Subgroups — #35297325

Order 21248.

398 conjugacy classes of subgroups.

The subgroup lattice has 262 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	#24907456	C₂	1	2	yes	maximal
3	#24906219	C₂	1	1	yes	maximal, characteristic
4	#24905337	C₂	1	2	yes	maximal
5	C₈₃×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₆₆	—	2	1	no	—
8	C₈×Dic₁₆₆	—	2	1	no	—
9	C₈×Dic₁₆₆	—	2	1	no	—
10	C₈×Dic₁₆₆	—	2	1	no	—
11	#19127725	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₈₃⋊C₈	C₄	1	2	yes	—
14	C₂×#14581710	—	2	2	no	—
15	C₂×C₈×Dic₈₃	C₄	1	2	yes	—
16	#19127469	—	2	2	no	—
17	#19127452	—	2	1	no	—
18	#19127452	—	2	1	no	—
19	#423×C₈₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	#334×C₈₃	—	2	1	no	—
22	#334×C₈₃	—	2	1	no	—
23	C₂²×Dic₁₆₆	D₄	1	1	yes	characteristic
24	C₂×#13050574	—	2	2	no	—
25	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₆₆	—	2	1	no	—
27	C₄×Dic₁₆₆	—	2	1	no	—
28	C₄×Dic₁₆₆	—	2	1	no	—
29	C₄×Dic₁₆₆	—	2	1	no	—
30	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
31	C₂×Dic₈₃⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
33	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
34	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
35	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
36	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
37	Dic₈₃⋊C₈	—	2	2	no	—
38	Dic₈₃⋊C₈	—	2	2	no	—
39	#14581710	—	4	4	no	—
40	C₈×Dic₈₃	—	2	2	no	—
41	C₈×Dic₈₃	—	2	2	no	—
42	C₂×M₁₆×C₈₃	—	2	2	no	—
43	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₈₃	—	2	2	no	—
47	C₄⋊C₈×C₈₃	—	2	2	no	—
48	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
52	C₂×Dic₁₆₆	—	2	1	no	—
53	C₂×Dic₁₆₆	—	2	1	no	—
54	C₂×Dic₁₆₆	—	2	2	no	—
55	C₂×Dic₁₆₆	SD₁₆	1	4	yes	—
56	C₂²×Dic₈₃	C₄⋊C₄	1	2	yes	—
57	C₂²×Dic₈₃	C₄⋊C₄	1	2	yes	—
58	#13050574	—	4	2	no	—
59	#13050574	—	4	2	no	—
60	C₂×C₈₃⋊C₈	—	2	2	no	—
61	C₂×C₈₃⋊C₈	—	4	2	no	—
62	C₂×C₈₃⋊C₈	—	4	2	no	—
63	C₄⋊Dic₈₃	—	2	1	no	—
64	C₄⋊Dic₈₃	—	2	1	no	—
65	Dic₈₃⋊C₄	—	2	2	no	—
66	Dic₈₃⋊C₄	—	2	2	no	—
67	C₄×Dic₈₃	—	2	2	no	—
68	C₄×Dic₈₃	—	2	2	no	—
69	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
72	M₁₆×C₈₃	—	4	2	no	—
73	M₁₆×C₈₃	—	4	2	no	—
74	C₂×C₈×C₈₃	—	2	2	no	—
75	C₂×C₈×C₈₃	—	4	1	no	—
76	C₂×C₈×C₈₃	—	4	2	no	—
77	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₃	#41	1	1	yes	characteristic
84	C₄²×C₈₃	#41	1	1	yes	characteristic
85	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₈₃	—	2	1	no	—
87	C₂×C₄×C₈₃	—	2	1	no	—
88	C₂×C₄×C₈₃	—	4	1	no	—
89	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₈₃	#135	1	2	yes	—
97	C₂×Dic₈₃	—	2	2	no	—
98	C₂×Dic₈₃	—	2	2	no	—
99	C₂×Dic₈₃	—	2	2	no	—
100	C₂×Dic₈₃	—	4	2	no	—
101	C₂×Dic₈₃	#106	1	4	yes	—
102	Dic₁₆₆	—	2	4	no	—
103	Dic₁₆₆	—	2	4	no	—
104	C₈×C₈₃	—	2	1	no	—
105	C₈×C₈₃	—	2	1	no	—
106	C₈×C₈₃	—	2	1	no	—
107	C₈×C₈₃	—	8	2	no	—
108	C₈×C₈₃	#101	1	2	yes	complemented
109	C₈₃⋊C₈	—	4	2	no	—
110	C₈₃⋊C₈	—	8	2	no	—
111	C₂²×C₈₃	—	2	1	no	—
112	C₂²×C₈₃	—	2	1	no	—
113	C₂²×C₈₃	#380	1	1	yes	characteristic
114	C₂²×C₈₃	C₂×#106	1	1	yes	characteristic
115	C₂²×C₈₃	#427	1	1	yes	characteristic
116	C₄×C₈₃	—	2	1	no	—
117	C₄×C₈₃	—	2	1	no	—
118	C₄×C₈₃	—	2	1	no	—
119	C₄×C₈₃	C₂×#101	1	1	yes	characteristic
120	C₄×C₈₃	#413	1	1	yes	characteristic
121	C₄×C₈₃	#465	1	2	yes	—
122	C₄×C₈₃	#465	1	2	yes	—
123	Dic₈₃	—	2	4	no	—
124	Dic₈₃	—	4	2	no	—
125	#8615	—	83	1	no	maximal, Sylow
126	C₂×C₈₃	—	2	1	no	—
127	C₂×C₈₃	—	2	1	no	—
128	C₂×C₈₃	#1801	1	1	yes	characteristic
129	C₂×C₈₃	#1807	1	1	yes	characteristic
130	C₂×C₈₃	#1813	1	1	yes	characteristic
131	C₂×C₈×Q₈	—	83	1	no	—
132	#1801	—	83	2	no	—
133	#1566	—	83	1	no	—
134	#1566	D₈₃	1	1	yes	characteristic
135	#1292	—	83	2	no	—
136	C₈₃	#8615	1	1	yes	characteristic, complemented, Sylow
137	C₂×C₄×Q₈	—	83	1	no	—
