Subgroups — #35322084

Order 21248.

262 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908212	C₂	1	1	yes	maximal, characteristic
2	#24907344	C₂	1	1	yes	maximal, characteristic
3	#24905558	C₂	1	1	yes	maximal, characteristic
4	#24905558	C₂	1	1	yes	maximal, characteristic
5	#24905557	C₂	1	1	yes	maximal, characteristic
6	#24905540	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#14581728	C₂²	1	1	yes	characteristic
10	C₂×#14581728	C₂²	1	1	yes	characteristic
11	#19127728	—	2	1	no	—
12	#19127657	—	2	1	no	—
13	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
14	#19127460	—	2	1	no	—
15	#19127460	—	2	1	no	—
16	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
17	#427×C₈₃	C₂²	1	1	yes	characteristic
18	#427×C₈₃	C₂²	1	1	yes	characteristic
19	#367×C₈₃	C₄	1	1	yes	characteristic, complemented
20	#367×C₈₃	C₄	1	1	yes	characteristic, complemented
21	#366×C₈₃	C₄	1	1	yes	characteristic, complemented
22	#366×C₈₃	C₄	1	1	yes	characteristic, complemented
23	C₂×#13050574	—	2	1	no	—
24	C₂×#13050574	—	2	1	no	—
25	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
27	C₂×C₈₃⋊C₁₆	—	2	1	no	—
28	C₂×C₈₃⋊C₁₆	—	2	1	no	—
29	C₂×C₈₃⋊C₁₆	—	2	1	no	—
30	#14581728	—	2	1	no	—
31	#14581728	—	2	1	no	—
32	#14581728	—	2	1	no	—
33	#14581728	Q₈	1	2	yes	—
34	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
35	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
36	#14581703	—	4	1	no	—
37	C₄×C₈₃⋊C₈	—	4	1	no	—
38	C₈×Dic₈₃	—	2	1	no	—
39	C₈×Dic₈₃	—	2	1	no	—
40	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
41	#117×C₈₃	—	2	1	no	—
42	#117×C₈₃	—	2	1	no	—
43	C₂×C₁₆×C₈₃	—	2	1	no	—
44	C₂×C₁₆×C₈₃	—	2	1	no	—
45	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
47	#107×C₈₃	C₂×C₄	1	1	yes	characteristic
48	#107×C₈₃	C₂×C₄	1	1	yes	characteristic
49	#107×C₈₃	C₂×C₄	1	1	yes	characteristic
50	#107×C₈₃	C₂×C₄	1	1	yes	characteristic
51	#106×C₈₃	—	2	1	no	—
52	#106×C₈₃	—	2	1	no	—
53	C₂²×Dic₈₃	—	2	1	no	—
54	#13050574	—	2	2	no	—
55	#13050574	—	4	1	no	—
56	#13050574	—	4	1	no	—
57	#13050574	—	4	1	no	—
58	C₂×C₈₃⋊C₈	—	2	1	no	—
59	C₂×C₈₃⋊C₈	—	2	1	no	—
60	C₂×C₈₃⋊C₈	—	4	1	no	—
61	C₂×C₈₃⋊C₈	—	4	1	no	—
62	C₂×C₈₃⋊C₈	D₈	1	1	yes	characteristic
63	C₂×C₈₃⋊C₈	D₈	1	1	yes	characteristic
64	C₄×Dic₈₃	—	2	1	no	—
65	C₄×Dic₈₃	—	2	1	no	—
66	C₈₃⋊C₁₆	—	4	1	no	—
67	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
71	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
74	C₄⋊C₄×C₈₃	—	2	1	no	—
75	C₄⋊C₄×C₈₃	—	2	1	no	—
76	C₄⋊C₄×C₈₃	—	2	1	no	—
77	C₄⋊C₄×C₈₃	—	2	1	no	—
78	C₂²⋊C₄×C₈₃	—	4	1	no	—
79	C₂²⋊C₄×C₈₃	—	4	1	no	—
80	C₄²×C₈₃	—	4	1	no	—
81	C₄²×C₈₃	—	4	1	no	—
82	C₁₆×C₈₃	—	2	1	no	—
83	C₁₆×C₈₃	—	2	1	no	—
84	C₂³×C₈₃	#135	1	1	yes	characteristic
85	C₂×C₄×C₈₃	—	4	1	no	—
86	C₂×C₄×C₈₃	—	4	1	no	—
