Subgroups — #35558136

Order 21248.

484 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916438	C₂	1	1	yes	maximal, characteristic
2	#24909369	C₂	1	2	yes	maximal, complemented
3	#24909292	C₂	1	2	yes	maximal, complemented
4	#24909291	C₂	1	2	yes	maximal, complemented
5	#24909290	C₂	1	2	yes	maximal, complemented
6	#24909269	C₂	1	2	yes	maximal, complemented
7	#24909220	C₂	1	2	yes	maximal, complemented
8	#24909219	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	#19128235	C₄	1	1	yes	characteristic
11	C₂×#14581732	C₂²	1	1	yes	characteristic
12	C₂×#14581732	C₂²	1	1	yes	characteristic
13	C₂×#14581731	C₂²	1	2	yes	—
14	C₂×C₁₆×D₈₃	C₂²	1	2	yes	—
15	#19127831	C₄	1	1	yes	characteristic
16	C₈×Dic₁₆₆	C₂²	1	1	yes	characteristic
17	#19127780	C₄	1	1	yes	characteristic
18	#19127772	C₄	1	2	yes	—
19	C₈×D₃₃₂	C₂²	1	1	yes	characteristic
20	C₈×C₂²⋊D₈₃	C₂²	1	2	yes	—
21	#19127760	C₄	1	2	yes	—
22	#19127758	C₄	1	1	yes	characteristic
23	#19127655	C₂²	1	2	yes	—
24	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
25	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
26	Dic₈₃⋊C₁₆	C₂²	1	2	yes	—
27	Dic₈₃⋊C₁₆	C₂²	1	2	yes	—
28	(C₂×D₈₃)⋊C₁₆	C₂²	1	2	yes	—
29	(C₂×D₈₃)⋊C₁₆	C₂²	1	2	yes	—
30	#19127476	C₂²	1	2	yes	—
31	C₁₆×Dic₈₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
34	C₄⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
35	C₂²⋊C₁₆×C₈₃	C₂²	1	2	yes	—
36	#346×C₈₃	C₂²	1	2	yes	—
37	C₂×#13050573	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₃	C₂³	1	2	yes	—
39	#14581786	C₈	1	2	yes	—
40	#14581784	C₈	1	1	yes	characteristic
41	#14581780	C₈	1	1	yes	characteristic
42	#14581775	C₈	1	2	yes	—
43	#14581771	C₈	1	1	yes	characteristic
44	#14581766	C₈	1	1	yes	characteristic
45	C₄×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
46	C₄×D₃₃₂	C₂×C₄	1	1	yes	characteristic
47	C₄×C₂²⋊D₈₃	C₂×C₄	1	2	yes	—
48	#14581749	C₂×C₄	1	2	yes	—
49	#14581732	—	4	1	no	—
50	#14581732	—	4	1	no	—
51	#14581731	—	4	2	no	—
52	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
53	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
54	C₁₆×D₈₃	—	4	2	no	—
55	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
56	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
57	Dic₈₃⋊C₈	C₂×C₄	1	2	yes	—
58	Dic₈₃⋊C₈	C₂×C₄	1	2	yes	—
59	C₂²⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
60	(C₂×D₈₃)⋊C₈	C₂×C₄	1	2	yes	—
61	(C₂×D₈₃)⋊C₈	C₂×C₄	1	2	yes	—
62	#14581703	C₂×C₄	1	2	yes	—
63	#14581702	C₂×C₄	1	2	yes	—
64	C₈×Dic₈₃	C₂³	1	2	yes	—
65	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
69	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
70	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
74	C₂×Dic₁₆₆	C₂×C₈	1	1	yes	characteristic
75	C₂×D₃₃₂	C₂×C₈	1	1	yes	characteristic
76	C₂×C₂²⋊D₈₃	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
78	#13050573	—	4	2	no	—
79	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
80	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
81	C₈×D₈₃	—	2	2	no	—
82	C₈×D₈₃	—	2	2	no	—
83	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
84	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
85	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
86	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
87	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
