Subgroups — #35295366

Order 21248.

516 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581792	C₂	1	1	yes	maximal, characteristic
2	#24907316	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127446	C₂	1	2	yes	maximal
4	#24905347	C₂	1	2	yes	maximal
5	C₈₃×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₆₆	—	2	1	no	—
8	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14581703	—	2	1	no	—
10	#19127607	—	2	1	no	—
11	#19127599	—	2	1	no	—
12	#19127599	C₄	1	2	yes	—
13	#19127571	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊Dic₈₃	—	2	1	no	—
15	C₄×C₄⋊Dic₈₃	C₄	1	2	yes	—
16	#19127456	—	2	1	no	—
17	#19127446	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
19	C₂×#97×C₈₃	—	2	1	no	—
20	C₄³×C₈₃	C₂²	1	1	yes	characteristic
21	#336×C₈₃	—	2	1	no	—
22	C₂²×Dic₁₆₆	—	2	1	no	—
23	C₂²×C₈₃⋊C₈	—	2	2	no	—
24	#14581792	—	2	2	no	—
25	#14581792	C₂×C₄	1	4	yes	—
26	C₄×Dic₁₆₆	—	4	4	no	—
27	C₂×C₄⋊Dic₈₃	—	2	1	no	—
28	C₂×C₄⋊Dic₈₃	—	2	1	no	—
29	C₂×C₄⋊Dic₈₃	—	2	1	no	—
30	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
32	C₂×Dic₈₃⋊C₄	—	2	2	no	—
33	C₂×C₄×Dic₈₃	—	2	2	no	—
34	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
35	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	4	yes	—
36	#14581703	—	4	2	no	—
37	C₈₃⋊#95	—	2	2	no	—
38	C₈₃⋊#95	—	2	2	no	—
39	C₂²×C₈×C₈₃	—	2	2	no	—
40	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₈₃	—	2	2	no	—
48	C₄⋊C₈×C₈₃	C₂×C₄	1	4	yes	—
49	#97×C₈₃	—	4	2	no	—
50	C₂×Dic₁₆₆	—	2	4	no	—
51	C₂×Dic₁₆₆	—	4	1	no	—
52	C₂×Dic₁₆₆	—	4	2	no	—
53	C₂×Dic₁₆₆	—	4	1	no	—
54	C₂²×Dic₈₃	—	2	2	no	—
55	C₂²×Dic₈₃	—	2	2	no	—
56	C₂×C₈₃⋊C₈	—	2	4	no	—
57	C₂×C₈₃⋊C₈	—	4	2	no	—
58	C₂×C₈₃⋊C₈	—	4	2	no	—
59	C₄⋊Dic₈₃	—	2	2	no	—
60	C₄⋊Dic₈₃	—	2	4	no	—
61	C₄⋊Dic₈₃	—	4	1	no	—
62	C₄⋊Dic₈₃	—	4	2	no	—
63	C₄⋊Dic₈₃	—	4	1	no	—
64	C₄⋊Dic₈₃	—	4	2	no	—
65	C₄⋊Dic₈₃	C₄²	1	4	yes	—
66	Dic₈₃⋊C₄	—	4	4	no	—
67	C₄×Dic₈₃	—	4	4	no	—
68	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₃	—	2	4	no	—
76	C₂×C₈×C₈₃	—	4	2	no	—
77	C₂×C₈×C₈₃	—	4	2	no	—
78	C₄²×C₈₃	—	2	2	no	—
79	C₄²×C₈₃	—	2	1	no	—
80	C₄²×C₈₃	—	2	1	no	—
81	C₄²×C₈₃	—	2	2	no	—
82	C₄²×C₈₃	—	2	2	no	—
83	C₄²×C₈₃	—	2	1	no	—
84	C₄²×C₈₃	—	2	1	no	—
85	C₄²×C₈₃	—	4	1	no	—
86	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₈₃	#41	1	2	yes	—
88	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
94	C₂×C₄×C₈₃	—	2	1	no	—
95	C₂×C₄×C₈₃	—	2	1	no	—
96	C₂×C₄×C₈₃	—	2	1	no	—
97	C₂×C₄×C₈₃	—	2	1	no	—
98	C₂×C₄×C₈₃	—	2	1	no	—
99	C₂×C₄×C₈₃	—	2	1	no	—
100	C₂×C₄×C₈₃	—	2	1	no	—
101	C₂×C₄×C₈₃	—	2	1	no	—
102	C₂×C₄×C₈₃	#101	1	2	yes	—
