Subgroups — C₁₁×#11374 #14588567

Order 2816.

598 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#430	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#1733	C₂	1	2	yes	maximal
3	C₂×C₁₁×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1729	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#1729	C₂	1	2	yes	maximal
6	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×#117	C₂²	1	2	yes	—
11	C₁₁×#430	—	2	2	no	—
12	C₁₁×#430	C₄	1	4	yes	complemented
13	C₂×C₁₁×#108	—	2	2	no	—
14	C₁₁×#339	—	2	4	no	—
15	C₁₁×H(3,ℤ/4)	—	2	2	no	—
16	C₁₁×#336	—	2	2	no	—
17	C₁₁×#336	—	2	4	no	—
18	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
19	C₂×C₁₁×M₁₆	—	2	2	no	—
20	C₂×C₁₁×M₁₆	—	2	2	no	—
21	C₂×C₁₁×M₁₆	—	2	2	no	—
22	C₂×C₁₁×M₁₆	—	2	2	no	—
23	C₂×C₁₁×M₁₆	—	2	2	no	—
24	C₂×C₁₁×M₁₆	—	2	2	no	—
25	C₂×C₁₁×M₁₆	—	2	2	no	—
26	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
29	C₂²×C₈×C₁₁	—	2	1	no	—
30	C₂²×C₈×C₁₁	—	2	2	no	—
31	C₂²×C₈×C₁₁	—	2	2	no	—
32	C₂²×C₈×C₁₁	—	2	2	no	—
33	C₂²×C₈×C₁₁	—	2	1	no	—
34	C₂²×C₈×C₁₁	D₄	1	2	yes	—
35	C₂²×C₈×C₁₁	Q₈	1	2	yes	—
36	C₁₁×#117	—	2	2	no	—
37	C₁₁×#117	—	2	2	no	—
38	C₁₁×#117	—	2	2	no	—
39	C₁₁×#117	—	2	2	no	—
40	C₁₁×#117	—	2	2	no	—
41	C₁₁×#117	—	2	2	no	—
42	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₁×C₄⋊C₄	Q₈	1	2	yes	—
44	C₂×C₁₁×C₂²⋊C₄	—	2	1	no	—
45	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₄²×C₁₁	—	2	1	no	—
47	C₂×C₄²×C₁₁	D₄	1	4	yes	—
48	C₁₁×#108	—	2	8	no	—
49	#11374	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₁	#41	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	—	2	2	no	—
52	C₂²×C₄×C₁₁	—	2	4	no	—
53	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	#41	1	2	yes	—
60	C₂²×C₄×C₁₁	#41	1	2	yes	—
61	C₂²×C₄×C₁₁	#41	1	2	yes	—
62	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
63	C₁₁×M₁₆	—	2	4	no	—
64	C₁₁×M₁₆	—	2	4	no	—
65	C₁₁×M₁₆	—	2	4	no	—
66	C₁₁×M₁₆	—	2	4	no	—
67	C₁₁×M₁₆	—	4	2	no	—
68	C₁₁×M₁₆	—	4	2	no	—
69	C₂×C₈×C₁₁	—	2	4	no	—
70	C₂×C₈×C₁₁	—	2	4	no	—
71	C₂×C₈×C₁₁	—	2	2	no	—
72	C₂×C₈×C₁₁	—	2	2	no	—
73	C₂×C₈×C₁₁	—	2	2	no	—
74	C₂×C₈×C₁₁	—	4	2	no	—
75	C₂×C₈×C₁₁	—	4	1	no	—
76	C₂×C₈×C₁₁	—	4	2	no	—
77	C₂×C₈×C₁₁	—	4	1	no	—
78	C₂×C₈×C₁₁	—	4	2	no	—
79	C₂×C₈×C₁₁	C₂²⋊C₄	1	4	yes	—
80	C₁₁×C₄⋊C₄	—	2	4	no	—
81	C₁₁×C₄⋊C₄	—	4	2	no	—
82	C₁₁×C₄⋊C₄	—	4	2	no	—
83	C₁₁×C₂²⋊C₄	—	4	2	no	—
84	C₁₁×C₂²⋊C₄	—	4	2	no	—
85	C₁₁×C₂²⋊C₄	—	4	2	no	—
86	C₄²×C₁₁	—	2	4	no	—
87	C₄²×C₁₁	—	2	4	no	—
88	C₄²×C₁₁	—	4	1	no	—
89	C₄²×C₁₁	—	4	1	no	—
90	C₂×#430	C₂×C₁₁	1	1	yes	characteristic
91	#1733	C₂×C₁₁	1	2	yes	—
92	C₂×H(3,ℤ/4)	C₂×C₁₁	1	1	yes	characteristic
93	#1729	C₂×C₁₁	1	1	yes	characteristic
94	#1729	C₂×C₁₁	1	2	yes	—
95	C₂³×C₁₁	—	4	2	no	—
96	C₂³×C₁₁	C₄×Q₈	1	1	yes	characteristic
