Subgroups — #35304551

Order 21248.

506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24915861	C₂	1	1	yes	maximal, characteristic
2	#24907580	C₂	1	1	yes	maximal, characteristic
3	#24905347	C₂	1	4	yes	maximal
4	C₈₃×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581792	—	2	1	no	—
6	C₂×#14581792	—	2	1	no	—
7	C₂×#14581792	C₄	1	2	yes	—
8	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
9	#19127571	—	2	1	no	—
10	#19127571	—	2	1	no	—
11	#19127571	C₄	1	2	yes	—
12	#19127571	C₂²	1	2	yes	—
13	#19127456	—	2	1	no	—
14	#19127456	—	2	1	no	—
15	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
16	C₂×#97×C₈₃	—	2	1	no	—
17	C₄³×C₈₃	C₂²	1	1	yes	characteristic
18	#336×C₈₃	—	2	1	no	—
19	C₂²×Dic₁₆₆	—	2	2	no	—
20	C₂²×C₈₃⋊C₈	—	2	2	no	—
21	#14581792	—	4	4	no	—
22	#14581792	—	4	4	no	—
23	#14581792	—	4	4	no	—
24	C₂×C₄⋊Dic₈₃	—	2	2	no	—
25	C₂×C₄⋊Dic₈₃	—	2	2	no	—
26	C₂×C₄⋊Dic₈₃	—	2	2	no	—
27	C₂×C₄⋊Dic₈₃	—	2	2	no	—
28	C₂×C₄⋊Dic₈₃	—	2	2	no	—
29	C₂×C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
30	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
31	C₂²×C₈×C₈₃	—	2	2	no	—
32	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₃	—	2	2	no	—
40	C₄⋊C₈×C₈₃	Q₈	1	4	yes	complemented
41	#97×C₈₃	—	4	2	no	—
42	C₂×Dic₁₆₆	—	4	4	no	—
43	C₂×Dic₁₆₆	—	4	4	no	—
44	C₂×Dic₁₆₆	—	4	4	no	—
45	C₂²×Dic₈₃	—	2	4	no	—
46	C₂×C₈₃⋊C₈	—	4	2	no	—
47	C₂×C₈₃⋊C₈	—	4	2	no	—
48	C₂×C₈₃⋊C₈	—	4	2	no	—
49	C₄⋊Dic₈₃	—	4	4	no	—
50	C₄⋊Dic₈₃	—	4	4	no	—
51	C₄⋊Dic₈₃	—	4	4	no	—
52	C₄⋊Dic₈₃	—	4	4	no	—
53	C₄⋊Dic₈₃	—	4	4	no	—
54	C₄⋊Dic₈₃	—	4	4	no	—
55	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₃	—	2	4	no	—
63	C₂×C₈×C₈₃	—	4	2	no	—
64	C₂×C₈×C₈₃	—	4	2	no	—
65	C₄²×C₈₃	—	2	2	no	—
66	C₄²×C₈₃	—	2	2	no	—
67	C₄²×C₈₃	—	2	2	no	—
68	C₄²×C₈₃	—	2	2	no	—
69	C₄²×C₈₃	—	2	2	no	—
70	C₄²×C₈₃	SD₁₆	1	2	yes	—
71	C₄²×C₈₃	Q₁₆	1	2	yes	—
72	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
73	C₄²×C₈₃	#41	1	2	yes	—
74	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₃	—	2	1	no	—
76	C₂×C₄×C₈₃	—	2	1	no	—
77	C₂×C₄×C₈₃	—	2	1	no	—
78	C₂×C₄×C₈₃	—	2	1	no	—
79	C₂×C₄×C₈₃	—	2	1	no	—
80	C₂×C₄×C₈₃	—	2	1	no	—
81	C₂×C₄×C₈₃	—	2	1	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	#100	1	2	yes	—
86	C₂×C₄×C₈₃	#101	1	2	yes	—
87	C₂×C₄×C₈₃	#103	1	2	yes	—
88	C₂×C₄×C₈₃	#103	1	2	yes	—
89	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
90	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₈₃	#137	1	2	yes	—
98	C₂×Dic₈₃	—	4	4	no	—
99	C₂×Dic₈₃	—	4	4	no	—
100	C₂×Dic₈₃	—	4	4	no	—
101	Dic₁₆₆	—	8	8	no	—
102	C₈×C₈₃	—	4	4	no	—
103	C₈₃⋊C₈	—	8	2	no	—
104	C₂²×C₈₃	#391	1	1	yes	characteristic
105	C₂²×C₈₃	C₂×#100	1	1	yes	characteristic
106	C₂²×C₈₃	C₂×#101	1	1	yes	characteristic
107	C₂²×C₈₃	C₂×#103	1	1	yes	characteristic
108	C₂²×C₈₃	#418	1	1	yes	characteristic
109	C₂²×C₈₃	#418	1	1	yes	characteristic
110	C₂²×C₈₃	#419	1	1	yes	characteristic
111	C₄×C₈₃	—	2	2	no	—
112	C₄×C₈₃	—	2	2	no	—
113	C₄×C₈₃	—	2	2	no	—
114	C₄×C₈₃	—	2	2	no	—
115	C₄×C₈₃	—	2	2	no	—
116	C₄×C₈₃	#476	1	2	yes	—
117	C₄×C₈₃	#477	1	2	yes	—
118	C₄×C₈₃	#489	1	2	yes	—
119	C₄×C₈₃	#501	1	2	yes	—
120	Dic₈₃	—	8	4	no	—
121	#9903	—	83	1	no	maximal, Sylow
122	C₂×C₈₃	#1548	1	1	yes	characteristic
123	C₂×C₈₃	#1549	1	1	yes	characteristic
124	C₂×C₈₃	#1555	1	1	yes	characteristic
125	C₂×C₈₃	#1556	1	1	yes	characteristic
126	C₂×C₈₃	#1987	1	1	yes	characteristic
127	C₂×C₈₃	#1993	1	1	yes	characteristic
128	C₂×C₈₃	#1997	1	1	yes	characteristic
129	#2868	—	83	1	no	—
130	#1831	—	83	1	no	—
