Subgroups — #14669698

Order 2816.

478 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal
3	#13066694	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₁×#405	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₁×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×#117	C₄	1	1	yes	characteristic
8	C₁₁×#405	—	2	4	no	—
9	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
10	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁⋊C₁₆	C₂²	1	4	yes	—
14	C₂²⋊(C₁₁⋊C₁₆)	—	2	4	no	—
15	C₄⋊(C₁₁⋊C₁₆)	—	2	4	no	—
16	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₁×M₁₆	—	2	2	no	—
18	C₂×C₁₁×M₁₆	—	2	2	no	—
19	C₂×C₁₁×M₁₆	—	2	2	no	—
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	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	D₄	1	2	yes	—
26	C₂²×C₈×C₁₁	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
28	C₁₁×#117	—	2	2	no	—
29	C₁₁×#117	—	2	2	no	—
30	C₂×C₁₁×C₄⋊C₄	C₈	1	2	yes	—
31	C₂×C₁₁×C₂²⋊C₄	C₈	1	2	yes	—
32	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₁₁×#97	—	2	2	no	—
35	C₁₁×#97	—	2	2	no	—
36	C₄×C₈×C₁₁	—	2	2	no	—
37	C₄×C₈×C₁₁	D₄	1	4	yes	—
38	C₂×C₁₁⋊C₁₆	—	2	4	no	—
39	C₂×C₁₁⋊C₁₆	—	2	4	no	—
40	C₂×C₁₁⋊C₁₆	—	2	4	no	—
41	#12784	—	11	1	no	maximal, Sylow
42	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	—	2	1	no	—
44	C₂²×C₄×C₁₁	—	2	1	no	—
45	C₂²×C₄×C₁₁	—	2	1	no	—
46	C₂²×C₄×C₁₁	—	2	1	no	—
47	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
52	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
53	C₁₁×M₁₆	—	2	8	no	—
54	C₂×C₈×C₁₁	—	2	4	no	—
55	C₂×C₈×C₁₁	—	2	4	no	—
56	C₂×C₈×C₁₁	—	2	1	no	—
57	C₂×C₈×C₁₁	—	2	1	no	—
58	C₂×C₈×C₁₁	—	2	1	no	—
59	C₂×C₈×C₁₁	—	2	1	no	—
60	C₂×C₈×C₁₁	—	2	1	no	—
61	C₂×C₈×C₁₁	—	2	1	no	—
62	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
65	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₁₁	#41	1	2	yes	—
69	C₁₁×C₄⋊C₄	—	2	4	no	—
70	C₁₁×C₂²⋊C₄	—	2	4	no	—
71	C₄²×C₁₁	—	2	1	no	—
72	C₄²×C₁₁	—	2	1	no	—
73	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₁	M₁₆	1	2	yes	—
75	C₁₁⋊C₁₆	—	4	4	no	—
76	C₂×#405	D₁₁	1	1	yes	characteristic
77	C₂×C₄⋊C₁₆	—	11	2	no	—
78	C₂×C₂²⋊C₁₆	—	11	2	no	—
79	#1374	—	11	2	no	—
80	C₂³×C₁₁	—	2	1	no	—
81	C₂³×C₁₁	—	2	1	no	—
82	C₂³×C₁₁	—	2	4	no	—
83	C₂³×C₁₁	C₄⋊C₈	1	2	yes	—
84	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₁	—	2	1	no	—
86	C₂×C₄×C₁₁	—	2	1	no	—
87	C₂×C₄×C₁₁	—	2	4	no	—
88	C₂×C₄×C₁₁	—	2	2	no	—
89	C₂×C₄×C₁₁	—	2	2	no	—
90	C₂×C₄×C₁₁	—	2	2	no	—
91	C₂×C₄×C₁₁	—	2	2	no	—
92	C₂×C₄×C₁₁	—	2	2	no	—
93	C₂×C₄×C₁₁	—	2	4	no	—
94	C₂×C₄×C₁₁	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
102	C₈×C₁₁	—	2	2	no	—
103	C₈×C₁₁	—	2	2	no	—
104	C₈×C₁₁	—	2	2	no	—
105	C₈×C₁₁	#121	1	2	yes	—
106	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
107	C₂²×M₁₆	Dic₁₁	1	1	yes	characteristic
108	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
109	C₂×#117	Dic₁₁	1	1	yes	characteristic
110	C₂²×C₁₆	—	11	4	no	—
111	#405	—	2	4	no	—
112	C₂×#97	Dic₁₁	1	1	yes	characteristic
113	C₂×#97	Dic₁₁	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
116	C₄⋊C₁₆	—	22	4	no	—
117	C₂²⋊C₁₆	—	22	4	no	—
118	C₂²×C₁₁	—	2	2	no	—
119	C₂²×C₁₁	—	2	4	no	—
120	C₂²×C₁₁	—	4	2	no	—
121	C₂²×C₁₁	—	4	2	no	—
122	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₂²×C₁₁	#431	1	1	yes	characteristic