97	C₂³×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₁₁	#123	1	2	yes	—
99	C₂³×C₁₁	#124	1	1	yes	characteristic
100	C₂³×C₁₁	#126	1	2	yes	—
101	C₂×C₄×C₁₁	—	4	8	no	—
102	C₂×C₄×C₁₁	—	4	4	no	—
103	C₂×C₄×C₁₁	—	4	4	no	—
104	C₂×C₄×C₁₁	—	4	2	no	—
105	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₁	#117	1	2	yes	—
107	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₁	C₄×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₁	#121	1	2	yes	—
110	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₁	#123	1	2	yes	—
114	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
115	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
116	C₂×C₄×C₁₁	#123	1	2	yes	—
117	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₁	#125	1	2	yes	—
119	C₂×C₄×C₁₁	#125	1	2	yes	—
120	C₂×C₄×C₁₁	#126	1	2	yes	—
121	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
123	C₈×C₁₁	—	2	4	no	—
124	C₈×C₁₁	—	4	4	no	—
125	C₈×C₁₁	—	4	2	no	—
126	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
127	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
128	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
129	C₂×#117	C₂²×C₁₁	1	1	yes	characteristic
130	C₂×#117	C₂²×C₁₁	1	2	yes	—
131	#430	—	2	2	no	—
132	#430	C₄×C₁₁	1	4	yes	complemented
133	C₂×#108	—	2	2	no	—
134	#339	—	2	4	no	—
135	H(3,ℤ/4)	—	2	2	no	—
136	#336	—	2	2	no	—
137	#336	—	2	4	no	—
138	C₂²×C₁₁	—	2	2	no	—
139	C₂²×C₁₁	—	2	4	no	—
140	C₂²×C₁₁	—	4	1	no	—
141	C₂²×C₁₁	—	4	1	no	—
142	C₂²×C₁₁	—	4	2	no	—
143	C₂²×C₁₁	#387	1	2	yes	—
144	C₂²×C₁₁	#391	1	1	yes	characteristic
145	C₂²×C₁₁	#397	1	2	yes	—
146	C₂²×C₁₁	#399	1	1	yes	characteristic
147	C₂²×C₁₁	#400	1	1	yes	characteristic
148	C₄×C₁₁	—	8	4	no	—
149	C₄×C₁₁	—	8	2	no	—
150	C₄×C₁₁	#381	1	1	yes	characteristic
151	C₄×C₁₁	#387	1	2	yes	—
152	C₄×C₁₁	#390	1	1	yes	characteristic
153	C₄×C₁₁	#398	1	2	yes	—
154	C₄×C₁₁	#399	1	1	yes	characteristic
155	C₄×C₁₁	#400	1	1	yes	characteristic
156	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	C₂×C₄×C₁₁	1	4	yes	—
165	C₂×M₁₆	C₂×C₄×C₁₁	1	2	yes	—
166	C₂×M₁₆	D₄×C₁₁	1	2	yes	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	D₄×C₁₁	1	2	yes	—
173	C₂²×C₈	Q₈×C₁₁	1	2	yes	—
174	#117	—	2	2	no	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	Q₈×C₁₁	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	1	no	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	D₄×C₁₁	1	4	yes	—
186	#108	—	2	8	no	—
187	C₂×C₁₁	—	2	2	no	—
188	C₂×C₁₁	—	2	2	no	—
189	C₂×C₁₁	—	2	1	no	—
190	C₂×C₁₁	—	2	1	no	—
191	C₂×C₁₁	#1445	1	1	yes	characteristic
192	C₂×C₁₁	#2098	1	2	yes	—
193	C₂⁴	C₁₁×#41	1	1	yes	characteristic
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
