Subgroups — #35559768

Order 21248.

598 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917977	C₂	1	1	yes	maximal, characteristic, complemented
2	#24909699	C₂	1	4	yes	maximal, complemented
3	#24909694	C₂	1	4	yes	maximal, complemented
4	#24909183	C₂	1	2	yes	maximal, complemented
5	#24909179	C₂	1	2	yes	maximal
6	C₄×#14581732	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#495×C₈₃	C₂	1	1	yes	maximal, characteristic
8	#19128258	—	2	1	no	—
9	C₂×#14581732	C₂²	1	2	yes	—
10	#19128106	C₂²	1	2	yes	complemented
11	#19128103	—	2	1	no	—
12	#19128103	C₂²	1	2	yes	complemented
13	#19128099	—	2	1	no	—
14	#19127739	—	2	1	no	—
15	#19127739	C₂²	1	2	yes	—
16	#19127505	C₂²	1	8	yes	complemented
17	#19127502	C₂²	1	8	yes	—
18	#19127477	C₂²	1	2	yes	complemented
19	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	complemented
20	C₂²×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
21	C₂²×SD₁₆×C₈₃	—	2	1	no	—
22	C₂²×D₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₂×#124×C₈₃	—	2	1	no	—
24	#495×C₈₃	—	2	1	no	—
25	#495×C₈₃	—	2	1	no	—
26	#495×C₈₃	C₂²	1	2	yes	—
27	#495×C₈₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
29	#14581792	—	2	2	no	—
30	#14581784	—	2	2	no	—
31	#14581771	—	4	2	no	—
32	C₄×Dic₁₆₆	—	4	2	no	—
33	#14581751	—	2	2	no	—
34	#14581732	—	2	4	no	—
35	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
36	#14581726	—	2	2	no	—
37	#14581726	C₂³	1	4	yes	—
38	#14581724	—	2	2	no	—
39	#14581724	—	2	2	no	—
40	C₂²×Q₈×C₈₃	—	2	1	no	—
41	C₂²×D₄×C₈₃	—	2	1	no	—
42	C₂×Q₁₆×C₈₃	—	2	1	no	—
43	C₂×Q₁₆×C₈₃	—	2	4	no	—
44	C₂×Q₁₆×C₈₃	C₂³	1	2	yes	—
45	C₂×SD₁₆×C₈₃	—	2	2	no	—
46	C₂×SD₁₆×C₈₃	—	2	2	no	—
47	C₂×SD₁₆×C₈₃	—	4	4	no	—
48	C₂×D₈×C₈₃	—	2	1	no	—
49	C₂×D₈×C₈₃	—	2	4	no	—
50	C₂×D₈×C₈₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
53	#124×C₈₃	—	2	2	no	—
54	#124×C₈₃	—	2	2	no	—
55	#124×C₈₃	—	4	2	no	—
56	C₂×C₂²⋊C₄×C₈₃	—	4	2	no	—
57	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₆₆	—	4	2	no	—
63	C₄⋊Dic₈₃	—	2	4	no	—
64	C₄⋊Dic₈₃	—	4	2	no	—
65	Dic₈₃⋊C₄	—	4	4	no	—
66	C₂²⋊Dic₈₃	—	4	2	no	—
67	C₄×Dic₈₃	—	4	2	no	—
68	C₈₃⋊C₁₆	—	2	4	no	—
69	C₂⁴×C₈₃	—	4	1	no	—
70	C₂×Q₈×C₈₃	—	2	2	no	—
71	C₂×Q₈×C₈₃	—	4	1	no	—
72	C₂×Q₈×C₈₃	—	4	2	no	—
73	C₂×Q₈×C₈₃	—	4	2	no	—
74	C₂×D₄×C₈₃	—	2	2	no	—
75	C₂×D₄×C₈₃	—	4	2	no	—
76	C₂×D₄×C₈₃	—	4	1	no	—
77	C₂×D₄×C₈₃	—	4	2	no	—
78	C₂²×C₄×C₈₃	—	4	1	no	—
79	C₂²×C₄×C₈₃	D₈	1	2	yes	—
80	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₈₃	—	2	4	no	—
82	Q₁₆×C₈₃	—	4	2	no	—
83	SD₁₆×C₈₃	—	4	4	no	—
84	SD₁₆×C₈₃	—	4	4	no	—
85	D₈×C₈₃	—	2	4	no	—
86	D₈×C₈₃	—	4	2	no	—
87	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
88	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₈₃	#41	1	4	yes	—
92	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
93	C₂²⋊C₄×C₈₃	—	4	4	no	—
