Subgroups — #35423248

Order 21248.

750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	#24910418	C₂	1	2	yes	maximal
3	#24908234	C₂	1	4	yes	maximal
4	#24908226	C₂	1	2	yes	maximal
5	#24907626	C₂	1	2	yes	maximal
6	#24907623	C₂	1	1	yes	maximal, characteristic
7	#107×Dic₈₃	C₂	1	2	yes	maximal
8	C₄×#107×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₆₆	C₄	1	8	yes	complemented
11	C₂×#14581726	C₂²	1	2	yes	—
12	C₂×#14581726	C₂²	1	2	yes	—
13	C₂×#14581723	C₂²	1	4	yes	—
14	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×Dic₈₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₈₃	C₂²	1	2	yes	—
17	#19127607	—	2	2	no	—
18	#19127588	—	2	4	no	—
19	#19127571	C₂²	1	2	yes	—
20	#19127566	C₂²	1	4	yes	—
21	C₄⋊C₄×Dic₈₃	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₈₃	—	2	2	no	—
23	#19127463	C₂²	1	4	yes	—
24	C₂×#107×C₈₃	C₂²	1	1	yes	characteristic
25	C₂×#107×C₈₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
28	#340×C₈₃	C₂²	1	2	yes	—
29	C₂²×Dic₁₆₆	D₄	1	1	yes	characteristic
30	C₂²×C₈₃⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₁₆₆	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₆₆	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₈₃	—	2	2	no	—
34	C₂×C₄⋊Dic₈₃	—	2	2	no	—
35	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₈₃	C₂³	1	4	yes	—
37	C₂×Dic₈₃⋊C₄	D₄	1	2	yes	—
38	C₂×C₄×Dic₈₃	—	2	4	no	—
39	C₂×C₄×Dic₈₃	C₂³	1	2	yes	—
40	#14581726	—	2	4	no	—
41	#14581726	—	4	2	no	—
42	#14581723	—	4	4	no	—
43	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	4	yes	—
44	Dic₈₃⋊C₈	C₂×C₄	1	8	yes	—
45	C₈×Dic₈₃	C₂×C₄	1	8	yes	—
46	C₈₃⋊#95	—	2	4	no	—
47	C₈₃⋊#95	—	2	4	no	—
48	C₈₃⋊#95	—	2	4	no	—
49	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₈₃	—	2	2	no	—
54	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
55	#107×C₈₃	—	2	2	no	—
56	#107×C₈₃	Q₈	1	4	yes	complemented
57	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
58	#95×C₈₃	—	2	2	no	—
59	C₂×Dic₁₆₆	—	2	2	no	—
60	C₂×Dic₁₆₆	—	2	2	no	—
61	C₂×Dic₁₆₆	C₄⋊C₄	1	4	yes	—
62	C₂²×Dic₈₃	—	2	4	no	—
63	C₂²×Dic₈₃	C₂×D₄	1	2	yes	—
64	C₂²×Dic₈₃	C₂×D₄	1	2	yes	—
65	C₂×C₈₃⋊C₈	—	2	2	no	—
66	C₂×C₈₃⋊C₈	—	2	2	no	—
67	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
68	C₄⋊Dic₈₃	—	2	4	no	—
69	C₄⋊Dic₈₃	—	2	4	no	—
70	C₄⋊Dic₈₃	—	4	4	no	—
71	C₄⋊Dic₈₃	—	8	2	no	—
72	C₄⋊Dic₈₃	C₄⋊C₄	1	4	yes	—
73	Dic₈₃⋊C₄	C₄⋊C₄	1	8	yes	—
74	C₄×Dic₈₃	—	8	4	no	—
75	C₄×Dic₈₃	C₂²×C₄	1	4	yes	—
76	C₄×Dic₈₃	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₈₃	—	2	2	no	—
78	C₂²×C₄×C₈₃	—	2	2	no	—
79	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₈₃	—	2	1	no	—
83	C₂×C₈×C₈₃	—	2	1	no	—
84	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₈₃	#41	1	2	yes	—
88	C₄⋊C₄×C₈₃	—	2	2	no	—
89	C₄⋊C₄×C₈₃	—	2	2	no	—
90	C₄⋊C₄×C₈₃	—	2	2	no	—
91	C₄⋊C₄×C₈₃	C₂×Q₈	1	4	yes	—
92	C₄²×C₈₃	—	4	4	no	—
93	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
95	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₃	—	2	1	no	—
97	C₂×C₄×C₈₃	—	2	1	no	—
98	C₂×C₄×C₈₃	—	2	4	no	—
99	C₂×C₄×C₈₃	—	4	2	no	—
100	C₂×C₄×C₈₃	—	4	2	no	—
101	C₂×C₄×C₈₃	—	4	2	no	—
102	C₂×C₄×C₈₃	—	4	2	no	—