200	C₂²×C₄	C₂×Q₈×C₁₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×Q₈×C₁₁	1	1	yes	characteristic
202	C₂²×C₄	C₁₁×#41	1	2	yes	—
203	C₂²×C₄	C₁₁×#41	1	2	yes	—
204	C₂²×C₄	C₁₁×#41	1	2	yes	—
205	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	C₁₁×C₂²⋊C₄	1	4	yes	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₄²	—	2	4	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	4	1	no	—
232	C₄²	—	4	1	no	—
233	C₁₁	#11374	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³	—	4	2	no	—
235	C₂³	C₄×Q₈×C₁₁	1	1	yes	characteristic
236	C₂³	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
237	C₂³	C₁₁×#123	1	2	yes	—
238	C₂³	C₁₁×#124	1	1	yes	characteristic
239	C₂³	C₁₁×#126	1	2	yes	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₁₁×#117	1	2	yes	—
246	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
247	C₂×C₄	C₄×Q₈×C₁₁	1	1	yes	characteristic
248	C₂×C₄	C₁₁×#121	1	2	yes	—
249	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
250	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
251	C₂×C₄	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₄	C₁₁×#123	1	2	yes	—
253	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
254	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
255	C₂×C₄	C₁₁×#123	1	2	yes	—
256	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
257	C₂×C₄	C₁₁×#125	1	2	yes	—
258	C₂×C₄	C₁₁×#125	1	2	yes	—
259	C₂×C₄	C₁₁×#126	1	2	yes	—
260	C₂×C₄	C₁₁×#127	1	1	yes	characteristic
261	C₂×C₄	C₁₁×C₄⋊Q₈	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	4	4	no	—
264	C₈	—	4	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	2	no	—
270	C₂²	C₁₁×#387	1	2	yes	—
271	C₂²	C₁₁×#391	1	1	yes	characteristic
272	C₂²	C₁₁×#397	1	2	yes	—
273	C₂²	C₁₁×#399	1	1	yes	characteristic
274	C₂²	C₁₁×#400	1	1	yes	characteristic
275	C₄	—	8	4	no	—
276	C₄	—	8	2	no	—
277	C₄	C₁₁×#381	1	1	yes	characteristic
278	C₄	C₁₁×#387	1	2	yes	—
279	C₄	C₁₁×#390	1	1	yes	characteristic
280	C₄	C₁₁×#398	1	2	yes	—
281	C₄	C₁₁×#399	1	1	yes	characteristic
282	C₄	C₁₁×#400	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	C₁₁×#1445	1	1	yes	characteristic
288	C₂	C₁₁×#2098	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₁	#127
commutator	C₂²×C₄	C₂²×C₄×C₁₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁
socle	C₂²×C₁₁	#399
prime radical	C₂⁴×C₁₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁ ◁ C₂³×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₁×#430	C₄	semidirect
49. #11374	C₁₁	direct (N × H)
132. #430	C₄×C₁₁	semidirect
233. C₁₁	#11374	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×#430	2
C₁₁×#1733	2
C₂×C₁₁×H(3,ℤ/4)	2
C₁₁×#1729	2
#11374	11

Subgroups — C11×#11374 #14588567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#11374 #14588567