Subgroups — #35322401

Order 21248.

350 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₈₃⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#24905541	C₂	1	1	yes	maximal, characteristic
6	#24905540	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×Dic₈₃⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
13	C₄×Dic₈₃⋊C₄	C₄	1	1	yes	characteristic
14	Dic₈₃⋊C₁₆	—	2	2	no	—
15	Dic₈₃⋊C₁₆	—	2	2	no	—
16	C₁₆×Dic₈₃	—	2	2	no	—
17	#19127456	C₄	1	1	yes	characteristic
18	#19127455	C₄	1	1	yes	characteristic
19	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₈₃	—	2	2	no	—
23	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₈₃⋊C₄	C₈	1	1	yes	characteristic
26	C₂×Dic₈₃⋊C₄	C₈	1	1	yes	characteristic
27	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈₃⋊C₁₆	—	2	1	no	—
30	C₂×C₈₃⋊C₁₆	—	2	1	no	—
31	C₂×C₈₃⋊C₁₆	—	2	1	no	—
32	C₂×C₈₃⋊C₁₆	—	2	1	no	—
33	C₂×C₈₃⋊C₁₆	—	2	1	no	—
34	C₂×C₈₃⋊C₁₆	Q₈	1	2	yes	—
35	Dic₈₃⋊C₈	—	2	2	no	—
36	C₈×Dic₈₃	—	2	1	no	—
37	C₈×Dic₈₃	—	2	1	no	—
38	C₈×Dic₈₃	D₄	1	2	yes	—
39	C₈×Dic₈₃	Q₈	1	2	yes	—
40	C₈₃⋊#95	C₈	1	1	yes	characteristic
41	C₈₃⋊#95	C₈	1	1	yes	characteristic
42	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₈₃	—	2	1	no	—
46	C₂×C₁₆×C₈₃	—	2	1	no	—
47	C₂×C₁₆×C₈₃	—	2	1	no	—
48	C₂×C₁₆×C₈₃	—	2	1	no	—
49	C₂×C₁₆×C₈₃	—	2	1	no	—
50	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
51	C₄×C₈×C₈₃	—	2	1	no	—
52	C₄×C₈×C₈₃	—	2	1	no	—
53	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈₃⋊C₈	—	2	1	no	—
58	C₂×C₈₃⋊C₈	—	2	1	no	—
59	C₂×C₈₃⋊C₈	—	2	1	no	—
60	C₂×C₈₃⋊C₈	—	2	1	no	—
61	C₂×C₈₃⋊C₈	—	2	1	no	—
62	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
63	Dic₈₃⋊C₄	—	2	2	no	—
64	Dic₈₃⋊C₄	C₁₆	1	4	yes	complemented
65	C₄×Dic₈₃	—	2	1	no	—
66	C₄×Dic₈₃	—	2	1	no	—
67	C₄×Dic₈₃	C₄⋊C₄	1	2	yes	—
68	C₄×Dic₈₃	M₁₆	1	2	yes	—
69	C₈₃⋊C₁₆	—	2	2	no	—
70	C₈₃⋊C₁₆	—	4	1	no	—
71	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₃	—	2	1	no	—
75	C₂×C₈×C₈₃	—	2	1	no	—
76	C₂×C₈×C₈₃	—	2	1	no	—
77	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
83	C₄²×C₈₃	—	2	1	no	—
84	C₄²×C₈₃	—	2	1	no	—
85	C₁₆×C₈₃	—	2	2	no	—
86	C₁₆×C₈₃	—	4	1	no	—
87	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₈₃	—	2	1	no	—
89	C₂×C₄×C₈₃	—	2	1	no	—
90	C₂×C₄×C₈₃	—	2	1	no	—
91	C₂×C₄×C₈₃	—	2	1	no	—
92	C₂×C₄×C₈₃	—	2	1	no	—
93	C₂×C₄×C₈₃	C₂×C₁₆	1	2	yes	—
94	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
99	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
100	C₂×Dic₈₃	—	2	1	no	—
101	C₂×Dic₈₃	—	2	1	no	—
102	C₂×Dic₈₃	—	2	1	no	—
103	C₂×Dic₈₃	—	2	1	no	—
104	C₂×Dic₈₃	—	2	1	no	—
105	C₂×Dic₈₃	—	2	1	no	—
106	C₂×Dic₈₃	—	2	1	no	—
107	C₂×Dic₈₃	C₄⋊C₈	1	2	yes	—
108	C₂×Dic₈₃	C₄⋊C₈	1	2	yes	—
109	C₂×Dic₈₃	C₂×C₁₆	1	2	yes	—
110	C₂×Dic₈₃	C₂×C₁₆	1	2	yes	—
111	C₂×Dic₈₃	M₃₂	1	2	yes	—