103	C₂×C₄×C₈₃	—	4	2	no	—
104	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₈₃	#135	1	2	yes	—
107	C₂×C₄×C₈₃	#135	1	2	yes	—
108	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×Dic₈₃	—	2	2	no	—
115	C₂×Dic₈₃	—	2	2	no	—
116	C₂×Dic₈₃	—	2	2	no	—
117	C₂×Dic₈₃	—	2	2	no	—
118	C₂×Dic₈₃	—	4	4	no	—
119	C₂×Dic₈₃	—	4	4	no	—
120	C₂×Dic₈₃	—	4	4	no	—
121	C₂×Dic₈₃	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×Dic₈₃	C₂×C₄⋊C₄	1	4	yes	—
123	Dic₁₆₆	—	2	8	no	—
124	C₈×C₈₃	—	2	2	no	—
125	C₈×C₈₃	C₄×Q₈	1	4	yes	—
126	C₈₃⋊C₈	—	2	4	no	—
127	C₂²×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₈₃	D₄×Q₈	1	1	yes	characteristic
129	C₂²×C₈₃	#550	1	1	yes	characteristic
130	C₂²×C₈₃	C₂×#135	1	1	yes	characteristic
131	C₂²×C₈₃	C₂×#135	1	1	yes	characteristic
132	C₂²×C₈₃	#576	1	1	yes	characteristic
133	C₂²×C₈₃	#578	1	1	yes	characteristic
134	C₄×C₈₃	—	2	2	no	—
135	C₄×C₈₃	—	4	4	no	—
136	C₄×C₈₃	—	8	2	no	—
137	C₄×C₈₃	#427	1	4	yes	—
138	C₄×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₄×C₈₃	#520	1	1	yes	characteristic
140	C₄×C₈₃	#557	1	1	yes	characteristic
141	C₄×C₈₃	Q₈²	1	1	yes	characteristic
142	Dic₈₃	—	2	4	no	—
143	Dic₈₃	—	2	4	no	—
144	Dic₈₃	—	8	4	no	—
145	#22727	—	83	1	no	maximal, Sylow
146	C₂×C₈₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₈₃	C₂×#427	1	1	yes	characteristic
148	C₂×C₈₃	#2913	1	1	yes	characteristic
149	C₂×C₈₃	#3380	1	1	yes	characteristic
150	C₂×C₈₃	#3383	1	1	yes	characteristic
151	C₂×C₈₃	#3393	1	1	yes	characteristic
152	C₂×C₈₃	#3396	1	1	yes	characteristic
153	C₂×C₈×Q₈	—	83	1	no	—
154	#2355	—	83	2	no	—
155	#1949	—	83	4	no	—
156	#1948	—	83	2	no	—
157	#1846	—	83	2	no	—
158	#1845	—	83	1	no	—
159	C₄×#107	—	83	2	no	—
160	C₄×#107	D₈₃	1	1	yes	characteristic
161	C₈₃	#22727	1	1	yes	characteristic, complemented, Sylow
162	C₂×C₄×Q₈	—	83	1	no	—
163	C₈×Q₈	—	83	8	no	—
164	C₂×#107	—	83	2	no	—
165	C₂×#107	—	83	2	no	—
166	C₂×#107	C₂×D₈₃	1	1	yes	characteristic
167	C₂×#107	C₂×D₈₃	1	1	yes	characteristic
168	C₂×#106	—	83	4	no	—
169	C₂×C₄⋊C₈	—	83	2	no	—
170	C₂×C₄⋊C₈	—	83	1	no	—
171	C₂×C₄×C₈	—	83	2	no	—
172	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
173	#391	—	166	2	no	—
174	#387	—	166	4	no	—
175	#384	—	83	2	no	—
176	#382	—	83	4	no	—
177	C₄×C₄⋊C₄	—	83	4	no	—
178	C₄×C₄⋊C₄	—	166	2	no	—
179	C₄×C₄⋊C₄	C₂×D₈₃	1	2	yes	—
180	#340	—	83	4	no	—
181	#340	C₂×D₈₃	1	2	yes	—
182	C₂²×Q₈	—	83	1	no	—
183	C₂²×C₈	—	83	2	no	—
184	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
186	C₄×Q₈	—	83	4	no	—
187	C₄×Q₈	—	83	4	no	—
188	C₂×C₄⋊C₄	—	83	1	no	—
189	C₂×C₄⋊C₄	—	83	2	no	—
190	C₂×C₄⋊C₄	—	83	4	no	—
191	C₂×C₄⋊C₄	—	166	2	no	—
192	C₂×C₄⋊C₄	—	166	2	no	—
193	C₂×C₄⋊C₄	C₂²×D₈₃	1	2	yes	—
194	C₂×C₄⋊C₄	C₂²×D₈₃	1	2	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	83	2	no	—
197	C₂×C₄²	—	166	4	no	—
198	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
199	#107	—	2	2	no	—
200	#107	—	332	2	no	—
201	#107	—	166	4	no	—
202	#107	Dic₁₆₆	1	4	yes	complemented
203	#106	—	332	4	no	—
204	C₄⋊C₈	—	83	8	no	—
205	C₄⋊C₈	—	83	4	no	—
206	C₄×C₈	—	83	8	no	—
207	C₄×C₈	C₄×D₈₃	1	4	yes	—
208	#95	—	2	2	no	—
209	#95	—	166	4	no	—
