Subgroups — #35320555

Order 21248.

396 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127524	C₂	1	2	yes	maximal, complemented
2	#24907501	C₂	1	1	yes	maximal, characteristic
3	#24907488	C₂	1	1	yes	maximal, characteristic
4	#24905562	C₂	1	2	yes	maximal, complemented
5	C₂²×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581728	C₂²	1	2	yes	—
7	C₂×#14581724	C₂²	1	2	yes	—
8	#19127665	—	2	1	no	—
9	#19127561	—	2	1	no	—
10	#19127524	—	2	4	no	—
11	#19127466	—	2	1	no	—
12	#19127464	—	2	1	no	—
13	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₈₃	—	2	4	no	—
15	C₂×M₃₂×C₈₃	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₈₃	C₄	1	2	yes	complemented
19	C₂×#13050574	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	—	2	2	no	—
21	C₂×C₂²⋊Dic₈₃	—	4	1	no	—
22	#14581728	—	2	2	no	—
23	#14581728	—	2	2	no	—
24	#14581724	—	2	2	no	—
25	#14581724	—	2	2	no	—
26	C₂²⋊(C₈₃⋊C₈)	—	4	1	no	—
27	C₂²⋊(C₈₃⋊C₈)	—	4	1	no	—
28	C₈₃⋊#95	—	4	1	no	—
29	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	—	2	2	no	—
31	C₂²×C₈×C₈₃	—	2	2	no	—
32	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
38	M₃₂×C₈₃	—	2	2	no	—
39	M₃₂×C₈₃	—	2	4	no	—
40	M₃₂×C₈₃	D₄	1	4	yes	—
41	C₂×C₁₆×C₈₃	—	2	4	no	—
42	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₈₃	—	4	2	no	—
45	#13050574	—	4	2	no	—
46	#13050574	—	4	2	no	—
47	C₂×C₈₃⋊C₈	—	4	2	no	—
48	C₄⋊Dic₈₃	—	4	2	no	—
49	C₄⋊Dic₈₃	—	4	2	no	—
50	C₂²⋊Dic₈₃	—	8	2	no	—
51	C₂⁴×C₈₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	—	2	2	no	—
53	C₂²×C₄×C₈₃	—	2	2	no	—
54	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	—	2	2	no	—
61	C₂×C₈×C₈₃	—	2	1	no	—
62	C₂×C₈×C₈₃	—	2	1	no	—
63	C₂×C₈×C₈₃	—	2	1	no	—
64	C₂×C₈×C₈₃	—	2	1	no	—
65	C₂×C₈×C₈₃	—	2	2	no	—
66	C₂×C₈×C₈₃	—	2	2	no	—
67	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₈₃	#41	1	2	yes	—
73	C₁₆×C₈₃	—	2	2	no	—
74	C₁₆×C₈₃	C₂²⋊C₄	1	4	yes	complemented
75	C₂³×C₈₃	—	2	2	no	—
76	C₂³×C₈₃	—	2	2	no	—
77	C₂³×C₈₃	—	4	1	no	—
78	C₂³×C₈₃	#106	1	1	yes	characteristic
79	C₂³×C₈₃	#106	1	1	yes	characteristic
80	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₈₃	—	2	2	no	—
82	C₂×C₄×C₈₃	—	2	1	no	—
83	C₂×C₄×C₈₃	—	2	1	no	—
84	C₂×C₄×C₈₃	—	2	1	no	—
85	C₂×C₄×C₈₃	—	2	1	no	—
86	C₂×C₄×C₈₃	—	2	2	no	—
87	C₂×C₄×C₈₃	—	2	2	no	—
88	C₂×C₄×C₈₃	#106	1	1	yes	characteristic
89	C₂×C₄×C₈₃	#106	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₃	#121	1	2	yes	—
92	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₈₃	#135	1	2	yes	—
94	C₂×Dic₈₃	—	8	2	no	—
95	C₂×Dic₈₃	—	8	2	no	—
96	C₂×Dic₈₃	—	8	2	no	—
97	C₈×C₈₃	—	2	1	no	—
98	C₈×C₈₃	—	2	1	no	—
99	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₈×C₈₃	#117	1	1	yes	characteristic
101	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
102	C₈₃⋊C₈	—	8	2	no	—
103	C₂²×C₈₃	—	2	1	no	—
104	C₂²×C₈₃	—	2	1	no	—
105	C₂²×C₈₃	—	2	1	no	—
106	C₂²×C₈₃	—	2	1	no	—
107	C₂²×C₈₃	—	2	1	no	—
108	C₂²×C₈₃	—	2	1	no	—
109	C₂²×C₈₃	—	2	1	no	—
110	C₂²×C₈₃	—	4	4	no	—
111	C₂²×C₈₃	#365	1	2	yes	complemented
112	C₂²×C₈₃	C₂×#106	1	1	yes	characteristic
113	C₂²×C₈₃	#465	1	2	yes	—
114	C₄×C₈₃	—	2	1	no	—
115	C₄×C₈₃	—	2	1	no	—
116	C₄×C₈₃	#380	1	1	yes	characteristic
117	C₄×C₈₃	#427	1	1	yes	characteristic
118	C₄×C₈₃	#465	1	2	yes	—
119	Dic₈₃	—	16	2	no	—
120	#12831	—	83	1	no	maximal, Sylow
121	C₂×C₈₃	—	2	2	no	—
122	C₂×C₈₃	—	2	2	no	—
123	C₂×C₈₃	—	4	1	no	—
124	C₂×C₈₃	#1807	1	1	yes	characteristic
125	C₂×C₈₃	C₂×#365	1	1	yes	characteristic
126	C₂×C₈₃	#2155	1	1	yes	characteristic
127	C₂²×M₃₂	D₈₃	1	1	yes	characteristic
128	C₂×#365	—	83	2	no	—
129	#1810	—	83	1	no	—
130	#1807	—	83	1	no	—
131	#1383	—	83	2	no	—
132	C₈₃	#12831	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
134	C₂×M₃₂	—	2	4	no	—
135	C₂×M₃₂	Dic₈₃	1	2	yes	complemented
136	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
