Subgroups — #14644752

Order 2816.

528 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#66922	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622120	C₂	1	2	yes	maximal
4	#13066500	C₂	1	2	yes	maximal
5	C₁₁×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×#66922	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₂	—	2	1	no	—
8	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₁₁×#97	—	2	1	no	—
10	C₄³×C₁₁	C₂²	1	1	yes	characteristic
11	#10622548	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₁₁	—	2	1	no	—
13	C₄×C₄⋊Dic₁₁	C₄	1	2	yes	—
14	#10622542	—	2	1	no	—
15	#10622541	—	2	1	no	—
16	#10622541	C₄	1	2	yes	—
17	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄×C₁₁⋊C₈	—	2	1	no	—
19	C₁₁×#336	—	2	1	no	—
20	#10622162	—	2	1	no	—
21	#10622120	C₄	1	8	yes	complemented
22	C₂²×Dic₂₂	—	2	1	no	—
23	C₂²×C₈×C₁₁	—	2	2	no	—
24	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₁	—	2	1	no	—
32	C₂×C₄⋊Dic₁₁	—	2	1	no	—
33	C₂×C₄⋊Dic₁₁	—	2	1	no	—
34	C₂×C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
36	C₂×Dic₁₁⋊C₄	—	2	2	no	—
37	C₂×C₄×Dic₁₁	—	2	2	no	—
38	C₂²×C₁₁⋊C₈	—	2	2	no	—
39	#66922	—	2	2	no	—
40	#66922	C₂×C₄	1	4	yes	—
41	C₄×Dic₂₂	—	4	2	no	—
42	C₄×Dic₂₂	—	4	2	no	—
43	C₁₁×C₄⋊C₈	—	2	2	no	—
44	C₁₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₁₁×#97	—	4	2	no	—
46	C₁₁⋊#95	—	2	2	no	—
47	C₁₁⋊#95	—	2	2	no	—
48	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
49	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
50	C₄×C₁₁⋊C₈	—	4	2	no	—
51	#8236	—	11	1	no	maximal, Sylow
52	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₁	—	2	2	no	—
60	C₂²×Dic₁₁	—	2	2	no	—
61	C₂×Dic₂₂	—	2	4	no	—
62	C₂×Dic₂₂	—	4	1	no	—
63	C₂×Dic₂₂	—	4	1	no	—
64	C₂×Dic₂₂	—	4	2	no	—
65	C₂×C₈×C₁₁	—	2	4	no	—
66	C₂×C₈×C₁₁	—	4	2	no	—
67	C₂×C₈×C₁₁	—	4	2	no	—
68	C₄²×C₁₁	—	2	2	no	—
69	C₄²×C₁₁	—	2	2	no	—
70	C₄²×C₁₁	—	2	1	no	—
71	C₄²×C₁₁	—	2	1	no	—
72	C₄²×C₁₁	—	2	2	no	—
73	C₄²×C₁₁	—	4	1	no	—
74	C₄²×C₁₁	D₈	1	2	yes	—
75	C₄²×C₁₁	SD₁₆	1	2	yes	—
76	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₁	#41	1	2	yes	—
78	C₄⋊Dic₁₁	—	2	2	no	—
79	C₄⋊Dic₁₁	—	2	4	no	—
80	C₄⋊Dic₁₁	—	4	2	no	—
81	C₄⋊Dic₁₁	—	4	1	no	—
82	C₄⋊Dic₁₁	—	4	1	no	—
83	C₄⋊Dic₁₁	—	4	1	no	—
84	C₄⋊Dic₁₁	—	4	1	no	—
85	C₄⋊Dic₁₁	C₄²	1	4	yes	—
86	Dic₁₁⋊C₄	—	4	4	no	—
87	C₄×Dic₁₁	—	4	4	no	—
88	C₂×C₁₁⋊C₈	—	2	4	no	—
89	C₂×C₁₁⋊C₈	—	4	2	no	—
90	C₂×C₁₁⋊C₈	—	4	2	no	—
91	C₄×C₄⋊Q₈	—	11	1	no	—
92	#1767	D₁₁	1	1	yes	characteristic
93	C₄×C₄⋊C₈	—	11	1	no	—
94	C₂×#332	—	11	2	no	—
95	#1296	—	11	2	no	—
96	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
105	C₂×C₄×C₁₁	—	2	1	no	—
106	C₂×C₄×C₁₁	—	2	1	no	—
