Subgroups — C₂×#23782172 #35227774

Order 20864.

1880 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#23782172	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19104716	C₂	1	2	yes	maximal, complemented
4	C₂×#19104711	C₂	1	2	yes	maximal, complemented
5	C₂×#19104706	C₂	1	2	yes	maximal, complemented
6	C₂×#19104703	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₂²⋊D₁₆₃	C₂	1	4	yes	maximal, complemented
8	C₂×#19104694	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₆₃	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₁₆₃	C₂²	1	4	yes	—
12	C₂³×Dic₁₆₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₆₃	—	2	8	no	—
14	C₂²×C₄×D₁₆₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₆₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₆₃	C₂²	1	8	yes	complemented
17	#19104716	C₂²	1	16	yes	complemented
18	#19104711	C₂²	1	16	yes	complemented
19	#19104706	C₂²	1	16	yes	complemented
20	#19104703	C₂²	1	16	yes	complemented
21	C₄×C₂²⋊D₁₆₃	C₂²	1	32	yes	complemented
22	#19104694	C₂²	1	8	yes	complemented
23	C₂×C₄⋊Dic₁₆₃	C₂²	1	2	yes	complemented
24	C₂×Dic₁₆₃⋊C₄	C₂²	1	4	yes	complemented
25	C₂×C₂²⋊Dic₁₆₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊Dic₁₆₃	C₂²	1	4	yes	complemented
27	C₂×(C₂×D₁₆₃)⋊C₄	C₂²	1	4	yes	—
28	C₂×C₄×Dic₁₆₃	C₂²	1	2	yes	complemented
29	C₂×#41×C₁₆₃	—	2	8	no	—
30	C₂×#41×C₁₆₃	C₂²	1	8	yes	complemented
31	C₂²×Q₈×C₁₆₃	C₂²	1	1	yes	characteristic, complemented
32	C₂²×D₄×C₁₆₃	C₂²	1	2	yes	—
33	C₂²×D₄×C₁₆₃	C₂²	1	1	yes	characteristic, complemented
34	C₂³×C₄×C₁₆₃	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₁₆₃	C₂²	1	2	yes	complemented
36	C₂³×D₁₆₃	—	2	8	no	—
37	C₂³×D₁₆₃	—	2	4	no	—
38	C₂³×D₁₆₃	C₂³	1	4	yes	—
39	C₂³×D₁₆₃	C₂³	1	2	yes	—
40	C₂×C₂²⋊D₁₆₃	—	2	16	no	—
41	C₂×C₂²⋊D₁₆₃	C₂³	1	16	yes	—
42	C₂²×Dic₁₆₃	—	2	4	no	—
43	C₂²×Dic₁₆₃	C₂³	1	2	yes	—
44	C₂²×Dic₁₆₃	C₂³	1	4	yes	—
45	C₂²×Dic₁₆₃	C₂³	1	4	yes	—
46	C₂×C₄×D₁₆₃	—	2	8	no	—
47	C₂×C₄×D₁₆₃	—	2	8	no	—
48	C₂×C₄×D₁₆₃	—	2	32	no	—
49	C₂×C₄×D₁₆₃	C₂³	1	8	yes	—
50	C₂×C₄×D₁₆₃	C₂³	1	8	yes	—
51	C₄⋊Dic₁₆₃	C₂³	1	8	yes	complemented
52	Dic₁₆₃⋊C₄	C₂³	1	16	yes	complemented
53	C₂²⋊Dic₁₆₃	C₂³	1	8	yes	—
54	C₂²⋊Dic₁₆₃	C₂³	1	16	yes	complemented
55	(C₂×D₁₆₃)⋊C₄	C₂³	1	16	yes	—
56	C₄×Dic₁₆₃	C₂³	1	8	yes	complemented
57	C₂⁴×C₁₆₃	C₂³	1	2	yes	—
58	C₂⁴×C₁₆₃	C₂³	1	1	yes	characteristic
59	#41×C₁₆₃	—	2	32	no	—
60	C₂×Q₈×C₁₆₃	—	2	4	no	—
61	C₂×Q₈×C₁₆₃	C₂³	1	4	yes	complemented
62	C₂×D₄×C₁₆₃	—	2	8	no	—
63	C₂×D₄×C₁₆₃	—	2	4	no	—
64	C₂×D₄×C₁₆₃	C₂³	1	8	yes	—
65	C₂×D₄×C₁₆₃	C₂³	1	4	yes	complemented