137	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
138	C₂²×C₁₆	Dic₈₃	1	2	yes	complemented
139	C₂×#108	—	83	2	no	—
140	C₂×#106	—	83	2	no	—
141	#407	—	166	1	no	—
142	#380	—	166	1	no	—
143	#365	—	166	4	no	—
144	#341	—	166	1	no	—
145	#340	—	166	1	no	—
146	C₂³×C₄	D₃₃₂	1	1	yes	characteristic
147	C₂×M₁₆	—	166	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
151	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
152	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
153	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	166	2	no	—
157	C₂×C₂²⋊C₄	—	332	1	no	—
158	M₃₂	—	2	2	no	—
159	M₃₂	—	2	4	no	—
160	M₃₂	C₂²⋊D₈₃	1	4	yes	—
161	C₂×C₁₆	—	2	4	no	—
162	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
163	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
164	#108	—	166	2	no	—
165	#108	—	166	2	no	—
166	#106	—	166	2	no	—
167	#106	—	166	2	no	—
168	C₂²⋊C₈	—	332	1	no	—
169	C₂²⋊C₈	—	332	1	no	—
170	#95	—	332	1	no	—
171	C₂⁴	#13050579	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	332	2	no	—
175	C₂²×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
177	C₂²×C₄	#13050579	1	1	yes	characteristic
178	C₂²×C₄	#13050579	1	1	yes	characteristic
179	C₂²×C₄	#13050579	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
181	M₁₆	—	332	2	no	—
182	M₁₆	—	332	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	332	2	no	—
191	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
192	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
193	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
194	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
195	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
196	C₂×C₈	#13050590	1	2	yes	—
197	C₄⋊C₄	—	332	2	no	—
198	C₄⋊C₄	—	332	2	no	—
199	C₂²⋊C₄	—	664	2	no	—
200	C₁₆	—	2	2	no	—
201	C₁₆	C₂²⋊Dic₈₃	1	4	yes	complemented
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	4	1	no	—
205	C₂³	#14581724	1	1	yes	characteristic
206	C₂³	#14581724	1	1	yes	characteristic
207	C₂³	C₂×#13050579	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	664	2	no	—
216	C₂×C₄	—	664	2	no	—
217	C₂×C₄	—	664	2	no	—
218	C₂×C₄	#14581724	1	1	yes	characteristic
219	C₂×C₄	#14581724	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
221	C₂×C₄	#14581766	1	2	yes	—
222	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
223	C₂×C₄	#14581814	1	2	yes	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	664	2	no	—
227	C₈	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
228	C₈	#14581751	1	1	yes	characteristic
229	C₈	#14581771	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	4	no	—
238	C₂²	#19127524	1	2	yes	complemented
239	C₂²	C₂×#14581724	1	1	yes	characteristic
240	C₂²	#19127930	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	1328	2	no	—
244	C₄	#19127561	1	1	yes	characteristic
245	C₄	#19127739	1	1	yes	characteristic
246	C₄	#19127930	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	1	no	—
250	C₂	#24907488	1	1	yes	characteristic
251	C₂	C₂×#19127524	1	1	yes	characteristic
252	C₂	#24909236	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14581724
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂²×M₃₂×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#106
prime radical	C₂⁴×C₈₃	SD₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂²×M₃₂	D₈₃
hypocenter	C₈₃	#12831
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#19127524	C₂	semidirect
4. #24905562	C₂	semidirect
15. C₂×M₃₂×C₈₃	C₄	semidirect
18. C₂²×C₁₆×C₈₃	C₄	semidirect
74. C₁₆×C₈₃	C₂²⋊C₄	semidirect
111. C₂²×C₈₃	#365	semidirect
132. C₈₃	#12831	semidirect
135. C₂×M₃₂	Dic₈₃	semidirect
138. C₂²×C₁₆	Dic₈₃	semidirect
201. C₁₆	C₂²⋊Dic₈₃	semidirect
238. C₂²	#19127524	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127524	2
#24907501	2
#24907488	2
#24905562	2
C₂²×M₃₂×C₈₃	2
#12831	83