Subgroups — #14644744

Order 2816.

528 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068469	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622184	C₂	1	2	yes	maximal
4	#13066499	C₂	1	2	yes	maximal
5	C₄×C₁₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₁	C₄	1	2	yes	—
8	C₁₁×#391	C₄	1	2	yes	—
9	C₁₁×#389	—	2	1	no	—
10	C₁₁×#389	—	2	1	no	—
11	C₁₁×#384	C₂²	1	1	yes	characteristic
12	C₄×C₁₁×C₄⋊C₄	—	2	1	no	—
13	C₄×C₁₁×C₄⋊C₄	—	2	1	no	—
14	C₄³×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
17	C₂×#66867	—	2	1	no	—
18	C₂×C₄×C₁₁⋊C₈	—	2	1	no	—
19	#10622184	C₄	1	8	yes	complemented
20	#10622162	—	2	1	no	—
21	#10622162	—	2	1	no	—
22	C₂²×Q₈×C₁₁	C₂×C₄	1	2	yes	—
23	C₁₁×C₄⋊Q₈	—	2	2	no	—
24	C₁₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₄×Q₈×C₁₁	—	4	4	no	—
26	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₁	—	2	2	no	—
33	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
40	C₂²×C₁₁⋊C₈	—	2	2	no	—
41	C₂²×C₁₁⋊C₈	—	2	2	no	—
42	C₁₁×#95	—	2	2	no	—
43	C₁₁×#95	—	2	2	no	—
44	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
45	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
47	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	4	yes	—
48	#66867	—	4	2	no	—
49	C₄×C₁₁⋊C₈	—	4	2	no	—
50	#8235	—	11	1	no	maximal, Sylow
51	C₂×Q₈×C₁₁	—	2	2	no	—
52	C₂×Q₈×C₁₁	—	4	2	no	—
53	C₂×Q₈×C₁₁	—	4	2	no	—
54	C₂×Q₈×C₁₁	C₄²	1	4	yes	—
55	C₂²×C₄×C₁₁	—	2	2	no	—
56	C₂²×C₄×C₁₁	—	2	2	no	—
57	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
64	C₁₁×C₄⋊C₄	—	2	4	no	—
65	C₁₁×C₄⋊C₄	—	2	4	no	—
66	C₁₁×C₄⋊C₄	—	4	2	no	—
67	C₁₁×C₄⋊C₄	—	4	4	no	—
68	C₁₁×C₄⋊C₄	—	4	2	no	—
69	C₁₁×C₄⋊C₄	—	4	1	no	—
70	C₁₁×C₄⋊C₄	—	4	1	no	—
71	C₄²×C₁₁	—	2	2	no	—
72	C₄²×C₁₁	—	2	2	no	—
73	C₄²×C₁₁	—	2	1	no	—
74	C₄²×C₁₁	—	2	1	no	—
75	C₄²×C₁₁	—	2	2	no	—
76	C₄²×C₁₁	—	4	1	no	—
77	C₄²×C₁₁	—	4	4	no	—
78	C₄²×C₁₁	SD₁₆	1	2	yes	—
79	C₄²×C₁₁	Q₁₆	1	2	yes	—
80	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₁	#41	1	2	yes	—
82	C₂×C₁₁⋊C₈	—	2	4	no	—
83	C₂×C₁₁⋊C₈	—	2	4	no	—
84	C₂×C₁₁⋊C₈	—	4	2	no	—
85	C₂×C₁₁⋊C₈	—	4	2	no	—
86	C₂×C₁₁⋊C₈	—	4	2	no	—
87	C₂×C₁₁⋊C₈	—	4	2	no	—
88	C₄×C₄⋊Q₈	D₁₁	1	1	yes	characteristic
89	#1767	—	11	1	no	—
90	C₄×C₄⋊C₈	—	11	1	no	—
91	C₂×#333	—	11	2	no	—
92	#1296	—	11	2	no	—
93	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₁₁	—	4	4	no	—
95	Q₈×C₁₁	—	4	4	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	—	2	4	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	—	4	2	no	—
109	C₂×C₄×C₁₁	—	4	2	no	—
110	C₂×C₄×C₁₁	—	4	2	no	—
111	C₂×C₄×C₁₁	—	4	2	no	—
112	C₂×C₄×C₁₁	—	4	2	no	—
113	C₂×C₄×C₁₁	#100	1	2	yes	—
114	C₂×C₄×C₁₁	#103	1	2	yes	—
115	C₂×C₄×C₁₁	#103	1	2	yes	—
116	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₁	#135	1	2	yes	—
123	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₁₁	#137	1	2	yes	—
125	C₂×C₄×C₁₁	#137	1	1	yes	characteristic
126	C₁₁⋊C₈	—	4	4	no	—
127	C₁₁⋊C₈	—	4	4	no	—
128	C₂×C₄⋊Q₈	C₂×D₁₁	1	1	yes	characteristic
129	C₂×C₄×Q₈	Dic₁₁	1	2	yes	—
130	C₂×C₄⋊C₈	—	11	2	no	—
131	C₂×C₄⋊C₈	—	11	2	no	—
132	C₂×#97	—	22	1	no	—
133	C₂×C₄×C₈	—	22	1	no	—
134	#391	Dic₁₁	1	2	yes	—
135	#389	—	2	1	no	—
136	#389	—	2	1	no	—
137	#384	C₂×D₁₁	1	1	yes	characteristic
138	C₄×C₄⋊C₄	—	2	1	no	—
139	C₄×C₄⋊C₄	—	2	1	no	—
140	C₄³	C₂×D₁₁	1	1	yes	characteristic
141	#336	—	22	1	no	—
142	#336	—	22	1	no	—
143	#333	—	11	8	no	—
144	C₂²×C₁₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₁₁	C₂×#100	1	1	yes	characteristic
146	C₂²×C₁₁	#413	1	1	yes	characteristic
147	C₂²×C₁₁	C₂×#103	1	1	yes	characteristic
148	C₂²×C₁₁	#416	1	1	yes	characteristic
