Subgroups — #35322921

Order 21248.

398 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	C₂×Dic₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	#24905541	C₂	1	1	yes	maximal, characteristic
5	#24905540	C₂	1	1	yes	maximal, characteristic
6	#24905540	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#1374	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₆₆	C₄	1	1	yes	characteristic
9	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×Dic₁₆₆	—	2	2	no	—
12	C₈×Dic₁₆₆	—	2	2	no	—
13	C₂×C₄⋊(C₈₃⋊C₈)	C₄	1	1	yes	characteristic
14	C₂×Dic₈₃⋊C₈	C₄	1	1	yes	characteristic
15	C₂×Dic₈₃⋊C₈	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
18	Dic₈₃⋊C₁₆	—	2	2	no	—
19	C₁₆×Dic₈₃	—	2	2	no	—
20	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₆₆	C₈	1	1	yes	characteristic
24	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₆₆	—	2	2	no	—
27	C₄×Dic₁₆₆	—	2	2	no	—
28	C₂×C₄⋊Dic₈₃	C₈	1	1	yes	characteristic
29	C₂×Dic₈₃⋊C₄	C₈	1	1	yes	characteristic
30	C₂×Dic₈₃⋊C₄	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈₃⋊C₁₆	—	2	1	no	—
34	C₂×C₈₃⋊C₁₆	—	2	1	no	—
35	C₂×C₈₃⋊C₁₆	—	2	1	no	—
36	C₂×C₈₃⋊C₁₆	—	2	1	no	—
37	C₂×C₈₃⋊C₁₆	—	2	1	no	—
38	C₂×C₈₃⋊C₁₆	Q₈	1	2	yes	complemented
39	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
40	Dic₈₃⋊C₈	—	2	2	no	—
41	Dic₈₃⋊C₈	—	2	2	no	—
42	C₈×Dic₈₃	—	2	1	no	—
43	C₈×Dic₈₃	—	2	1	no	—
44	C₈×Dic₈₃	D₄	1	2	yes	—
45	C₈×Dic₈₃	D₄	1	2	yes	—
46	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₈₃	—	2	1	no	—
50	C₂×C₁₆×C₈₃	—	2	1	no	—
51	C₂×C₁₆×C₈₃	—	2	1	no	—
52	C₂×C₁₆×C₈₃	—	2	1	no	—
53	C₂×C₁₆×C₈₃	—	2	1	no	—
54	C₂×C₁₆×C₈₃	Q₈	1	2	yes	complemented
55	C₄×C₈×C₈₃	—	2	1	no	—
56	C₄×C₈×C₈₃	—	2	1	no	—
57	C₂×Dic₁₆₆	—	2	2	no	—
58	C₂×Dic₁₆₆	—	2	2	no	—
59	C₂×Dic₁₆₆	C₁₆	1	4	yes	complemented
60	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₈₃	C₂×C₈	1	1	yes	characteristic
64	C₂×C₈₃⋊C₈	—	2	1	no	—
65	C₂×C₈₃⋊C₈	—	2	1	no	—
66	C₂×C₈₃⋊C₈	—	2	1	no	—
67	C₂×C₈₃⋊C₈	—	2	1	no	—
68	C₂×C₈₃⋊C₈	—	2	1	no	—
69	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₄⋊Dic₈₃	—	2	2	no	—
71	Dic₈₃⋊C₄	—	2	2	no	—
72	Dic₈₃⋊C₄	—	2	2	no	—
73	C₄×Dic₈₃	—	2	1	no	—
74	C₄×Dic₈₃	—	2	1	no	—
75	C₄×Dic₈₃	C₂²⋊C₄	1	2	yes	—
76	C₄×Dic₈₃	M₁₆	1	2	yes	—
77	C₈₃⋊C₁₆	—	2	2	no	—
78	C₈₃⋊C₁₆	—	4	1	no	—
79	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₈₃	—	2	1	no	—
83	C₂×C₈×C₈₃	—	2	1	no	—
84	C₂×C₈×C₈₃	—	2	1	no	—
85	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
90	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
91	C₄²×C₈₃	—	2	1	no	—
92	C₄²×C₈₃	—	2	1	no	—
93	C₁₆×C₈₃	—	2	2	no	—
94	C₁₆×C₈₃	—	4	1	no	—
95	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₈₃	—	2	1	no	—
97	C₂×C₄×C₈₃	—	2	1	no	—
98	C₂×C₄×C₈₃	—	2	1	no	—
99	C₂×C₄×C₈₃	—	2	1	no	—
100	C₂×C₄×C₈₃	—	2	1	no	—
101	C₂×C₄×C₈₃	C₂×C₁₆	1	2	yes	—
102	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
107	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
108	C₂×Dic₈₃	—	2	1	no	—
109	C₂×Dic₈₃	—	2	1	no	—
110	C₂×Dic₈₃	—	2	1	no	—
111	C₂×Dic₈₃	—	2	1	no	—
112	C₂×Dic₈₃	—	2	1	no	—
