Subgroups — C₂×C₁₁×#2641 #14611813

Order 2816.

3326 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×#2641	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₁×#394	C₂	1	8	yes	maximal, complemented
5	C₂×C₁₁×#390	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₁₁	C₂²	1	2	yes	—
8	C₂³×D₄×C₁₁	C₂²	1	4	yes	—
9	C₂×C₁₁×#127	—	2	8	no	—
10	C₂×C₁₁×#127	C₂²	1	8	yes	—
11	C₂×C₁₁×#125	—	2	8	no	—
12	C₂×C₁₁×#125	C₂²	1	8	yes	complemented
13	C₂²×C₁₁×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂²×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	16	yes	—
16	C₂²×C₄²×C₁₁	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
18	C₁₁×#394	C₂²	1	64	yes	complemented
19	C₁₁×#390	C₂²	1	32	yes	complemented
20	C₁₁×#376	C₂²	1	8	yes	complemented
21	C₂×C₁₁×#95	C₂²	1	4	yes	—
22	C₂⁵×C₁₁	C₂³	1	4	yes	—
23	C₂²×D₄×C₁₁	—	2	16	no	—
24	C₂²×D₄×C₁₁	—	2	8	no	—
25	C₂²×D₄×C₁₁	—	2	16	no	—
26	C₂²×D₄×C₁₁	—	4	16	no	—
27	C₂²×D₄×C₁₁	C₂³	1	8	yes	—
28	C₂²×D₄×C₁₁	C₂³	1	16	yes	—
29	C₂³×C₄×C₁₁	C₂³	1	4	yes	—
30	C₂³×C₄×C₁₁	C₂³	1	4	yes	—
31	C₂³×C₄×C₁₁	C₂³	1	2	yes	—
32	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
33	C₁₁×#127	—	2	32	no	—
34	C₁₁×#125	—	2	32	no	—
35	C₂×C₁₁×C₄⋊C₄	—	2	16	no	—
36	C₂×C₁₁×C₄⋊C₄	—	2	8	no	—
37	C₂×C₁₁×C₄⋊C₄	C₂³	1	8	yes	—
38	C₂×C₁₁×C₄⋊C₄	C₂³	1	16	yes	—
39	C₂×C₁₁×C₂²⋊C₄	—	4	32	no	—
40	C₂×C₁₁×C₂²⋊C₄	C₂³	1	64	yes	—
41	C₂×C₄²×C₁₁	—	2	8	no	—
42	C₂×C₄²×C₁₁	D₄	1	8	yes	—
43	C₂×C₄²×C₁₁	C₂³	1	8	yes	—
44	C₂×C₄²×C₁₁	C₂³	1	4	yes	—
45	C₁₁×#95	C₂³	1	16	yes	—
46	C₂×#2641	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂⁴×C₁₁	—	2	8	no	—
48	C₂⁴×C₁₁	—	2	16	no	—
49	C₂⁴×C₁₁	—	2	32	no	—
50	C₂⁴×C₁₁	C₂⁴	1	8	yes	—
51	C₂⁴×C₁₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₄×C₁₁	—	2	32	no	—
53	C₂×D₄×C₁₁	—	2	16	no	—
54	C₂×D₄×C₁₁	—	2	32	no	—
55	C₂×D₄×C₁₁	—	4	32	no	—
56	C₂×D₄×C₁₁	—	4	64	no	—
57	C₂×D₄×C₁₁	—	4	32	no	—
58	C₂²×C₄×C₁₁	—	2	8	no	—
59	C₂²×C₄×C₁₁	—	2	8	no	—
60	C₂²×C₄×C₁₁	—	2	8	no	—
61	C₂²×C₄×C₁₁	—	2	16	no	—
62	C₂²×C₄×C₁₁	—	2	16	no	—
