Subgroups — #35310032

Order 21248.

650 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈₃⋊#338	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127458	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127457	C₂	1	2	yes	maximal
4	#24907159	C₂	1	2	yes	maximal
5	C₂×#405×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13050574	C₂²	1	2	yes	—
7	C₂×#14581751	C₂²	1	2	yes	—
8	C₈₃⋊#338	—	2	2	no	—
9	#19127458	—	2	2	no	—
10	#19127457	—	2	4	no	—
11	#19127455	C₄	1	8	yes	complemented
12	C₂²×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×#117×C₈₃	C₂²	1	1	yes	characteristic
15	#405×C₈₃	—	2	2	no	—
16	#405×C₈₃	—	2	2	no	—
17	C₂×#97×C₈₃	C₄	1	2	yes	complemented
18	C₂×C₄×C₈×C₈₃	C₄	1	2	yes	complemented
19	C₂×#13050574	—	2	2	no	—
20	C₂×#13050574	—	2	2	no	—
21	C₂×#13050574	—	2	2	no	—
22	C₂×#13050574	—	2	2	no	—
23	C₂×#13050574	—	2	2	no	—
24	C₂×#13050574	—	2	2	no	—
25	C₂²×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
26	#14581751	—	2	2	no	—
27	#14581751	—	2	2	no	—
28	#14581751	—	2	2	no	—
29	#14581751	—	2	2	no	—
30	C₂×C₄⋊Dic₈₃	—	2	2	no	—
31	C₂×C₂²⋊Dic₈₃	—	2	2	no	—
32	C₂×C₄×Dic₈₃	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₈₃	—	2	2	no	—
35	C₂×M₁₆×C₈₃	—	2	2	no	—
36	C₂×M₁₆×C₈₃	D₄	1	2	yes	—
37	C₂×M₁₆×C₈₃	D₄	1	2	yes	—
38	C₂²×C₈×C₈₃	—	2	1	no	—
39	C₂²×C₈×C₈₃	—	2	1	no	—
40	C₂²×C₈×C₈₃	—	2	2	no	—
41	C₂²×C₈×C₈₃	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
45	#117×C₈₃	—	2	1	no	—
46	#117×C₈₃	—	2	1	no	—
47	#117×C₈₃	D₄	1	2	yes	—
48	#117×C₈₃	Q₈	1	2	yes	—
49	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
51	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
52	#97×C₈₃	—	2	4	no	—
53	C₄×C₈×C₈₃	—	2	4	no	—
54	C₂²×Dic₈₃	—	2	4	no	—
55	#13050574	—	4	2	no	—
56	#13050574	—	4	2	no	—
57	#13050574	—	4	2	no	—
58	#13050574	—	4	2	no	—
59	C₂×C₈₃⋊C₈	—	2	4	no	—
60	C₂×C₈₃⋊C₈	—	2	4	no	—
61	C₂×C₈₃⋊C₈	—	2	4	no	—
62	C₄⋊Dic₈₃	—	4	2	no	—
63	C₄⋊Dic₈₃	—	4	2	no	—
64	C₂²⋊Dic₈₃	—	4	2	no	—
65	C₂²⋊Dic₈₃	—	4	2	no	—
66	C₄×Dic₈₃	—	2	4	no	—
67	C₄×Dic₈₃	—	2	4	no	—
68	C₂⁴×C₈₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	—	2	2	no	—
70	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₃	C₂×D₄	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	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
77	C₂²×C₄×C₈₃	#41	1	2	yes	—
78	C₂²×C₄×C₈₃	#41	1	2	yes	—
79	C₂²×C₄×C₈₃	#41	1	2	yes	—
80	M₁₆×C₈₃	—	2	4	no	—
81	M₁₆×C₈₃	—	2	4	no	—
82	C₂×C₈×C₈₃	—	2	1	no	—
83	C₂×C₈×C₈₃	—	2	1	no	—
84	C₂×C₈×C₈₃	—	2	1	no	—
85	C₂×C₈×C₈₃	—	2	1	no	—
86	C₂×C₈×C₈₃	—	2	2	no	—
87	C₂×C₈×C₈₃	—	2	2	no	—
88	C₂×C₈×C₈₃	—	2	4	no	—
89	C₂×C₈×C₈₃	—	2	4	no	—
90	C₂×C₈×C₈₃	C₄²	1	2	yes	—
91	C₂×C₈×C₈₃	C₄²	1	2	yes	—
92	C₂×C₈×C₈₃	C₂²⋊C₄	1	4	yes	—
93	C₄⋊C₄×C₈₃	—	2	2	no	—
94	C₄⋊C₄×C₈₃	—	2	2	no	—
95	C₂²⋊C₄×C₈₃	—	2	2	no	—
96	C₂²⋊C₄×C₈₃	—	2	2	no	—
97	C₄²×C₈₃	—	2	2	no	—
98	C₄²×C₈₃	C₄⋊C₄	1	4	yes	—
99	C₂³×C₈₃	—	4	2	no	—
100	C₂³×C₈₃	#117	1	2	yes	—
101	C₂³×C₈₃	#117	1	1	yes	characteristic
102	C₂³×C₈₃	#121	1	1	yes	characteristic