103	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₈₃	#137	1	2	yes	—
111	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
112	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
113	C₂×Dic₈₃	—	2	4	no	—
114	C₂×Dic₈₃	—	4	2	no	—
115	C₂×Dic₈₃	—	4	2	no	—
116	C₂×Dic₈₃	—	4	2	no	—
117	C₂×Dic₈₃	—	4	2	no	—
118	C₂×Dic₈₃	—	4	2	no	—
119	Dic₁₆₆	—	4	4	no	—
120	Dic₁₆₆	—	4	4	no	—
121	C₈×C₈₃	—	4	4	no	—
122	C₈₃⋊C₈	—	4	4	no	—
123	C₂²×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₈₃	C₂×#101	1	1	yes	characteristic
125	C₂²×C₈₃	#413	1	1	yes	characteristic
126	C₂²×C₈₃	#417	1	1	yes	characteristic
127	C₂²×C₈₃	#417	1	1	yes	characteristic
128	C₂²×C₈₃	#418	1	1	yes	characteristic
129	C₂²×C₈₃	#419	1	1	yes	characteristic
130	C₄×C₈₃	—	2	2	no	—
131	C₄×C₈₃	—	2	1	no	—
132	C₄×C₈₃	—	2	1	no	—
133	C₄×C₈₃	—	4	1	no	—
134	C₄×C₈₃	—	4	1	no	—
135	C₄×C₈₃	—	4	1	no	—
136	C₄×C₈₃	#440	1	2	yes	—
137	C₄×C₈₃	#440	1	2	yes	—
138	C₄×C₈₃	#441	1	2	yes	—
139	C₄×C₈₃	#442	1	2	yes	—
140	Dic₈₃	—	4	4	no	—
141	Dic₈₃	—	8	2	no	—
142	#8233	—	83	1	no	maximal, Sylow
143	C₂×C₈₃	#1556	1	1	yes	characteristic
144	C₂×C₈₃	#1557	1	1	yes	characteristic
145	C₂×C₈₃	#1559	1	1	yes	characteristic
146	C₂×C₈₃	#1561	1	1	yes	characteristic
147	C₂×C₈₃	C₄×#101	1	1	yes	characteristic
148	C₂×C₈₃	#1762	1	1	yes	characteristic
149	C₂×C₈₃	#1763	1	1	yes	characteristic
150	C₄×C₄⋊Q₈	—	83	1	no	—
151	#1767	—	83	1	no	—
152	#1767	D₈₃	1	1	yes	characteristic
153	C₂×#332	—	83	2	no	—
154	#1296	—	83	2	no	—
155	C₈₃	#8233	1	1	yes	characteristic, complemented, Sylow
156	C₂×C₄⋊Q₈	—	83	1	no	—
157	C₂×C₄×Q₈	—	166	1	no	—
158	C₂×C₄⋊C₈	—	83	2	no	—
159	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
160	C₂×#97	—	2	1	no	—
161	C₂×#97	—	166	1	no	—
162	#391	—	166	1	no	—
163	#389	—	83	2	no	—
164	#389	—	166	1	no	—
165	#384	—	83	1	no	—
166	C₄×C₄⋊C₄	—	83	2	no	—
167	C₄×C₄⋊C₄	—	166	1	no	—
168	C₄³	C₂×D₈₃	1	1	yes	characteristic
169	#336	—	2	1	no	—
170	#336	—	166	1	no	—
171	#332	—	83	8	no	—
172	C₂²×Q₈	—	166	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	166	2	no	—
175	C₄⋊Q₈	—	83	4	no	—
176	C₄⋊Q₈	—	166	2	no	—
177	C₄×Q₈	—	332	4	no	—
178	C₂×C₄⋊C₄	—	83	2	no	—
179	C₂×C₄⋊C₄	—	83	2	no	—
180	C₂×C₄⋊C₄	—	166	2	no	—
181	C₂×C₄⋊C₄	—	166	1	no	—
182	C₂×C₄⋊C₄	—	166	1	no	—
183	C₂×C₄⋊C₄	—	166	1	no	—
184	C₂×C₄²	—	166	2	no	—
185	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
187	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
188	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	83	4	no	—
194	C₄⋊C₈	—	166	2	no	—
195	C₄⋊C₈	C₄×D₈₃	1	4	yes	—
196	#97	—	4	2	no	—
197	#97	—	332	2	no	—
