Subgroups — #35297474

Order 21248.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 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	#24907591	C₂	1	1	yes	maximal, characteristic
3	#24907469	C₂	1	2	yes	maximal
4	#24905347	C₂	1	2	yes	maximal
5	C₂×#334×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₆₆	—	2	2	no	—
8	C₈×Dic₁₆₆	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₈₃⋊C₈	—	2	1	no	—
11	C₂×#14581717	—	2	2	no	—
12	C₂×C₈×Dic₈₃	—	2	1	no	—
13	#19127571	C₂²	1	2	yes	—
14	#19127464	—	2	2	no	—
15	#19127464	—	2	1	no	—
16	#19127464	—	2	1	no	—
17	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	#334×C₈₃	—	2	2	no	—
20	C₂²×Dic₁₆₆	D₄	1	1	yes	characteristic
21	C₂²×C₈₃⋊C₈	—	2	1	no	—
22	C₄×Dic₁₆₆	—	2	1	no	—
23	C₄×Dic₁₆₆	—	2	1	no	—
24	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₆₆	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₈₃	—	2	2	no	—
27	C₂×C₄⋊Dic₈₃	—	2	2	no	—
28	C₂×C₄⋊Dic₈₃	—	2	2	no	—
29	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
30	C₂×Dic₈₃⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₈₃	—	2	1	no	—
32	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
33	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₈₃⋊C₈	—	2	4	no	—
35	#14581717	—	4	4	no	—
36	C₈×Dic₈₃	—	2	4	no	—
37	C₂²×C₈×C₈₃	—	2	2	no	—
38	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₈₃	—	2	2	no	—
42	C₄⋊C₈×C₈₃	—	2	2	no	—
43	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₈₃	D₄	1	2	yes	—
45	C₂×Dic₁₆₆	—	2	1	no	—
46	C₂×Dic₁₆₆	—	2	2	no	—
47	C₂×Dic₁₆₆	C₄⋊C₄	1	2	yes	—
48	C₂×Dic₁₆₆	SD₁₆	1	4	yes	—
49	C₂²×Dic₈₃	—	2	1	no	—
50	C₂²×Dic₈₃	—	2	1	no	—
51	C₂²×Dic₈₃	—	4	2	no	—
52	C₂×C₈₃⋊C₈	—	2	2	no	—
53	C₂×C₈₃⋊C₈	—	4	2	no	—
54	C₄⋊Dic₈₃	—	2	1	no	—
55	C₄⋊Dic₈₃	—	4	2	no	—
56	C₄⋊Dic₈₃	—	4	2	no	—
57	C₄⋊Dic₈₃	—	4	2	no	—
58	C₄⋊Dic₈₃	—	4	2	no	—
59	C₄⋊Dic₈₃	—	8	2	no	—
60	C₄⋊Dic₈₃	C₄⋊C₄	1	2	yes	—
61	Dic₈₃⋊C₄	—	2	2	no	—
62	Dic₈₃⋊C₄	—	2	2	no	—
63	C₄×Dic₈₃	—	2	2	no	—
64	C₄×Dic₈₃	—	2	2	no	—
65	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
68	C₂×C₈×C₈₃	—	2	2	no	—
69	C₂×C₈×C₈₃	—	2	2	no	—
70	C₂×C₈×C₈₃	—	4	2	no	—
71	C₂×C₈×C₈₃	—	4	2	no	—
72	C₂×C₈×C₈₃	—	4	2	no	—
73	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₈₃	#41	1	1	yes	characteristic
79	C₂³×C₈₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₈₃	—	2	1	no	—
81	C₂×C₄×C₈₃	—	2	1	no	—
82	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₈₃	#135	1	2	yes	—
90	C₂×C₄×C₈₃	#135	1	2	yes	—
91	C₂×C₄×C₈₃	#137	1	2	yes	—
92	C₂×Dic₈₃	—	2	2	no	—
93	C₂×Dic₈₃	—	2	4	no	—
94	C₂×Dic₈₃	—	2	2	no	—
95	C₂×Dic₈₃	—	4	1	no	—
96	C₂×Dic₈₃	—	4	1	no	—
97	C₂×Dic₈₃	—	8	2	no	—
98	C₂×Dic₈₃	—	8	2	no	—
99	C₂×Dic₈₃	—	8	2	no	—
100	Dic₁₆₆	—	2	2	no	—
101	Dic₁₆₆	—	2	4	no	—
102	Dic₁₆₆	#106	1	4	yes	—
103	C₈×C₈₃	—	2	2	no	—
104	C₈×C₈₃	—	2	2	no	—
105	C₈×C₈₃	—	8	2	no	—
106	C₈₃⋊C₈	—	4	2	no	—
107	C₂²×C₈₃	#380	1	1	yes	characteristic
108	C₂²×C₈₃	#464	1	2	yes	—
109	C₂²×C₈₃	#465	1	2	yes	—
110	C₂²×C₈₃	#472	1	1	yes	characteristic
111	C₂²×C₈₃	#473	1	1	yes	characteristic
112	C₄×C₈₃	—	2	1	no	—
113	C₄×C₈₃	—	2	1	no	—
114	C₄×C₈₃	#416	1	1	yes	characteristic
115	C₄×C₈₃	#419	1	1	yes	characteristic
116	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
117	C₄×C₈₃	#427	1	1	yes	characteristic
118	C₄×C₈₃	#465	1	2	yes	—
119	C₄×C₈₃	#479	1	2	yes	—
120	Dic₈₃	—	2	4	no	—
121	Dic₈₃	—	4	2	no	—
122	Dic₈₃	—	16	2	no	—
123	#8647	—	83	1	no	maximal, Sylow
124	C₂×C₈₃	#1678	1	2	yes	—
125	C₂×C₈₃	#1680	1	2	yes	—
126	C₂×C₈₃	#1805	1	1	yes	characteristic
127	C₂×C₈₃	#1807	1	1	yes	characteristic
128	C₂×C₈₃	#1816	1	1	yes	characteristic
129	C₂×C₈×Q₈	—	83	1	no	—
130	#1834	—	83	1	no	—
131	#1803	—	83	2	no	—
132	C₂×#334	D₈₃	1	1	yes	characteristic
133	#1296	—	83	2	no	—
134	C₈₃	#8647	1	1	yes	characteristic, complemented, Sylow
135	C₂×C₄×Q₈	—	83	1	no	—
136	C₈×Q₈	—	83	4	no	—
137	C₈×Q₈	—	166	2	no	—