87	C₂×C₄×C₈₃	—	4	1	no	—
88	C₂×C₄×C₈₃	—	4	1	no	—
89	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
94	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
95	C₂×Dic₈₃	—	2	2	no	—
96	C₂×Dic₈₃	—	4	1	no	—
97	C₂×Dic₈₃	—	4	1	no	—
98	C₈×C₈₃	#117	1	1	yes	characteristic
99	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
100	C₈×C₈₃	#125	1	1	yes	characteristic
101	C₈×C₈₃	#126	1	1	yes	characteristic
102	C₈₃⋊C₈	—	2	1	no	—
103	C₈₃⋊C₈	—	2	1	no	—
104	C₈₃⋊C₈	—	4	2	no	—
105	C₈₃⋊C₈	—	8	1	no	—
106	C₂²×C₈₃	—	2	1	no	—
107	C₂²×C₈₃	—	2	1	no	—
108	C₂²×C₈₃	C₄×D₈	1	1	yes	characteristic
109	C₂²×C₈₃	#479	1	1	yes	characteristic
110	C₂²×C₈₃	#485	1	1	yes	characteristic
111	C₄×C₈₃	—	8	1	no	—
112	C₄×C₈₃	—	8	1	no	—
113	C₄×C₈₃	—	8	1	no	—
114	C₄×C₈₃	—	8	1	no	—
115	C₄×C₈₃	#387	1	1	yes	characteristic
116	C₄×C₈₃	#441	1	1	yes	characteristic
117	C₄×C₈₃	#474	1	1	yes	characteristic
118	C₄×C₈₃	#480	1	1	yes	characteristic
119	Dic₈₃	—	4	2	no	—
120	#13113	—	83	1	no	maximal, Sylow
121	C₂×C₈₃	—	2	1	no	—
122	C₂×C₈₃	—	2	1	no	—
123	C₂×C₈₃	#1918	1	1	yes	characteristic
124	C₂×C₈₃	#2178	1	1	yes	characteristic
125	C₂×C₈₃	#2179	1	1	yes	characteristic
126	#2158	D₈₃	1	1	yes	characteristic
127	#1945	—	83	1	no	—
128	#1772	—	83	1	no	—
129	#1382	—	83	1	no	—
130	#1382	—	83	1	no	—
131	#1381	—	83	1	no	—
132	#1374	—	83	1	no	—
133	C₈₃	#13113	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₁₆	—	83	1	no	—
135	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
136	C₂×#108	—	83	1	no	—
137	C₂×#108	—	83	1	no	—
138	#427	C₂×D₈₃	1	1	yes	characteristic
139	#427	C₂×D₈₃	1	1	yes	characteristic
140	#424	—	166	1	no	—
141	#405	—	166	1	no	—
142	C₂×C₄×C₈	—	83	1	no	—
143	#367	Dic₈₃	1	1	yes	characteristic, complemented
144	#367	Dic₈₃	1	1	yes	characteristic, complemented
145	#366	Dic₈₃	1	1	yes	characteristic, complemented
146	#366	Dic₈₃	1	1	yes	characteristic, complemented
147	#339	—	166	1	no	—
148	#339	—	166	1	no	—
149	C₂×M₁₆	—	166	1	no	—
150	C₂×M₁₆	—	166	1	no	—
151	C₂²×C₈	—	83	1	no	—
152	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
153	#117	—	2	1	no	—
154	#117	—	2	1	no	—
155	C₂×C₄²	—	83	1	no	—
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	166	1	no	—
159	C₂×C₁₆	—	166	1	no	—
160	C₂×C₁₆	—	166	1	no	—
161	C₂×C₁₆	C₂×Dic₈₃	1	1	yes	characteristic
162	C₂×C₁₆	C₂×Dic₈₃	1	1	yes	characteristic
163	#108	—	83	2	no	—
164	#108	—	166	1	no	—
165	#108	—	166	1	no	—
166	#108	—	166	1	no	—
167	#107	C₂×Dic₈₃	1	1	yes	characteristic
168	#107	C₂×Dic₈₃	1	1	yes	characteristic
169	#107	C₂×Dic₈₃	1	1	yes	characteristic
170	#107	C₂×Dic₈₃	1	1	yes	characteristic
171	#106	—	2	1	no	—
172	#106	—	2	1	no	—
173	C₄⋊C₈	—	332	1	no	—
174	C₄⋊C₈	—	332	1	no	—
175	#97	—	332	1	no	—
176	C₄×C₈	—	332	1	no	—
177	C₄×C₈	—	166	1	no	—
178	C₄×C₈	—	166	1	no	—
