Subgroups — C₂×C₁₁×#1694 #14596341

Order 2816.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 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	1	yes	maximal, characteristic
2	C₂×C₁₁×#462	C₂	1	2	yes	maximal
3	C₂×C₁₁×#446	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1694	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₁×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#332	C₂	1	2	yes	maximal
7	C₂²×C₁₁×Q₁₆	—	2	2	no	—
8	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₁₁×#500	—	2	2	no	—
11	C₁₁×#500	C₂²	1	4	yes	—
12	C₁₁×#462	—	2	4	no	—
13	C₁₁×#462	C₂²	1	8	yes	—
14	C₁₁×#446	—	2	2	no	—
15	C₁₁×#446	C₂²	1	4	yes	—
16	C₂×C₁₁×#107	—	2	1	no	—
17	C₂×C₁₁×#107	—	2	1	no	—
18	C₂×C₁₁×C₄⋊C₈	—	2	1	no	—
19	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₁×#103	—	2	2	no	—
22	C₂×C₁₁×#97	—	2	1	no	—
23	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
24	C₁₁×#334	C₂²	1	4	yes	—
25	C₁₁×#332	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₁	—	2	2	no	—
27	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
28	C₂×C₁₁×Q₁₆	—	2	8	no	—
29	C₂×C₁₁×Q₁₆	—	4	8	no	—
30	C₂²×C₈×C₁₁	—	2	1	no	—
31	C₂²×C₈×C₁₁	—	2	2	no	—
32	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
34	C₁₁×C₄⋊Q₈	—	2	2	no	—
35	C₁₁×C₄⋊Q₈	—	2	2	no	—
36	C₁₁×C₄⋊Q₈	—	2	2	no	—
37	C₁₁×C₄⋊Q₈	C₂³	1	4	yes	—
38	C₄×Q₈×C₁₁	—	2	1	no	—
39	C₄×Q₈×C₁₁	—	2	1	no	—
40	C₄×Q₈×C₁₁	D₄	1	2	yes	—
41	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
42	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₁	—	2	1	no	—
47	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
48	C₁₁×#107	—	2	4	no	—
49	C₁₁×#107	—	2	4	no	—
50	C₁₁×C₄⋊C₈	—	2	4	no	—
51	C₁₁×C₄⋊C₈	—	2	1	no	—
52	C₁₁×C₄⋊C₈	—	2	2	no	—
53	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
54	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
55	C₁₁×#103	—	2	8	no	—
56	C₁₁×#97	—	2	4	no	—
57	C₄×C₈×C₁₁	D₄	1	2	yes	—
58	C₄×C₈×C₁₁	C₂³	1	2	yes	—
59	C₂×#1694	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×Q₈×C₁₁	—	2	1	no	—
61	C₂×Q₈×C₁₁	—	2	4	no	—
62	C₂×Q₈×C₁₁	—	2	4	no	—
63	C₂×Q₈×C₁₁	—	4	2	no	—
64	C₂×Q₈×C₁₁	—	4	4	no	—
65	C₂×Q₈×C₁₁	—	4	4	no	—
66	C₂×Q₈×C₁₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₁	—	2	1	no	—
68	C₂²×C₄×C₁₁	—	2	1	no	—
69	C₂²×C₄×C₁₁	—	4	2	no	—
70	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
73	C₁₁×Q₁₆	—	4	16	no	—
74	C₂×C₈×C₁₁	—	2	2	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	—	2	4	no	—
77	C₂×C₈×C₁₁	—	4	4	no	—
78	C₂×C₈×C₁₁	—	4	2	no	—
79	C₂×C₈×C₁₁	Q₁₆	1	4	yes	complemented
80	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
82	C₁₁×C₄⋊C₄	—	2	2	no	—
83	C₁₁×C₄⋊C₄	—	2	1	no	—
84	C₁₁×C₄⋊C₄	—	2	2	no	—
85	C₁₁×C₄⋊C₄	—	2	4	no	—