63	C₂²×C₄×C₁₁	C₂×D₄	1	8	yes	—
64	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₁	#41	1	8	yes	—
66	C₂²×C₄×C₁₁	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₁₁	C₂⁴	1	8	yes	—
68	C₂²×C₄×C₁₁	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₁₁	C₂⁴	1	2	yes	—
70	C₁₁×C₄⋊C₄	—	2	32	no	—
71	C₁₁×C₄⋊C₄	—	2	16	no	—
72	C₁₁×C₂²⋊C₄	—	4	64	no	—
73	C₄²×C₁₁	—	2	16	no	—
74	C₄²×C₁₁	C₂×D₄	1	16	yes	—
75	C₂²×#127	C₂×C₁₁	1	1	yes	characteristic
76	C₂²×#125	C₂×C₁₁	1	1	yes	characteristic, complemented
77	#2641	C₂×C₁₁	1	16	yes	direct product factor
78	C₂×#394	C₂×C₁₁	1	8	yes	complemented
79	C₂×#390	C₂×C₁₁	1	4	yes	complemented
80	C₂×#376	C₂×C₁₁	1	1	yes	characteristic, complemented
81	C₂³×C₁₁	—	2	8	no	—
82	C₂³×C₁₁	—	2	16	no	—
83	C₂³×C₁₁	—	2	32	no	—
84	C₂³×C₁₁	—	4	16	no	—
85	C₂³×C₁₁	—	4	32	no	—
86	C₂³×C₁₁	—	4	64	no	—
87	C₂³×C₁₁	C₂²×D₄	1	2	yes	—
88	C₂³×C₁₁	C₂×#41	1	4	yes	—
89	C₂³×C₁₁	H(5,𝔽₂)	1	8	yes	—
90	C₂³×C₁₁	C₂⁵	1	1	yes	characteristic
91	D₄×C₁₁	—	4	64	no	—
92	D₄×C₁₁	—	4	32	no	—
93	C₂×C₄×C₁₁	—	2	16	no	—
94	C₂×C₄×C₁₁	—	2	8	no	—
95	C₂×C₄×C₁₁	—	2	8	no	—
96	C₂×C₄×C₁₁	—	2	8	no	—
97	C₂×C₄×C₁₁	—	2	16	no	—
98	C₂×C₄×C₁₁	—	2	16	no	—
99	C₂×C₄×C₁₁	—	4	16	no	—
100	C₂×C₄×C₁₁	—	4	16	no	—
101	C₂×C₄×C₁₁	#124	1	16	yes	—
102	C₂×C₄×C₁₁	C₂²×D₄	1	8	yes	—
103	C₂×C₄×C₁₁	C₂²×D₄	1	4	yes	—
104	C₂×C₄×C₁₁	C₂×#41	1	8	yes	—
105	C₂³×D₄	C₂²×C₁₁	1	2	yes	—
106	C₂³×D₄	C₂²×C₁₁	1	4	yes	—
107	C₂×#127	—	2	8	no	—
108	C₂×#127	C₂²×C₁₁	1	8	yes	—
109	C₂×#125	—	2	8	no	—
110	C₂×#125	C₂²×C₁₁	1	8	yes	complemented
111	C₂²×C₄⋊C₄	C₂²×C₁₁	1	4	yes	—
112	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
113	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	16	yes	—
114	C₂²×C₄²	C₂²×C₁₁	1	2	yes	—
115	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
116	#394	C₂²×C₁₁	1	64	yes	complemented
117	#390	C₂²×C₁₁	1	32	yes	complemented
118	#376	C₂²×C₁₁	1	8	yes	complemented
119	C₂×#95	C₂²×C₁₁	1	4	yes	—
120	C₂²×C₁₁	—	4	8	no	—
121	C₂²×C₁₁	—	4	16	no	—
122	C₂²×C₁₁	—	4	32	no	—
