Subgroups — #35303333

Order 21248.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
2	#24905859	C₂	1	2	yes	maximal
3	#24905786	C₂	1	2	yes	maximal
4	#24905746	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₃₂	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₆₆	C₂²	1	2	yes	—
8	C₄×Dic₃₃₂	—	2	1	no	—
9	C₄×Dic₃₃₂	—	2	1	no	—
10	C₄×Dic₃₃₂	—	2	1	no	—
11	C₄×Dic₃₃₂	—	2	1	no	—
12	#19127754	C₂²	1	1	yes	characteristic
13	#19127754	C₂²	1	1	yes	characteristic
14	C₂×#14581726	C₄	1	1	yes	characteristic
15	C₂×#14581717	C₄	1	2	yes	—
16	#19127433	—	2	4	no	—
17	#19127426	—	2	4	no	—
18	#19127422	—	2	1	no	—
19	C₈×C₈₃⋊C₈	—	2	1	no	—
20	#431×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	#335×C₈₃	—	2	1	no	—
23	#334×C₈₃	—	2	1	no	—
24	C₂²×Dic₁₆₆	C₂×C₄	1	2	yes	—
25	C₂×Dic₃₃₂	—	2	1	no	—
26	C₂×Dic₃₃₂	—	2	1	no	—
27	C₂×Dic₃₃₂	—	2	2	no	—
28	C₂×Dic₃₃₂	C₈	1	4	yes	complemented
29	C₄×Dic₁₆₆	—	2	2	no	—
30	C₄×Dic₁₆₆	—	2	2	no	—
31	C₄×Dic₁₆₆	—	2	2	no	—
32	C₄×Dic₁₆₆	—	2	2	no	—
33	C₂×C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
34	C₂×Dic₈₃⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₈₃	—	2	2	no	—
36	#14581726	—	2	1	no	—
37	#14581726	—	2	1	no	—
38	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
39	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
40	#14581717	—	2	2	no	—
41	#14581717	—	2	2	no	—
42	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
43	C₂²⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
44	C₄×C₈₃⋊C₈	—	2	1	no	—
45	C₄×C₈₃⋊C₈	—	2	1	no	—
46	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₈₃	—	2	2	no	—
50	C₂²⋊C₈×C₈₃	—	2	2	no	—
51	C₄×C₈×C₈₃	—	2	2	no	—
52	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
56	C₂×Dic₁₆₆	—	2	2	no	—
57	C₂×Dic₁₆₆	—	2	2	no	—
58	C₂×Dic₁₆₆	—	2	2	no	—
59	C₂×Dic₁₆₆	—	2	2	no	—
60	C₂×Dic₁₆₆	—	2	2	no	—
61	C₂×Dic₁₆₆	C₂×C₈	1	4	yes	—
62	C₂²×Dic₈₃	—	2	2	no	—
63	C₂²×Dic₈₃	—	2	2	no	—
64	Dic₃₃₂	—	2	4	no	—
65	Dic₃₃₂	—	2	4	no	—
66	C₂×C₈₃⋊C₈	—	2	2	no	—
67	C₂×C₈₃⋊C₈	—	2	2	no	—
68	C₄⋊Dic₈₃	—	2	2	no	—
69	C₄⋊Dic₈₃	—	2	2	no	—
70	Dic₈₃⋊C₄	—	4	2	no	—
71	Dic₈₃⋊C₄	—	4	2	no	—
72	C₄×Dic₈₃	—	4	2	no	—
73	C₄×Dic₈₃	—	4	2	no	—
74	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₈₃	—	2	1	no	—
78	C₂×C₈×C₈₃	—	2	1	no	—
79	C₂×C₈×C₈₃	—	4	2	no	—
80	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
86	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₃	#41	1	1	yes	characteristic
90	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₈₃	—	2	2	no	—
92	C₂×C₄×C₈₃	—	4	1	no	—
93	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
102	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
103	C₂×Dic₈₃	—	2	4	no	—
104	C₂×Dic₈₃	—	4	2	no	—
105	C₂×Dic₈₃	—	4	2	no	—
106	C₂×Dic₈₃	—	4	2	no	—
107	C₂×Dic₈₃	—	8	2	no	—