113	C₂×Dic₈₃	—	2	1	no	—
114	C₂×Dic₈₃	—	2	1	no	—
115	C₂×Dic₈₃	C₂²⋊C₈	1	2	yes	—
116	C₂×Dic₈₃	C₂²⋊C₈	1	2	yes	—
117	C₂×Dic₈₃	C₂×C₁₆	1	2	yes	—
118	C₂×Dic₈₃	C₂×C₁₆	1	2	yes	—
119	C₂×Dic₈₃	M₃₂	1	2	yes	—
120	Dic₁₆₆	—	2	8	no	—
121	C₈×C₈₃	—	4	1	no	—
122	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₈₃	#124	1	1	yes	characteristic
125	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
126	C₈₃⋊C₈	—	2	2	no	—
127	C₈₃⋊C₈	—	4	1	no	—
128	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₂²×C₈₃	C₈×Q₈	1	1	yes	characteristic
130	C₂²×C₈₃	C₈×Q₈	1	1	yes	characteristic
131	C₂²×C₈₃	C₂²×C₁₆	1	1	yes	characteristic
132	C₂²×C₈₃	C₂×M₃₂	1	1	yes	characteristic
133	C₂²×C₈₃	#504	1	1	yes	characteristic
134	C₂²×C₈₃	#504	1	1	yes	characteristic
135	C₄×C₈₃	—	2	2	no	—
136	C₄×C₈₃	—	4	1	no	—
137	C₄×C₈₃	#391	1	1	yes	characteristic
138	C₄×C₈₃	#408	1	1	yes	characteristic
139	C₄×C₈₃	#446	1	1	yes	characteristic
140	C₄×C₈₃	#446	1	1	yes	characteristic
141	Dic₈₃	—	2	2	no	—
142	Dic₈₃	—	2	2	no	—
143	Dic₈₃	—	4	1	no	—
144	Dic₈₃	C₂²⋊C₁₆	1	4	yes	—
145	#13263	—	83	1	no	maximal, Sylow
146	C₂×C₈₃	#1984	1	1	yes	characteristic
147	C₂×C₈₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
148	C₂×C₈₃	#2113	1	1	yes	characteristic
149	C₂×C₈₃	Q₈×C₁₆	1	1	yes	characteristic
150	C₂×C₈₃	Q₈×C₁₆	1	1	yes	characteristic
151	C₂×C₈₃	#2183	1	1	yes	characteristic
152	C₂×C₈₃	#2183	1	1	yes	characteristic
153	C₂×C₈×Q₈	—	83	1	no	—
154	C₂×C₄⋊C₁₆	—	83	1	no	—
155	C₂×C₄×C₁₆	—	83	1	no	—
156	#1374	—	83	1	no	—
157	#1374	—	83	1	no	—
158	#1374	—	83	1	no	—
159	#1374	D₈₃	1	1	yes	characteristic
160	C₈₃	#13263	1	1	yes	characteristic, complemented, Sylow
161	C₂×C₄×Q₈	—	83	1	no	—
162	C₂²×C₁₆	—	83	1	no	—
163	C₂²×C₁₆	—	83	1	no	—
164	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
165	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
166	C₈×Q₈	—	166	2	no	—
167	C₈×Q₈	—	166	2	no	—
168	C₂×C₄⋊C₈	—	83	1	no	—
169	C₂×C₄⋊C₈	—	83	1	no	—
170	C₂×C₄⋊C₈	—	83	1	no	—
171	C₂×C₄×C₈	—	83	1	no	—
172	C₂×C₄×C₈	—	83	1	no	—
173	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
174	C₄⋊C₁₆	—	166	2	no	—
175	C₄×C₁₆	—	166	2	no	—
176	C₂²×Q₈	—	83	1	no	—
177	C₂²×C₈	—	83	1	no	—
178	C₂²×C₈	—	83	1	no	—
179	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
181	C₄×Q₈	—	166	2	no	—
182	C₄×Q₈	—	166	2	no	—
183	C₂×C₄⋊C₄	—	83	1	no	—
184	C₂×C₄⋊C₄	—	83	1	no	—
185	C₂×C₄⋊C₄	—	83	1	no	—
186	C₂×C₄²	—	83	1	no	—
187	C₂×C₄²	—	83	1	no	—
188	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	83	2	no	—
195	C₂×C₁₆	—	166	1	no	—
196	C₂×C₁₆	—	166	1	no	—
197	C₂×C₁₆	—	166	1	no	—
198	C₂×C₁₆	—	166	1	no	—
199	C₂×C₁₆	—	166	1	no	—
200	C₂×C₁₆	Dic₁₆₆	1	2	yes	complemented
201	C₄⋊C₈	—	166	2	no	—
202	C₄⋊C₈	—	166	2	no	—
203	C₄⋊C₈	—	166	2	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	83	2	no	—
207	C₄×C₈	—	83	2	no	—
208	C₄×C₈	—	166	1	no	—
209	C₄×C₈	—	166	1	no	—
210	C₂×Q₈	—	83	4	no	—
211	C₂×Q₈	—	166	2	no	—
212	C₂×Q₈	—	166	2	no	—
213	C₂²×C₄	—	83	1	no	—
214	C₂²×C₄	—	83	1	no	—
215	C₂²×C₄	—	83	1	no	—
216	C₂²×C₄	—	83	1	no	—