103	C₂³×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₈₃	#126	1	2	yes	—
105	C₂×C₄×C₈₃	—	2	4	no	—
106	C₂×C₄×C₈₃	—	4	4	no	—
107	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₃	#117	1	2	yes	—
111	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₈₃	#123	1	2	yes	—
121	C₂×C₄×C₈₃	#123	1	2	yes	—
122	C₂×C₄×C₈₃	#124	1	2	yes	—
123	C₂×C₄×C₈₃	#125	1	2	yes	—
124	C₂×C₄×C₈₃	#126	1	2	yes	—
125	C₂×Dic₈₃	—	4	8	no	—
126	C₂×Dic₈₃	—	4	4	no	—
127	C₈×C₈₃	—	2	2	no	—
128	C₈×C₈₃	—	2	2	no	—
129	C₈×C₈₃	—	2	4	no	—
130	C₈₃⋊C₈	—	4	4	no	—
131	C₂²×C₈₃	—	2	4	no	—
132	C₂²×C₈₃	—	2	2	no	—
133	C₂²×C₈₃	—	4	1	no	—
134	C₂²×C₈₃	—	4	1	no	—
135	C₂²×C₈₃	—	4	2	no	—
136	C₂²×C₈₃	#376	1	1	yes	characteristic
137	C₂²×C₈₃	#380	1	1	yes	characteristic
138	C₂²×C₈₃	#383	1	1	yes	characteristic
139	C₂²×C₈₃	#387	1	2	yes	—
140	C₂²×C₈₃	#388	1	2	yes	—
141	C₄×C₈₃	—	4	4	no	—
142	C₄×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₈₃	#381	1	1	yes	characteristic
145	C₄×C₈₃	#382	1	1	yes	characteristic
146	C₄×C₈₃	#387	1	2	yes	—
147	C₄×C₈₃	#388	1	2	yes	—
148	Dic₈₃	—	8	4	no	—
149	#10800	—	83	1	no	maximal, Sylow
150	C₂×C₈₃	—	2	2	no	—
151	C₂×C₈₃	—	2	1	no	—
152	C₂×C₈₃	—	2	1	no	—
153	C₂×C₈₃	—	2	2	no	—
154	C₂×C₈₃	#1434	1	1	yes	characteristic
155	C₂×C₈₃	#1747	1	2	yes	—
156	C₂×#405	D₈₃	1	1	yes	characteristic
157	C₂×#338	—	83	1	no	—
158	C₂×H(3,ℤ/4)	—	83	1	no	—
159	C₂×H(3,ℤ/4)	—	83	2	no	—
160	#1729	—	83	2	no	—
161	C₈₃	#10800	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	—	83	2	no	—
163	C₂²×M₁₆	C₂×D₈₃	1	1	yes	characteristic
164	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
165	C₂×#117	—	83	2	no	—
166	C₂×#117	C₂×D₈₃	1	1	yes	characteristic
167	#405	—	2	2	no	—
168	#405	—	2	2	no	—
169	C₂×#97	Dic₈₃	1	2	yes	complemented
170	C₂×C₄×C₈	Dic₈₃	1	2	yes	complemented
171	#338	—	166	2	no	—
172	H(3,ℤ/4)	—	166	2	no	—
173	H(3,ℤ/4)	—	166	4	no	—
174	#336	—	83	8	no	—
175	C₂³×C₄	C₂²×D₈₃	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	166	2	no	—
179	C₂×M₁₆	—	166	2	no	—
180	C₂×M₁₆	—	166	2	no	—
181	C₂×M₁₆	—	166	2	no	—
182	C₂×M₁₆	—	166	2	no	—
183	C₂×M₁₆	—	166	2	no	—
184	C₂×M₁₆	C₂²⋊D₈₃	1	2	yes	—
185	C₂×M₁₆	C₂²⋊D₈₃	1	2	yes	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	83	4	no	—
190	C₂²×C₈	C₄×D₈₃	1	4	yes	—
191	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
193	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
194	#117	—	2	1	no	—
195	#117	—	2	1	no	—
196	#117	—	166	2	no	—
197	#117	—	166	2	no	—
198	#117	—	166	2	no	—
199	#117	—	166	2	no	—
200	#117	Dic₁₆₆	1	2	yes	—
201	#117	D₃₃₂	1	2	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	166	2	no	—
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₂²⋊C₄	—	166	2	no	—
206	C₂×C₄²	—	83	4	no	—
207	C₂×C₄²	C₂×Dic₈₃	1	2	yes	—
208	#97	—	2	4	no	—
209	C₄×C₈	—	2	4	no	—
210	C₂⁴	#13050590	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	166	4	no	—
213	C₂²×C₄	C₂×C₄×D₈₃	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
218	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
