Subgroups — #14670327

Order 2816.

420 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#66883	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622151	C₂	1	2	yes	maximal, complemented
3	#13068641	C₂	1	1	yes	maximal, characteristic
4	#13066715	C₂	1	2	yes	maximal, complemented
5	C₁₁×#2112	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂²×#4318	—	2	1	no	—
8	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
9	#10622825	—	2	1	no	—
10	C₂×#66883	—	2	1	no	—
11	C₂×#66883	—	2	4	no	—
12	C₂×#66883	C₂²	1	2	yes	—
13	C₂×#66881	C₂²	1	1	yes	characteristic
14	C₂×#66881	C₂²	1	1	yes	characteristic
15	C₁₁×C₂²⋊C₁₆	C₄	1	2	yes	complemented
16	C₁₁×C₂²⋊C₁₆	C₄	1	2	yes	complemented
17	#10622189	—	2	1	no	—
18	#10622151	—	2	4	no	—
19	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	—	2	2	no	—
21	C₂²×C₈×C₁₁	—	2	2	no	—
22	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₁₁	—	4	1	no	—
29	C₂×C₄⋊Dic₁₁	—	2	1	no	—
30	C₂×C₄⋊Dic₁₁	—	2	1	no	—
31	C₂×#4318	—	2	4	no	—
32	C₂×#4318	—	2	2	no	—
33	C₂×#4318	—	4	2	no	—
34	C₂×#4318	—	4	1	no	—
35	C₂²×C₁₁⋊C₈	—	4	1	no	—
36	C₁₁×M₃₂	—	2	2	no	—
37	C₁₁×M₃₂	—	2	2	no	—
38	C₂×C₁₁×C₁₆	C₂×C₄	1	4	yes	—
39	C₁₁⋊#95	—	4	1	no	—
40	#66883	—	2	4	no	—
41	#66883	—	2	2	no	—
42	#66883	D₄	1	4	yes	—
43	#66881	—	2	2	no	—
44	#66881	—	2	2	no	—
45	#12928	—	11	1	no	maximal, Sylow
46	C₂⁴×C₁₁	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₁₁	—	2	1	no	—
48	C₂²×C₄×C₁₁	—	2	1	no	—
49	C₂²×C₄×C₁₁	—	2	1	no	—
50	C₂²×C₄×C₁₁	—	2	1	no	—
51	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₁	—	4	1	no	—
58	C₂²×Dic₁₁	—	4	1	no	—
59	C₂×C₈×C₁₁	—	2	2	no	—
60	C₂×C₈×C₁₁	—	2	2	no	—
61	C₂×C₈×C₁₁	—	2	2	no	—
62	C₂×C₈×C₁₁	—	2	2	no	—
63	C₂×C₈×C₁₁	—	2	1	no	—
64	C₂×C₈×C₁₁	—	2	1	no	—
65	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₁₁	#41	1	2	yes	—
71	C₂²⋊Dic₁₁	—	8	2	no	—
72	C₄⋊Dic₁₁	—	4	1	no	—
73	C₄⋊Dic₁₁	—	4	1	no	—
74	C₄⋊Dic₁₁	—	4	1	no	—
75	C₄⋊Dic₁₁	—	4	1	no	—
76	#4318	—	2	4	no	—
77	#4318	—	4	4	no	—
78	#4318	—	4	2	no	—
79	C₂×C₁₁⋊C₈	—	4	4	no	—
80	C₂×C₁₁⋊C₈	—	4	2	no	—
81	C₁₁×C₁₆	—	2	4	no	—
82	C₂²×#108	—	11	1	no	—
83	C₂×#365	—	11	2	no	—
84	#2112	D₁₁	1	1	yes	characteristic
85	#1807	—	11	1	no	—
86	#1383	—	11	2	no	—
87	C₂³×C₁₁	—	2	2	no	—
88	C₂³×C₁₁	—	2	1	no	—
89	C₂³×C₁₁	—	2	1	no	—
90	C₂³×C₁₁	—	4	1	no	—
91	C₂³×C₁₁	#106	1	1	yes	characteristic
92	C₂³×C₁₁	#106	1	1	yes	characteristic
93	C₂³×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₁	—	2	1	no	—
95	C₂×C₄×C₁₁	—	2	1	no	—
96	C₂×C₄×C₁₁	—	2	1	no	—
97	C₂×C₄×C₁₁	—	2	1	no	—
98	C₂×C₄×C₁₁	—	2	1	no	—
99	C₂×C₄×C₁₁	—	2	1	no	—
100	C₂×C₄×C₁₁	—	2	1	no	—
101	C₂×C₄×C₁₁	—	2	1	no	—
102	C₂×C₄×C₁₁	—	2	1	no	—
103	C₂×C₄×C₁₁	—	2	1	no	—
104	C₂×C₄×C₁₁	#106	1	1	yes	characteristic