217	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	83	2	no	—
224	C₂×C₈	—	166	1	no	—
225	C₂×C₈	—	166	1	no	—
226	C₂×C₈	—	166	1	no	—
227	C₂×C₈	—	166	1	no	—
228	C₂×C₈	—	166	1	no	—
229	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
230	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
231	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
232	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
233	C₂×C₈	#13050589	1	1	yes	characteristic
234	C₂×C₈	#13050590	1	1	yes	characteristic
235	C₄⋊C₄	—	166	2	no	—
236	C₄⋊C₄	—	166	2	no	—
237	C₄⋊C₄	—	166	2	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	83	2	no	—
241	C₄²	—	83	2	no	—
242	C₄²	—	166	1	no	—
243	C₄²	—	166	1	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	4	1	no	—
246	C₁₆	—	332	1	no	—
247	C₁₆	—	166	2	no	—
248	C₂³	C₂×C₈×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	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	83	2	no	—
256	C₂×C₄	—	83	2	no	—
257	C₂×C₄	—	83	2	no	—
258	C₂×C₄	—	83	2	no	—
259	C₂×C₄	—	83	2	no	—
260	C₂×C₄	—	166	1	no	—
261	C₂×C₄	—	166	1	no	—
262	C₂×C₄	—	166	1	no	—
263	C₂×C₄	—	166	1	no	—
264	C₂×C₄	—	166	1	no	—
265	C₂×C₄	—	166	1	no	—
266	C₂×C₄	—	166	1	no	—
267	C₂×C₄	C₁₆×D₈₃	1	2	yes	—
268	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
269	C₂×C₄	#14581757	1	1	yes	characteristic
270	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
271	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
272	C₂×C₄	#14581798	1	1	yes	characteristic
273	C₂×C₄	#14581800	1	1	yes	characteristic
274	C₈	—	4	1	no	—
275	C₈	—	332	1	no	—
276	C₈	—	166	2	no	—
277	C₈	#14581769	1	1	yes	characteristic
278	C₈	#14581770	1	1	yes	characteristic
279	C₈	#14581778	1	1	yes	characteristic
280	C₈	#14581791	1	1	yes	characteristic
281	C₂²	C₂²⋊C₈×D₈₃	1	1	yes	characteristic
282	C₂²	#19127825	1	1	yes	characteristic
283	C₂²	C₈×Dic₁₆₆	1	1	yes	characteristic
284	C₂²	C₂×C₁₆×D₈₃	1	1	yes	characteristic
285	C₂²	M₃₂×D₈₃	1	1	yes	characteristic
286	C₂²	#19128117	1	1	yes	characteristic
287	C₂²	#19128119	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	332	1	no	—
291	C₄	—	83	4	no	—
292	C₄	—	166	2	no	—
293	C₄	—	166	2	no	—
294	C₄	#19127605	1	1	yes	characteristic
295	C₄	#19127666	1	1	yes	characteristic
296	C₄	#19127829	1	1	yes	characteristic
297	C₄	#19127830	1	1	yes	characteristic
298	C₂	#24908397	1	1	yes	characteristic
299	C₂	C₂²⋊C₁₆×D₈₃	1	1	yes	characteristic
300	C₂	#24909075	1	1	yes	characteristic
301	C₂	#24909365	1	1	yes	characteristic
302	C₂	C₁₆×Dic₁₆₆	1	1	yes	characteristic
303	C₂	#24909369	1	1	yes	characteristic
304	C₂	#24909370	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂²×C₁₆
Fitting	C₈₃×#1374	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1374	D₈₃
hypocenter	C₈₃	#13263
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
38. C₂×C₈₃⋊C₁₆	Q₈	semidirect
54. C₂×C₁₆×C₈₃	Q₈	semidirect
59. C₂×Dic₁₆₆	C₁₆	semidirect
160. C₈₃	#13263	semidirect
200. C₂×C₁₆	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
C₂×Dic₈₃⋊C₁₆	2
C₂×C₁₆×Dic₈₃	2
#24905541	2
#24905540	2
C₈₃×#1374	2
#13263	83