107	C₂×C₄×C₁₁	—	2	1	no	—
108	C₂×C₄×C₁₁	#101	1	2	yes	—
109	C₂×C₄×C₁₁	#102	1	2	yes	—
110	C₂×C₄×C₁₁	#102	1	2	yes	—
111	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₁	#135	1	2	yes	—
118	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₁₁	#137	1	2	yes	—
120	C₂×C₄×C₁₁	#137	1	1	yes	characteristic
121	C₂×Dic₁₁	—	2	4	no	—
122	C₂×Dic₁₁	—	4	2	no	—
123	C₂×Dic₁₁	—	4	2	no	—
124	C₂×Dic₁₁	—	4	2	no	—
125	C₂×Dic₁₁	—	4	2	no	—
126	C₂×Dic₁₁	—	4	2	no	—
127	Dic₂₂	—	4	4	no	—
128	Dic₂₂	—	4	4	no	—
129	C₈×C₁₁	—	4	4	no	—
130	C₁₁⋊C₈	—	4	4	no	—
131	C₂×C₄⋊Q₈	—	11	1	no	—
132	C₂×C₄×Q₈	—	22	1	no	—
133	C₂×C₄⋊C₈	—	11	2	no	—
134	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
135	C₂×#97	—	2	1	no	—
136	C₂×C₄×C₈	—	22	1	no	—
137	#391	—	22	1	no	—
138	#389	—	11	2	no	—
139	#389	—	22	1	no	—
140	#384	—	11	1	no	—
141	C₄×C₄⋊C₄	—	11	2	no	—
142	C₄×C₄⋊C₄	—	22	1	no	—
143	C₄³	C₂×D₁₁	1	1	yes	characteristic
144	#336	—	2	1	no	—
145	#336	—	22	1	no	—
146	#332	—	11	8	no	—
147	C₂²×C₁₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₁₁	C₂×#101	1	1	yes	characteristic
149	C₂²×C₁₁	#413	1	1	yes	characteristic
150	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
151	C₂²×C₁₁	#416	1	1	yes	characteristic
152	C₂²×C₁₁	#417	1	1	yes	characteristic
153	C₂²×C₁₁	#419	1	1	yes	characteristic
154	C₄×C₁₁	—	2	2	no	—
155	C₄×C₁₁	—	2	2	no	—
156	C₄×C₁₁	—	2	2	no	—
157	C₄×C₁₁	—	2	1	no	—
158	C₄×C₁₁	—	2	1	no	—
159	C₄×C₁₁	—	4	1	no	—
160	C₄×C₁₁	C₄×D₈	1	2	yes	—
161	C₄×C₁₁	C₄×SD₁₆	1	2	yes	—
162	C₄×C₁₁	#440	1	2	yes	—
163	C₄×C₁₁	#441	1	2	yes	—
164	Dic₁₁	—	4	4	no	—
165	Dic₁₁	—	8	2	no	—
166	C₂²×Q₈	—	22	1	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	22	2	no	—
169	C₄⋊Q₈	—	11	4	no	—
170	C₄⋊Q₈	—	22	2	no	—
171	C₄×Q₈	—	44	2	no	—
172	C₄×Q₈	—	44	2	no	—
173	C₂×C₄⋊C₄	—	11	2	no	—
174	C₂×C₄⋊C₄	—	11	2	no	—
175	C₂×C₄⋊C₄	—	22	1	no	—
176	C₂×C₄⋊C₄	—	22	2	no	—
177	C₂×C₄⋊C₄	—	22	1	no	—
178	C₂×C₄⋊C₄	—	22	1	no	—
179	C₂×C₄²	—	22	2	no	—
180	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
182	C₂×C₄²	D₄₄	1	1	yes	characteristic
183	C₂×C₄²	D₄₄	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	—	11	4	no	—
189	C₄⋊C₈	—	22	2	no	—
190	C₄⋊C₈	C₄×D₁₁	1	4	yes	—
191	#97	—	4	2	no	—
192	C₄×C₈	—	44	2	no	—
193	#95	—	22	2	no	—
194	#95	—	22	2	no	—
195	C₂×C₁₁	#1546	1	1	yes	characteristic
196	C₂×C₁₁	#1547	1	1	yes	characteristic
197	C₂×C₁₁	#1552	1	1	yes	characteristic
198	C₂×C₁₁	#1554	1	1	yes	characteristic
199	C₂×C₁₁	C₄×#101	1	1	yes	characteristic
200	C₂×C₁₁	C₄×#102	1	1	yes	characteristic
201	C₂×C₁₁	#1763	1	1	yes	characteristic
202	C₂×Q₈	—	22	4	no	—
203	C₂×Q₈	—	44	1	no	—
204	C₂×Q₈	—	44	1	no	—
205	C₂×Q₈	—	44	2	no	—
