Subgroups — C₂×C₁₁×#1722 #14596808

Order 2816.

914 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#500	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₁×#492	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁×#486	C₂	1	2	yes	maximal
5	C₁₁×#1722	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₁×Q₁₆	—	2	2	no	—
8	C₂²×C₁₁×SD₁₆	—	2	2	no	—
9	C₂×C₁₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₁₁×#127	C₂²	1	1	yes	characteristic
12	C₁₁×#500	—	2	4	no	—
13	C₁₁×#500	C₂²	1	8	yes	complemented
14	C₁₁×#494	—	2	2	no	—
15	C₁₁×#494	C₂²	1	4	yes	complemented
16	C₁₁×#492	—	2	2	no	—
17	C₁₁×#492	C₂²	1	4	yes	—
18	C₁₁×#486	—	2	4	no	—
19	C₁₁×#486	C₂²	1	8	yes	—
20	C₂×C₁₁×#107	—	2	4	no	—
21	C₂×C₁₁×#102	—	2	4	no	—
22	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
24	C₁₁×#322	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₁₁	—	2	1	no	—
26	C₂²×Q₈×C₁₁	—	2	2	no	—
27	C₂²×D₄×C₁₁	—	2	2	no	—
28	C₂²×D₄×C₁₁	—	2	1	no	—
29	C₂×C₁₁×Q₁₆	—	2	8	no	—
30	C₂×C₁₁×Q₁₆	—	4	8	no	—
31	C₂×C₁₁×SD₁₆	—	2	8	no	—
32	C₂×C₁₁×SD₁₆	—	4	8	no	—
33	C₂²×C₈×C₁₁	D₄	1	4	yes	—
34	C₂²×C₈×C₁₁	D₄	1	2	yes	—
35	C₁₁×C₄⋊Q₈	—	2	2	no	—
36	C₁₁×C₄⋊Q₈	—	2	2	no	—
37	C₁₁×C₄⋊Q₈	—	2	2	no	—
38	C₁₁×C₄⋊Q₈	—	2	1	no	—
39	C₁₁×C₄⋊Q₈	—	2	2	no	—
40	C₁₁×C₄⋊Q₈	C₂³	1	2	yes	—
41	C₁₁×C₄⋊Q₈	C₂³	1	4	yes	—
42	C₁₁×#127	—	2	1	no	—
43	C₁₁×#127	—	2	2	no	—
44	C₁₁×#127	C₂³	1	2	yes	—
45	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
49	C₁₁×#107	—	2	16	no	—
50	C₁₁×#102	—	2	16	no	—
51	C₄×C₈×C₁₁	D₄	1	2	yes	—
52	C₄×C₈×C₁₁	D₄	1	4	yes	—
53	C₄×C₈×C₁₁	C₂³	1	2	yes	—
54	C₄×C₈×C₁₁	C₂³	1	4	yes	—
55	C₂×#1722	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₁	—	4	1	no	—
57	C₂×Q₈×C₁₁	—	2	2	no	—
58	C₂×Q₈×C₁₁	—	2	4	no	—
59	C₂×Q₈×C₁₁	—	4	1	no	—
60	C₂×Q₈×C₁₁	—	4	2	no	—
61	C₂×Q₈×C₁₁	—	4	4	no	—
62	C₂×Q₈×C₁₁	—	4	8	no	—
63	C₂×D₄×C₁₁	—	2	4	no	—
64	C₂×D₄×C₁₁	—	2	2	no	—
65	C₂×D₄×C₁₁	—	4	1	no	—
66	C₂×D₄×C₁₁	—	4	2	no	—
67	C₂×D₄×C₁₁	—	4	4	no	—
68	C₂×D₄×C₁₁	—	4	8	no	—
69	C₂²×C₄×C₁₁	—	4	2	no	—
70	C₂²×C₄×C₁₁	—	4	1	no	—
71	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
73	C₁₁×Q₁₆	—	4	16	no	—
74	C₁₁×SD₁₆	—	4	16	no	—
75	C₂×C₈×C₁₁	—	2	8	no	—
76	C₂×C₈×C₁₁	D₈	1	8	yes	—
77	C₂×C₈×C₁₁	C₂×D₄	1	8	yes	—
78	C₂×C₈×C₁₁	C₂×D₄	1	4	yes	—
79	C₁₁×C₄⋊C₄	—	2	4	no	—
80	C₁₁×C₄⋊C₄	—	2	4	no	—
81	C₁₁×C₄⋊C₄	—	2	4	no	—
82	C₁₁×C₄⋊C₄	—	4	2	no	—
83	C₁₁×C₄⋊C₄	—	4	2	no	—
84	C₁₁×C₄⋊C₄	—	4	2	no	—
85	C₄²×C₁₁	C₂×D₄	1	2	yes	—
86	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
88	C₂×#500	C₂×C₁₁	1	2	yes	complemented
89	C₂×#494	C₂×C₁₁	1	1	yes	characteristic, complemented
90	C₂×#492	C₂×C₁₁	1	1	yes	characteristic