112	C₈×C₈₃	—	4	1	no	—
113	C₈×C₈₃	#121	1	1	yes	characteristic
114	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₈×C₈₃	#125	1	1	yes	characteristic
116	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
117	C₈₃⋊C₈	—	2	2	no	—
118	C₈₃⋊C₈	—	4	1	no	—
119	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₈₃	C₈×Q₈	1	1	yes	characteristic
122	C₂²×C₈₃	C₂²×C₁₆	1	1	yes	characteristic
123	C₂²×C₈₃	C₂×M₃₂	1	1	yes	characteristic
124	C₂²×C₈₃	#504	1	1	yes	characteristic
125	C₂²×C₈₃	#504	1	1	yes	characteristic
126	C₄×C₈₃	—	2	2	no	—
127	C₄×C₈₃	—	4	1	no	—
128	C₄×C₈₃	#389	1	1	yes	characteristic
129	C₄×C₈₃	#424	1	1	yes	characteristic
130	C₄×C₈₃	#436	1	1	yes	characteristic
131	C₄×C₈₃	#446	1	1	yes	characteristic
132	Dic₈₃	—	2	2	no	—
133	Dic₈₃	—	2	2	no	—
134	Dic₈₃	—	4	1	no	—
135	Dic₈₃	C₄⋊C₁₆	1	4	yes	complemented
136	#13159	—	83	1	no	maximal, Sylow
137	C₂×C₈₃	#1939	1	1	yes	characteristic
138	C₂×C₈₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
139	C₂×C₈₃	#2151	1	1	yes	characteristic
140	C₂×C₈₃	D₄×C₁₆	1	1	yes	characteristic
141	C₂×C₈₃	#2169	1	1	yes	characteristic
142	C₂×C₈₃	Q₈×C₁₆	1	1	yes	characteristic
143	C₂×C₈₃	#2183	1	1	yes	characteristic
144	C₂×C₄⋊C₁₆	—	83	1	no	—
145	C₂×C₄⋊C₁₆	—	83	1	no	—
146	C₂×C₄⋊C₁₆	D₈₃	1	1	yes	characteristic
147	C₂×C₄×C₁₆	—	83	1	no	—
148	C₈×C₄⋊C₄	—	83	1	no	—
149	#1374	—	83	1	no	—
150	#1374	—	83	1	no	—
151	C₈₃	#13159	1	1	yes	characteristic, complemented, Sylow
152	C₂²×C₁₆	—	83	1	no	—
153	C₂²×C₁₆	—	83	1	no	—
154	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
155	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
156	C₂×C₄⋊C₈	—	83	1	no	—
157	C₂×C₄×C₈	—	83	1	no	—
158	C₂×C₄×C₈	—	83	1	no	—
159	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
160	C₄×C₄⋊C₄	—	83	1	no	—
161	C₄⋊C₁₆	—	2	2	no	—
162	C₄⋊C₁₆	—	166	2	no	—
163	C₄⋊C₁₆	—	166	2	no	—
164	C₄×C₁₆	—	166	2	no	—
165	#336	—	83	1	no	—
166	#336	—	83	1	no	—
167	C₂²×C₈	—	83	1	no	—
168	C₂²×C₈	—	83	1	no	—
169	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	—	83	1	no	—
172	C₂×C₄⋊C₄	—	83	1	no	—
173	C₂×C₄²	—	83	1	no	—
174	C₂×C₄²	—	83	1	no	—
175	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	83	2	no	—
182	C₂×C₁₆	—	166	1	no	—
183	C₂×C₁₆	—	166	1	no	—
184	C₂×C₁₆	—	166	1	no	—
185	C₂×C₁₆	—	166	1	no	—
186	C₂×C₁₆	—	166	1	no	—
187	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
188	C₄⋊C₈	—	166	2	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	83	2	no	—
192	C₄×C₈	—	83	2	no	—
193	C₄×C₈	—	166	1	no	—
194	C₄×C₈	—	166	1	no	—
195	#95	—	83	1	no	—
196	#95	—	83	1	no	—
197	C₂²×C₄	—	83	1	no	—
198	C₂²×C₄	—	83	1	no	—
199	C₂²×C₄	—	83	1	no	—
200	C₂²×C₄	—	83	1	no	—
201	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
202	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	83	2	no	—
208	C₂×C₈	—	166	1	no	—