206	C₂²×C₄	—	22	2	no	—
207	C₂²×C₄	—	22	2	no	—
208	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	22	4	no	—
219	C₂×C₈	—	44	2	no	—
220	C₂×C₈	—	44	2	no	—
221	C₄⋊C₄	—	11	4	no	—
222	C₄⋊C₄	—	22	2	no	—
223	C₄⋊C₄	—	22	4	no	—
224	C₄⋊C₄	—	44	2	no	—
225	C₄⋊C₄	—	44	4	no	—
226	C₄⋊C₄	—	44	1	no	—
227	C₄⋊C₄	—	44	1	no	—
228	C₄⋊C₄	—	44	1	no	—
229	C₄⋊C₄	—	44	1	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	44	4	no	—
237	C₄²	#4323	1	2	yes	—
238	C₄²	#4325	1	2	yes	—
239	C₄²	C₂×C₄×D₁₁	1	2	yes	—
240	C₄²	#4339	1	2	yes	—
241	C₁₁	#8236	1	1	yes	characteristic, complemented, Sylow
242	C₂³	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
243	Q₈	—	44	4	no	—
244	Q₈	—	44	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	22	4	no	—
257	C₂×C₄	—	44	2	no	—
258	C₂×C₄	—	44	2	no	—
259	C₂×C₄	—	44	2	no	—
260	C₂×C₄	—	44	2	no	—
261	C₂×C₄	—	44	2	no	—
262	C₂×C₄	#66873	1	2	yes	—
263	C₂×C₄	#66873	1	2	yes	—
264	C₂×C₄	#66888	1	2	yes	—
265	C₂×C₄	C₄²×D₁₁	1	1	yes	characteristic
266	C₂×C₄	#66925	1	1	yes	characteristic
267	C₂×C₄	C₄×D₄₄	1	2	yes	—
268	C₂×C₄	#66961	1	1	yes	characteristic
269	C₂×C₄	#66962	1	2	yes	—
270	C₂×C₄	#66962	1	1	yes	characteristic
271	C₂×C₄	C₄×C₂²⋊D₁₁	1	2	yes	—
272	C₂×C₄	C₂×#4323	1	1	yes	characteristic
273	C₂×C₄	C₂×#4325	1	1	yes	characteristic
274	C₂×C₄	#67003	1	2	yes	—
275	C₈	—	4	4	no	—
276	C₈	—	44	4	no	—
277	C₂²	C₄×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
278	C₂²	C₂×#66873	1	1	yes	characteristic
279	C₂²	#10622615	1	1	yes	characteristic
280	C₂²	#10622733	1	1	yes	characteristic
281	C₂²	#10622745	1	1	yes	characteristic
282	C₂²	#10622746	1	1	yes	characteristic
283	C₂²	C₂×#66888	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	44	4	no	—
291	C₄	—	88	2	no	—
292	C₄	#10622323	1	2	yes	—
293	C₄	#10622329	1	2	yes	—
294	C₄	C₄×#4323	1	2	yes	—
295	C₄	C₄×#4325	1	2	yes	—
296	C₂	#13067289	1	1	yes	characteristic
297	C₂	#13067294	1	1	yes	characteristic
298	C₂	#13067317	1	1	yes	characteristic
299	C₂	#13067327	1	1	yes	characteristic
300	C₂	C₄×#66888	1	1	yes	characteristic
301	C₂	C₄×#66873	1	1	yes	characteristic
302	C₂	#13068447	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₁)⋊C₄
commutator	C₂×C₄×C₁₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₁₁×#1767	C₂
socle, prime radical	C₂³×C₁₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	#1767	D₁₁
hypocenter	C₁₁	#8236
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
21. #10622120	C₄	semidirect
241. C₁₁	#8236	semidirect

Maximal subgroups

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

subgroup	index
C₄×#66922	2
C₄×C₄⋊(C₁₁⋊C₈)	2
C₂×#10622120	2
#13066500	2
C₁₁×#1767	2
#8236	11