Subgroups — #35304529

Order 21248.

538 conjugacy classes of subgroups.

The subgroup lattice has 282 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	#24907615	C₂	1	1	yes	maximal, characteristic
3	#24905347	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₈₃	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)	—	2	1	no	—
9	C₂×C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
11	#19127571	—	2	1	no	—
12	#19127571	—	2	1	no	—
13	#19127571	C₄	1	2	yes	—
14	#19127571	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×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	2	no	—
23	#14581792	—	4	2	no	—
24	#14581792	—	4	2	no	—
25	#14581792	—	4	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	—	2	2	no	—
30	C₂×C₄⋊Dic₈₃	—	2	2	no	—
31	C₂×C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
32	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
33	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
34	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
35	C₂²×C₈×C₈₃	—	2	2	no	—
36	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₈₃	—	2	2	no	—
44	C₄⋊C₈×C₈₃	Q₈	1	4	yes	complemented
45	C₄×C₈×C₈₃	—	4	2	no	—
46	C₂×Dic₁₆₆	—	4	4	no	—
47	C₂×Dic₁₆₆	—	4	4	no	—
48	C₂×Dic₁₆₆	—	4	4	no	—
49	C₂²×Dic₈₃	—	2	4	no	—
50	C₂×C₈₃⋊C₈	—	4	2	no	—
51	C₂×C₈₃⋊C₈	—	4	2	no	—
52	C₂×C₈₃⋊C₈	—	4	2	no	—
53	C₄⋊Dic₈₃	—	4	4	no	—
54	C₄⋊Dic₈₃	—	4	4	no	—
55	C₄⋊Dic₈₃	—	4	4	no	—
56	C₄⋊Dic₈₃	—	4	2	no	—
57	C₄⋊Dic₈₃	—	4	2	no	—
58	C₄⋊Dic₈₃	—	4	2	no	—
59	C₄⋊Dic₈₃	—	4	2	no	—
60	C₄⋊Dic₈₃	—	4	4	no	—
61	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₃	—	2	4	no	—
69	C₂×C₈×C₈₃	—	4	2	no	—
70	C₂×C₈×C₈₃	—	4	2	no	—
71	C₄²×C₈₃	—	2	2	no	—
72	C₄²×C₈₃	—	2	2	no	—
73	C₄²×C₈₃	—	2	2	no	—
74	C₄²×C₈₃	D₈	1	2	yes	—
75	C₄²×C₈₃	D₈	1	2	yes	—
76	C₄²×C₈₃	SD₁₆	1	2	yes	—
77	C₄²×C₈₃	SD₁₆	1	2	yes	—
78	C₄²×C₈₃	SD₁₆	1	2	yes	—
79	C₄²×C₈₃	Q₁₆	1	2	yes	—
80	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
81	C₄²×C₈₃	#41	1	2	yes	—
82	C₂³×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
87	C₂×C₄×C₈₃	—	2	1	no	—
88	C₂×C₄×C₈₃	—	2	1	no	—
89	C₂×C₄×C₈₃	#100	1	2	yes	—
90	C₂×C₄×C₈₃	#101	1	2	yes	—
91	C₂×C₄×C₈₃	#102	1	2	yes	—
92	C₂×C₄×C₈₃	#102	1	2	yes	—
93	C₂×C₄×C₈₃	#102	1	2	yes	—
94	C₂×C₄×C₈₃	#102	1	2	yes	—
95	C₂×C₄×C₈₃	#103	1	2	yes	—
96	C₂×C₄×C₈₃	#103	1	2	yes	—
97	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
98	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₃	#137	1	2	yes	—
108	C₂×Dic₈₃	—	4	4	no	—
109	C₂×Dic₈₃	—	4	4	no	—
110	C₂×Dic₈₃	—	4	4	no	—
111	Dic₁₆₆	—	8	8	no	—
112	C₈×C₈₃	—	4	4	no	—
113	C₈₃⋊C₈	—	8	2	no	—
114	C₂²×C₈₃	#391	1	1	yes	characteristic
115	C₂²×C₈₃	C₂×#100	1	1	yes	characteristic
116	C₂²×C₈₃	C₂×#101	1	1	yes	characteristic
117	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
118	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
119	C₂²×C₈₃	C₂×#103	1	1	yes	characteristic
120	C₂²×C₈₃	#419	1	1	yes	characteristic
121	C₄×C₈₃	—	2	2	no	—
122	C₄×C₈₃	—	2	2	no	—
123	C₄×C₈₃	—	2	2	no	—
124	C₄×C₈₃	#331	1	4	yes	—
125	C₄×C₈₃	#332	1	4	yes	—
126	C₄×C₈₃	#474	1	2	yes	—
127	C₄×C₈₃	#475	1	2	yes	—
128	C₄×C₈₃	#486	1	1	yes	characteristic
129	C₄×C₈₃	#487	1	1	yes	characteristic
130	C₄×C₈₃	#498	1	1	yes	characteristic
131	C₄×C₈₃	#500	1	1	yes	characteristic
132	Dic₈₃	—	8	4	no	—
133	#9898	—	83	1	no	maximal, Sylow
134	C₂×C₈₃	C₂×#331	1	1	yes	characteristic
135	C₂×C₈₃	C₂×#332	1	1	yes	characteristic
136	C₂×C₈₃	#1546	1	1	yes	characteristic
137	C₂×C₈₃	#1548	1	1	yes	characteristic
138	C₂×C₈₃	#1988	1	1	yes	characteristic
139	C₂×C₈₃	#1990	1	1	yes	characteristic
140	C₂×C₈₃	#1997	1	1	yes	characteristic
141	#2868	—	83	1	no	—
142	#1842	—	83	1	no	—
143	C₄×C₄⋊C₈	D₈₃	1	1	yes	characteristic
144	#1296	—	83	4	no	—
145	C₈₃	#9898	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄⋊Q₈	—	83	2	no	—
147	C₂×C₄⋊Q₈	—	166	1	no	—
148	C₂×C₄⋊Q₈	—	166	1	no	—
149	C₂×C₄⋊C₈	—	83	2	no	—