66	C₂²×C₄×C₁₆₃	—	2	8	no	—
67	C₂²×C₄×C₁₆₃	D₄	1	8	yes	complemented
68	C₂²×C₄×C₁₆₃	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₆₃	C₂³	1	4	yes	—
71	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₆₃	C₂³	1	8	yes	complemented
73	C₂²×D₁₆₃	—	2	32	no	—
74	C₂²×D₁₆₃	—	2	8	no	—
75	C₂²×D₁₆₃	—	2	4	no	—
76	C₂²×D₁₆₃	—	2	4	no	—
77	C₂²×D₁₆₃	—	2	16	no	—
78	C₂²×D₁₆₃	#41	1	8	yes	complemented
79	C₂²×D₁₆₃	C₂⁴	1	4	yes	—
80	C₂²⋊D₁₆₃	—	2	32	no	—
81	C₂×Dic₁₆₃	—	2	8	no	—
82	C₂×Dic₁₆₃	—	2	4	no	—
83	C₂×Dic₁₆₃	—	2	8	no	—
84	C₂×Dic₁₆₃	C₂⁴	1	8	yes	—
85	C₂×Dic₁₆₃	C₂⁴	1	4	yes	—
86	C₄×D₁₆₃	—	2	16	no	—
87	C₄×D₁₆₃	—	2	16	no	—
88	C₂³×C₁₆₃	—	2	4	no	—
89	C₂³×C₁₆₃	—	2	4	no	—
90	C₂³×C₁₆₃	—	2	8	no	—
91	C₂³×C₁₆₃	C₂×D₄	1	4	yes	—
92	C₂³×C₁₆₃	C₂⁴	1	4	yes	—
93	C₂³×C₁₆₃	C₂⁴	1	2	yes	—
94	C₂³×C₁₆₃	C₂⁴	1	1	yes	characteristic
95	Q₈×C₁₆₃	—	2	8	no	—
96	D₄×C₁₆₃	—	2	16	no	—
97	D₄×C₁₆₃	—	2	8	no	—
98	C₂×C₄×C₁₆₃	—	2	4	no	—
99	C₂×C₄×C₁₆₃	—	2	16	no	—
100	C₂×C₄×C₁₆₃	C₂×D₄	1	4	yes	—
101	C₂×C₄×C₁₆₃	C₂×D₄	1	4	yes	—
102	C₂×C₄×C₁₆₃	C₂×D₄	1	16	yes	complemented
103	C₂×C₄×C₁₆₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₆₃	C₂⁴	1	2	yes	—
105	C₂×C₄×C₁₆₃	C₂⁴	1	2	yes	—
106	C₂²×C₁₆₃	—	2	8	no	—
107	C₂²×C₁₆₃	—	2	4	no	—
108	C₂²×C₁₆₃	—	2	4	no	—
109	C₂²×C₁₆₃	—	2	8	no	—
110	C₂²×C₁₆₃	—	4	8	no	—
111	C₂²×C₁₆₃	C₂²×D₄	1	4	yes	—
112	C₂²×C₁₆₃	C₂²×D₄	1	2	yes	—
113	C₂²×C₁₆₃	C₂×#41	1	4	yes	—
114	C₂²×C₁₆₃	C₂⁵	1	1	yes	characteristic
115	C₂×D₁₆₃	—	2	32	no	—
116	C₂×D₁₆₃	—	2	16	no	—
117	C₂×D₁₆₃	—	2	4	no	—
118	C₂×D₁₆₃	—	2	4	no	—
119	C₂×D₁₆₃	C₂×#41	1	8	yes	complemented
120	C₄×C₁₆₃	—	2	4	no	—
121	C₄×C₁₆₃	C₂²×D₄	1	4	yes	—
122	C₄×C₁₆₃	C₂²×D₄	1	4	yes	—
123	Dic₁₆₃	—	2	4	no	—
124	Dic₁₆₃	—	2	8	no	—
125	C₂×C₁₆₃	—	2	4	no	—
126	C₂×C₁₆₃	—	4	4	no	—
127	C₂×C₁₆₃	#525	1	4	yes	complemented
128	C₂×C₁₆₃	C₂³×D₄	1	1	yes	characteristic
129	C₂×C₁₆₃	C₂²×#41	1	2	yes	—
130	D₁₆₃	—	2	8	no	—
131	C₁₆₃	C₂×#525	1	1	yes	characteristic, complemented, Sylow
132	C₂×#525	—	163	1	no	maximal, Sylow
133	C₂²×#41	D₁₆₃	1	1	yes	characteristic, complemented
134	C₂⁴×C₄	—	163	1	no	—
135	#525	—	163	16	no	—
136	C₂×#123	—	163	2	no	—
137	C₂×C₂²⋊Q₈	—	163	2	no	—
138	C₂×#121	—	163	2	no	—
139	C₂×#120	—	163	2	no	—
140	C₂×C₄×D₄	—	163	4	no	—
141	C₂×#117	—	163	1	no	—
142	C₂⁵	—	163	1	no	—
143	C₂×#41	—	2	8	no	—
144	C₂×#41	C₂×D₁₆₃	1	8	yes	complemented
145	C₂²×Q₈	C₂×D₁₆₃	1	1	yes	characteristic, complemented
146	C₂²×D₄	—	163	4	no	—
147	C₂²×D₄	C₂×D₁₆₃	1	2	yes	—
148	C₂²×D₄	C₂×D₁₆₃	1	1	yes	characteristic, complemented
149	C₂³×C₄	—	326	8	no	—
