—	2	2	no	—
100	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₈×C₈₃	#117	1	1	yes	characteristic
102	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₈₃⋊C₈	—	2	2	no	—
105	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
106	C₂²×C₈₃	#466	1	1	yes	characteristic
107	C₂²×C₈₃	#466	1	1	yes	characteristic
108	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
109	C₂²×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
110	C₂²×C₈₃	#508	1	1	yes	characteristic
111	C₂²×C₈₃	#508	1	1	yes	characteristic
112	C₂×D₈₃	—	2	2	no	—
113	C₂×D₈₃	—	2	2	no	—
114	C₂×D₈₃	—	2	2	no	—
115	C₂×D₈₃	—	2	2	no	—
116	C₂×D₈₃	—	2	2	no	—
117	C₂×D₈₃	—	2	2	no	—
118	C₂×D₈₃	#366	1	4	yes	complemented
119	C₄×C₈₃	—	16	2	no	—
120	C₄×C₈₃	#380	1	1	yes	characteristic
121	C₄×C₈₃	#427	1	1	yes	characteristic
122	C₄×C₈₃	#467	1	1	yes	characteristic
123	C₄×C₈₃	#467	1	1	yes	characteristic
124	Dic₈₃	—	2	2	no	—
125	Dic₈₃	—	16	2	no	—
126	#12834	—	83	1	no	maximal, Sylow
127	C₂×C₈₃	#1808	1	1	yes	characteristic
128	C₂×C₈₃	C₂×#366	1	1	yes	characteristic
129	C₂×C₈₃	#2158	1	1	yes	characteristic
130	C₂×C₈₃	#2215	1	1	yes	characteristic
131	C₂×C₈₃	#2215	1	1	yes	characteristic
132	C₂×C₈₃	#2216	1	1	yes	characteristic
133	C₂×C₈₃	#2216	1	1	yes	characteristic
134	D₈₃	—	2	4	no	—
135	C₂³×C₁₆	—	83	1	no	—
136	C₂×#366	—	83	1	no	—
137	C₂×#366	D₈₃	1	1	yes	characteristic, complemented
138	#1808	—	83	2	no	—
139	#1380	—	83	2	no	—
140	C₈₃	#12834	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₈	—	83	1	no	—
142	C₂²×C₁₆	—	83	1	no	—
143	C₂²×C₁₆	—	83	4	no	—
144	C₂²×C₁₆	—	166	2	no	—
145	C₂²×C₁₆	—	166	2	no	—
146	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
147	C₂×#107	—	83	2	no	—
148	C₂×#107	C₂×D₈₃	1	2	yes	—
149	#380	—	166	2	no	—
150	#366	—	2	2	no	—
151	#366	—	166	2	no	—
152	#340	—	166	2	no	—
153	C₂³×C₄	—	83	1	no	—
154	C₂²×C₈	—	83	1	no	—
155	C₂²×C₈	—	83	2	no	—
156	C₂²×C₈	—	83	2	no	—
157	C₂²×C₈	—	166	1	no	—
158	C₂²×C₈	—	166	1	no	—
159	C₂²×C₈	—	166	1	no	—
160	C₂²×C₈	—	166	1	no	—
161	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	166	2	no	—
164	C₂×C₂²⋊C₄	—	332	2	no	—
165	C₂×C₁₆	—	83	2	no	—
166	C₂×C₁₆	—	166	1	no	—
167	C₂×C₁₆	—	166	1	no	—
168	C₂×C₁₆	—	166	8	no	—
169	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
170	C₂×C₁₆	D₃₃₂	1	2	yes	—
171	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
172	#107	—	2	2	no	—
173	#107	—	2	2	no	—
174	#107	—	166	2	no	—
175	#107	—	166	2	no	—
176	#95	—	332	2	no	—
177	C₂⁴	—	83	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	332	2	no	—
180	C₂²×C₄	—	83	1	no	—
181	C₂²×C₄	—	83	2	no	—
182	C₂²×C₄	—	83	2	no	—
183	C₂²×C₄	—	166	1	no	—
184	C₂²×C₄	—	166	1	no	—
185	C₂²×C₄	—	166	1	no	—
186	C₂²×C₄	—	166	1	no	—
187	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
188	C₂×C₈	—	83	2	no	—
189	C₂×C₈	—	166	1	no	—
190	C₂×C₈	—	166	1	no	—
191	C₂×C₈	—	166	2	no	—
192	C₂×C₈	—	166	2	no	—
193	C₂×C₈	—	166	2	no	—
194	C₂×C₈	—	166	2	no	—
195	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
