Subgroups — #35297530

Order 21248.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 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	#24906211	C₂	1	1	yes	maximal, characteristic
3	#24905891	C₂	1	2	yes	maximal
4	#24905358	C₂	1	2	yes	maximal
5	C₈₃×#1848	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	C₄	1	2	yes	—
10	C₈×Dic₁₆₆	C₄	1	2	yes	—
11	#19127754	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₈₃⋊C₈	—	2	1	no	—
14	C₂×C₈×Dic₈₃	—	2	1	no	—
15	#19127454	—	2	1	no	—
16	#19127452	—	2	1	no	—
17	#19127431	—	2	4	no	—
18	C₂×#108×C₈₃	—	2	1	no	—
19	C₂×#108×C₈₃	—	2	1	no	—
20	#423×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₆₆	Q₈	1	1	yes	characteristic
23	C₂²×C₈₃⋊C₈	—	2	1	no	—
24	C₄×Dic₁₆₆	—	2	1	no	—
25	C₄×Dic₁₆₆	—	2	1	no	—
26	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₆₆	D₄	1	2	yes	—
28	C₂×C₄⋊Dic₈₃	Q₈	1	1	yes	characteristic
29	C₂×Dic₈₃⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₈₃	—	2	1	no	—
31	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
32	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
33	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₈₃⋊C₈	—	2	2	no	—
35	Dic₈₃⋊C₈	—	2	2	no	—
36	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
37	C₄×C₈₃⋊C₈	—	2	2	no	—
38	C₈×Dic₈₃	—	2	2	no	—
39	C₈×Dic₈₃	—	2	2	no	—
40	C₂×M₁₆×C₈₃	—	2	2	no	—
41	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
44	#108×C₈₃	—	4	2	no	—
45	#108×C₈₃	—	4	2	no	—
46	C₄⋊C₈×C₈₃	—	2	2	no	—
47	C₄⋊C₈×C₈₃	—	2	2	no	—
48	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₈₃	C₂×C₄	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	4	no	—
54	C₂×Dic₁₆₆	C₄⋊C₄	1	2	yes	—
55	C₂²×Dic₈₃	—	2	1	no	—
56	C₂²×Dic₈₃	—	2	1	no	—
57	C₂×C₈₃⋊C₈	—	2	2	no	—
58	C₂×C₈₃⋊C₈	—	4	1	no	—
59	C₂×C₈₃⋊C₈	—	4	1	no	—
60	C₂×C₈₃⋊C₈	—	4	2	no	—
61	C₄⋊Dic₈₃	—	2	1	no	—
62	C₄⋊Dic₈₃	C₄⋊C₄	1	2	yes	—
63	Dic₈₃⋊C₄	—	2	2	no	—
64	Dic₈₃⋊C₄	—	2	2	no	—
65	C₄×Dic₈₃	—	2	2	no	—
66	C₄×Dic₈₃	—	2	2	no	—
67	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
70	M₁₆×C₈₃	—	4	2	no	—
71	M₁₆×C₈₃	—	4	2	no	—
72	C₂×C₈×C₈₃	—	2	1	no	—
73	C₂×C₈×C₈₃	—	2	1	no	—
74	C₂×C₈×C₈₃	—	4	2	no	—
75	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
80	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
81	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₈₃	#41	1	1	yes	characteristic
85	C₂³×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₃	—	2	2	no	—
87	C₂×C₄×C₈₃	—	4	1	no	—
88	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₈₃	C₂×C₄⋊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	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
98	C₂×Dic₈₃	—	2	4	no	—
99	C₂×Dic₈₃	—	2	2	no	—
100	C₂×Dic₈₃	—	2	2	no	—
101	C₂×Dic₈₃	—	4	2	no	—
102	Dic₁₆₆	—	2	4	no	—
103	Dic₁₆₆	—	2	2	no	—
104	Dic₁₆₆	#108	1	4	yes	complemented
105	C₈×C₈₃	—	2	1	no	—
106	C₈×C₈₃	—	2	1	no	—
107	C₈×C₈₃	—	8	2	no	—
108	C₈×C₈₃	#103	1	2	yes	—
109	C₈×C₈₃	#103	1	2	yes	—
110	C₈₃⋊C₈	—	4	2	no	—
111	C₈₃⋊C₈	—	8	2	no	—
112	C₂²×C₈₃	—	2	1	no	—
113	C₂²×C₈₃	—	2	1	no	—
114	C₂²×C₈₃	#380	1	1	yes	characteristic
115	C₂²×C₈₃	#475	1	1	yes	characteristic
116	C₂²×C₈₃	#477	1	1	yes	characteristic
117	C₄×C₈₃	—	2	1	no	—
118	C₄×C₈₃	—	2	1	no	—
119	C₄×C₈₃	—	2	1	no	—
120	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
121	C₄×C₈₃	#417	1	1	yes	characteristic
122	C₄×C₈₃	C₂×#108	1	2	yes	—
123	C₄×C₈₃	#460	1	1	yes	characteristic
124	C₄×C₈₃	#462	1	1	yes	characteristic
125	Dic₈₃	—	2	4	no	—
126	Dic₈₃	—	4	2	no	—
127	#8659	—	83	1	no	maximal, Sylow
128	C₂×C₈₃	—	2	1	no	—
129	C₂×C₈₃	—	2	1	no	—
130	C₂×C₈₃	#1803	1	1	yes	characteristic
131	C₂×C₈₃	#1810	1	1	yes	characteristic
132	C₂×C₈₃	#1814	1	1	yes	characteristic
133	C₂×C₈×Q₈	—	83	1	no	—
134	#1848	D₈₃	1	1	yes	characteristic
135	#1564	—	83	1	no	—
136	#1487	—	83	2	no	—
137	#1301	—	83	2	no	—
138	C₈₃	#8659	1	1	yes	characteristic, complemented, Sylow
139	C₂×C₄×Q₈	—	83	1	no	—
140	C₈×Q₈	—	83	2	no	—