150	C₂³×C₄	—	163	1	no	—
151	C₂³×C₄	—	163	2	no	—
152	C₂³×C₄	—	163	8	no	—
153	C₂³×C₄	—	163	2	no	—
154	C₂³×C₄	C₂×D₁₆₃	1	1	yes	characteristic
155	C₂³×C₄	C₂×D₁₆₃	1	2	yes	complemented
156	#123	—	163	16	no	—
157	C₂²⋊Q₈	—	163	16	no	—
158	#121	—	163	16	no	—
159	#120	—	163	16	no	—
160	C₄×D₄	—	163	32	no	—
161	#117	—	163	8	no	—
162	C₂×C₄⋊C₄	—	163	2	no	—
163	C₂×C₄⋊C₄	—	163	4	no	—
164	C₂×C₂²⋊C₄	—	163	2	no	—
165	C₂×C₂²⋊C₄	—	163	4	no	—
166	C₂×C₂²⋊C₄	—	163	4	no	—
167	C₂×C₄²	—	163	2	no	—
168	C₂⁴	—	326	8	no	—
169	C₂⁴	—	326	4	no	—
170	C₂⁴	—	163	4	no	—
171	C₂⁴	—	163	2	no	—
172	C₂⁴	C₂²×D₁₆₃	1	2	yes	—
173	C₂⁴	C₂²×D₁₆₃	1	1	yes	characteristic
174	#41	—	2	32	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	C₂²×D₁₆₃	1	4	yes	complemented
177	C₂×D₄	—	2	8	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	326	16	no	—
180	C₂×D₄	—	163	16	no	—
181	C₂×D₄	C₂²×D₁₆₃	1	8	yes	—
182	C₂×D₄	C₂²×D₁₆₃	1	4	yes	complemented
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	326	4	no	—
185	C₂²×C₄	—	326	8	no	—
186	C₂²×C₄	—	326	8	no	—
187	C₂²×C₄	—	326	32	no	—
188	C₂²×C₄	—	163	4	no	—
189	C₂²×C₄	—	163	2	no	—
190	C₂²×C₄	—	163	4	no	—
191	C₂²×C₄	—	163	8	no	—
192	C₂²×C₄	—	163	8	no	—
193	C₂²×C₄	C₂²⋊D₁₆₃	1	8	yes	complemented
194	C₂²×C₄	C₂²×D₁₆₃	1	2	yes	—
195	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₁₆₃	1	4	yes	—
197	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₆₃	1	8	yes	complemented
199	C₄⋊C₄	—	163	8	no	—
200	C₄⋊C₄	—	163	16	no	—
201	C₂²⋊C₄	—	163	8	no	—
202	C₂²⋊C₄	—	163	16	no	—
203	C₂²⋊C₄	—	163	16	no	—
204	C₄²	—	163	8	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	4	no	—
207	C₂³	—	2	8	no	—
208	C₂³	—	326	32	no	—
209	C₂³	—	326	16	no	—
210	C₂³	—	326	8	no	—
211	C₂³	—	326	4	no	—
212	C₂³	—	326	4	no	—
213	C₂³	—	163	8	no	—
214	C₂³	—	163	4	no	—
215	C₂³	C₂×C₂²⋊D₁₆₃	1	4	yes	—
216	C₂³	C₂³×D₁₆₃	1	4	yes	—
217	C₂³	C₂³×D₁₆₃	1	2	yes	—
218	C₂³	C₂³×D₁₆₃	1	1	yes	characteristic
219	Q₈	—	2	8	no	—
220	D₄	—	2	16	no	—
221	D₄	—	2	8	no	—
222	D₄	—	326	32	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	326	8	no	—
226	C₂×C₄	—	326	4	no	—
227	C₂×C₄	—	326	8	no	—
228	C₂×C₄	—	326	16	no	—
229	C₂×C₄	—	326	16	no	—
230	C₂×C₄	—	163	8	no	—
231	C₂×C₄	—	163	4	no	—
232	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	16	yes	complemented
235	C₂×C₄	C₂³×D₁₆₃	1	4	yes	—
236	C₂×C₄	C₂³×D₁₆₃	1	2	yes	—
237	C₂×C₄	C₂³×D₁₆₃	1	2	yes	—
238	C₂²	—	2	8	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	8	no	—
243	C₂²	—	326	32	no	—
244	C₂²	—	326	16	no	—
245	C₂²	—	326	4	no	—
246	C₂²	—	326	4	no	—