91	C₂×#486	C₂×C₁₁	1	2	yes	—
92	#1722	C₂×C₁₁	1	8	yes	direct product factor
93	C₂×#322	C₂×C₁₁	1	1	yes	characteristic, complemented
94	C₂³×C₁₁	—	4	2	no	—
95	C₂³×C₁₁	—	8	2	no	—
96	C₂³×C₁₁	—	8	4	no	—
97	C₂³×C₁₁	#127	1	1	yes	characteristic
98	Q₈×C₁₁	—	4	4	no	—
99	Q₈×C₁₁	—	4	8	no	—
100	Q₈×C₁₁	—	8	2	no	—
101	Q₈×C₁₁	—	8	4	no	—
102	D₄×C₁₁	—	4	4	no	—
103	D₄×C₁₁	—	4	8	no	—
104	D₄×C₁₁	—	8	2	no	—
105	D₄×C₁₁	—	8	4	no	—
106	C₂×C₄×C₁₁	—	4	4	no	—
107	C₂×C₄×C₁₁	—	4	2	no	—
108	C₂×C₄×C₁₁	—	8	2	no	—
109	C₂×C₄×C₁₁	—	8	4	no	—
110	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
111	C₂×C₄×C₁₁	#127	1	2	yes	—
112	C₂×C₄×C₁₁	C₂×D₈	1	4	yes	—
113	C₂×C₄×C₁₁	#137	1	8	yes	—
114	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂²×D₄	1	2	yes	—
116	C₈×C₁₁	—	2	8	no	—
117	C₈×C₁₁	C₂×D₈	1	8	yes	—
118	C₂²×Q₁₆	—	2	2	no	—
119	C₂²×SD₁₆	—	2	2	no	—
120	C₂×C₄⋊Q₈	C₂²×C₁₁	1	2	yes	—
121	C₂×C₄⋊Q₈	C₂²×C₁₁	1	1	yes	characteristic
122	C₂×#127	C₂²×C₁₁	1	1	yes	characteristic
123	#500	—	2	4	no	—
124	#500	C₂²×C₁₁	1	8	yes	complemented
125	#494	—	2	2	no	—
126	#494	C₂²×C₁₁	1	4	yes	complemented
127	#492	—	2	2	no	—
128	#492	C₂²×C₁₁	1	4	yes	—
129	#486	—	2	4	no	—
130	#486	C₂²×C₁₁	1	8	yes	—
131	C₂×#107	—	2	4	no	—
132	C₂×#102	—	2	4	no	—
133	C₂×C₄×C₈	C₂²×C₁₁	1	2	yes	—
134	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
135	#322	C₂²×C₁₁	1	4	yes	complemented
136	C₂²×C₁₁	—	8	2	no	—
137	C₂²×C₁₁	—	8	4	no	—
138	C₂²×C₁₁	—	16	4	no	—
139	C₂²×C₁₁	#493	1	2	yes	—
140	C₂²×C₁₁	#497	1	4	yes	—
141	C₂²×C₁₁	C₂×#127	1	1	yes	characteristic
142	C₄×C₁₁	—	8	2	no	—
143	C₄×C₁₁	—	8	4	no	—
144	C₄×C₁₁	#493	1	2	yes	—
145	C₄×C₁₁	#497	1	4	yes	—
146	C₄×C₁₁	C₂²×D₈	1	2	yes	—
147	C₄×C₁₁	C₂×#137	1	4	yes	—
148	C₂²×Q₈	—	2	1	no	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×D₄	—	2	1	no	—
151	C₂²×D₄	—	2	2	no	—
152	C₂×Q₁₆	—	2	8	no	—
153	C₂×Q₁₆	—	4	8	no	—
154	C₂×SD₁₆	—	2	8	no	—
155	C₂×SD₁₆	—	4	8	no	—
156	C₂²×C₈	D₄×C₁₁	1	4	yes	—
157	C₂²×C₈	D₄×C₁₁	1	2	yes	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	1	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₂³×C₁₁	1	4	yes	—
164	C₄⋊Q₈	C₂³×C₁₁	1	2	yes	—
165	#127	—	2	2	no	—
166	#127	—	2	1	no	—
167	#127	C₂³×C₁₁	1	2	yes	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
172	#107	—	2	16	no	—
173	#102	—	2	16	no	—
174	C₄×C₈	D₄×C₁₁	1	4	yes	—
175	C₄×C₈	D₄×C₁₁	1	2	yes	—
176	C₄×C₈	C₂³×C₁₁	1	4	yes	—
177	C₄×C₈	C₂³×C₁₁	1	2	yes	—
178	C₂×C₁₁	—	16	2	no	—
179	C₂×C₁₁	#1722	1	4	yes	direct product factor
180	C₂×C₁₁	C₂×#493	1	1	yes	characteristic
181	C₂×C₁₁	C₂×#497	1	2	yes	—