197	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
198	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
199	C₂×C₈	#13050589	1	1	yes	characteristic
200	C₂×C₈	#13050591	1	1	yes	characteristic
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	332	2	no	—
204	C₄⋊C₄	—	332	2	no	—
205	C₂²⋊C₄	—	664	4	no	—
206	C₁₆	—	166	2	no	—
207	C₁₆	(C₂×D₈₃)⋊C₄	1	4	yes	complemented
208	C₂³	—	83	2	no	—
209	C₂³	—	83	2	no	—
210	C₂³	—	83	2	no	—
211	C₂³	—	166	1	no	—
212	C₂³	—	166	1	no	—
213	C₂³	—	166	1	no	—
214	C₂³	—	166	1	no	—
215	C₂³	D₈×D₈₃	1	1	yes	characteristic
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	83	2	no	—
220	C₂×C₄	—	664	2	no	—
221	C₂×C₄	—	664	2	no	—
222	C₂×C₄	—	664	2	no	—
223	C₂×C₄	—	166	1	no	—
224	C₂×C₄	—	166	1	no	—
225	C₂×C₄	—	166	2	no	—
226	C₂×C₄	—	166	2	no	—
227	C₂×C₄	—	166	2	no	—
228	C₂×C₄	—	166	2	no	—
229	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
230	C₂×C₄	#14581763	1	1	yes	characteristic
231	C₂×C₄	#14581767	1	1	yes	characteristic
232	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
233	C₂×C₄	#14581811	1	1	yes	characteristic
234	C₂×C₄	#14581815	1	1	yes	characteristic
235	C₈	—	166	2	no	—
236	C₈	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
237	C₈	#14581750	1	1	yes	characteristic
238	C₈	#14581768	1	1	yes	characteristic
239	C₈	#14581772	1	1	yes	characteristic
240	C₂²	—	83	4	no	—
241	C₂²	—	166	2	no	—
242	C₂²	—	166	2	no	—
243	C₂²	—	166	2	no	—
244	C₂²	—	166	2	no	—
245	C₂²	—	166	2	no	—
246	C₂²	—	166	2	no	—
247	C₂²	#107×D₈₃	1	1	yes	characteristic
248	C₂²	#19127934	1	1	yes	characteristic
249	C₂²	#19127938	1	1	yes	characteristic
250	C₂²	D₁₆×D₈₃	1	1	yes	characteristic
251	C₂²	Q₃₂×D₈₃	1	1	yes	characteristic
252	C₂²	#19128130	1	1	yes	characteristic
253	C₂²	#19128134	1	1	yes	characteristic
254	C₄	—	16	2	no	—
255	C₄	—	1328	2	no	—
256	C₄	—	166	2	no	—
257	C₄	#19127558	1	1	yes	characteristic
258	C₄	#19127738	1	1	yes	characteristic
259	C₄	#19127939	1	1	yes	characteristic
260	C₄	#19127943	1	1	yes	characteristic
261	C₂	—	166	4	no	—
262	C₂	#24907489	1	1	yes	characteristic
263	C₂	#366×D₈₃	1	1	yes	characteristic
264	C₂	#24909244	1	1	yes	characteristic
265	C₂	#24909546	1	1	yes	characteristic
266	C₂	#24909550	1	1	yes	characteristic
267	C₂	#24909551	1	1	yes	characteristic
268	C₂	#24909555	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₃
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#366×C₈₃	C₂
socle	C₂³×C₈₃	C₂×D₈
prime radical	C₂³×D₈₃	D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#366	D₈₃
hypocenter	C₈₃	#12834
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. C₂×#19127527	C₂	semidirect
4. #24905554	C₂	semidirect
5. C₂×#366×C₈₃	C₂	semidirect
10. C₂×C₁₆×D₈₃	C₄	semidirect
72. C₁₆×C₈₃	C₂²⋊C₄	semidirect
118. C₂×D₈₃	#366	semidirect
137. C₂×#366	D₈₃	semidirect
140. C₈₃	#12834	semidirect
207. C₁₆	(C₂×D₈₃)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₈₃	2
C₂×#19127527	2
#24907489	2
#24905554	2
C₂×#366×C₈₃	2
#12834	83