179	C₂²×C₄	—	166	1	no	—
180	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
181	M₁₆	—	332	1	no	—
182	M₁₆	—	332	1	no	—
183	M₁₆	—	332	1	no	—
184	M₁₆	—	166	2	no	—
185	C₂×C₈	—	332	1	no	—
186	C₂×C₈	—	332	1	no	—
187	C₂×C₈	—	83	1	no	—
188	C₂×C₈	—	83	1	no	—
189	C₂×C₈	—	166	1	no	—
190	C₂×C₈	—	166	1	no	—
191	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
192	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
193	C₂×C₈	#13050589	1	1	yes	characteristic
194	C₂×C₈	#13050590	1	1	yes	characteristic
195	C₂×C₈	#13050590	1	1	yes	characteristic
196	C₂×C₈	#13050591	1	1	yes	characteristic
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	1	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	166	1	no	—
206	C₄²	—	166	1	no	—
207	C₁₆	—	2	1	no	—
208	C₁₆	—	2	1	no	—
209	C₁₆	—	332	1	no	—
210	C₂³	#14581811	1	1	yes	characteristic
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	332	1	no	—
216	C₂×C₄	—	332	1	no	—
217	C₂×C₄	—	166	2	no	—
218	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
219	C₂×C₄	#14581770	1	1	yes	characteristic
220	C₂×C₄	#14581777	1	1	yes	characteristic
221	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
222	C₂×C₄	#14581824	1	1	yes	characteristic
223	C₂×C₄	#14581824	1	1	yes	characteristic
224	C₈	—	332	2	no	—
225	C₈	—	664	1	no	—
226	C₈	—	166	1	no	—
227	C₈	—	166	1	no	—
228	C₈	#14581751	1	1	yes	characteristic
229	C₈	Q₈×Dic₈₃	1	1	yes	characteristic
230	C₈	#14581782	1	1	yes	characteristic
231	C₈	#14581787	1	1	yes	characteristic
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	D₈×Dic₈₃	1	1	yes	characteristic
235	C₂²	#19128007	1	1	yes	characteristic
236	C₂²	#19128044	1	1	yes	characteristic
237	C₄	—	8	1	no	—
238	C₄	—	8	1	no	—
239	C₄	—	8	1	no	—
240	C₄	—	8	1	no	—
241	C₄	—	332	2	no	—
242	C₄	#19127587	1	1	yes	characteristic
243	C₄	#19127808	1	1	yes	characteristic
244	C₄	#19127980	1	1	yes	characteristic
245	C₄	#19128016	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	#24908036	1	1	yes	characteristic
249	C₂	#24909344	1	1	yes	characteristic
250	C₂	#24909349	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₈₃
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#2158	C₂
socle	C₂²×C₈₃	C₄×D₈
prime radical	C₂³×C₈₃	#135
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#2158	D₈₃
hypocenter	C₈₃	#13113
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
19. #367×C₈₃	C₄	semidirect
20. #367×C₈₃	C₄	semidirect
21. #366×C₈₃	C₄	semidirect
22. #366×C₈₃	C₄	semidirect
133. C₈₃	#13113	semidirect
143. #367	Dic₈₃	semidirect
144. #367	Dic₈₃	semidirect
145. #366	Dic₈₃	semidirect
146. #366	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24908212	2
#24907344	2
#24905558	2
#24905557	2
#24905540	2
C₈₃×#2158	2
#13113	83