141	C₈×Q₈	—	83	2	no	—
142	C₈×Q₈	—	166	1	no	—
143	C₈×Q₈	—	166	1	no	—
144	#431	—	83	2	no	—
145	C₂×#108	—	2	1	no	—
146	C₂×#108	—	2	1	no	—
147	#423	C₂×D₈₃	1	2	yes	—
148	C₂×C₄⋊C₈	—	83	1	no	—
149	C₂×C₄⋊C₈	—	166	1	no	—
150	C₂×C₄×C₈	—	166	1	no	—
151	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
152	#335	—	166	1	no	—
153	#334	—	166	1	no	—
154	Q₈⋊C₈	—	166	4	no	—
155	C₂²×Q₈	—	83	1	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂²×C₈	—	166	1	no	—
158	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
160	C₄×Q₈	—	83	2	no	—
161	C₄×Q₈	—	83	2	no	—
162	C₄×Q₈	—	166	1	no	—
163	C₄×Q₈	—	166	1	no	—
164	C₂×C₄⋊C₄	—	83	1	no	—
165	C₂×C₄⋊C₄	—	166	1	no	—
166	C₂×C₄²	—	166	1	no	—
167	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
168	#108	—	4	2	no	—
169	#108	—	4	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	83	2	no	—
173	C₄⋊C₈	—	166	2	no	—
174	C₄⋊C₈	—	166	2	no	—
175	C₄⋊C₈	—	166	2	no	—
176	C₄⋊C₈	—	166	1	no	—
177	C₂²⋊C₈	—	166	2	no	—
178	C₄×C₈	—	166	2	no	—
179	C₄×C₈	—	166	2	no	—
180	C₄×C₈	—	166	2	no	—
181	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
182	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
183	C₄×C₈	D₃₃₂	1	1	yes	characteristic
184	C₄×C₈	D₃₃₂	1	1	yes	characteristic
185	C₂×Q₈	—	83	2	no	—
186	C₂×Q₈	—	166	4	no	—
187	C₂×Q₈	—	166	1	no	—
188	C₂²×C₄	—	166	1	no	—
189	C₂²×C₄	—	166	1	no	—
190	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
191	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
192	C₂²×C₄	#13050591	1	1	yes	characteristic
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	332	1	no	—
199	C₂×C₈	—	332	1	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	C₂×C₈	#13050578	1	2	yes	—
207	C₂×C₈	#13050580	1	2	yes	—
208	C₄⋊C₄	—	83	2	no	—
209	C₄⋊C₄	—	166	2	no	—
210	C₄⋊C₄	—	166	1	no	—
211	C₄⋊C₄	—	166	2	no	—
212	C₄²	—	166	2	no	—
213	C₄²	—	166	2	no	—
214	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
215	C₄²	D₄×D₈₃	1	1	yes	characteristic
216	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
217	C₄²	#13050589	1	1	yes	characteristic
218	C₂³	#14581768	1	1	yes	characteristic
219	Q₈	—	83	4	no	—
220	Q₈	—	166	4	no	—
221	Q₈	—	166	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	332	2	no	—
225	C₂×C₄	—	166	2	no	—
226	C₂×C₄	—	166	4	no	—
227	C₂×C₄	—	166	2	no	—
228	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
230	C₂×C₄	#14581750	1	1	yes	characteristic
231	C₂×C₄	#14581763	1	1	yes	characteristic
232	C₂×C₄	#14581767	1	1	yes	characteristic
233	C₂×C₄	#14581772	1	1	yes	characteristic
234	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
235	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
236	C₂×C₄	#14581818	1	1	yes	characteristic
237	C₂×C₄	#14581823	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	8	2	no	—
241	C₈	—	332	2	no	—
242	C₈	—	664	2	no	—
243	C₈	#14581715	1	2	yes	—
244	C₈	#14581715	1	2	yes	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	#19127558	1	1	yes	characteristic
248	C₂²	#19127983	1	1	yes	characteristic
249	C₂²	#19127993	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	332	2	no	—
254	C₄	—	166	4	no	—
255	C₄	C₂×#14581715	1	1	yes	characteristic
256	C₄	#19127700	1	1	yes	characteristic
257	C₄	#108×D₈₃	1	2	yes	—
258	C₄	#19127898	1	1	yes	characteristic
259	C₄	#19127908	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#24907465	1	1	yes	characteristic
263	C₂	#24907498	1	1	yes	characteristic
264	C₂	#24907518	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581763
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1848	C₂
socle	C₂²×C₈₃	#380
prime radical	C₂³×C₈₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1848	D₈₃
hypocenter	C₈₃	#8659
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
104. Dic₁₆₆	#108	semidirect
138. C₈₃	#8659	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
#24906211	2
#24905891	2
#24905358	2
C₈₃×#1848	2
#8659	83