86	C₁₁×C₄⋊C₄	—	2	4	no	—
87	C₁₁×C₄⋊C₄	—	4	2	no	—
88	C₁₁×C₄⋊C₄	—	4	2	no	—
89	C₁₁×C₄⋊C₄	C₂×D₄	1	2	yes	—
90	C₄²×C₁₁	—	2	2	no	—
91	C₄²×C₁₁	—	2	2	no	—
92	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₁	#41	1	1	yes	characteristic
95	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
96	C₂×#500	C₂×C₁₁	1	1	yes	characteristic
97	C₂×#462	C₂×C₁₁	1	2	yes	—
98	C₂×#446	C₂×C₁₁	1	1	yes	characteristic
99	#1694	C₂×C₁₁	1	8	yes	direct product factor
100	C₂×#334	C₂×C₁₁	1	1	yes	characteristic
101	C₂×#332	C₂×C₁₁	1	2	yes	—
102	C₂³×C₁₁	#121	1	1	yes	characteristic
103	Q₈×C₁₁	—	2	4	no	—
104	Q₈×C₁₁	—	4	2	no	—
105	Q₈×C₁₁	—	4	4	no	—
106	Q₈×C₁₁	—	4	4	no	—
107	Q₈×C₁₁	—	8	4	no	—
108	C₂×C₄×C₁₁	—	2	2	no	—
109	C₂×C₄×C₁₁	—	2	2	no	—
110	C₂×C₄×C₁₁	—	2	2	no	—
111	C₂×C₄×C₁₁	—	4	2	no	—
112	C₂×C₄×C₁₁	—	4	2	no	—
113	C₂×C₄×C₁₁	—	4	4	no	—
114	C₂×C₄×C₁₁	—	8	4	no	—
115	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	4	yes	—
120	C₂×C₄×C₁₁	#137	1	2	yes	—
121	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
124	C₈×C₁₁	—	2	2	no	—
125	C₈×C₁₁	—	4	4	no	—
126	C₈×C₁₁	—	4	2	no	—
127	C₈×C₁₁	C₂×Q₁₆	1	4	yes	complemented
128	C₂²×Q₁₆	—	2	2	no	—
129	C₂×C₄⋊Q₈	C₂²×C₁₁	1	2	yes	—
130	C₂×C₄×Q₈	C₂²×C₁₁	1	1	yes	characteristic
131	#500	—	2	2	no	—
132	#500	C₂²×C₁₁	1	4	yes	—
133	#462	—	2	4	no	—
134	#462	C₂²×C₁₁	1	8	yes	—
135	#446	—	2	2	no	—
136	#446	C₂²×C₁₁	1	4	yes	—
137	C₂×#107	—	2	1	no	—
138	C₂×#107	—	2	1	no	—
139	C₂×C₄⋊C₈	—	2	1	no	—
140	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
141	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
142	C₂×#103	—	2	2	no	—
143	C₂×#97	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
145	#334	C₂²×C₁₁	1	4	yes	—
146	#332	C₂²×C₁₁	1	8	yes	—
147	C₂²×C₁₁	#462	1	2	yes	—
148	C₂²×C₁₁	#469	1	2	yes	—
149	C₂²×C₁₁	#473	1	2	yes	—
150	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
151	C₄×C₁₁	—	2	1	no	—
152	C₄×C₁₁	—	2	1	no	—
153	C₄×C₁₁	—	4	2	no	—
154	C₄×C₁₁	—	4	2	no	—
155	C₄×C₁₁	—	8	4	no	—
156	C₄×C₁₁	#462	1	2	yes	—
157	C₄×C₁₁	#469	1	2	yes	—
158	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
159	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
160	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
161	C₂²×Q₈	—	2	2	no	—
162	C₂²×Q₈	D₄×C₁₁	1	1	yes	characteristic
163	C₂×Q₁₆	—	2	8	no	—
164	C₂×Q₁₆	—	4	8	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	C₂³×C₁₁	1	4	yes	—
173	C₄×Q₈	—	2	1	no	—
174	C₄×Q₈	—	2	1	no	—
175	C₄×Q₈	D₄×C₁₁	1	2	yes	—
176	C₄×Q₈	C₂³×C₁₁	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
181	C₂×C₄²	—	2	1	no	—
182	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
183	#107	—	2	4	no	—
184	#107	—	2	4	no	—
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	C₂³×C₁₁	1	2	yes	—