105	C₂×C₄×C₁₁	#106	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₁	#120	1	1	yes	characteristic
108	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₁₁	#137	1	1	yes	characteristic
112	C₂×Dic₁₁	—	8	1	no	—
113	C₂×Dic₁₁	—	8	1	no	—
114	C₂×Dic₁₁	—	8	1	no	—
115	C₂×Dic₁₁	—	8	1	no	—
116	C₂×Dic₁₁	—	8	1	no	—
117	C₂×Dic₁₁	—	8	1	no	—
118	C₈×C₁₁	—	2	1	no	—
119	C₈×C₁₁	—	2	1	no	—
120	C₈×C₁₁	C₄×D₄	1	2	yes	—
121	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
122	C₁₁⋊C₈	—	4	4	no	—
123	C₂²×M₁₆	—	22	1	no	—
124	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
125	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
126	C₂×#108	—	11	2	no	—
127	C₂×#108	—	22	4	no	—
128	C₂×#108	—	22	1	no	—
129	C₂×#106	—	11	1	no	—
130	C₂×#106	—	11	1	no	—
131	#380	—	22	1	no	—
132	#365	—	22	4	no	—
133	C₂²⋊C₁₆	Dic₁₁	1	2	yes	complemented
134	C₂²⋊C₁₆	Dic₁₁	1	2	yes	complemented
135	#340	—	22	1	no	—
136	C₂²×C₁₁	—	2	1	no	—
137	C₂²×C₁₁	—	2	1	no	—
138	C₂²×C₁₁	—	2	1	no	—
139	C₂²×C₁₁	—	2	1	no	—
140	C₂²×C₁₁	—	2	1	no	—
141	C₂²×C₁₁	—	2	1	no	—
142	C₂²×C₁₁	—	2	1	no	—
143	C₂²×C₁₁	—	4	2	no	—
144	C₂²×C₁₁	—	4	2	no	—
145	C₂²×C₁₁	#365	1	2	yes	complemented
146	C₂²×C₁₁	C₂×#106	1	1	yes	characteristic
147	C₂²×C₁₁	#448	1	1	yes	characteristic
148	C₂²×C₁₁	#481	1	1	yes	characteristic
149	C₄×C₁₁	—	2	1	no	—
150	C₄×C₁₁	—	2	1	no	—
151	C₄×C₁₁	#385	1	1	yes	characteristic
152	C₄×C₁₁	#428	1	1	yes	characteristic
153	C₄×C₁₁	#450	1	1	yes	characteristic
154	C₄×C₁₁	#483	1	1	yes	characteristic
155	Dic₁₁	—	16	1	no	—
156	Dic₁₁	—	16	1	no	—
157	C₂³×C₄	D₄₄	1	1	yes	characteristic
158	C₂×M₁₆	—	22	4	no	—
159	C₂×M₁₆	—	22	2	no	—
160	C₂×M₁₆	—	44	2	no	—
161	C₂×M₁₆	—	44	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	44	1	no	—
165	C₂²×C₈	Dic₂₂	1	1	yes	characteristic
166	C₂²×C₈	Dic₂₂	1	1	yes	characteristic
167	C₂²×C₈	D₄₄	1	1	yes	characteristic
168	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
169	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
171	C₂×C₄⋊C₄	—	22	1	no	—
172	C₂×C₄⋊C₄	—	22	1	no	—
173	C₂×C₂²⋊C₄	—	44	1	no	—
174	M₃₂	—	2	2	no	—
175	M₃₂	—	2	2	no	—
176	C₂×C₁₆	C₂×Dic₁₁	1	4	yes	—
177	#108	—	11	4	no	—
178	#108	—	22	4	no	—
179	#108	—	22	2	no	—
180	#106	—	22	2	no	—
181	#106	—	22	2	no	—
182	#95	—	44	1	no	—
183	C₂×C₁₁	—	2	1	no	—
184	C₂×C₁₁	—	2	1	no	—
185	C₂×C₁₁	—	2	2	no	—
186	C₂×C₁₁	—	4	1	no	—
187	C₂×C₁₁	#1853	1	1	yes	characteristic
188	C₂×C₁₁	C₂×#365	1	1	yes	characteristic
189	C₂×C₁₁	#2155	1	1	yes	characteristic
190	C₂⁴	#4314	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	44	1	no	—
196	C₂²×C₄	—	44	1	no	—
197	C₂²×C₄	#4314	1	1	yes	characteristic
198	C₂²×C₄	#4314	1	1	yes	characteristic