210	#95	—	166	4	no	—
211	#95	—	166	4	no	—
212	C₂×Q₈	—	83	4	no	—
213	C₂×Q₈	—	166	2	no	—
214	C₂×Q₈	—	166	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	83	2	no	—
218	C₂²×C₄	—	83	2	no	—
219	C₂²×C₄	—	166	4	no	—
220	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
221	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
222	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	83	4	no	—
226	C₂×C₈	—	166	2	no	—
227	C₂×C₈	—	166	2	no	—
228	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
230	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
231	C₂×C₈	#13050590	1	2	yes	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	332	4	no	—
236	C₄⋊C₄	—	83	8	no	—
237	C₄⋊C₄	—	83	4	no	—
238	C₄⋊C₄	—	664	2	no	—
239	C₄⋊C₄	—	166	4	no	—
240	C₄⋊C₄	—	166	4	no	—
241	C₄⋊C₄	C₂×Dic₁₆₆	1	4	yes	—
242	C₄²	—	4	4	no	—
243	C₄²	—	83	4	no	—
244	C₄²	—	83	4	no	—
245	C₄²	—	664	4	no	—
246	C₄²	C₂×C₄×D₈₃	1	2	yes	—
247	C₄²	Q₈×D₈₃	1	2	yes	—
248	C₂³	C₂×D₄×D₈₃	1	1	yes	characteristic
249	Q₈	—	166	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	332	4	no	—
259	C₂×C₄	—	332	4	no	—
260	C₂×C₄	—	332	4	no	—
261	C₂×C₄	—	83	4	no	—
262	C₂×C₄	—	83	4	no	—
263	C₂×C₄	—	166	2	no	—
264	C₂×C₄	—	166	2	no	—
265	C₂×C₄	—	166	2	no	—
266	C₂×C₄	—	166	2	no	—
267	C₂×C₄	C₄⋊C₄×D₈₃	1	2	yes	—
268	C₂×C₄	C₄⋊C₄×D₈₃	1	2	yes	—
269	C₂×C₄	#14581811	1	2	yes	—
270	C₂×C₄	#14581815	1	2	yes	—
271	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
272	C₂×C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
274	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
275	C₂×C₄	#14581838	1	1	yes	characteristic
276	C₂×C₄	#14581839	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	166	4	no	—
279	C₈	C₄×Dic₁₆₆	1	4	yes	—
280	C₂²	C₂×C₄⋊C₄×D₈₃	1	1	yes	characteristic
281	C₂²	D₄×Dic₁₆₆	1	1	yes	characteristic
282	C₂²	#19128372	1	1	yes	characteristic
283	C₂²	C₂×#14581811	1	1	yes	characteristic
284	C₂²	C₂×#14581815	1	1	yes	characteristic
285	C₂²	#19128503	1	1	yes	characteristic
286	C₂²	#19128515	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	2	no	—
290	C₄	—	664	4	no	—
291	C₄	—	166	4	no	—
292	C₄	—	166	4	no	—
293	C₄	#19127738	1	4	yes	—
294	C₄	C₂×C₄×Dic₁₆₆	1	1	yes	characteristic
295	C₄	#19128181	1	1	yes	characteristic
296	C₄	#19128429	1	1	yes	characteristic
297	C₄	Q₈×Dic₁₆₆	1	1	yes	characteristic
298	C₂	C₄⋊C₄×Dic₁₆₆	1	1	yes	characteristic
299	C₂	C₂×#19127738	1	1	yes	characteristic
300	C₂	#24916060	1	1	yes	characteristic
301	C₂	#24920328	1	1	yes	characteristic
302	C₂	#24920352	1	1	yes	characteristic
303	C₂	#24920449	1	1	yes	characteristic
304	C₂	#24920473	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₄×#107×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×D₄
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	C₄×#107	D₈₃
hypocenter	C₈₃	#22727
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₁₆₆	C₄	semidirect
56. #107×C₈₃	Q₈	semidirect
161. C₈₃	#22727	semidirect
202. #107	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₆₆	2
#24910418	2
#24908234	2
#24908226	2
#24907626	2
#24907623	2
#107×Dic₈₃	2
C₄×#107×C₈₃	2
#22727	83