209	C₂×C₈	—	166	1	no	—
210	C₂×C₈	—	166	1	no	—
211	C₂×C₈	—	166	1	no	—
212	C₂×C₈	—	166	1	no	—
213	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
215	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
216	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
217	C₂×C₈	#13050589	1	1	yes	characteristic
218	C₂×C₈	#13050590	1	1	yes	characteristic
219	C₄⋊C₄	—	83	4	no	—
220	C₄⋊C₄	—	166	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	83	2	no	—
224	C₄²	—	83	2	no	—
225	C₄²	—	166	1	no	—
226	C₄²	—	166	1	no	—
227	C₁₆	—	2	2	no	—
228	C₁₆	—	4	1	no	—
229	C₁₆	—	332	1	no	—
230	C₁₆	—	166	2	no	—
231	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	83	2	no	—
238	C₂×C₄	—	83	2	no	—
239	C₂×C₄	—	83	2	no	—
240	C₂×C₄	—	83	2	no	—
241	C₂×C₄	—	83	2	no	—
242	C₂×C₄	—	166	1	no	—
243	C₂×C₄	—	166	1	no	—
244	C₂×C₄	—	166	1	no	—
245	C₂×C₄	—	166	1	no	—
246	C₂×C₄	—	166	1	no	—
247	C₂×C₄	—	166	1	no	—
248	C₂×C₄	—	166	1	no	—
249	C₂×C₄	C₁₆×D₈₃	1	2	yes	—
250	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
251	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
252	C₂×C₄	#14581757	1	1	yes	characteristic
253	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
254	C₂×C₄	#14581798	1	1	yes	characteristic
255	C₂×C₄	#14581800	1	1	yes	characteristic
256	C₈	—	4	1	no	—
257	C₈	—	332	1	no	—
258	C₈	—	166	2	no	—
259	C₈	#14581764	1	1	yes	characteristic
260	C₈	#14581770	1	1	yes	characteristic
261	C₈	#14581782	1	1	yes	characteristic
262	C₈	#14581790	1	1	yes	characteristic
263	C₂²	C₄⋊C₈×D₈₃	1	1	yes	characteristic
264	C₂²	C₈×C₂²⋊D₈₃	1	1	yes	characteristic
265	C₂²	#19127825	1	1	yes	characteristic
266	C₂²	C₂×C₁₆×D₈₃	1	1	yes	characteristic
267	C₂²	M₃₂×D₈₃	1	1	yes	characteristic
268	C₂²	#19128117	1	1	yes	characteristic
269	C₂²	#19128119	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	332	1	no	—
273	C₄	—	83	4	no	—
274	C₄	—	166	2	no	—
275	C₄	—	166	2	no	—
276	C₄	#19127596	1	1	yes	characteristic
277	C₄	#19127726	1	1	yes	characteristic
278	C₄	#19127777	1	1	yes	characteristic
279	C₄	#19127830	1	1	yes	characteristic
280	C₂	#24908178	1	1	yes	characteristic
281	C₂	C₄⋊C₁₆×D₈₃	1	1	yes	characteristic
282	C₂	#24909220	1	1	yes	characteristic
283	C₂	C₁₆×C₂²⋊D₈₃	1	1	yes	characteristic
284	C₂	#24909293	1	1	yes	characteristic
285	C₂	#24909365	1	1	yes	characteristic
286	C₂	#24909370	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂²×C₁₆
Fitting	C₂×C₄⋊C₁₆×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₃
hypocenter	C₈₃	#13159
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
64. Dic₈₃⋊C₄	C₁₆	semidirect
135. Dic₈₃	C₄⋊C₁₆	semidirect
151. C₈₃	#13159	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₃⋊C₁₆	2
C₂×C₁₆×Dic₈₃	2
C₈×Dic₈₃⋊C₄	2
#24905541	2
#24905540	2
C₂×C₄⋊C₁₆×C₈₃	2
#13159	83