123	C₂²×C₁₁	—	8	32	no	—
124	C₂²×C₁₁	C₂×#124	1	4	yes	—
125	C₂²×C₁₁	#535	1	8	yes	—
126	C₂²×C₁₁	#540	1	16	yes	—
127	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
128	C₂²×C₁₁	C₂²×#41	1	2	yes	—
129	C₂²×C₁₁	C₂×H(5,𝔽₂)	1	4	yes	—
130	C₄×C₁₁	—	2	8	no	—
131	C₄×C₁₁	—	4	8	no	—
132	C₄×C₁₁	—	4	8	no	—
133	C₄×C₁₁	C₂×#124	1	8	yes	—
134	C₂⁵	C₂³×C₁₁	1	4	yes	—
135	C₂²×D₄	—	2	16	no	—
136	C₂²×D₄	—	2	8	no	—
137	C₂²×D₄	—	2	16	no	—
138	C₂²×D₄	—	4	16	no	—
139	C₂²×D₄	C₂³×C₁₁	1	8	yes	—
140	C₂²×D₄	C₂³×C₁₁	1	16	yes	—
141	C₂³×C₄	C₂³×C₁₁	1	4	yes	—
142	C₂³×C₄	C₂³×C₁₁	1	4	yes	—
143	C₂³×C₄	C₂³×C₁₁	1	2	yes	—
144	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
145	#127	—	2	32	no	—
146	#125	—	2	32	no	—
147	C₂×C₄⋊C₄	—	2	16	no	—
148	C₂×C₄⋊C₄	—	2	8	no	—
149	C₂×C₄⋊C₄	C₂³×C₁₁	1	8	yes	—
150	C₂×C₄⋊C₄	C₂³×C₁₁	1	16	yes	—
151	C₂×C₂²⋊C₄	—	4	32	no	—
152	C₂×C₂²⋊C₄	C₂³×C₁₁	1	64	yes	—
153	C₂×C₄²	—	2	8	no	—
154	C₂×C₄²	D₄×C₁₁	1	8	yes	—
155	C₂×C₄²	C₂³×C₁₁	1	8	yes	—
156	C₂×C₄²	C₂³×C₁₁	1	4	yes	—
157	#95	C₂³×C₁₁	1	16	yes	—
158	C₂×C₁₁	—	8	8	no	—
159	C₂×C₁₁	#2641	1	8	yes	direct product factor
160	C₂×C₁₁	C₂²×#124	1	1	yes	characteristic
161	C₂×C₁₁	C₂×#535	1	2	yes	—
162	C₂×C₁₁	C₂×#540	1	4	yes	—
163	C₂⁴	—	2	8	no	—
164	C₂⁴	—	2	16	no	—
165	C₂⁴	—	2	32	no	—
166	C₂⁴	C₂⁴×C₁₁	1	8	yes	—
167	C₂⁴	C₂⁴×C₁₁	1	1	yes	characteristic
168	C₂×D₄	—	2	32	no	—
169	C₂×D₄	—	2	16	no	—
170	C₂×D₄	—	2	32	no	—
171	C₂×D₄	—	4	32	no	—
172	C₂×D₄	—	4	64	no	—
173	C₂×D₄	—	4	32	no	—
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	16	no	—
178	C₂²×C₄	—	2	16	no	—
179	C₂²×C₄	C₂×D₄×C₁₁	1	8	yes	—
180	C₂²×C₄	C₂×D₄×C₁₁	1	4	yes	—
181	C₂²×C₄	C₁₁×#41	1	8	yes	—
182	C₂²×C₄	C₂⁴×C₁₁	1	8	yes	—
183	C₂²×C₄	C₂⁴×C₁₁	1	8	yes	—
184	C₂²×C₄	C₂⁴×C₁₁	1	4	yes	—
185	C₂²×C₄	C₂⁴×C₁₁	1	2	yes	—
186	C₄⋊C₄	—	2	32	no	—
187	C₄⋊C₄	—	2	16	no	—
188	C₂²⋊C₄	—	4	64	no	—
189	C₄²	—	2	16	no	—
190	C₄²	C₂×D₄×C₁₁	1	16	yes	—
191	C₁₁	C₂×#2641	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³	—	2	8	no	—
193	C₂³	—	2	16	no	—
194	C₂³	—	2	32	no	—