182	C₂⁴	—	4	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	1	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×Q₈	—	4	8	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	4	no	—
194	C₂×D₄	—	4	8	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
199	Q₁₆	—	4	16	no	—
200	SD₁₆	—	4	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	C₁₁×D₈	1	8	yes	—
203	C₂×C₈	C₂×D₄×C₁₁	1	8	yes	—
204	C₂×C₈	C₂×D₄×C₁₁	1	4	yes	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄²	C₂×D₄×C₁₁	1	2	yes	—
212	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
213	C₄²	C₂⁴×C₁₁	1	1	yes	characteristic
214	C₁₁	C₂×#1722	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	4	2	no	—
216	C₂³	—	8	2	no	—
217	C₂³	—	8	4	no	—
218	C₂³	C₁₁×#127	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	Q₈	—	4	8	no	—
221	Q₈	—	8	2	no	—
222	Q₈	—	8	4	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	8	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	C₁₁×#127	1	1	yes	characteristic
232	C₂×C₄	C₁₁×#127	1	2	yes	—
233	C₂×C₄	C₂×C₁₁×D₈	1	4	yes	—
234	C₂×C₄	C₁₁×#137	1	8	yes	—
235	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₁₁	1	2	yes	—
237	C₈	—	2	8	no	—
238	C₈	C₂×C₁₁×D₈	1	8	yes	—
239	C₂²	—	8	4	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	16	4	no	—
242	C₂²	C₁₁×#493	1	2	yes	—
243	C₂²	C₁₁×#497	1	4	yes	—
244	C₂²	C₂×C₁₁×#127	1	1	yes	characteristic
245	C₄	—	8	2	no	—
246	C₄	—	8	4	no	—
247	C₄	C₁₁×#493	1	2	yes	—
248	C₄	C₁₁×#497	1	4	yes	—
249	C₄	C₂²×C₁₁×D₈	1	2	yes	—
250	C₄	C₂×C₁₁×#137	1	4	yes	—
251	C₂	—	16	2	no	—
252	C₂	C₁₁×#1722	1	4	yes	direct product factor
253	C₂	C₂×C₁₁×#493	1	1	yes	characteristic
254	C₂	C₂×C₁₁×#497	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₁
prime radical	C₂×C₁₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₂×C₄²×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₁×#500	C₂	semidirect
2. C₂×C₁₁×#494	C₂	semidirect
5. C₁₁×#1722	C₂	direct (N × H)
6. C₂×C₁₁×#322	C₂	semidirect
13. C₁₁×#500	C₂²	semidirect
15. C₁₁×#494	C₂²	semidirect
24. C₁₁×#322	C₂²	semidirect
55. C₂×#1722	C₁₁	direct (N × H)
88. C₂×#500	C₂×C₁₁	semidirect
89. C₂×#494	C₂×C₁₁	semidirect
92. #1722	C₂×C₁₁	direct (N × H)
93. C₂×#322	C₂×C₁₁	semidirect
124. #500	C₂²×C₁₁	semidirect
126. #494	C₂²×C₁₁	semidirect
135. #322	C₂²×C₁₁	semidirect
179. C₂×C₁₁	#1722	direct (N × H)
214. C₁₁	C₂×#1722	direct (N × H)
252. C₂	C₁₁×#1722	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#500	2
C₂×C₁₁×#494	2
C₂×C₁₁×#492	2
C₂×C₁₁×#486	2
C₁₁×#1722	2
C₂×C₁₁×#322	2
C₂×#1722	11

Subgroups — C2×C11×#1722 #14596808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#1722 #14596808