247	C₂²	—	163	8	no	—
248	C₂²	C₂²×C₂²⋊D₁₆₃	1	4	yes	—
249	C₂²	C₂²×C₂²⋊D₁₆₃	1	2	yes	—
250	C₂²	#41×D₁₆₃	1	4	yes	—
251	C₂²	C₂⁴×D₁₆₃	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	326	4	no	—
254	C₄	—	326	8	no	—
255	C₄	C₂²×C₂²⋊D₁₆₃	1	4	yes	—
256	C₄	C₂²×C₂²⋊D₁₆₃	1	4	yes	—
257	C₂	—	2	4	no	—
258	C₂	—	4	4	no	—
259	C₂	—	326	8	no	—
260	C₂	#23782172	1	4	yes	direct product factor
261	C₂	C₂³×C₂²⋊D₁₆₃	1	1	yes	characteristic
262	C₂	C₂×#41×D₁₆₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₆₃
commutator, cosocle	C₂²×C₁₆₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₆₃
Fitting	C₂²×#41×C₁₆₃	C₂
socle	C₂³×C₁₆₃	C₂⁴
hypercenter	C₂²×#41	D₁₆₃
hypocenter	C₁₆₃	C₂×#525
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄×D₁₆₃	C₂	semidirect
2. #23782172	C₂	direct (N × H)
3. C₂×#19104716	C₂	semidirect
4. C₂×#19104711	C₂	semidirect
5. C₂×#19104706	C₂	semidirect
6. C₂×#19104703	C₂	semidirect
7. C₂×C₄×C₂²⋊D₁₆₃	C₂	semidirect
8. C₂×#19104694	C₂	semidirect
9. C₂²×#41×C₁₆₃	C₂	semidirect
16. C₂²×C₄×D₁₆₃	C₂²	semidirect
17. #19104716	C₂²	semidirect
18. #19104711	C₂²	semidirect
19. #19104706	C₂²	semidirect
20. #19104703	C₂²	semidirect
21. C₄×C₂²⋊D₁₆₃	C₂²	semidirect
22. #19104694	C₂²	semidirect
23. C₂×C₄⋊Dic₁₆₃	C₂²	semidirect
24. C₂×Dic₁₆₃⋊C₄	C₂²	semidirect
26. C₂×C₂²⋊Dic₁₆₃	C₂²	semidirect
28. C₂×C₄×Dic₁₆₃	C₂²	semidirect
30. C₂×#41×C₁₆₃	C₂²	semidirect
31. C₂²×Q₈×C₁₆₃	C₂²	semidirect
33. C₂²×D₄×C₁₆₃	C₂²	semidirect
35. C₂³×C₄×C₁₆₃	C₂²	semidirect
51. C₄⋊Dic₁₆₃	C₂³	semidirect
52. Dic₁₆₃⋊C₄	C₂³	semidirect
54. C₂²⋊Dic₁₆₃	C₂³	semidirect
56. C₄×Dic₁₆₃	C₂³	semidirect
61. C₂×Q₈×C₁₆₃	C₂³	semidirect
65. C₂×D₄×C₁₆₃	C₂³	semidirect
67. C₂²×C₄×C₁₆₃	D₄	semidirect
72. C₂²×C₄×C₁₆₃	C₂³	semidirect
78. C₂²×D₁₆₃	#41	semidirect
102. C₂×C₄×C₁₆₃	C₂×D₄	semidirect
119. C₂×D₁₆₃	C₂×#41	semidirect
127. C₂×C₁₆₃	#525	semidirect
131. C₁₆₃	C₂×#525	semidirect
133. C₂²×#41	D₁₆₃	semidirect
144. C₂×#41	C₂×D₁₆₃	semidirect
145. C₂²×Q₈	C₂×D₁₆₃	semidirect
148. C₂²×D₄	C₂×D₁₆₃	semidirect
155. C₂³×C₄	C₂×D₁₆₃	semidirect
176. C₂×Q₈	C₂²×D₁₆₃	semidirect
182. C₂×D₄	C₂²×D₁₆₃	semidirect
193. C₂²×C₄	C₂²⋊D₁₆₃	semidirect
198. C₂²×C₄	C₂²×D₁₆₃	semidirect
234. C₂×C₄	C₂×C₂²⋊D₁₆₃	semidirect
260. C₂	#23782172	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₁₆₃	2
#23782172	2
C₂×#19104716	2
C₂×#19104711	2
C₂×#19104706	2
C₂×#19104703	2
C₂×C₄×C₂²⋊D₁₆₃	2
C₂×#19104694	2
C₂²×#41×C₁₆₃	2
C₂×#525	163

Subgroups — C2×#23782172 #35227774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23782172 #35227774