Subgroups — #35223993

Order 20864.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104757	C₂	1	2	yes	maximal, complemented
2	#23782215	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14579699	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781888	C₂	1	2	yes	maximal, complemented
5	#23781881	C₂	1	2	yes	maximal, complemented
6	#23781876	C₂	1	1	yes	maximal, characteristic, complemented
7	#23781871	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781866	C₂	1	1	yes	maximal, characteristic, complemented
9	#23781861	C₂	1	1	yes	maximal, characteristic, complemented
10	#23781662	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	2	yes	—
13	C₂×#14579699	C₂²	1	1	yes	characteristic
14	#19104757	—	2	8	no	—
15	C₂×Dic₆₅₂	C₂²	1	2	yes	—
16	C₂×#14579688	C₂²	1	2	yes	—
17	C₂×#14579688	C₂²	1	2	yes	—
18	C₂×D₁₃₀₄	C₂²	1	2	yes	—
19	#19104735	C₂²	1	1	yes	characteristic
20	#19104732	C₂²	1	1	yes	characteristic
21	#19104723	C₂²	1	2	yes	—
22	#19104709	—	2	1	no	—
23	#19104709	—	2	1	no	—
24	C₄×Dic₃₂₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₃₂₆	C₂²	1	1	yes	characteristic
26	C₄×D₆₅₂	—	2	1	no	—
27	C₄×D₆₅₂	—	2	1	no	—
28	C₄×D₆₅₂	C₂²	1	1	yes	characteristic
29	C₄×D₆₅₂	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₆₃	—	2	1	no	—
31	C₄×C₂²⋊D₁₆₃	—	2	1	no	—
32	#19104694	C₂²	1	1	yes	characteristic
33	#19104692	—	2	1	no	—
34	C₄²×D₁₆₃	—	2	1	no	—
35	#19104660	C₂²	1	2	yes	—
36	#19104660	C₂²	1	2	yes	—
37	#19104657	C₂²	1	2	yes	—
38	#19104657	C₂²	1	2	yes	—
39	C₂²×C₈×C₁₆₃	C₂²	1	2	yes	—
40	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₁₆₃	C₂²	1	2	yes	—
42	C₄⋊C₈×C₁₆₃	C₂²	1	1	yes	characteristic
43	C₄⋊C₈×C₁₆₃	C₂²	1	1	yes	characteristic
44	#14579699	—	2	4	no	—
45	#14579699	—	2	2	no	—
46	#14579699	—	2	4	no	—
47	#14579699	D₄	1	4	yes	complemented
48	C₂×Dic₃₂₆	C₂³	1	2	yes	—
49	C₂×Dic₃₂₆	C₂³	1	1	yes	characteristic
50	C₂×D₆₅₂	—	2	1	no	—
51	C₂×D₆₅₂	—	2	1	no	—
52	C₂×D₆₅₂	C₂³	1	2	yes	—
53	C₂×D₆₅₂	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₆₃	—	2	1	no	—
55	C₂×C₂²⋊D₁₆₃	—	2	2	no	—
56	C₂×C₄×D₁₆₃	—	2	1	no	—
57	C₂×C₄×D₁₆₃	—	2	2	no	—
58	C₂×C₄×D₁₆₃	—	2	1	no	—
59	C₂×C₄×D₁₆₃	—	2	1	no	—
60	Dic₆₅₂	—	2	4	no	—
61	#14579688	—	2	4	no	—
62	#14579688	—	2	4	no	—
63	D₁₃₀₄	—	2	4	no	—
64	C₄⋊Dic₁₆₃	—	2	1	no	—
65	C₄⋊Dic₁₆₃	—	2	1	no	—
66	C₄⋊Dic₁₆₃	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₁₆₃	C₂³	1	1	yes	characteristic
68	Dic₁₆₃⋊C₄	—	2	1	no	—
69	Dic₁₆₃⋊C₄	—	2	1	no	—
70	C₂²⋊Dic₁₆₃	—	2	1	no	—
71	(C₂×D₁₆₃)⋊C₄	—	2	1	no	—
72	(C₂×D₁₆₃)⋊C₄	—	2	1	no	—
73	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
74	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
75	C₄×Dic₁₆₃	—	2	1	no	—
76	C₄×Dic₁₆₃	—	2	1	no	—
77	C₄×Dic₁₆₃	—	2	1	no	—
78	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₆₃	—	2	4	no	—
82	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
83	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
84	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
85	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
86	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
87	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
88	C₂²×D₁₆₃	—	2	1	no	—
89	C₂²×D₁₆₃	—	2	2	no	—
90	Dic₃₂₆	—	2	1	no	—
91	Dic₃₂₆	—	2	2	no	—