189	C₄⋊C₈	C₂³×C₁₁	1	4	yes	—
190	#103	—	2	8	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	D₄×C₁₁	1	2	yes	—
193	C₄×C₈	C₂³×C₁₁	1	2	yes	—
194	C₂×C₁₁	#1694	1	4	yes	direct product factor
195	C₂×C₁₁	C₂×#462	1	1	yes	characteristic
196	C₂×C₁₁	C₂×#469	1	1	yes	characteristic
197	C₂×C₁₁	C₂×#473	1	1	yes	characteristic
198	C₂×Q₈	—	2	1	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	C₂×D₄×C₁₁	1	2	yes	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
210	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
211	Q₁₆	—	4	16	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	C₁₁×Q₁₆	1	4	yes	complemented
218	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
219	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	C₂×D₄×C₁₁	1	2	yes	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
231	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
232	C₄²	C₁₁×#41	1	1	yes	characteristic
233	C₄²	C₂⁴×C₁₁	1	1	yes	characteristic
234	C₁₁	C₂×#1694	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	C₁₁×#121	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	4	4	no	—
239	Q₈	—	4	4	no	—
240	Q₈	—	8	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
249	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
250	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
251	C₂×C₄	C₂×C₁₁×Q₁₆	1	2	yes	—
252	C₂×C₄	C₁₁×AGL(1,ℤ/8)	1	4	yes	—
253	C₂×C₄	C₁₁×#137	1	2	yes	—
254	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	4	no	—
259	C₈	—	4	2	no	—
260	C₈	C₂×C₁₁×Q₁₆	1	4	yes	complemented
261	C₂²	C₁₁×#462	1	2	yes	—
262	C₂²	C₁₁×#469	1	2	yes	—
263	C₂²	C₁₁×#473	1	2	yes	—
264	C₂²	C₂×C₁₁×#121	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	4	no	—
270	C₄	C₁₁×#462	1	2	yes	—
271	C₄	C₁₁×#469	1	2	yes	—
272	C₄	C₂²×C₁₁×Q₁₆	1	1	yes	characteristic
273	C₄	C₂×C₁₁×AGL(1,ℤ/8)	1	2	yes	—
274	C₄	C₂×C₁₁×#137	1	1	yes	characteristic
275	C₂	C₁₁×#1694	1	4	yes	direct product factor
276	C₂	C₂×C₁₁×#462	1	1	yes	characteristic
277	C₂	C₂×C₁₁×#469	1	1	yes	characteristic
278	C₂	C₂×C₁₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₁
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
4. C₁₁×#1694	C₂	direct (N × H)
59. C₂×#1694	C₁₁	direct (N × H)
79. C₂×C₈×C₁₁	Q₁₆	semidirect
99. #1694	C₂×C₁₁	direct (N × H)
127. C₈×C₁₁	C₂×Q₁₆	semidirect
194. C₂×C₁₁	#1694	direct (N × H)
217. C₂×C₈	C₁₁×Q₁₆	semidirect
234. C₁₁	C₂×#1694	direct (N × H)
260. C₈	C₂×C₁₁×Q₁₆	semidirect
275. C₂	C₁₁×#1694	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×#462	2
C₂×C₁₁×#446	2
C₁₁×#1694	2
C₂×C₁₁×#334	2
C₂×C₁₁×#332	2
C₂×#1694	11

Subgroups — C2×C11×#1694 #14596341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#1694 #14596341