125	C₂²×C₁₁	#504	1	4	yes	—
126	C₄×C₁₁	—	2	2	no	—
127	C₄×C₁₁	—	2	4	no	—
128	C₄×C₁₁	—	4	2	no	—
129	C₄×C₁₁	#380	1	1	yes	characteristic
130	C₄×C₁₁	#407	1	1	yes	characteristic
131	C₄×C₁₁	#423	1	1	yes	characteristic
132	C₄×C₁₁	#432	1	1	yes	characteristic
133	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	Dic₂₂	1	2	yes	—
140	C₂²×C₈	D₄₄	1	2	yes	—
141	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
142	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
145	#117	—	2	2	no	—
146	#117	—	2	2	no	—
147	C₂×C₄⋊C₄	C₁₁⋊C₈	1	2	yes	—
148	C₂×C₂²⋊C₄	C₁₁⋊C₈	1	2	yes	—
149	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
150	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
151	C₂×C₁₆	—	22	4	no	—
152	C₂×C₁₆	—	22	4	no	—
153	C₂×C₁₆	—	22	4	no	—
154	#97	—	2	2	no	—
155	#97	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	C₂²⋊D₁₁	1	4	yes	—
158	C₂×C₁₁	—	4	2	no	—
159	C₂×C₁₁	#1797	1	1	yes	characteristic
160	C₂×C₁₁	#2113	1	2	yes	—
161	C₂×C₁₁	#2151	1	2	yes	—
162	C₂×C₁₁	#2164	1	2	yes	—
163	C₂⁴	C₂×C₁₁⋊C₈	1	1	yes	characteristic
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
169	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
171	C₂²×C₄	#4318	1	2	yes	—
172	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
173	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
174	M₁₆	—	2	8	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	#4318	1	2	yes	—
184	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
185	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
186	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
188	C₂×C₈	#4339	1	2	yes	—
189	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	#4318	1	2	yes	—
195	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
196	C₁₆	—	44	4	no	—
197	C₁₁	#12784	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	2	4	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
202	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
213	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	4	yes	—
214	C₂×C₄	C₂×#4318	1	1	yes	characteristic
215	C₂×C₄	C₂×#4318	1	1	yes	characteristic
216	C₂×C₄	C₂×#4318	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
218	C₂×C₄	#66979	1	1	yes	characteristic
219	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	#66977	1	2	yes	—
224	C₈	#66983	1	2	yes	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
230	C₂²	#10622538	1	1	yes	characteristic
231	C₂²	#10622753	1	4	yes	—
232	C₂²	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	#10622537	1	1	yes	characteristic
237	C₄	#10622539	1	1	yes	characteristic
238	C₄	#10622820	1	1	yes	characteristic
239	C₄	#10622825	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	#13068596	1	1	yes	characteristic
242	C₂	#13070230	1	2	yes	—
243	C₂	#13070375	1	2	yes	—
244	C₂	#13070423	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator, socle	C₂³×C₁₁	C₂²×C₈
Fitting	C₂×C₁₁×#405	C₂
prime radical	C₂⁴×C₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#405	D₁₁
hypocenter	C₁₁	#12784
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
3. #13066694	C₂	semidirect
197. C₁₁	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×C₂²⋊(C₁₁⋊C₁₆)	2
#13066694	2
C₂×C₁₁×#405	2
#12784	11