198	#95	—	166	2	no	—
199	#95	—	166	2	no	—
200	C₂×Q₈	—	332	2	no	—
201	C₂×Q₈	—	332	1	no	—
202	C₂×Q₈	—	332	1	no	—
203	C₂×Q₈	—	166	4	no	—
204	C₂²×C₄	—	166	2	no	—
205	C₂²×C₄	—	166	2	no	—
206	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	332	2	no	—
217	C₂×C₈	—	332	2	no	—
218	C₂×C₈	—	166	4	no	—
219	C₄⋊C₄	—	332	2	no	—
220	C₄⋊C₄	—	332	4	no	—
221	C₄⋊C₄	—	332	1	no	—
222	C₄⋊C₄	—	332	2	no	—
223	C₄⋊C₄	—	332	1	no	—
224	C₄⋊C₄	—	83	4	no	—
225	C₄⋊C₄	—	166	2	no	—
226	C₄⋊C₄	—	166	4	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	332	4	no	—
236	C₄²	C₂×C₄×D₈₃	1	2	yes	—
237	C₄²	#13050590	1	2	yes	—
238	C₂³	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
239	Q₈	—	332	4	no	—
240	Q₈	—	332	4	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	332	2	no	—
255	C₂×C₄	—	332	2	no	—
256	C₂×C₄	—	332	2	no	—
257	C₂×C₄	—	332	2	no	—
258	C₂×C₄	—	332	2	no	—
259	C₂×C₄	—	166	4	no	—
260	C₂×C₄	#14581710	1	2	yes	—
261	C₂×C₄	C₄²×D₈₃	1	1	yes	characteristic
262	C₂×C₄	#14581749	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₈₃	1	2	yes	—
264	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
265	C₂×C₄	#14581818	1	1	yes	characteristic
266	C₂×C₄	#14581820	1	1	yes	characteristic
267	C₂×C₄	#14581820	1	2	yes	—
268	C₂×C₄	#14581822	1	2	yes	—
269	C₂×C₄	#14581823	1	1	yes	characteristic
270	C₂×C₄	#14581825	1	1	yes	characteristic
271	C₈	—	4	4	no	—
272	C₈	—	332	4	no	—
273	C₂²	C₄×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
274	C₂²	C₂×#14581710	1	1	yes	characteristic
275	C₂²	#19127683	1	1	yes	characteristic
276	C₂²	#19127700	1	1	yes	characteristic
277	C₂²	#19127703	1	1	yes	characteristic
278	C₂²	#19127707	1	1	yes	characteristic
279	C₂²	#19127709	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	332	4	no	—
287	C₄	—	664	2	no	—
288	C₄	#19127801	1	2	yes	—
289	C₄	#19127803	1	2	yes	—
290	C₄	#19127806	1	2	yes	—
291	C₄	#19127814	1	2	yes	—
292	C₂	#24906179	1	1	yes	characteristic
293	C₂	#24906183	1	1	yes	characteristic
294	C₂	#24906194	1	1	yes	characteristic
295	C₂	#24906200	1	1	yes	characteristic
296	C₂	C₄×#14581710	1	1	yes	characteristic
297	C₂	#24907289	1	1	yes	characteristic
298	C₂	#24907296	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₈₃)⋊C₄
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1767	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1767	D₈₃
hypocenter	C₈₃	#8233
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
17. #19127446	C₄	semidirect
155. C₈₃	#8233	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581792	2
#24907316	2
C₂×#19127446	2
#24905347	2
C₈₃×#1767	2
#8233	83