108	Dic₁₆₆	—	2	4	no	—
109	Dic₁₆₆	—	2	4	no	—
110	Dic₁₆₆	—	4	2	no	—
111	Dic₁₆₆	—	4	2	no	—
112	C₈×C₈₃	—	2	1	no	—
113	C₈×C₈₃	—	2	1	no	—
114	C₈×C₈₃	—	8	2	no	—
115	C₈×C₈₃	C₂²⋊C₈	1	2	yes	—
116	C₈×C₈₃	C₂²⋊C₈	1	2	yes	—
117	C₈₃⋊C₈	—	4	2	no	—
118	C₈₃⋊C₈	—	4	2	no	—
119	C₂²×C₈₃	—	2	1	no	—
120	C₂²×C₈₃	—	2	1	no	—
121	C₂²×C₈₃	#390	1	1	yes	characteristic
122	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
123	C₂²×C₈₃	#436	1	1	yes	characteristic
124	C₄×C₈₃	—	2	1	no	—
125	C₄×C₈₃	—	2	1	no	—
126	C₄×C₈₃	—	2	1	no	—
127	C₄×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₄×C₈₃	#408	1	1	yes	characteristic
129	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
130	C₄×C₈₃	#436	1	1	yes	characteristic
131	C₄×C₈₃	#473	1	2	yes	—
132	Dic₈₃	—	4	4	no	—
133	Dic₈₃	—	8	2	no	—
134	#9669	—	83	1	no	maximal, Sylow
135	C₂×C₈₃	—	2	1	no	—
136	C₂×C₈₃	—	2	1	no	—
137	C₂×C₈₃	#1954	1	1	yes	characteristic
138	C₂×C₈₃	#1976	1	1	yes	characteristic
139	C₂×C₈₃	#1979	1	1	yes	characteristic
140	C₂×C₄×Q₁₆	—	83	1	no	—
141	#1564	D₈₃	1	1	yes	characteristic
142	#1479	—	83	2	no	—
143	#1461	—	83	2	no	—
144	#1450	—	83	1	no	—
145	C₈₃	#9669	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₁₆	—	83	1	no	—
147	C₂×C₄×Q₈	—	83	2	no	—
148	C₄×Q₁₆	—	166	1	no	—
149	C₄×Q₁₆	—	166	1	no	—
150	C₄×Q₁₆	—	166	1	no	—
151	C₄×Q₁₆	—	166	1	no	—
152	#431	—	83	1	no	—
153	#431	—	83	1	no	—
154	#431	C₂×D₈₃	1	2	yes	—
155	C₂×#107	—	83	1	no	—
156	C₂×#103	—	83	2	no	—
157	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
158	#335	—	2	1	no	—
159	#334	—	2	1	no	—
160	Q₈⋊C₈	—	166	4	no	—
161	#324	—	166	4	no	—
162	#322	—	166	1	no	—
163	C₈²	—	166	1	no	—
164	C₂²×Q₈	—	83	2	no	—
165	C₂×Q₁₆	—	83	4	no	—
166	C₂×Q₁₆	—	166	1	no	—
167	C₂×Q₁₆	—	166	1	no	—
168	C₂×Q₁₆	—	166	2	no	—
169	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
170	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
171	C₄×Q₈	—	166	2	no	—
172	C₄×Q₈	—	166	2	no	—
173	C₄×Q₈	—	166	2	no	—
174	C₄×Q₈	—	166	2	no	—
175	C₂×C₄⋊C₄	—	83	2	no	—
176	C₂×C₄⋊C₄	—	166	2	no	—
177	C₂×C₄²	—	166	2	no	—
178	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
179	#107	—	166	1	no	—
180	#107	—	166	1	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	83	2	no	—
183	C₄⋊C₈	—	166	1	no	—
184	#103	—	166	2	no	—
185	#103	—	166	2	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	83	2	no	—
188	C₂²⋊C₈	—	166	1	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	166	1	no	—
191	C₄×C₈	—	166	1	no	—
192	C₄×C₈	D₃₃₂	1	1	yes	characteristic
193	C₄×C₈	D₃₃₂	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
196	C₂×Q₈	—	83	4	no	—
197	C₂×Q₈	—	166	2	no	—
198	C₂×Q₈	—	166	2	no	—
199	C₂×Q₈	—	166	2	no	—
200	C₂×Q₈	—	166	2	no	—
201	C₂×Q₈	—	166	2	no	—
202	C₂²×C₄	—	166	2	no	—
203	C₂²×C₄	—	166	2	no	—
