Subgroups — #35310045

Order 21248.

650 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127472	C₂	1	2	yes	maximal
2	#24907202	C₂	1	1	yes	maximal, characteristic
3	#24907196	C₂	1	1	yes	maximal, characteristic
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	#19127472	—	2	4	no	—
9	#19127470	C₄	1	4	yes	complemented
10	#19127468	C₄	1	4	yes	—
11	#19127455	—	2	4	no	—
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	4	no	—
16	C₂×#97×C₈₃	C₄	1	2	yes	complemented
17	C₂×C₄×C₈×C₈₃	C₄	1	2	yes	complemented
18	C₂×#13050574	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
24	C₂²×C₈₃⋊C₈	—	2	2	no	—
25	#14581751	—	2	2	no	—
26	#14581751	—	2	2	no	—
27	#14581751	—	2	2	no	—
28	#14581751	—	2	2	no	—
29	C₂×C₄⋊Dic₈₃	—	2	2	no	—
30	C₂×C₂²⋊Dic₈₃	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₈₃	—	2	2	no	—
32	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₈₃	—	2	2	no	—
34	C₂×M₁₆×C₈₃	—	2	2	no	—
35	C₂×M₁₆×C₈₃	D₄	1	2	yes	—
36	C₂×M₁₆×C₈₃	Q₈	1	2	yes	—
37	C₂²×C₈×C₈₃	—	2	2	no	—
38	C₂²×C₈×C₈₃	—	2	1	no	—
39	C₂²×C₈×C₈₃	—	2	1	no	—
40	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₈₃	D₄	1	4	yes	—
44	#117×C₈₃	—	2	2	no	—
45	#117×C₈₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
49	#97×C₈₃	—	2	4	no	—
50	C₄×C₈×C₈₃	—	2	4	no	—
51	C₂²×Dic₈₃	—	2	4	no	—
52	#13050574	—	2	4	no	—
53	#13050574	—	2	4	no	—
54	#13050574	—	4	2	no	—
55	#13050574	—	4	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	—	4	2	no	—
60	C₄⋊Dic₈₃	—	4	2	no	—
61	C₂²⋊Dic₈₃	—	2	4	no	—
62	C₂²⋊Dic₈₃	—	2	4	no	—
63	C₄×Dic₈₃	—	4	2	no	—
64	C₄×Dic₈₃	—	4	2	no	—
65	C₂⁴×C₈₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	C₄²	1	4	yes	—
67	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₃	#41	1	2	yes	—
73	C₂²×C₄×C₈₃	#41	1	2	yes	—
74	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
75	C₂²×C₄×C₈₃	#41	1	2	yes	—
76	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
77	M₁₆×C₈₃	—	2	4	no	—
78	M₁₆×C₈₃	—	2	4	no	—
79	C₂×C₈×C₈₃	—	2	2	no	—
80	C₂×C₈×C₈₃	—	2	1	no	—
81	C₂×C₈×C₈₃	—	2	1	no	—
82	C₂×C₈×C₈₃	—	2	1	no	—
83	C₂×C₈×C₈₃	—	2	1	no	—
84	C₂×C₈×C₈₃	—	2	2	no	—
85	C₂×C₈×C₈₃	—	2	4	no	—
86	C₂×C₈×C₈₃	—	2	4	no	—
87	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
90	C₄⋊C₄×C₈₃	—	2	4	no	—
91	C₂²⋊C₄×C₈₃	—	2	4	no	—
92	C₄²×C₈₃	—	2	2	no	—
93	C₄²×C₈₃	—	2	2	no	—
94	C₂³×C₈₃	—	4	2	no	—
95	C₂³×C₈₃	C₂×C₄²	1	1	yes	characteristic
96	C₂³×C₈₃	#117	1	2	yes	—
97	C₂³×C₈₃	C₄×D₄	1	2	yes	—
98	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₈₃	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₈₃	—	2	4	no	—
101	C₂×C₄×C₈₃	—	4	4	no	—
102	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₃	#117	1	2	yes	—
105	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₈₃	#121	1	2	yes	—
113	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₈₃	#123	1	2	yes	—
115	C₂×C₄×C₈₃	#123	1	2	yes	—