92	Dic₃₂₆	—	2	2	no	—
93	Dic₃₂₆	C₂×D₄	1	2	yes	—
94	D₆₅₂	—	2	2	no	—
95	D₆₅₂	—	2	1	no	—
96	D₆₅₂	—	2	2	no	—
97	D₆₅₂	—	4	2	no	—
98	D₆₅₂	—	4	2	no	—
99	D₆₅₂	C₂×D₄	1	2	yes	—
100	C₂²⋊D₁₆₃	—	2	4	no	—
101	C₂²⋊D₁₆₃	—	4	4	no	—
102	C₂×Dic₁₆₃	—	2	1	no	—
103	C₂×Dic₁₆₃	—	2	1	no	—
104	C₂×Dic₁₆₃	—	2	1	no	—
105	C₂×Dic₁₆₃	—	2	2	no	—
106	C₄×D₁₆₃	—	2	4	no	—
107	C₄×D₁₆₃	—	4	2	no	—
108	C₄×D₁₆₃	—	4	4	no	—
109	C₄×D₁₆₃	—	4	2	no	—
110	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₆₃	—	2	2	no	—
112	C₂×C₄×C₁₆₃	—	4	1	no	—
113	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₆₃	#41	1	2	yes	—
120	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
121	C₈×C₁₆₃	—	2	2	no	—
122	C₈×C₁₆₃	—	2	2	no	—
123	C₂²×C₁₆₃	—	2	1	no	—
124	C₂²×C₁₆₃	—	2	1	no	—
125	C₂²×C₁₆₃	#121	1	2	yes	—
126	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×D₁₆₃	—	2	2	no	—
128	C₂×D₁₆₃	—	4	1	no	—
129	C₂×D₁₆₃	—	4	1	no	—
130	C₂×D₁₆₃	—	4	2	no	—
131	C₂×D₁₆₃	—	4	2	no	—
132	C₂×D₁₆₃	—	4	2	no	—
133	C₄×C₁₆₃	—	2	1	no	—
134	C₄×C₁₆₃	—	2	1	no	—
135	C₄×C₁₆₃	—	2	1	no	—
136	C₄×C₁₆₃	#121	1	2	yes	—
137	C₄×C₁₆₃	#135	1	2	yes	complemented
138	C₄×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
140	Dic₁₆₃	—	2	2	no	—
141	Dic₁₆₃	—	4	2	no	—
142	Dic₁₆₃	—	4	1	no	—
143	Dic₁₆₃	—	4	1	no	—
144	C₂×C₁₆₃	—	2	2	no	—
145	C₂×C₁₆₃	C₂×#121	1	1	yes	characteristic
146	C₂×C₁₆₃	C₂×#135	1	1	yes	characteristic
147	C₂×C₁₆₃	#575	1	1	yes	characteristic
148	D₁₆₃	—	4	2	no	—
149	D₁₆₃	—	8	2	no	—
150	C₁₆₃	#3035	1	1	yes	characteristic, complemented, Sylow
151	#3035	—	163	1	no	maximal, Sylow
152	C₂×#135	—	163	2	no	—
153	#532	—	163	1	no	—
154	C₄×#41	—	163	1	no	—
155	#464	—	163	2	no	—
156	#463	—	163	2	no	—
157	#462	—	163	1	no	—
158	#461	—	163	1	no	—
159	#460	—	163	1	no	—
160	#459	—	163	1	no	—
161	C₂×C₄⋊C₈	D₁₆₃	1	1	yes	characteristic, complemented
162	#416	—	163	2	no	—
163	C₂×#41	—	163	2	no	—
164	C₂×#41	—	163	1	no	—
165	#135	—	326	8	no	—
166	C₂×Q₁₆	—	163	2	no	—
167	C₂×SD₁₆	—	163	2	no	—
168	C₂×SD₁₆	—	163	2	no	—
169	C₂×D₈	—	163	2	no	—
170	C₂²×C₈	C₂×D₁₆₃	1	2	yes	—
171	C₄⋊Q₈	—	163	1	no	—
172	#127	—	163	1	no	—
173	#124	—	163	2	no	—
174	#121	—	326	1	no	—
175	#121	—	326	1	no	—
176	C₄×Q₈	—	163	1	no	—
177	C₄×Q₈	—	163	1	no	—
178	C₄×D₄	—	326	1	no	—
179	C₄×D₄	—	326	1	no	—
180	C₄×D₄	—	326	1	no	—
181	C₄×D₄	—	326	1	no	—
182	C₄×D₄	—	163	1	no	—
183	C₄×D₄	—	163	1	no	—
184	#117	—	326	1	no	—
185	#117	—	163	1	no	—
186	C₂×C₄²	—	326	1	no	—
187	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
188	C₄⋊C₈	C₂×D₁₆₃	1	1	yes	characteristic
189	C₄⋊C₈	C₂×D₁₆₃	1	1	yes	characteristic
190	C₄⋊C₈	C₂×D₁₆₃	1	2	yes	—
191	#103	—	163	2	no	—
192	#103	—	163	2	no	—
193	#102	—	163	2	no	—
194	#102	—	163	2	no	—
195	#41	—	326	2	no	—
196	#41	—	326	4	no	—
197	#41	—	326	4	no	—
198	#41	—	163	4	no	—
199	C₂×Q₈	—	163	2	no	—
200	C₂×Q₈	—	163	1	no	—
201	C₂×D₄	—	326	1	no	—
202	C₂×D₄	—	326	2	no	—
203	C₂×D₄	—	326	1	no	—
204	C₂×D₄	—	326	1	no	—
205	C₂×D₄	—	163	2	no	—
206	C₂×D₄	—	163	1	no	—
207	C₂²×C₄	—	326	1	no	—
208	C₂²×C₄	—	326	1	no	—
209	C₂²×C₄	—	326	1	no	—
210	C₂²×C₄	—	326	2	no	—
211	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
212	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
214	Q₁₆	—	326	4	no	—
215	SD₁₆	—	326	4	no	—
216	SD₁₆	—	326	4	no	—
