Subgroups — #14653865

Order 2816.

484 conjugacy classes of subgroups.

The subgroup lattice has 276 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×#66867	C₂	1	1	yes	maximal, characteristic
3	#13066497	C₂	1	4	yes	maximal
4	C₁₁×#2862	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₁₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₁₁×#391	—	2	1	no	—
8	C₁₁×#391	—	2	1	no	—
9	C₁₁×#391	—	2	1	no	—
10	C₁₁×#391	—	2	1	no	—
11	C₁₁×#382	C₄	1	2	yes	—
12	C₁₁×#382	C₂²	1	2	yes	—
13	C₄³×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
15	C₂×#66867	—	2	1	no	—
16	C₂×#66867	C₂²	1	2	yes	—
17	C₂×#66867	C₂²	1	2	yes	—
18	C₂×C₄×C₁₁⋊C₈	—	2	1	no	—
19	#10622162	—	2	1	no	—
20	C₂²×Q₈×C₁₁	—	2	1	no	—
21	C₂²×Q₈×C₁₁	—	2	1	no	—
22	C₁₁×C₄⋊Q₈	—	4	2	no	—
23	C₁₁×C₄⋊Q₈	—	4	2	no	—
24	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
25	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
35	C₂²×C₁₁⋊C₈	—	2	2	no	—
36	C₂²×C₁₁⋊C₈	—	2	2	no	—
37	C₁₁×#95	—	2	2	no	—
38	C₁₁×#95	—	2	2	no	—
39	C₁₁×#95	—	2	2	no	—
40	C₁₁×#95	—	2	2	no	—
41	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
42	#66867	—	2	4	no	—
43	#66867	—	2	2	no	—
44	#66867	—	2	4	no	—
45	#66867	Q₈	1	4	yes	complemented
46	C₄×C₁₁⋊C₈	—	2	4	no	—
47	#9890	—	11	1	no	maximal, Sylow
48	C₂×Q₈×C₁₁	—	4	2	no	—
49	C₂×Q₈×C₁₁	—	4	2	no	—
50	C₂×Q₈×C₁₁	—	4	2	no	—
51	C₂×Q₈×C₁₁	—	4	2	no	—
52	C₂×Q₈×C₁₁	—	4	2	no	—
53	C₂×Q₈×C₁₁	—	4	2	no	—
54	C₂²×C₄×C₁₁	—	2	2	no	—
55	C₂²×C₄×C₁₁	—	2	2	no	—
56	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₁₁×C₄⋊C₄	—	4	2	no	—
64	C₁₁×C₄⋊C₄	—	4	2	no	—
65	C₁₁×C₄⋊C₄	—	4	2	no	—
66	C₁₁×C₄⋊C₄	—	4	2	no	—
67	C₄²×C₁₁	—	2	2	no	—
68	C₄²×C₁₁	—	2	2	no	—
69	C₄²×C₁₁	—	2	2	no	—
70	C₄²×C₁₁	—	2	2	no	—
71	C₄²×C₁₁	—	2	2	no	—
72	C₄²×C₁₁	—	4	1	no	—
73	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
74	C₄²×C₁₁	#41	1	2	yes	—
75	C₂×C₁₁⋊C₈	—	2	4	no	—
76	C₂×C₁₁⋊C₈	—	2	4	no	—
77	C₂×C₁₁⋊C₈	—	2	4	no	—
78	C₂×C₁₁⋊C₈	—	2	4	no	—
79	C₂×C₁₁⋊C₈	—	4	2	no	—
80	C₂×C₁₁⋊C₈	—	4	2	no	—
81	#2862	D₁₁	1	1	yes	characteristic
82	#1767	—	11	1	no	—
83	C₄×#97	—	11	1	no	—
84	#1295	—	11	4	no	—
85	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₁₁	—	8	4	no	—
87	Q₈×C₁₁	—	8	4	no	—
88	C₂×C₄×C₁₁	—	2	1	no	—
89	C₂×C₄×C₁₁	—	2	1	no	—
90	C₂×C₄×C₁₁	—	2	1	no	—
91	C₂×C₄×C₁₁	—	2	1	no	—
92	C₂×C₄×C₁₁	—	2	1	no	—
93	C₂×C₄×C₁₁	—	2	1	no	—
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	—	4	2	no	—
102	C₂×C₄×C₁₁	—	4	2	no	—
103	C₂×C₄×C₁₁	—	4	2	no	—
104	C₂×C₄×C₁₁	—	4	2	no	—
105	C₂×C₄×C₁₁	—	4	2	no	—
106	C₂×C₄×C₁₁	—	4	2	no	—
107	C₂×C₄×C₁₁	#100	1	2	yes	—
108	C₂×C₄×C₁₁	#101	1	2	yes	—
109	C₂×C₄×C₁₁	#104	1	2	yes	—
110	C₂×C₄×C₁₁	#104	1	2	yes	—
111	C₂×C₄×C₁₁	#104	1	2	yes	—
112	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
116	C₁₁⋊C₈	—	2	8	no	—
117	C₁₁⋊C₈	—	4	4	no	—
118	C₂×C₄⋊Q₈	Dic₁₁	1	1	yes	characteristic
119	C₂×C₄⋊Q₈	Dic₁₁	1	1	yes	characteristic
120	C₂×C₄⋊C₈	—	22	1	no	—
121	C₂×#97	—	11	2	no	—
122	C₂×#97	—	11	2	no	—
123	C₂×#97	—	22	1	no	—
124	C₂×C₄×C₈	—	22	1	no	—
125	#391	—	2	1	no	—
126	#391	—	2	1	no	—
127	#391	—	2	1	no	—
128	#391	—	2	1	no	—
129	#382	Dic₁₁	1	2	yes	—
130	#382	C₂×D₁₁	1	2	yes	—
131	C₄³	C₂×D₁₁	1	1	yes	characteristic
132	#336	—	22	1	no	—
133	C₂²×C₁₁	#391	1	1	yes	characteristic
134	C₂²×C₁₁	C₂×#100	1	1	yes	characteristic
135	C₂²×C₁₁	C₂×#101	1	1	yes	characteristic