116	C₂×C₄×C₈₃	#124	1	2	yes	—
117	C₂×C₄×C₈₃	#125	1	2	yes	—
118	C₂×C₄×C₈₃	#126	1	2	yes	—
119	C₂×C₄×C₈₃	#126	1	2	yes	—
120	C₂×Dic₈₃	—	4	8	no	—
121	C₂×Dic₈₃	—	4	4	no	—
122	C₈×C₈₃	—	2	4	no	—
123	C₈×C₈₃	—	2	2	no	—
124	C₈×C₈₃	—	2	2	no	—
125	C₈₃⋊C₈	—	4	4	no	—
126	C₂²×C₈₃	—	2	4	no	—
127	C₂²×C₈₃	—	2	2	no	—
128	C₂²×C₈₃	—	4	2	no	—
129	C₂²×C₈₃	—	4	1	no	—
130	C₂²×C₈₃	—	4	1	no	—
131	C₂²×C₈₃	#376	1	1	yes	characteristic
132	C₂²×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₈₃	#387	1	2	yes	—
135	C₂²×C₈₃	#388	1	2	yes	—
136	C₄×C₈₃	—	4	4	no	—
137	C₄×C₈₃	#380	1	1	yes	characteristic
138	C₄×C₈₃	#381	1	1	yes	characteristic
139	C₄×C₈₃	#382	1	1	yes	characteristic
140	C₄×C₈₃	#383	1	1	yes	characteristic
141	C₄×C₈₃	#387	1	2	yes	—
142	C₄×C₈₃	#388	1	2	yes	—
143	Dic₈₃	—	8	4	no	—
144	#10803	—	83	1	no	maximal, Sylow
145	C₂×C₈₃	—	2	2	no	—
146	C₂×C₈₃	—	2	1	no	—
147	C₂×C₈₃	—	2	1	no	—
148	C₂×C₈₃	—	2	2	no	—
149	C₂×C₈₃	#1434	1	1	yes	characteristic
150	C₂×C₈₃	#1757	1	2	yes	—
151	C₂×#405	D₈₃	1	1	yes	characteristic
152	C₂×#344	—	83	2	no	—
153	#1742	—	83	1	no	—
154	#1740	—	83	1	no	—
155	#1729	—	83	2	no	—
156	C₈₃	#10803	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	83	2	no	—
158	C₂²×M₁₆	C₂×D₈₃	1	1	yes	characteristic
159	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
160	C₂×#117	—	83	2	no	—
161	C₂×#117	C₂×D₈₃	1	1	yes	characteristic
162	#405	—	2	4	no	—
163	C₂×#97	Dic₈₃	1	2	yes	complemented
164	C₂×C₄×C₈	Dic₈₃	1	2	yes	complemented
165	#344	—	166	4	no	—
166	#343	—	83	4	no	—
167	#342	—	83	4	no	—
168	#336	—	166	4	no	—
169	C₂³×C₄	C₂²×D₈₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	83	4	no	—
173	C₂×M₁₆	—	166	2	no	—
174	C₂×M₁₆	—	166	2	no	—
175	C₂×M₁₆	—	166	2	no	—
176	C₂×M₁₆	—	166	2	no	—
177	C₂×M₁₆	—	166	2	no	—
178	C₂×M₁₆	Dic₁₆₆	1	2	yes	—
179	C₂×M₁₆	D₃₃₂	1	2	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	166	2	no	—
184	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
187	C₂²×C₈	C₂²⋊D₈₃	1	4	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	166	2	no	—
191	#117	—	166	2	no	—
192	#117	—	166	2	no	—
193	#117	—	166	2	no	—
194	C₂×C₄⋊C₄	—	166	2	no	—
195	C₂×C₄⋊C₄	C₄×D₈₃	1	2	yes	—
196	C₂×C₂²⋊C₄	—	83	4	no	—
197	C₂×C₂²⋊C₄	C₄×D₈₃	1	2	yes	—
198	C₂×C₄²	—	166	2	no	—
199	C₂×C₄²	C₂×Dic₈₃	1	2	yes	—
200	#97	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	C₂⁴	C₂×C₄×D₈₃	1	1	yes	characteristic
203	C₂²×C₄	—	166	4	no	—
204	C₂²×C₄	C₄×Dic₈₃	1	4	yes	—
205	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
208	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
209	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
210	C₂²×C₄	#13050590	1	2	yes	—
211	C₂²×C₄	#13050590	1	2	yes	—
212	C₂²×C₄	#13050590	1	1	yes	characteristic
213	C₂²×C₄	#13050590	1	1	yes	characteristic
214	C₂²×C₄	#13050590	1	2	yes	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	332	2	no	—