94	C₂²⋊C₄×C₈₃	—	4	4	no	—
95	C₄²×C₈₃	D₈	1	2	yes	—
96	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₈₃	—	4	2	no	—
99	C₂³×C₈₃	—	8	4	no	—
100	C₂³×C₈₃	—	8	2	no	—
101	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
102	Q₈×C₈₃	—	4	2	no	—
103	Q₈×C₈₃	—	4	2	no	—
104	Q₈×C₈₃	—	8	2	no	—
105	D₄×C₈₃	—	4	2	no	—
106	D₄×C₈₃	—	4	2	no	—
107	D₄×C₈₃	—	8	2	no	—
108	C₂×C₄×C₈₃	—	4	2	no	—
109	C₂×C₄×C₈₃	—	4	2	no	—
110	C₂×C₄×C₈₃	—	8	2	no	—
111	C₂×C₄×C₈₃	#124	1	2	yes	—
112	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₈₃	—	4	4	no	—
118	Dic₁₆₆	—	8	4	no	—
119	C₈×C₈₃	#124	1	2	yes	—
120	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
121	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
122	C₈×C₈₃	C₂×#41	1	2	yes	—
123	C₂²×C₈₃	—	2	2	no	—
124	C₂²×C₈₃	—	8	2	no	—
125	C₂²×C₈₃	—	8	4	no	—
126	C₂²×C₈₃	—	16	4	no	—
127	C₂²×C₈₃	#486	1	2	yes	—
128	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
129	C₄×C₈₃	—	2	2	no	—
130	C₄×C₈₃	—	2	2	no	—
131	C₄×C₈₃	—	8	2	no	—
132	C₄×C₈₃	#486	1	2	yes	—
133	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
134	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
135	Dic₈₃	—	8	4	no	—
136	#33706	—	83	1	no	maximal, Sylow
137	C₂×C₈₃	—	2	2	no	—
138	C₂×C₈₃	—	16	2	no	—
139	C₂×C₈₃	C₂×#486	1	1	yes	characteristic
140	C₂×C₈₃	#3414	1	2	yes	—
141	#3160	—	83	1	no	—
142	C₂×#495	D₈₃	1	1	yes	characteristic
143	#2243	—	83	4	no	—
144	#2242	—	83	4	no	—
145	#2141	—	83	2	no	—
146	#2140	—	83	2	no	—
147	C₄×M₃₂	—	83	1	no	—
148	C₈₃	#33706	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
150	C₂²×SD₁₆	—	2	1	no	—
151	C₂²×D₈	C₂×D₈₃	1	1	yes	characteristic
152	#533	—	166	1	no	—
153	C₂×#124	—	2	1	no	—
154	C₂×M₃₂	—	83	2	no	—
155	#500	—	83	2	no	—
156	#499	—	83	2	no	—
157	#499	—	166	1	no	—
158	#498	—	166	1	no	—
159	#495	—	2	1	no	—
160	#495	—	2	1	no	—
161	#495	C₂×D₈₃	1	2	yes	—
162	#495	C₂×D₈₃	1	2	yes	—
163	#427	—	83	2	no	—
164	#427	—	166	1	no	—
165	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
166	#358	—	83	8	no	—
167	#357	—	83	8	no	—
168	#346	—	83	2	no	—
169	C₄×C₁₆	—	83	2	no	—
170	C₂²×Q₈	—	2	1	no	—
171	C₂²×D₄	—	2	1	no	—
172	C₂×Q₁₆	—	2	1	no	—
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×Q₁₆	C₂²×D₈₃	1	2	yes	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×SD₁₆	—	4	4	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	—	2	4	no	—
180	C₂×D₈	C₂²×D₈₃	1	2	yes	—
181	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	166	2	no	—
184	#125	—	166	2	no	—
185	#124	—	2	2	no	—
186	#124	—	2	2	no	—
187	#124	—	4	2	no	—
188	C₂²⋊Q₈	—	332	2	no	—
189	C₄×Q₈	—	332	2	no	—
190	#117	—	166	2	no	—
191	C₂×C₂²⋊C₄	—	4	2	no	—
192	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
193	M₃₂	—	166	4	no	—
194	C₂×C₁₆	—	83	4	no	—
195	#107	—	83	4	no	—
196	#107	—	166	2	no	—
197	#106	—	166	2	no	—