150	C₂×C₄⋊C₈	—	166	1	no	—
151	C₂×C₄⋊C₈	—	166	1	no	—
152	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
153	C₂×C₄×C₈	—	2	1	no	—
154	#384	—	83	2	no	—
155	#384	—	83	2	no	—
156	#384	—	166	1	no	—
157	#384	—	166	1	no	—
158	C₄³	C₂×D₈₃	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	C₂²×Q₈	—	166	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	166	2	no	—
163	C₄⋊Q₈	—	332	4	no	—
164	C₄⋊Q₈	—	332	2	no	—
165	C₄⋊Q₈	—	332	2	no	—
166	C₄⋊Q₈	—	332	2	no	—
167	C₄⋊Q₈	—	332	2	no	—
168	C₂×C₄⋊C₄	—	83	4	no	—
169	C₂×C₄⋊C₄	—	166	2	no	—
170	C₂×C₄⋊C₄	—	166	2	no	—
171	C₂×C₄⋊C₄	—	166	2	no	—
172	C₂×C₄⋊C₄	—	166	2	no	—
173	C₂×C₄⋊C₄	—	166	2	no	—
174	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
176	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
177	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	332	2	no	—
183	C₄⋊C₈	—	332	2	no	—
184	C₄⋊C₈	—	166	4	no	—
185	C₄⋊C₈	Dic₁₆₆	1	4	yes	complemented
186	C₄×C₈	—	4	2	no	—
187	C₂×Q₈	—	332	4	no	—
188	C₂×Q₈	—	332	4	no	—
189	C₂×Q₈	—	332	4	no	—
190	C₂²×C₄	—	166	4	no	—
191	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	332	2	no	—
202	C₂×C₈	—	332	2	no	—
203	C₂×C₈	—	332	2	no	—
204	C₄⋊C₄	—	332	4	no	—
205	C₄⋊C₄	—	332	4	no	—
206	C₄⋊C₄	—	332	2	no	—
207	C₄⋊C₄	—	332	2	no	—
208	C₄⋊C₄	—	332	2	no	—
209	C₄⋊C₄	—	332	2	no	—
210	C₄⋊C₄	—	332	4	no	—
211	C₄⋊C₄	—	332	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	#13050575	1	2	yes	—
216	C₄²	D₆₆₄	1	2	yes	—
217	C₄²	#13050577	1	2	yes	—
218	C₄²	#13050579	1	2	yes	—
219	C₄²	#13050579	1	2	yes	—
220	C₄²	Dic₃₃₂	1	2	yes	—
221	C₄²	C₂×Dic₁₆₆	1	2	yes	—
222	C₄²	#13050590	1	2	yes	—
223	C₂³	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
224	Q₈	—	664	8	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	332	4	no	—
232	C₂×C₄	—	332	4	no	—
233	C₂×C₄	—	332	4	no	—
234	C₂×C₄	#14581708	1	2	yes	—
235	C₂×C₄	#14581710	1	2	yes	—
236	C₂×C₄	#14581712	1	2	yes	—
237	C₂×C₄	#14581712	1	2	yes	—
238	C₂×C₄	#14581714	1	2	yes	—
239	C₂×C₄	#14581714	1	2	yes	—
240	C₂×C₄	#14581717	1	2	yes	—
241	C₂×C₄	#14581717	1	2	yes	—
242	C₂×C₄	C₄×Dic₁₆₆	1	2	yes	—
243	C₂×C₄	#14581771	1	2	yes	—
244	C₂×C₄	#14581780	1	1	yes	characteristic
245	C₂×C₄	#14581792	1	1	yes	characteristic
246	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
247	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
248	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
249	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
250	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
251	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
252	C₂×C₄	#14581822	1	2	yes	—
253	C₈	—	4	4	no	—
254	C₈	—	664	2	no	—
255	C₂²	#19127607	1	1	yes	characteristic
256	C₂²	C₂×#14581708	1	1	yes	characteristic
257	C₂²	C₂×#14581710	1	1	yes	characteristic
258	C₂²	C₂×#14581712	1	1	yes	characteristic
259	C₂²	C₂×#14581714	1	1	yes	characteristic
260	C₂²	C₂×#14581717	1	1	yes	characteristic
261	C₂²	#19127709	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	664	4	no	—
266	C₄	#19127444	1	4	yes	—
267	C₄	#19127448	1	4	yes	—
268	C₄	#19127981	1	2	yes	—
269	C₄	#19127986	1	2	yes	—
270	C₄	#19128048	1	1	yes	characteristic
271	C₄	#19128052	1	1	yes	characteristic
272	C₄	#19128099	1	1	yes	characteristic
273	C₄	#19128106	1	1	yes	characteristic
274	C₂	C₂×#19127444	1	1	yes	characteristic
275	C₂	C₂×#19127448	1	1	yes	characteristic
276	C₂	#24906134	1	1	yes	characteristic
277	C₂	#24906143	1	1	yes	characteristic
278	C₂	#24908415	1	1	yes	characteristic
279	C₂	#24908424	1	1	yes	characteristic
280	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	C₄×C₄⋊C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₃
hypocenter	C₈₃	#9898
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
44. C₄⋊C₈×C₈₃	Q₈	semidirect
145. C₈₃	#9898	semidirect
185. 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
#24907615	2
#24905347	2
C₄×C₄⋊C₈×C₈₃	2
#9898	83