Subgroups — #14670245

Order 2816.

420 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#66876	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622101	C₂	1	2	yes	maximal, complemented
3	#13070227	C₂	1	1	yes	maximal, characteristic
4	#13066698	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
9	C₄×C₁₁×C₂²⋊C₄	C₄	1	1	yes	characteristic
10	C₂×#66876	—	2	1	no	—
11	C₂×#66876	—	2	4	no	—
12	C₂×#66876	C₂²	1	1	yes	characteristic
13	C₂×#66876	C₂²	1	1	yes	characteristic
14	C₂×#66876	C₂²	1	2	yes	—
15	C₂²×C₁₁⋊C₁₆	—	2	1	no	—
16	C₁₁×#336	C₄	1	2	yes	—
17	C₂²⋊(C₁₁⋊C₁₆)	—	2	1	no	—
18	C₂²⋊(C₁₁⋊C₁₆)	—	2	1	no	—
19	#10622101	—	2	4	no	—
20	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	—	2	2	no	—
22	C₂²×C₈×C₁₁	—	2	2	no	—
23	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
30	C₂×C₁₁×C₂²⋊C₄	—	2	1	no	—
31	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
32	C₁₁×C₂²⋊C₈	C₈	1	2	yes	—
33	C₁₁×C₂²⋊C₈	C₈	1	2	yes	—
34	C₄×C₈×C₁₁	—	2	2	no	—
35	C₄×C₈×C₁₁	—	2	2	no	—
36	C₁₁×#95	—	2	1	no	—
37	#66876	—	2	4	no	—
38	#66876	—	2	2	no	—
39	#66876	—	2	2	no	—
40	#66876	—	2	2	no	—
41	#66876	D₄	1	4	yes	—
42	C₂×C₁₁⋊C₁₆	—	2	4	no	—
43	C₂×C₁₁⋊C₁₆	—	2	2	no	—
44	C₂×C₁₁⋊C₁₆	—	2	1	no	—
45	C₂×C₁₁⋊C₁₆	—	2	1	no	—
46	#12914	—	11	1	no	maximal, Sylow
47	C₂⁴×C₁₁	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	—	2	2	no	—
49	C₂²×C₄×C₁₁	—	2	2	no	—
50	C₂²×C₄×C₁₁	—	2	2	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	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₁	—	2	1	no	—
58	C₂×C₈×C₁₁	—	2	1	no	—
59	C₂×C₈×C₁₁	—	2	2	no	—
60	C₂×C₈×C₁₁	—	2	2	no	—
61	C₂×C₈×C₁₁	—	2	1	no	—
62	C₂×C₈×C₁₁	—	2	1	no	—
63	C₂×C₈×C₁₁	—	2	2	no	—
64	C₂×C₈×C₁₁	—	2	2	no	—
65	C₂×C₈×C₁₁	—	2	2	no	—
66	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₁	C₂×C₈	1	4	yes	—
69	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₁₁	#41	1	2	yes	—
73	C₁₁×C₂²⋊C₄	—	2	4	no	—
74	C₄²×C₁₁	—	2	2	no	—
75	C₄²×C₁₁	—	2	2	no	—
76	C₁₁⋊C₁₆	—	2	4	no	—
77	C₁₁⋊C₁₆	—	4	1	no	—
78	C₁₁⋊C₁₆	—	4	1	no	—
79	C₂²×M₃₂	—	11	1	no	—
80	C₂×#364	—	11	2	no	—
81	#2112	—	11	1	no	—
82	C₈×C₂²⋊C₄	D₁₁	1	1	yes	characteristic
83	#1376	—	11	2	no	—
84	C₂³×C₁₁	—	2	2	no	—
85	C₂³×C₁₁	—	2	2	no	—
86	C₂³×C₁₁	—	4	1	no	—
87	C₂³×C₁₁	C₄⋊C₈	1	1	yes	characteristic
88	C₂³×C₁₁	C₄⋊C₈	1	1	yes	characteristic
89	C₂³×C₁₁	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₁	—	2	2	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	2	no	—
96	C₂×C₄×C₁₁	—	2	2	no	—
97	C₂×C₄×C₁₁	—	2	4	no	—
98	C₂×C₄×C₁₁	—	4	2	no	—
99	C₂×C₄×C₁₁	—	4	2	no	—
100	C₂×C₄×C₁₁	C₄⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₁	C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₁	#131	1	2	yes	—
106	C₈×C₁₁	—	2	1	no	—
107	C₈×C₁₁	—	2	1	no	—
108	C₈×C₁₁	—	2	4	no	—
109	C₈×C₁₁	#120	1	1	yes	characteristic
110	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
111	C₈×C₁₁	#123	1	1	yes	characteristic
112	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
113	C₂×M₃₂	—	11	2	no	—
114	C₂×M₃₂	—	11	1	no	—
115	C₂×M₃₂	—	11	1	no	—
116	C₂×M₃₂	—	22	4	no	—
117	C₂×M₃₂	—	22	1	no	—
118	C₂²×C₁₆	—	22	1	no	—
119	C₂×C₂²⋊C₈	Dic₁₁	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
121	C₄×C₂²⋊C₄	Dic₁₁	1	1	yes	characteristic
122	#364	—	22	4	no	—
123	C₂²⋊C₁₆	—	22	1	no	—
124	C₂²⋊C₁₆	—	22	1	no	—
125	#336	Dic₁₁	1	2	yes	—
126	C₂²×C₁₁	—	2	1	no	—
127	C₂²×C₁₁	—	2	1	no	—
128	C₂²×C₁₁	—	2	1	no	—
129	C₂²×C₁₁	—	2	1	no	—