149	C₂²×C₁₁	#417	1	1	yes	characteristic
150	C₂²×C₁₁	#419	1	1	yes	characteristic
151	C₄×C₁₁	—	2	2	no	—
152	C₄×C₁₁	—	2	2	no	—
153	C₄×C₁₁	—	2	2	no	—
154	C₄×C₁₁	—	2	1	no	—
155	C₄×C₁₁	—	2	1	no	—
156	C₄×C₁₁	—	4	1	no	—
157	C₄×C₁₁	—	4	4	no	—
158	C₄×C₁₁	—	8	2	no	—
159	C₄×C₁₁	C₄×SD₁₆	1	2	yes	—
160	C₄×C₁₁	C₄×Q₁₆	1	2	yes	—
161	C₄×C₁₁	#440	1	2	yes	—
162	C₄×C₁₁	#441	1	2	yes	—
163	C₂²×Q₈	C₂×Dic₁₁	1	2	yes	—
164	C₂²×C₈	—	22	2	no	—
165	C₂²×C₈	—	22	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	C₄×D₁₁	1	4	yes	—
168	C₄×Q₈	—	4	4	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	C₂×Dic₁₁	1	2	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
183	C₄⋊C₈	—	11	4	no	—
184	C₄⋊C₈	—	11	4	no	—
185	C₄⋊C₈	—	22	2	no	—
186	C₄⋊C₈	—	22	2	no	—
187	#97	—	44	2	no	—
188	C₄×C₈	—	44	2	no	—
189	#95	—	2	2	no	—
190	#95	—	2	2	no	—
191	C₂×C₁₁	#1548	1	1	yes	characteristic
192	C₂×C₁₁	#1550	1	1	yes	characteristic
193	C₂×C₁₁	#1551	1	1	yes	characteristic
194	C₂×C₁₁	#1554	1	1	yes	characteristic
195	C₂×C₁₁	C₄×#100	1	1	yes	characteristic
196	C₂×C₁₁	C₄×#103	1	1	yes	characteristic
197	C₂×C₁₁	#1763	1	1	yes	characteristic
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	C₄×Dic₁₁	1	4	yes	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
211	C₂×C₈	—	22	4	no	—
212	C₂×C₈	—	22	4	no	—
213	C₂×C₈	—	44	2	no	—
214	C₂×C₈	—	44	2	no	—
215	C₂×C₈	—	44	2	no	—
216	C₂×C₈	—	44	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	4	4	no	—
231	C₄²	#4325	1	2	yes	—
232	C₄²	#4326	1	2	yes	—
233	C₄²	C₂×C₄×D₁₁	1	2	yes	—
234	C₄²	#4339	1	2	yes	—
235	C₁₁	#8235	1	1	yes	characteristic, complemented, Sylow
236	C₂³	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
237	Q₈	—	4	4	no	—
238	Q₈	—	4	4	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	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	#66897	1	2	yes	—
257	C₂×C₄	#66899	1	2	yes	—
258	C₂×C₄	#66899	1	2	yes	—
259	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
260	C₂×C₄	#66979	1	1	yes	characteristic
261	C₂×C₄	C₄×C₂²⋊D₁₁	1	2	yes	—
262	C₂×C₄	C₄×C₂²⋊D₁₁	1	2	yes	—
263	C₂×C₄	C₂×#4325	1	1	yes	characteristic
264	C₂×C₄	#66995	1	2	yes	—
265	C₂×C₄	C₂×#4326	1	1	yes	characteristic
266	C₂×C₄	#67002	1	1	yes	characteristic
267	C₂×C₄	#67003	1	2	yes	—
268	C₂×C₄	#67004	1	1	yes	characteristic
269	C₈	—	44	4	no	—
270	C₈	—	44	4	no	—
271	C₂²	C₄×C₂²⋊Dic₁₁	1	1	yes	characteristic
272	C₂²	C₂×#66897	1	1	yes	characteristic
273	C₂²	C₂×#66899	1	1	yes	characteristic
274	C₂²	#10622838	1	1	yes	characteristic
275	C₂²	#10622845	1	1	yes	characteristic
276	C₂²	#10622846	1	1	yes	characteristic
277	C₂²	#10622850	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	C₄×#4325	1	2	yes	—
287	C₄	#10622639	1	2	yes	—
288	C₄	C₄×#4326	1	2	yes	—
289	C₄	#10622643	1	2	yes	—
290	C₂	#13067297	1	1	yes	characteristic
291	C₂	#13067306	1	1	yes	characteristic
292	C₂	#13067311	1	1	yes	characteristic
293	C₂	#13067326	1	1	yes	characteristic
294	C₂	C₄×#66897	1	1	yes	characteristic
295	C₂	C₄×#66899	1	1	yes	characteristic
296	C₂	#13068446	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₁
commutator	C₂×C₄×C₁₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₄×C₁₁×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₄×C₄⋊Q₈	D₁₁
hypocenter	C₁₁	#8235
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
19. #10622184	C₄	semidirect
235. C₁₁	#8235	semidirect

Maximal subgroups

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

subgroup	index
#13068469	2
C₄×C₄⋊(C₁₁⋊C₈)	2
C₂×#10622184	2
#13066499	2
C₄×C₁₁×C₄⋊Q₈	2
#8235	11