Subgroups — #2742×C₂₆₃ #46779711

Order 33664.

416 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₂₆₃	C₂	1	2	yes	maximal, complemented
2	#399×C₂₆₃	C₂	1	1	yes	maximal, characteristic
3	#395×C₂₆₃	C₂	1	2	yes	maximal, complemented
4	#393×C₂₆₃	C₂	1	1	yes	maximal, characteristic
5	#389×C₂₆₃	C₂	1	2	yes	maximal, complemented
6	#388×C₂₆₃	C₂	1	2	yes	maximal
7	#387×C₂₆₃	C₂	1	2	yes	maximal, complemented
8	#385×C₂₆₃	C₂	1	1	yes	maximal, characteristic
9	#380×C₂₆₃	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₂₆₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₆₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₆₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₆₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₆₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₆₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₆₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₆₃	C₂²	1	2	yes	—
23	#95×C₂₆₃	C₂²	1	2	yes	—
24	#95×C₂₆₃	C₂²	1	2	yes	—
25	#95×C₂₆₃	C₂²	1	2	yes	—
26	#95×C₂₆₃	C₂²	1	1	yes	characteristic
27	#95×C₂₆₃	C₂²	1	2	yes	—
28	#95×C₂₆₃	C₂²	1	2	yes	—
29	#95×C₂₆₃	C₂²	1	2	yes	—
30	#95×C₂₆₃	C₂²	1	1	yes	characteristic
31	C₂⁴×C₂₆₃	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₂₆₃	—	4	2	no	—
33	C₂×Q₈×C₂₆₃	—	4	1	no	—
34	C₂²×C₄×C₂₆₃	—	2	4	no	—
35	C₂²×C₄×C₂₆₃	—	4	1	no	—
36	C₂²×C₄×C₂₆₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₆₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₆₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₆₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₆₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₆₃	C₂³	1	2	yes	—
44	C₄⋊C₄×C₂₆₃	—	2	4	no	—
45	C₄⋊C₄×C₂₆₃	—	2	4	no	—
46	C₄⋊C₄×C₂₆₃	—	4	2	no	—
47	C₄⋊C₄×C₂₆₃	—	4	2	no	—
48	C₄⋊C₄×C₂₆₃	—	4	2	no	—
49	C₄⋊C₄×C₂₆₃	—	4	1	no	—
50	C₂²⋊C₄×C₂₆₃	—	4	2	no	—
51	C₂²⋊C₄×C₂₆₃	—	4	2	no	—
52	C₂²⋊C₄×C₂₆₃	—	4	1	no	—
53	C₂²⋊C₄×C₂₆₃	Q₈	1	4	yes	complemented
54	C₄²×C₂₆₃	—	4	2	no	—
55	C₂³×C₂₆₃	—	2	2	no	—
56	C₂³×C₂₆₃	—	2	2	no	—
57	C₂³×C₂₆₃	—	2	1	no	—
58	C₂³×C₂₆₃	—	2	1	no	—
59	C₂³×C₂₆₃	C₂×Q₈	1	2	yes	—
60	C₂³×C₂₆₃	C₂⁴	1	1	yes	characteristic
61	Q₈×C₂₆₃	—	4	4	no	—
62	C₂×C₄×C₂₆₃	—	2	2	no	—
63	C₂×C₄×C₂₆₃	—	2	2	no	—
64	C₂×C₄×C₂₆₃	—	2	2	no	—
65	C₂×C₄×C₂₆₃	—	2	2	no	—
66	C₂×C₄×C₂₆₃	—	2	2	no	—
67	C₂×C₄×C₂₆₃	—	2	2	no	—
68	C₂×C₄×C₂₆₃	—	2	2	no	—
69	C₂×C₄×C₂₆₃	—	2	2	no	—
70	C₂×C₄×C₂₆₃	—	2	2	no	—
71	C₂×C₄×C₂₆₃	—	2	2	no	—
72	C₂×C₄×C₂₆₃	—	2	1	no	—
73	C₂×C₄×C₂₆₃	—	2	2	no	—
74	C₂×C₄×C₂₆₃	—	2	2	no	—
75	C₂×C₄×C₂₆₃	—	2	1	no	—
76	C₂×C₄×C₂₆₃	—	2	2	no	—
77	C₂×C₄×C₂₆₃	—	2	2	no	—
78	C₂×C₄×C₂₆₃	—	2	2	no	—
79	C₂×C₄×C₂₆₃	—	2	2	no	—
80	C₂×C₄×C₂₆₃	—	2	2	no	—
81	C₂×C₄×C₂₆₃	—	2	2	no	—
82	C₂×C₄×C₂₆₃	—	4	4	no	—
83	C₂×C₄×C₂₆₃	C₂×Q₈	1	4	yes	—
84	C₂×C₄×C₂₆₃	#41	1	4	yes	—
85	C₂²×C₂₆₃	—	2	2	no	—
86	C₂²×C₂₆₃	—	2	4	no	—
87	C₂²×C₂₆₃	—	4	2	no	—
88	C₂²×C₂₆₃	—	4	1	no	—
89	C₂²×C₂₆₃	—	4	1	no	—
90	C₂²×C₂₆₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂²×C₂₆₃	C₂×#41	1	2	yes	—
92	C₂²×C₂₆₃	H(5,𝔽₂)	1	2	yes	—
93	C₂²×C₂₆₃	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₂₆₃	2₋¹⁺⁴	1	1	yes	characteristic
95	C₄×C₂₆₃	—	2	4	no	—
96	C₄×C₂₆₃	—	2	4	no	—
97	C₄×C₂₆₃	—	4	2	no	—
98	C₄×C₂₆₃	—	4	2	no	—
99	C₄×C₂₆₃	—	4	1	no	—
100	C₄×C₂₆₃	—	4	2	no	—
101	C₄×C₂₆₃	—	4	2	no	—
102	C₄×C₂₆₃	—	4	1	no	—
103	C₂×C₂₆₃	—	4	2	no	—
104	C₂×C₂₆₃	#542	1	2	yes	—
105	C₂×C₂₆₃	#544	1	1	yes	characteristic
106	C₂×C₂₆₃	#551	1	1	yes	characteristic
107	C₂×C₂₆₃	#554	1	2	yes	—
108	C₂×C₂₆₃	#560	1	1	yes	characteristic
109	C₂₆₃	#2742	1	1	yes	characteristic, direct product factor, Sylow
110	#2742	C₂₆₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	#400	C₂×C₂₆₃	1	2	yes	complemented
112	#399	C₂×C₂₆₃	1	1	yes	characteristic