130	C₂²×C₁₁	—	2	1	no	—
131	C₂²×C₁₁	—	2	1	no	—
132	C₂²×C₁₁	—	2	1	no	—
133	C₂²×C₁₁	—	4	4	no	—
134	C₂²×C₁₁	#364	1	2	yes	complemented
135	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
136	C₂²×C₁₁	#435	1	2	yes	—
137	C₄×C₁₁	—	2	1	no	—
138	C₄×C₁₁	—	2	1	no	—
139	C₄×C₁₁	—	4	4	no	—
140	C₄×C₁₁	#385	1	1	yes	characteristic
141	C₄×C₁₁	#424	1	1	yes	characteristic
142	C₄×C₁₁	C₈×D₄	1	2	yes	—
143	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	Dic₂₂	1	1	yes	characteristic
147	C₂²×C₈	Dic₂₂	1	1	yes	characteristic
148	C₂²×C₈	D₄₄	1	1	yes	characteristic
149	C₂²×C₈	D₄₄	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
151	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	—	2	1	no	—
154	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
155	M₃₂	—	11	4	no	—
156	M₃₂	—	22	4	no	—
157	M₃₂	—	22	2	no	—
158	M₃₂	—	22	2	no	—
159	M₃₂	—	22	2	no	—
160	C₂×C₁₆	—	22	4	no	—
161	C₂×C₁₆	—	22	2	no	—
162	C₂×C₁₆	—	22	1	no	—
163	C₂×C₁₆	—	22	1	no	—
164	C₂²⋊C₈	C₁₁⋊C₈	1	2	yes	—
165	C₂²⋊C₈	C₁₁⋊C₈	1	2	yes	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	#95	—	2	1	no	—
169	C₂×C₁₁	—	2	2	no	—
170	C₂×C₁₁	—	2	2	no	—
171	C₂×C₁₁	—	4	1	no	—
172	C₂×C₁₁	#1850	1	1	yes	characteristic
173	C₂×C₁₁	C₂×#364	1	1	yes	characteristic
174	C₂×C₁₁	#2153	1	1	yes	characteristic
175	C₂⁴	#4318	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
181	C₂²×C₄	#4318	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
184	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	C₂×C₁₁⋊C₈	1	4	yes	—
195	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
196	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
197	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
198	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
199	C₂×C₈	D₄×D₁₁	1	2	yes	—
200	C₂×C₈	#4341	1	2	yes	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₁₆	—	22	4	no	—
205	C₁₆	—	44	1	no	—
206	C₁₆	—	44	1	no	—
207	C₁₁	#12914	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
212	C₂³	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
213	C₂³	C₂×#4318	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
225	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
228	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
229	C₂×C₄	#67001	1	2	yes	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	4	no	—
233	C₈	#66931	1	2	yes	—
234	C₈	#66935	1	1	yes	characteristic
235	C₈	#66939	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	#10622101	1	2	yes	complemented
245	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
246	C₂²	#10622624	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	4	no	—
250	C₄	#10622563	1	1	yes	characteristic
251	C₄	#10622612	1	1	yes	characteristic
252	C₄	D₄×C₁₁⋊C₈	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	#13068796	1	1	yes	characteristic
257	C₂	C₂×#10622101	1	1	yes	characteristic
258	C₂	#13070382	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₈×C₁₁×C₂²⋊C₄	C₂
socle	C₂²×C₁₁	C₂×C₄⋊C₈
prime radical	C₂⁴×C₁₁	M₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₈×C₂²⋊C₄	D₁₁
hypocenter	C₁₁	#12914
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#10622101	C₂	semidirect
4. #13066698	C₂	semidirect
134. C₂²×C₁₁	#364	semidirect
207. C₁₁	#12914	semidirect
244. C₂²	#10622101	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#66876	2
C₂×#10622101	2
#13070227	2
#13066698	2
C₈×C₁₁×C₂²⋊C₄	2
#12914	11