138	C₂×C₄⋊C₈	—	83	1	no	—
139	C₂×C₄⋊C₈	—	166	1	no	—
140	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
141	C₂×#103	—	166	2	no	—
142	C₂×C₄×C₈	—	166	1	no	—
143	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
144	#384	—	83	2	no	—
145	#340	—	166	2	no	—
146	#340	—	166	1	no	—
147	#340	—	166	1	no	—
148	#334	—	2	2	no	—
149	C₂²×Q₈	—	83	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	166	1	no	—
152	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
153	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
154	C₄×Q₈	—	83	2	no	—
155	C₄×Q₈	—	83	2	no	—
156	C₄×Q₈	—	166	1	no	—
157	C₄×Q₈	—	166	1	no	—
158	C₂×C₄⋊C₄	—	83	1	no	—
159	C₂×C₄⋊C₄	—	166	2	no	—
160	C₂×C₄⋊C₄	—	166	2	no	—
161	C₂×C₄⋊C₄	—	166	2	no	—
162	C₂×C₄⋊C₄	—	166	1	no	—
163	C₂×C₄²	—	166	1	no	—
164	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	83	2	no	—
168	C₄⋊C₈	—	166	4	no	—
169	C₄⋊C₈	—	166	1	no	—
170	#103	—	332	4	no	—
171	C₄×C₈	—	166	4	no	—
172	C₄×C₈	C₄×D₈₃	1	2	yes	—
173	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
174	C₂×Q₈	—	83	4	no	—
175	C₂×Q₈	—	83	2	no	—
176	C₂×Q₈	—	166	1	no	—
177	C₂×Q₈	—	166	2	no	—
178	C₂²×C₄	—	332	2	no	—
179	C₂²×C₄	—	166	1	no	—
180	C₂²×C₄	—	166	1	no	—
181	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
183	C₂²×C₄	#13050591	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	332	2	no	—
190	C₂×C₈	—	166	2	no	—
191	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
192	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
193	C₄⋊C₄	—	332	2	no	—
194	C₄⋊C₄	—	332	2	no	—
195	C₄⋊C₄	—	332	2	no	—
196	C₄⋊C₄	—	332	2	no	—
197	C₄⋊C₄	—	83	2	no	—
198	C₄⋊C₄	—	664	2	no	—
199	C₄⋊C₄	—	166	1	no	—
200	C₄⋊C₄	—	166	2	no	—
201	C₄⋊C₄	—	166	2	no	—
202	C₄²	—	166	2	no	—
203	C₄²	—	166	2	no	—
204	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
205	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
206	C₄²	Q₈×D₈₃	1	1	yes	characteristic
207	C₄²	#13050589	1	1	yes	characteristic
208	C₂³	#14581767	1	1	yes	characteristic
209	Q₈	—	83	4	no	—
210	Q₈	—	166	4	no	—
211	Q₈	—	166	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	332	1	no	—
215	C₂×C₄	—	332	1	no	—
216	C₂×C₄	—	664	2	no	—
217	C₂×C₄	—	664	2	no	—
218	C₂×C₄	—	664	2	no	—
219	C₂×C₄	—	166	2	no	—
220	C₂×C₄	—	166	2	no	—
221	C₂×C₄	—	166	4	no	—
222	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
224	C₂×C₄	#14581750	1	1	yes	characteristic
225	C₂×C₄	#14581763	1	1	yes	characteristic
226	C₂×C₄	#14581768	1	1	yes	characteristic
227	C₂×C₄	#14581772	1	1	yes	characteristic
228	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
229	C₂×C₄	#14581813	1	2	yes	—
230	C₂×C₄	#14581814	1	2	yes	—
231	C₂×C₄	#14581825	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	8	2	no	—
235	C₈	—	332	2	no	—
236	C₂²	#19127558	1	1	yes	characteristic
237	C₂²	#19127924	1	2	yes	—
238	C₂²	#19127931	1	2	yes	—
239	C₂²	#19127972	1	1	yes	characteristic
240	C₂²	#19127977	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	1328	2	no	—
244	C₄	—	332	2	no	—
245	C₄	—	166	4	no	—
246	C₄	#19127698	1	1	yes	characteristic
247	C₄	#19127711	1	1	yes	characteristic
248	C₄	#106×D₈₃	1	1	yes	characteristic
249	C₄	#19127736	1	1	yes	characteristic
250	C₄	#19127927	1	2	yes	—
251	C₄	#19128009	1	2	yes	—
252	C₂	#24906880	1	2	yes	—
253	C₂	#24906894	1	2	yes	—
254	C₂	#24907479	1	1	yes	characteristic
255	C₂	#24907485	1	1	yes	characteristic
256	C₂	#24907524	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581767
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#334×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#121
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#334	D₈₃
hypocenter	C₈₃	#8647
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₆₆	C₄	semidirect
134. C₈₃	#8647	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
#24907591	2
#24907469	2
#24905347	2
C₂×#334×C₈₃	2
#8647	83