218	M₁₆	—	332	2	no	—
219	M₁₆	—	166	4	no	—
220	M₁₆	—	166	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	332	2	no	—
230	C₂×C₈	—	332	2	no	—
231	C₂×C₈	—	166	4	no	—
232	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
233	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
234	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	332	2	no	—
237	C₄⋊C₄	—	332	2	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	166	4	no	—
240	C₂²⋊C₄	—	166	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	332	2	no	—
244	C₄²	—	332	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₄×Dic₈₃	1	1	yes	characteristic
247	C₂³	#14581749	1	2	yes	—
248	C₂³	C₄×C₂²⋊D₈₃	1	2	yes	—
249	C₂³	C₄×D₃₃₂	1	1	yes	characteristic
250	C₂³	C₄×Dic₁₆₆	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	332	8	no	—
254	C₂×C₄	—	332	4	no	—
255	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
257	C₂×C₄	#14581749	1	2	yes	—
258	C₂×C₄	#14581751	1	1	yes	characteristic
259	C₂×C₄	#14581751	1	1	yes	characteristic
260	C₂×C₄	#14581751	1	1	yes	characteristic
261	C₂×C₄	#14581751	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₈₃	1	2	yes	—
263	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
264	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
265	C₂×C₄	#14581766	1	2	yes	—
266	C₂×C₄	#14581771	1	2	yes	—
267	C₂×C₄	#14581775	1	2	yes	—
268	C₂×C₄	#14581775	1	2	yes	—
269	C₂×C₄	#14581780	1	2	yes	—
270	C₂×C₄	#14581784	1	2	yes	—
271	C₂×C₄	#14581786	1	2	yes	—
272	C₂×C₄	#14581786	1	2	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₈	—	332	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	#19127547	1	1	yes	characteristic
283	C₂²	C₄×C₂²⋊Dic₈₃	1	1	yes	characteristic
284	C₂²	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
285	C₂²	#19127585	1	2	yes	—
286	C₂²	#19127592	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	664	4	no	—
289	C₄	#19127561	1	1	yes	characteristic
290	C₄	#19127564	1	1	yes	characteristic
291	C₄	#19127567	1	1	yes	characteristic
292	C₄	#19127569	1	1	yes	characteristic
293	C₄	#19127585	1	2	yes	—
294	C₄	#19127592	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	#24905690	1	1	yes	characteristic
300	C₂	#24907263	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₃
commutator	C₂³×C₈₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₃
Fitting	C₂×#405×C₈₃	C₂
socle	C₂²×C₈₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₈₃	C₂²×C₄
cosocle	C₂³×C₄×C₈₃	C₂³
hypercenter	C₂×#405	D₈₃
hypocenter	C₈₃	#10803
LCS	G ▷ C₂³×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
9. #19127470	C₄	semidirect
16. C₂×#97×C₈₃	C₄	semidirect
17. C₂×C₄×C₈×C₈₃	C₄	semidirect
156. C₈₃	#10803	semidirect
163. C₂×#97	Dic₈₃	semidirect
164. 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×#19127472	2
#24907202	2
#24907196	2
#24907159	2
C₂×#405×C₈₃	2
#10803	83