204	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
206	C₂²×C₄	#13050591	1	1	yes	characteristic
207	Q₁₆	—	166	4	no	—
208	Q₁₆	—	166	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	166	2	no	—
213	C₂×C₈	—	166	2	no	—
214	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
215	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
216	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
217	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
218	C₂×C₈	C₈×D₈₃	1	2	yes	—
219	C₂×C₈	#13050573	1	2	yes	—
220	C₄⋊C₄	—	332	2	no	—
221	C₄⋊C₄	—	332	2	no	—
222	C₄⋊C₄	—	166	2	no	—
223	C₄⋊C₄	—	166	2	no	—
224	C₄²	—	332	2	no	—
225	C₄²	—	332	2	no	—
226	C₄²	D₄×D₈₃	1	1	yes	characteristic
227	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
228	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
229	C₄²	#13050591	1	1	yes	characteristic
230	C₂³	#14581754	1	1	yes	characteristic
231	Q₈	—	332	2	no	—
232	Q₈	—	332	2	no	—
233	Q₈	—	166	4	no	—
234	Q₈	—	166	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	332	2	no	—
238	C₂×C₄	—	332	2	no	—
239	C₂×C₄	—	332	2	no	—
240	C₂×C₄	—	664	2	no	—
241	C₂×C₄	—	166	4	no	—
242	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	#14581754	1	1	yes	characteristic
244	C₂×C₄	#14581767	1	1	yes	characteristic
245	C₂×C₄	#14581767	1	1	yes	characteristic
246	C₂×C₄	#14581781	1	1	yes	characteristic
247	C₂×C₄	#14581788	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
249	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
250	C₂×C₄	#14581800	1	1	yes	characteristic
251	C₂×C₄	#14581800	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	8	2	no	—
255	C₈	—	332	2	no	—
256	C₈	—	332	2	no	—
257	C₈	(C₂×D₈₃)⋊C₈	1	2	yes	—
258	C₈	(C₂×D₈₃)⋊C₈	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	#19127602	1	1	yes	characteristic
262	C₂²	#19127766	1	1	yes	characteristic
263	C₂²	#19127778	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	332	4	no	—
268	C₄	—	664	2	no	—
269	C₄	C₂×(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
270	C₄	#19127669	1	1	yes	characteristic
271	C₄	#19127766	1	1	yes	characteristic
272	C₄	#19127778	1	1	yes	characteristic
273	C₄	#19127977	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#24908256	1	1	yes	characteristic
277	C₂	#24908359	1	1	yes	characteristic
278	C₂	#24908376	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581767
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1564	C₂
socle	C₂²×C₈₃	#390
prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1564	D₈₃
hypocenter	C₈₃	#9669
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
28. C₂×Dic₃₃₂	C₈	semidirect
145. C₈₃	#9669	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₃₂	2
#24905859	2
#24905786	2
#24905746	2
C₈₃×#1564	2
#9669	83