88	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
89	C₂²⋊Dic₈₃	C₂×C₈	1	2	yes	—
90	(C₂×D₈₃)⋊C₄	C₂×C₈	1	2	yes	—
91	(C₂×D₈₃)⋊C₄	C₂×C₈	1	2	yes	—
92	(C₂×D₈₃)⋊C₄	C₂×C₈	1	2	yes	—
93	(C₂×D₈₃)⋊C₄	C₂×C₈	1	2	yes	—
94	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
95	C₈₃⋊C₁₆	—	2	2	no	—
96	C₈₃⋊C₁₆	—	2	2	no	—
97	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₃	—	2	1	no	—
101	C₂×C₈×C₈₃	—	2	1	no	—
102	C₂×C₈×C₈₃	—	4	1	no	—
103	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
107	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
108	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
109	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
111	C₁₆×C₈₃	—	2	2	no	—
112	C₁₆×C₈₃	—	2	2	no	—
113	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
114	C₂²×D₈₃	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	4	1	no	—
118	C₂×C₄×C₈₃	—	4	1	no	—
119	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
125	C₂²⋊D₈₃	—	4	2	no	—
126	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
127	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
129	D₃₃₂	—	4	1	no	—
130	C₄×D₈₃	—	2	2	no	—
131	C₄×D₈₃	—	2	2	no	—
132	Dic₁₆₆	—	4	1	no	—
133	C₈×C₈₃	—	2	1	no	—
134	C₈×C₈₃	—	2	1	no	—
135	C₈×C₈₃	—	2	1	no	—
136	C₈×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
138	C₈₃⋊C₈	—	2	2	no	—
139	C₈₃⋊C₈	—	2	2	no	—
140	C₂²×C₈₃	—	2	1	no	—
141	C₂²×C₈₃	—	2	1	no	—
142	C₂²×C₈₃	—	2	1	no	—
143	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
144	C₂×D₈₃	—	2	2	no	—
145	C₂×D₈₃	—	2	2	no	—
146	C₂×D₈₃	—	2	2	no	—
147	C₄×C₈₃	—	2	1	no	—
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	#520	1	1	yes	characteristic
153	C₄×C₈₃	#568	1	1	yes	characteristic
154	Dic₈₃	—	2	2	no	—
155	Dic₈₃	—	2	2	no	—
156	Dic₈₃	—	2	2	no	—
157	#33509	—	83	1	no	maximal, Sylow
158	C₂×C₈₃	—	4	1	no	—
159	C₂×C₈₃	#2988	1	1	yes	characteristic
160	C₂×C₈₃	#3407	1	1	yes	characteristic
161	C₂×C₈₃	#3407	1	1	yes	characteristic
162	D₈₃	—	4	2	no	—
163	#2966	—	83	1	no	—
164	#2183	—	83	2	no	—
165	#2169	—	83	2	no	—
166	#2168	—	83	2	no	—
167	#2168	—	83	2	no	—
168	#2164	—	83	2	no	—
169	#2163	D₈₃	1	1	yes	characteristic, complemented
170	#2151	—	83	2	no	—
171	#2150	—	83	1	no	—
172	C₈₃	#33509	1	1	yes	characteristic, complemented, Sylow
173	#529	—	83	1	no	—
174	C₂×M₃₂	—	83	1	no	—
175	C₂×M₃₂	—	83	1	no	—
176	C₂×M₃₂	—	83	2	no	—
177	C₂²×C₁₆	—	83	2	no	—
178	#446	—	83	1	no	—
179	C₈×Q₈	—	83	1	no	—
180	#436	—	83	1	no	—
181	#435	—	83	2	no	—
182	C₈×D₄	—	83	2	no	—
183	C₈×D₄	—	83	1	no	—
184	#433	—	83	2	no	—
185	#432	—	83	1	no	—
186	#405	—	83	2	no	—
187	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
188	C₄⋊C₁₆	—	83	2	no	—
189	C₄⋊C₁₆	—	83	2	no	—
190	C₄⋊C₁₆	—	83	2	no	—
191	C₄⋊C₁₆	—	83	2	no	—
192	C₄⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
193	C₄⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
194	C₂²⋊C₁₆	—	83	2	no	—
195	C₂²⋊C₁₆	—	83	2	no	—
196	C₂²⋊C₁₆	C₂×D₈₃	1	2	yes	—
197	#346	—	83	2	no	—
198	#346	C₂×D₈₃	1	2	yes	—
199	C₄×C₁₆	—	83	2	no	—
200	C₂×M₁₆	—	83	2	no	—
201	C₂²×C₈	—	83	2	no	—
202	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
204	#126	—	83	2	no	—
205	#125	—	83	1	no	—
206	#124	—	83	1	no	—
207	#123	—	83	2	no	—
208	C₂²⋊Q₈	—	83	1	no	—
209	#121	—	83	1	no	—
210	C₄×Q₈	—	83	1	no	—
211	C₄×D₄	—	83	2	no	—
212	C₄×D₄	—	83	1	no	—