199	C₂²×C₄	#4314	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
201	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
203	M₁₆	—	22	4	no	—
204	M₁₆	—	44	4	no	—
205	M₁₆	—	44	2	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	44	4	no	—
213	C₂×C₈	—	44	2	no	—
214	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
215	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
216	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
217	C₂×C₈	D₄×D₁₁	1	2	yes	—
218	C₂×C₈	#4341	1	2	yes	—
219	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
220	C₄⋊C₄	—	44	1	no	—
221	C₄⋊C₄	—	44	1	no	—
222	C₄⋊C₄	—	44	1	no	—
223	C₄⋊C₄	—	44	1	no	—
224	C₂²⋊C₄	—	88	2	no	—
225	C₁₆	—	2	4	no	—
226	C₁₁	#12928	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	#66881	1	1	yes	characteristic
232	C₂³	#66881	1	1	yes	characteristic
233	C₂³	C₂×#4314	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	88	1	no	—
245	C₂×C₄	—	88	1	no	—
246	C₂×C₄	—	88	1	no	—
247	C₂×C₄	—	88	1	no	—
248	C₂×C₄	—	88	1	no	—
249	C₂×C₄	—	88	1	no	—
250	C₂×C₄	#66881	1	1	yes	characteristic
251	C₂×C₄	#66881	1	1	yes	characteristic
252	C₂×C₄	#66935	1	1	yes	characteristic
253	C₂×C₄	#66939	1	1	yes	characteristic
254	C₂×C₄	C₂×#4314	1	1	yes	characteristic
255	C₂×C₄	#66961	1	1	yes	characteristic
256	C₂×C₄	#66962	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	44	4	no	—
261	C₈	#66931	1	2	yes	—
262	C₈	D₄×Dic₁₁	1	2	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	#10622151	1	2	yes	complemented
273	C₂²	#10622348	1	1	yes	characteristic
274	C₂²	#10622351	1	1	yes	characteristic
275	C₂²	C₂×#66881	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	176	1	no	—
279	C₄	—	176	1	no	—
280	C₄	#10622334	1	1	yes	characteristic
281	C₄	#10622344	1	1	yes	characteristic
282	C₄	#10622563	1	1	yes	characteristic
283	C₄	#10622734	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	—	4	1	no	—
288	C₂	#13068811	1	1	yes	characteristic
289	C₂	C₂×#10622151	1	1	yes	characteristic
290	C₂	#13070389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#66881
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#2112	C₂
socle	C₂²×C₁₁	C₂×#106
prime radical	C₂⁴×C₁₁	SD₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#2112	D₁₁
hypocenter	C₁₁	#12928
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
2. C₂×#10622151	C₂	semidirect
4. #13066715	C₂	semidirect
15. C₁₁×C₂²⋊C₁₆	C₄	semidirect
16. C₁₁×C₂²⋊C₁₆	C₄	semidirect
133. C₂²⋊C₁₆	Dic₁₁	semidirect
134. C₂²⋊C₁₆	Dic₁₁	semidirect
145. C₂²×C₁₁	#365	semidirect
226. C₁₁	#12928	semidirect
272. C₂²	#10622151	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#66883	2
C₂×#10622151	2
#13068641	2
#13066715	2
C₁₁×#2112	2
#12928	11