136	C₂²×C₁₁	C₂×#104	1	1	yes	characteristic
137	C₂²×C₁₁	C₂×#104	1	1	yes	characteristic
138	C₂²×C₁₁	C₂×#104	1	1	yes	characteristic
139	C₂²×C₁₁	#421	1	1	yes	characteristic
140	C₄×C₁₁	—	2	1	no	—
141	C₄×C₁₁	—	2	1	no	—
142	C₄×C₁₁	—	2	2	no	—
143	C₄×C₁₁	—	2	2	no	—
144	C₄×C₁₁	—	2	2	no	—
145	C₄×C₁₁	—	2	2	no	—
146	C₄×C₁₁	—	4	1	no	—
147	C₄×C₁₁	—	8	2	no	—
148	C₄×C₁₁	—	8	2	no	—
149	C₄×C₁₁	#329	1	4	yes	—
150	C₂²×Q₈	—	2	1	no	—
151	C₂²×Q₈	—	2	1	no	—
152	C₂²×C₈	—	22	2	no	—
153	C₂²×C₈	—	22	2	no	—
154	C₄⋊Q₈	—	4	2	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	C₂×Dic₁₁	1	2	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₁₁	1	2	yes	—
160	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
161	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
163	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
167	C₄⋊C₈	—	22	4	no	—
168	#97	—	11	4	no	—
169	#97	—	22	4	no	—
170	#97	—	22	2	no	—
171	#97	—	22	4	no	—
172	C₄×C₈	—	22	4	no	—
173	#95	—	2	2	no	—
174	#95	—	2	2	no	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	C₂×C₁₁	C₂×#329	1	1	yes	characteristic
178	C₂×C₁₁	#1526	1	1	yes	characteristic
179	C₂×C₁₁	#1527	1	1	yes	characteristic
180	C₂×C₁₁	#1531	1	1	yes	characteristic
181	C₂×C₁₁	#1986	1	1	yes	characteristic
182	C₂×C₁₁	#1994	1	1	yes	characteristic
183	C₂×C₁₁	#1997	1	1	yes	characteristic
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
199	C₂×C₈	—	22	4	no	—
200	C₂×C₈	—	22	4	no	—
201	C₂×C₈	—	22	4	no	—
202	C₂×C₈	—	22	4	no	—
203	C₂×C₈	—	44	2	no	—
204	C₂×C₈	—	44	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	1	no	—
215	C₄²	Q₈×D₁₁	1	2	yes	—
216	C₄²	#4343	1	2	yes	—
217	C₁₁	#9890	1	1	yes	characteristic, complemented, Sylow
218	C₂³	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
219	Q₈	—	8	4	no	—
220	Q₈	—	8	4	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	#66897	1	2	yes	—
241	C₂×C₄	#66900	1	2	yes	—
242	C₂×C₄	#66901	1	2	yes	—
243	C₂×C₄	#66901	1	2	yes	—
244	C₂×C₄	#66901	1	2	yes	—
245	C₂×C₄	#66997	1	1	yes	characteristic
246	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
247	C₂×C₄	#66999	1	2	yes	—
248	C₂×C₄	#67000	1	1	yes	characteristic
249	C₈	—	22	8	no	—
250	C₈	—	44	4	no	—
251	C₂²	C₂×#66897	1	1	yes	characteristic
252	C₂²	C₂×#66900	1	1	yes	characteristic
253	C₂²	#10622843	1	1	yes	characteristic
254	C₂²	C₂×#66901	1	1	yes	characteristic
255	C₂²	C₂×#66901	1	1	yes	characteristic
256	C₂²	C₂×#66901	1	1	yes	characteristic
257	C₂²	#10622849	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	8	2	no	—
266	C₄	—	8	2	no	—
267	C₄	#10622178	1	4	yes	—
268	C₂	C₂×#10622178	1	1	yes	characteristic
269	C₂	#13067211	1	1	yes	characteristic
270	C₂	#13067214	1	1	yes	characteristic
271	C₂	#13067231	1	1	yes	characteristic
272	C₂	#13069561	1	1	yes	characteristic
273	C₂	#13069594	1	1	yes	characteristic
274	C₂	#13069606	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₁
commutator	C₂²×C₄×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₁₁×#2862	C₂
socle, prime radical	C₂³×C₁₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	#2862	D₁₁
hypocenter	C₁₁	#9890
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
45. #66867	Q₈	semidirect
217. C₁₁	#9890	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×#66867	2
#13066497	2
C₁₁×#2862	2
#9890	11