213	#117	—	83	2	no	—
214	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
215	M₃₂	—	332	2	no	—
216	M₃₂	—	332	1	no	—
217	M₃₂	—	332	1	no	—
218	C₂×C₁₆	—	332	2	no	—
219	C₂×C₁₆	—	83	2	no	—
220	C₂×C₁₆	—	83	2	no	—
221	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
222	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
223	C₄⋊C₈	—	83	2	no	—
224	C₄⋊C₈	—	83	2	no	—
225	C₄⋊C₈	—	83	1	no	—
226	C₄⋊C₈	—	83	1	no	—
227	C₂²⋊C₈	—	83	2	no	—
228	C₂²⋊C₈	—	83	2	no	—
229	C₂²⋊C₈	—	83	2	no	—
230	#97	—	83	2	no	—
231	#97	—	83	2	no	—
232	C₄×C₈	—	83	2	no	—
233	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
234	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
237	C₂×Q₈	—	83	1	no	—
238	C₂×D₄	—	83	2	no	—
239	C₂×D₄	—	83	1	no	—
240	C₂²×C₄	—	83	2	no	—
241	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
242	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
244	M₁₆	—	332	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	83	2	no	—
249	C₂×C₈	—	83	2	no	—
250	C₂×C₈	—	166	2	no	—
251	C₂×C₈	—	166	2	no	—
252	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
256	C₄⋊C₄	—	83	2	no	—
257	C₄⋊C₄	—	83	2	no	—
258	C₄⋊C₄	—	83	1	no	—
259	C₄⋊C₄	—	83	2	no	—
260	C₄⋊C₄	—	83	1	no	—
261	C₄⋊C₄	—	83	2	no	—
262	C₂²⋊C₄	—	83	2	no	—
263	C₂²⋊C₄	—	83	2	no	—
264	C₂²⋊C₄	—	83	2	no	—
265	C₂²⋊C₄	—	83	2	no	—
266	C₂²⋊C₄	—	83	2	no	—
267	C₄²	—	83	2	no	—
268	C₄²	C₈×D₈₃	1	1	yes	characteristic
269	C₄²	C₈×D₈₃	1	1	yes	characteristic
270	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
271	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	—	2	2	no	—
274	C₁₆	—	166	2	no	—
275	C₁₆	—	166	2	no	—
276	C₂³	—	83	2	no	—
277	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
278	Q₈	—	332	1	no	—
279	D₄	—	332	1	no	—
280	D₄	—	332	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	83	2	no	—
286	C₂×C₄	—	83	2	no	—
287	C₂×C₄	—	83	2	no	—
288	C₂×C₄	—	166	2	no	—
289	C₂×C₄	—	166	2	no	—
290	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
291	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	166	2	no	—
300	C₈	—	166	2	no	—
301	C₈	#14581838	1	1	yes	characteristic
302	C₈	#14581839	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	166	2	no	—
307	C₂²	—	166	2	no	—
308	C₂²	—	166	2	no	—
309	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	166	2	no	—
316	C₄	—	166	2	no	—
317	C₄	—	166	2	no	—
318	C₄	#19128181	1	1	yes	characteristic
319	C₄	#19128467	1	1	yes	characteristic
320	C₂	—	4	1	no	—
321	C₂	—	332	2	no	—
322	C₂	#24916571	1	1	yes	characteristic
323	C₂	#24920565	1	1	yes	characteristic
324	C₂	#24920565	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₈
Fitting	C₈₃×#2163	C₂
prime radical	#14581766	C₈
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#2163	D₈₃
hypocenter	C₈₃	#33509
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #24909369	C₂	semidirect
3. #24909292	C₂	semidirect
4. #24909291	C₂	semidirect
5. #24909290	C₂	semidirect
6. #24909269	C₂	semidirect
7. #24909220	C₂	semidirect
8. #24909219	C₂	semidirect
9. C₈₃×#2163	C₂	semidirect
169. #2163	D₈₃	semidirect
172. C₈₃	#33509	semidirect

Maximal subgroups

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

subgroup	index
#24916438	2
#24909369	2
#24909292	2
#24909291	2
#24909290	2
#24909269	2
#24909220	2
#24909219	2
C₈₃×#2163	2
#33509	83