217	D₈	—	326	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
220	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
221	C₄⋊C₄	—	326	1	no	—
222	C₄⋊C₄	—	326	1	no	—
223	C₄⋊C₄	—	326	1	no	—
224	C₄⋊C₄	—	326	1	no	—
225	C₄⋊C₄	—	163	1	no	—
226	C₄⋊C₄	—	163	1	no	—
227	C₂²⋊C₄	—	326	1	no	—
228	C₂²⋊C₄	—	326	1	no	—
229	C₂²⋊C₄	—	326	1	no	—
230	C₂²⋊C₄	—	326	2	no	—
231	C₂²⋊C₄	—	326	2	no	—
232	C₄²	—	326	1	no	—
233	C₄²	—	326	1	no	—
234	C₄²	—	326	1	no	—
235	C₄²	D₆₅₂	1	1	yes	characteristic
236	C₄²	D₆₅₂	1	1	yes	characteristic
237	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
238	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
239	C₂³	—	326	1	no	—
240	C₂³	—	326	2	no	—
241	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
242	Q₈	—	326	1	no	—
243	Q₈	—	326	2	no	—
244	Q₈	—	326	2	no	—
245	Q₈	—	163	2	no	—
246	D₄	—	326	4	no	—
247	D₄	—	326	2	no	—
248	D₄	—	326	1	no	—
249	D₄	—	326	2	no	—
250	D₄	—	652	2	no	—
251	D₄	—	652	4	no	—
252	D₄	—	652	2	no	—
253	D₄	—	163	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	326	1	no	—
257	C₂×C₄	—	326	4	no	—
258	C₂×C₄	—	326	1	no	—
259	C₂×C₄	—	326	1	no	—
260	C₂×C₄	—	326	2	no	—
261	C₂×C₄	—	652	2	no	—
262	C₂×C₄	—	652	4	no	—
263	C₂×C₄	—	652	2	no	—
264	C₂×C₄	D₄×D₁₆₃	1	2	yes	—
265	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
266	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
267	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
268	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
269	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
270	C₂×C₄	#14579700	1	2	yes	—
271	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	326	2	no	—
277	C₂²	—	652	2	no	—
278	C₂²	—	652	2	no	—
279	C₂²	—	652	2	no	—
280	C₂²	—	652	1	no	—
281	C₂²	—	652	1	no	—
282	C₂²	#19104710	1	2	yes	—
283	C₂²	C₂²×D₆₅₂	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	326	2	no	—
288	C₄	—	652	2	no	—
289	C₄	—	652	1	no	—
290	C₄	—	652	1	no	—
291	C₄	#19104710	1	2	yes	—
292	C₄	#19104757	1	2	yes	complemented
293	C₄	C₂×D₄×D₁₆₃	1	1	yes	characteristic
294	C₄	C₂×#14579700	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	1304	2	no	—
297	C₂	—	652	2	no	—
298	C₂	C₂×#19104710	1	1	yes	characteristic
299	C₂	C₂×#19104757	1	1	yes	characteristic
300	C₂	#23782465	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₅₂
commutator, cosocle	C₂×C₄×C₁₆₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	C₂×C₄⋊C₈×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
hypercenter	C₂×C₄⋊C₈	D₁₆₃
hypocenter	C₁₆₃	#3035
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19104757	C₂	semidirect
2. #23782215	C₂	semidirect
3. C₄×#14579699	C₂	semidirect
4. #23781888	C₂	semidirect
5. #23781881	C₂	semidirect
6. #23781876	C₂	semidirect
7. #23781871	C₂	semidirect
8. #23781866	C₂	semidirect
9. #23781861	C₂	semidirect
10. #23781662	C₂	semidirect
11. C₂×C₄⋊C₈×C₁₆₃	C₂	semidirect
47. #14579699	D₄	semidirect
137. C₄×C₁₆₃	#135	semidirect
150. C₁₆₃	#3035	semidirect
161. C₂×C₄⋊C₈	D₁₆₃	semidirect
292. C₄	#19104757	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19104757	2
#23782215	2
C₄×#14579699	2
#23781888	2
#23781881	2
#23781876	2
#23781871	2
#23781866	2
#23781861	2
#23781662	2
C₂×C₄⋊C₈×C₁₆₃	2
#3035	163