195	C₂³	—	4	16	no	—
196	C₂³	—	4	32	no	—
197	C₂³	—	4	64	no	—
198	C₂³	C₂²×D₄×C₁₁	1	2	yes	—
199	C₂³	C₂×C₁₁×#41	1	4	yes	—
200	C₂³	C₁₁×H(5,𝔽₂)	1	8	yes	—
201	C₂³	C₂⁵×C₁₁	1	1	yes	characteristic
202	D₄	—	4	64	no	—
203	D₄	—	4	32	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	4	16	no	—
211	C₂×C₄	—	4	16	no	—
212	C₂×C₄	C₁₁×#124	1	16	yes	—
213	C₂×C₄	C₂²×D₄×C₁₁	1	8	yes	—
214	C₂×C₄	C₂²×D₄×C₁₁	1	4	yes	—
215	C₂×C₄	C₂×C₁₁×#41	1	8	yes	—
216	C₂²	—	4	8	no	—
217	C₂²	—	4	16	no	—
218	C₂²	—	4	32	no	—
219	C₂²	—	8	32	no	—
220	C₂²	C₂×C₁₁×#124	1	4	yes	—
221	C₂²	C₁₁×#535	1	8	yes	—
222	C₂²	C₁₁×#540	1	16	yes	—
223	C₂²	C₂³×D₄×C₁₁	1	1	yes	characteristic
224	C₂²	C₂²×C₁₁×#41	1	2	yes	—
225	C₂²	C₂×C₁₁×H(5,𝔽₂)	1	4	yes	—
226	C₄	—	2	8	no	—
227	C₄	—	4	8	no	—
228	C₄	—	4	8	no	—
229	C₄	C₂×C₁₁×#124	1	8	yes	—
230	C₂	—	8	8	no	—
231	C₂	C₁₁×#2641	1	8	yes	direct product factor
232	C₂	C₂²×C₁₁×#124	1	1	yes	characteristic
233	C₂	C₂×C₁₁×#535	1	2	yes	—
234	C₂	C₂×C₁₁×#540	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₁
prime radical	C₂²×C₁₁×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₁×#125	C₂	semidirect
3. C₁₁×#2641	C₂	direct (N × H)
4. C₂×C₁₁×#394	C₂	semidirect
5. C₂×C₁₁×#390	C₂	semidirect
6. C₂×C₁₁×#376	C₂	semidirect
12. C₂×C₁₁×#125	C₂²	semidirect
18. C₁₁×#394	C₂²	semidirect
19. C₁₁×#390	C₂²	semidirect
20. C₁₁×#376	C₂²	semidirect
46. C₂×#2641	C₁₁	direct (N × H)
76. C₂²×#125	C₂×C₁₁	semidirect
77. #2641	C₂×C₁₁	direct (N × H)
78. C₂×#394	C₂×C₁₁	semidirect
79. C₂×#390	C₂×C₁₁	semidirect
80. C₂×#376	C₂×C₁₁	semidirect
110. C₂×#125	C₂²×C₁₁	semidirect
116. #394	C₂²×C₁₁	semidirect
117. #390	C₂²×C₁₁	semidirect
118. #376	C₂²×C₁₁	semidirect
159. C₂×C₁₁	#2641	direct (N × H)
191. C₁₁	C₂×#2641	direct (N × H)
231. C₂	C₁₁×#2641	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₁×#127	2
C₂²×C₁₁×#125	2
C₁₁×#2641	2
C₂×C₁₁×#394	2
C₂×C₁₁×#390	2
C₂×C₁₁×#376	2
C₂×#2641	11

Subgroups — C2×C11×#2641 #14611813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#2641 #14611813