198	#106	—	166	2	no	—
199	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
203	C₂⁴	—	4	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	332	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	#13050575	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
216	Q₁₆	—	2	4	no	—
217	Q₁₆	—	4	2	no	—
218	SD₁₆	—	4	4	no	—
219	SD₁₆	—	4	4	no	—
220	D₈	—	2	4	no	—
221	D₈	—	4	2	no	—
222	C₂×C₈	#13050578	1	4	yes	—
223	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
224	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
226	C₂×C₈	#13050589	1	4	yes	—
227	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
228	C₄⋊C₄	—	332	4	no	—
229	C₄⋊C₄	—	332	2	no	—
230	C₄⋊C₄	—	166	4	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₂²⋊C₄	—	332	2	no	—
234	C₄²	—	332	2	no	—
235	C₄²	#13050575	1	2	yes	—
236	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
237	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
238	C₁₆	—	166	4	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	2	no	—
242	C₂³	C₂×#13050575	1	1	yes	characteristic
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	8	2	no	—
246	Q₈	—	664	4	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	332	4	no	—
254	C₂×C₄	#14581779	1	2	yes	—
255	C₂×C₄	C₂×#13050575	1	2	yes	—
256	C₂×C₄	C₂×#13050575	1	2	yes	—
257	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
258	C₂×C₄	C₂×#13050578	1	2	yes	—
259	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
260	C₈	#14581779	1	2	yes	—
261	C₈	C₂×#13050578	1	2	yes	—
262	C₈	C₂×#13050578	1	2	yes	—
263	C₈	C₂×#13050589	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	16	4	no	—
268	C₂²	#19128047	1	2	yes	—
269	C₂²	C₂²×#13050575	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	8	2	no	—
273	C₄	—	664	4	no	—
274	C₄	#19128047	1	2	yes	—
275	C₄	C₂×#14581779	1	1	yes	characteristic
276	C₄	C₂²×#13050578	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×#19128047	1	1	yes	characteristic
280	C₂	#24920597	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#13050575
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#495×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂²×D₈×C₈₃	C₂²
hypercenter	C₂×#495	D₈₃
hypocenter	C₈₃	#33706
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. #24917977	C₂	semidirect
2. #24909699	C₂	semidirect
3. #24909694	C₂	semidirect
4. #24909183	C₂	semidirect
6. C₄×#14581732	C₂	semidirect
10. #19128106	C₂²	semidirect
12. #19128103	C₂²	semidirect
16. #19127505	C₂²	semidirect
18. #19127477	C₂²	semidirect
19. C₄×C₈₃⋊C₁₆	C₂²	semidirect
148. C₈₃	#33706	semidirect

Maximal subgroups

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

subgroup	index
#24917977	2
#24909699	2
#24909694	2
#24909183	2
#24909179	2
C₄×#14581732	2
C₂×#495×C₈₃	2
#33706	83