138	C₈×Q₈	—	166	1	no	—
139	C₈×Q₈	—	166	1	no	—
140	C₈×Q₈	—	166	1	no	—
141	C₈×Q₈	—	166	1	no	—
142	#423	—	83	2	no	—
143	#423	C₂×D₈₃	1	2	yes	—
144	C₂×C₄⋊C₈	—	83	2	no	—
145	C₂×C₄⋊C₈	—	83	1	no	—
146	C₂×#101	—	166	2	no	—
147	C₂×C₄×C₈	—	83	2	no	—
148	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
149	#343	—	166	2	no	—
150	#334	—	2	1	no	—
151	#334	—	2	1	no	—
152	#334	—	166	1	no	—
153	#334	—	166	1	no	—
154	C₂²×Q₈	—	83	1	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	166	2	no	—
157	C₂²×C₈	—	83	2	no	—
158	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
160	C₄×Q₈	—	166	1	no	—
161	C₄×Q₈	—	166	1	no	—
162	C₄×Q₈	—	166	1	no	—
163	C₄×Q₈	—	166	1	no	—
164	C₂×C₄⋊C₄	—	83	1	no	—
165	C₂×C₄⋊C₄	—	83	2	no	—
166	C₂×C₄²	—	83	2	no	—
167	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	166	2	no	—
171	C₄⋊C₈	—	166	2	no	—
172	C₄⋊C₈	—	166	1	no	—
173	C₄⋊C₈	—	166	2	no	—
174	C₄⋊C₈	—	166	1	no	—
175	C₄⋊C₈	—	166	2	no	—
176	#101	—	332	4	no	—
177	C₄×C₈	—	166	2	no	—
178	C₄×C₈	—	166	2	no	—
179	C₄×C₈	D₃₃₂	1	1	yes	characteristic
180	C₄×C₈	D₃₃₂	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
183	C₂×Q₈	—	83	4	no	—
184	C₂×Q₈	—	166	1	no	—
185	C₂×Q₈	—	166	1	no	—
186	C₂×Q₈	—	166	2	no	—
187	C₂²×C₄	—	83	2	no	—
188	C₂²×C₄	—	83	2	no	—
189	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
190	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
191	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	332	2	no	—
195	M₁₆	—	332	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	332	2	no	—
200	C₂×C₈	—	332	2	no	—
201	C₂×C₈	—	166	2	no	—
202	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
206	C₄⋊C₄	—	166	1	no	—
207	C₄⋊C₄	—	166	1	no	—
208	C₄⋊C₄	—	166	2	no	—
209	C₄⋊C₄	—	166	2	no	—
210	C₄²	—	166	2	no	—
211	C₄²	—	166	2	no	—
212	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
213	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
214	C₄²	#13050589	1	1	yes	characteristic
215	C₄²	#13050591	1	1	yes	characteristic
216	C₂³	C₄⋊C₄×D₈₃	1	1	yes	characteristic
217	Q₈	—	166	4	no	—
218	Q₈	—	166	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	332	2	no	—
223	C₂×C₄	—	83	4	no	—
224	C₂×C₄	—	166	2	no	—
225	C₂×C₄	—	166	2	no	—
226	C₂×C₄	—	166	2	no	—
227	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
228	C₂×C₄	#14581750	1	1	yes	characteristic
229	C₂×C₄	#14581763	1	1	yes	characteristic
230	C₂×C₄	#14581767	1	1	yes	characteristic
231	C₂×C₄	#14581768	1	1	yes	characteristic
232	C₂×C₄	#14581772	1	1	yes	characteristic
233	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
234	C₂×C₄	#14581813	1	2	yes	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	332	2	no	—
240	C₈	—	664	2	no	—
241	C₈	#14581710	1	2	yes	complemented
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#19127558	1	1	yes	characteristic
245	C₂²	#106×D₈₃	1	1	yes	characteristic
246	C₂²	#19127736	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	332	2	no	—
251	C₄	—	166	4	no	—
252	C₄	C₂×#14581710	1	1	yes	characteristic
253	C₄	#19127683	1	1	yes	characteristic
254	C₄	#19127927	1	2	yes	—
255	C₄	#19127931	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#24907456	1	1	yes	characteristic
259	C₂	#24907485	1	1	yes	characteristic
260	C₂	#24907510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19127558
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1566	C₂
socle	C₂²×C₈₃	#380
prime radical	C₂³×C₈₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1566	D₈₃
hypocenter	C₈₃	#8615
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
108. C₈×C₈₃	#101	semidirect
136. C₈₃	#8615	semidirect
241. C₈	#14581710	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
#24907456	2
#24906219	2
#24905337	2
C₈₃×#1566	2
#8615	83