131	#1767	D₈₃	1	1	yes	characteristic
132	#1296	—	83	4	no	—
133	C₈₃	#9903	1	1	yes	characteristic, complemented, Sylow
134	C₂×C₄⋊Q₈	—	83	2	no	—
135	C₂×C₄⋊Q₈	—	166	1	no	—
136	C₂×C₄⋊Q₈	—	166	1	no	—
137	C₂×C₄⋊C₈	—	83	2	no	—
138	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
139	C₂×#97	—	2	1	no	—
140	#384	—	83	2	no	—
141	#384	—	83	2	no	—
142	#384	—	166	1	no	—
143	#384	—	166	1	no	—
144	C₄³	C₂×D₈₃	1	1	yes	characteristic
145	#336	—	2	1	no	—
146	#336	—	166	1	no	—
147	#336	—	166	1	no	—
148	C₂²×Q₈	—	166	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	166	2	no	—
151	C₄⋊Q₈	—	332	4	no	—
152	C₄⋊Q₈	—	332	4	no	—
153	C₄⋊Q₈	—	332	4	no	—
154	C₂×C₄⋊C₄	—	83	4	no	—
155	C₂×C₄⋊C₄	—	166	2	no	—
156	C₂×C₄⋊C₄	—	166	2	no	—
157	C₂×C₄⋊C₄	—	166	2	no	—
158	C₂×C₄⋊C₄	—	166	2	no	—
159	C₂×C₄⋊C₄	—	166	2	no	—
160	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
161	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
162	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
163	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	166	4	no	—
169	C₄⋊C₈	Dic₁₆₆	1	4	yes	complemented
170	#97	—	4	2	no	—
171	C₂×Q₈	—	332	4	no	—
172	C₂×Q₈	—	332	4	no	—
173	C₂×Q₈	—	332	4	no	—
174	C₂²×C₄	—	166	4	no	—
175	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	332	2	no	—
186	C₂×C₈	—	332	2	no	—
187	C₂×C₈	—	332	2	no	—
188	C₄⋊C₄	—	332	4	no	—
189	C₄⋊C₄	—	332	4	no	—
190	C₄⋊C₄	—	332	4	no	—
191	C₄⋊C₄	—	332	4	no	—
192	C₄⋊C₄	—	332	4	no	—
193	C₄⋊C₄	—	332	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	#13050578	1	2	yes	—
200	C₄²	#13050580	1	2	yes	—
201	C₄²	C₂×Dic₁₆₆	1	2	yes	—
202	C₄²	#13050590	1	2	yes	—
203	C₂³	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
204	Q₈	—	664	8	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	332	4	no	—
216	C₂×C₄	—	332	4	no	—
217	C₂×C₄	—	332	4	no	—
218	C₂×C₄	#14581708	1	2	yes	—
219	C₂×C₄	#14581710	1	2	yes	—
220	C₂×C₄	#14581715	1	2	yes	—
221	C₂×C₄	#14581715	1	2	yes	—
222	C₂×C₄	C₄×Dic₁₆₆	1	2	yes	—
223	C₂×C₄	#14581771	1	2	yes	—
224	C₂×C₄	#14581780	1	1	yes	characteristic
225	C₂×C₄	#14581792	1	1	yes	characteristic
226	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
227	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
228	C₂×C₄	#14581817	1	2	yes	—
229	C₂×C₄	#14581820	1	2	yes	—
230	C₂×C₄	#14581825	1	2	yes	—
231	C₈	—	4	4	no	—
232	C₈	—	664	2	no	—
233	C₂²	#19127607	1	1	yes	characteristic
234	C₂²	C₂×#14581708	1	1	yes	characteristic
235	C₂²	C₂×#14581710	1	1	yes	characteristic
236	C₂²	C₂×#14581715	1	1	yes	characteristic
237	C₂²	#19127705	1	1	yes	characteristic
238	C₂²	#19127707	1	1	yes	characteristic
239	C₂²	#19127711	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	664	4	no	—
246	C₄	#19127991	1	2	yes	—
247	C₄	#19127996	1	2	yes	—
248	C₄	#19128061	1	2	yes	—
249	C₄	#19128110	1	2	yes	—
250	C₂	#24906145	1	1	yes	characteristic
251	C₂	#24906149	1	1	yes	characteristic
252	C₂	#24906176	1	1	yes	characteristic
253	C₂	#24906181	1	1	yes	characteristic
254	C₂	#24908411	1	1	yes	characteristic
255	C₂	#24908436	1	1	yes	characteristic
256	C₂	#24908452	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₈₃)⋊C₄
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1767	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1767	D₈₃
hypocenter	C₈₃	#9903
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
40. C₄⋊C₈×C₈₃	Q₈	semidirect
133. C₈₃	#9903	semidirect
169. C₄⋊C₈	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
#24915861	2
#24907580	2
#24905347	2
C₈₃×#1767	2
#9903	83