113	#395	C₂×C₂₆₃	1	2	yes	complemented
114	#393	C₂×C₂₆₃	1	1	yes	characteristic
115	#389	C₂×C₂₆₃	1	2	yes	complemented
116	#388	C₂×C₂₆₃	1	2	yes	—
117	#387	C₂×C₂₆₃	1	2	yes	complemented
118	#385	C₂×C₂₆₃	1	1	yes	characteristic
119	#380	C₂×C₂₆₃	1	2	yes	—
120	C₂²×Q₈	C₂²×C₂₆₃	1	1	yes	characteristic
121	C₂³×C₄	C₂²×C₂₆₃	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂²×C₂₆₃	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₂₆₃	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₂₆₃	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₂₆₃	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₂₆₃	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₂₆₃	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	C₂²×C₂₆₃	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₂₆₃	1	2	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₂₆₃	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₂²×C₂₆₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₂₆₃	1	2	yes	—
133	#95	C₂²×C₂₆₃	1	2	yes	—
134	#95	C₂²×C₂₆₃	1	2	yes	—
135	#95	C₂²×C₂₆₃	1	2	yes	—
136	#95	C₂²×C₂₆₃	1	1	yes	characteristic
137	#95	C₂²×C₂₆₃	1	2	yes	—
138	#95	C₂²×C₂₆₃	1	2	yes	—
139	#95	C₂²×C₂₆₃	1	2	yes	—
140	#95	C₂²×C₂₆₃	1	1	yes	characteristic
141	C₂⁴	C₂³×C₂₆₃	1	1	yes	characteristic
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	4	1	no	—
146	C₂²×C₄	C₂³×C₂₆₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₂₆₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₂₆₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₂₆₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₂₆₃	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₂₆₃	1	2	yes	—
152	C₂²×C₄	C₂³×C₂₆₃	1	2	yes	—
153	C₂²×C₄	C₂³×C₂₆₃	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	1	no	—
160	C₂²⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	1	no	—
163	C₂²⋊C₄	Q₈×C₂₆₃	1	4	yes	complemented
164	C₄²	—	4	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	1	no	—
169	C₂³	C₂×Q₈×C₂₆₃	1	2	yes	—
170	C₂³	C₂⁴×C₂₆₃	1	1	yes	characteristic
171	Q₈	—	4	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×Q₈×C₂₆₃	1	4	yes	—
194	C₂×C₄	#41×C₂₆₃	1	4	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	1	no	—
199	C₂²	—	4	1	no	—
200	C₂²	C₂²×Q₈×C₂₆₃	1	1	yes	characteristic
201	C₂²	C₂×#41×C₂₆₃	1	2	yes	—
202	C₂²	H(5,𝔽₂)×C₂₆₃	1	2	yes	—
203	C₂²	H(5,𝔽₂)×C₂₆₃	1	1	yes	characteristic
204	C₂²	2₋¹⁺⁴×C₂₆₃	1	1	yes	characteristic
205	C₄	—	2	4	no	—
206	C₄	—	2	4	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	1	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	1	no	—
212	C₄	—	4	2	no	—
213	C₂	—	4	2	no	—
214	C₂	#542×C₂₆₃	1	2	yes	—
215	C₂	#544×C₂₆₃	1	1	yes	characteristic
216	C₂	#551×C₂₆₃	1	1	yes	characteristic
217	C₂	#554×C₂₆₃	1	2	yes	—
218	C₂	#560×C₂₆₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₆₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₆₃
prime radical	C₂⁴×C₂₆₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₆₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #400×C₂₆₃	C₂	semidirect
3. #395×C₂₆₃	C₂	semidirect
5. #389×C₂₆₃	C₂	semidirect
7. #387×C₂₆₃	C₂	semidirect
53. C₂²⋊C₄×C₂₆₃	Q₈	semidirect
109. C₂₆₃	#2742	direct (N × H)
110. #2742	C₂₆₃	direct (N × H)
111. #400	C₂×C₂₆₃	semidirect
113. #395	C₂×C₂₆₃	semidirect
115. #389	C₂×C₂₆₃	semidirect
117. #387	C₂×C₂₆₃	semidirect
163. C₂²⋊C₄	Q₈×C₂₆₃	semidirect

Maximal subgroups

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

subgroup	index
#400×C₂₆₃	2
#399×C₂₆₃	2
#395×C₂₆₃	2
#393×C₂₆₃	2
#389×C₂₆₃	2
#388×C₂₆₃	2
#387×C₂₆₃	2
#385×C₂₆₃	2
#380×C₂₆₃	2
#2742	263

Subgroups — #2742×C263 #46779711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2742×C₂₆₃ #46779711