219	C₂²×C₄	#13050590	1	1	yes	characteristic
220	C₂²×C₄	#13050590	1	2	yes	—
221	C₂²×C₄	#13050590	1	2	yes	—
222	C₂²×C₄	#13050590	1	2	yes	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	332	2	no	—
226	M₁₆	—	332	2	no	—
227	M₁₆	—	332	2	no	—
228	M₁₆	—	332	2	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	166	4	no	—
238	C₂×C₈	—	166	4	no	—
239	C₂×C₈	—	166	4	no	—
240	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
241	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₈₃	1	4	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	332	2	no	—
246	C₄⋊C₄	—	332	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	332	2	no	—
250	C₂²⋊C₄	—	332	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	166	4	no	—
253	C₄²	—	166	4	no	—
254	C₄²	C₄⋊Dic₈₃	1	4	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	#14581749	1	2	yes	—
257	C₂³	#14581751	1	1	yes	characteristic
258	C₂³	#14581766	1	1	yes	characteristic
259	C₂³	#14581771	1	1	yes	characteristic
260	C₂³	#14581786	1	2	yes	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	332	8	no	—
264	C₂×C₄	—	332	4	no	—
265	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
268	C₂×C₄	#14581749	1	2	yes	—
269	C₂×C₄	#14581751	1	1	yes	characteristic
270	C₂×C₄	#14581751	1	1	yes	characteristic
271	C₂×C₄	#14581751	1	1	yes	characteristic
272	C₂×C₄	C₄×C₂²⋊D₈₃	1	2	yes	—
273	C₂×C₄	C₄×C₂²⋊D₈₃	1	2	yes	—
274	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
275	C₂×C₄	C₄×Dic₁₆₆	1	2	yes	—
276	C₂×C₄	#14581766	1	1	yes	characteristic
277	C₂×C₄	#14581771	1	1	yes	characteristic
278	C₂×C₄	#14581775	1	2	yes	—
279	C₂×C₄	#14581775	1	2	yes	—
280	C₂×C₄	#14581780	1	2	yes	—
281	C₂×C₄	#14581784	1	2	yes	—
282	C₂×C₄	#14581786	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	332	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	#19127547	1	1	yes	characteristic
293	C₂²	#19127561	1	1	yes	characteristic
294	C₂²	#19127569	1	1	yes	characteristic
295	C₂²	#19127585	1	2	yes	—
296	C₂²	#19127592	1	2	yes	—
297	C₄	—	4	4	no	—
298	C₄	—	664	4	no	—
299	C₄	C₄×C₂²⋊Dic₈₃	1	1	yes	characteristic
300	C₄	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
301	C₄	#19127564	1	1	yes	characteristic
302	C₄	#19127567	1	1	yes	characteristic
303	C₄	#19127585	1	2	yes	—
304	C₄	#19127592	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	#24905690	1	1	yes	characteristic
310	C₂	#24907218	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₈₃
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₃
Fitting	C₂×#405×C₈₃	C₂
socle	C₂²×C₈₃	#376
prime radical	C₂⁴×C₈₃	#41
cosocle	C₂³×C₄×C₈₃	C₂³
hypercenter	C₂×#405	D₈₃
hypocenter	C₈₃	#10800
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
11. #19127455	C₄	semidirect
17. C₂×#97×C₈₃	C₄	semidirect
18. C₂×C₄×C₈×C₈₃	C₄	semidirect
161. C₈₃	#10800	semidirect
169. C₂×#97	Dic₈₃	semidirect
170. C₂×C₄×C₈	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈₃⋊#338	2
C₂×#19127458	2
C₂×#19127457	2
#24907159	2
C₂×#405×C₈₃	2
#10800	83