Subgroups — #14643239

Order 2816.

758 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622537	C₂	1	2	yes	maximal
2	#13066451	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×#4318	—	2	2	no	—
7	C₂×C₄×C₈×C₁₁	—	2	4	no	—
8	C₂×C₄×C₈×C₁₁	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
10	#10622537	—	2	4	no	—
11	#10622537	C₄	1	8	yes	—
12	C₂×C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
13	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	—	2	4	no	—
18	C₂²×C₈×C₁₁	—	2	8	no	—
19	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₁₁	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₁	D₄	1	4	yes	—
24	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
26	C₂×#4318	—	2	8	no	—
27	C₂×#4318	—	4	4	no	—
28	C₂×#4318	—	4	4	no	—
29	C₂²×C₁₁⋊C₈	—	2	4	no	—
30	C₄×C₈×C₁₁	—	2	4	no	—
31	C₄×C₈×C₁₁	—	2	4	no	—
32	C₄⋊(C₁₁⋊C₈)	—	2	8	no	—
33	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
34	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
35	#7874	—	11	1	no	maximal, Sylow
36	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₁	—	2	4	no	—
38	C₂²×C₄×C₁₁	—	2	4	no	—
39	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
40	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₁	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₁	D₈	1	2	yes	—
47	C₂²×C₄×C₁₁	SD₁₆	1	4	yes	—
48	C₂²×C₄×C₁₁	Q₁₆	1	2	yes	—
49	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₁	—	2	16	no	—
53	C₂×C₈×C₁₁	—	2	4	no	—
54	C₂×C₈×C₁₁	—	2	4	no	—
55	C₂×C₈×C₁₁	M₁₆	1	8	yes	complemented
56	C₄²×C₁₁	C₄²	1	2	yes	—
57	C₄²×C₁₁	C₂²⋊C₄	1	4	yes	—
58	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₁₁	C₄⋊C₄	1	4	yes	—
60	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
61	C₄²×C₁₁	C₂×D₄	1	2	yes	—
62	#4318	—	4	16	no	—
63	C₂×C₁₁⋊C₈	—	2	8	no	—
64	C₂×C₁₁⋊C₈	—	4	4	no	—
65	C₂×C₁₁⋊C₈	—	4	4	no	—
66	C₂×#423	—	11	2	no	—
67	C₂²×C₄×C₈	D₁₁	1	1	yes	characteristic
68	#1277	—	11	4	no	—
69	C₂³×C₁₁	—	2	2	no	—
70	C₂³×C₁₁	—	2	2	no	—
71	C₂³×C₁₁	#95	1	2	yes	—
72	C₂³×C₁₁	#106	1	2	yes	—
73	C₂³×C₁₁	#107	1	2	yes	—
74	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₁	—	2	2	no	—
76	C₂×C₄×C₁₁	—	2	2	no	—
77	C₂×C₄×C₁₁	—	2	8	no	—
78	C₂×C₄×C₁₁	—	2	4	no	—
79	C₂×C₄×C₁₁	—	2	4	no	—
80	C₂×C₄×C₁₁	#95	1	2	yes	—
81	C₂×C₄×C₁₁	#95	1	2	yes	—
82	C₂×C₄×C₁₁	#95	1	2	yes	—
83	C₂×C₄×C₁₁	#102	1	8	yes	—
84	C₂×C₄×C₁₁	#103	1	8	yes	—
85	C₂×C₄×C₁₁	#106	1	2	yes	—
86	C₂×C₄×C₁₁	#107	1	2	yes	—
87	C₂×C₄×C₁₁	#108	1	4	yes	—
88	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
93	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
96	C₈×C₁₁	—	2	8	no	—
97	C₁₁⋊C₈	—	4	8	no	—
98	C₂²×M₁₆	—	22	2	no	—
99	C₂³×C₈	Dic₁₁	1	2	yes	—
100	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
101	#423	—	11	8	no	—
102	#423	—	22	4	no	—
103	C₂×C₄⋊C₈	—	11	4	no	—
104	C₂×C₄⋊C₈	—	22	2	no	—
105	C₂×C₄×C₈	—	2	4	no	—
106	C₂×C₄×C₈	Dic₁₁	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
108	C₂²×C₁₁	—	2	4	no	—
109	C₂²×C₁₁	—	2	4	no	—
110	C₂²×C₁₁	—	2	2	no	—
111	C₂²×C₁₁	—	2	2	no	—
112	C₂²×C₁₁	#340	1	4	yes	—
113	C₂²×C₁₁	C₂×#95	1	1	yes	characteristic
114	C₂²×C₁₁	#423	1	2	yes	—
115	C₂²×C₁₁	C₂×#106	1	1	yes	characteristic
116	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
117	C₂²×C₁₁	C₂×#108	1	2	yes	—
118	C₄×C₁₁	—	2	2	no	—
119	C₄×C₁₁	—	2	2	no	—
120	C₄×C₁₁	#340	1	4	yes	—
121	C₄×C₁₁	#341	1	4	yes	—
122	C₄×C₁₁	C₄×M₁₆	1	2	yes	—
123	C₄×C₁₁	#407	1	2	yes	—
124	C₄×C₁₁	C₂×#102	1	2	yes	—
125	C₄×C₁₁	C₂×#103	1	2	yes	—
126	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
127	C₂³×C₄	D₄₄	1	1	yes	characteristic
128	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
129	C₂×M₁₆	—	22	8	no	—
130	C₂×M₁₆	—	44	4	no	—
131	C₂×M₁₆	—	44	4	no	—
132	C₂²×C₈	—	2	4	no	—
133	C₂²×C₈	—	2	8	no	—
134	C₂²×C₈	—	22	4	no	—
135	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
136	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
137	C₂×C₄²	C₄×D₁₁	1	4	yes	—
138	C₂×C₄²	D₄₄	1	1	yes	characteristic
139	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₁₁	1	4	yes	—
141	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
142	C₄⋊C₈	—	11	8	no	—
143	C₄⋊C₈	—	22	8	no	—
144	C₄⋊C₈	—	22	4	no	—
145	C₄×C₈	—	2	4	no	—
146	C₄×C₈	—	2	4	no	—
147	C₂×C₁₁	—	2	4	no	—
148	C₂×C₁₁	#1566	1	2	yes	—
149	C₂×C₁₁	#1567	1	2	yes	—
150	C₂×C₁₁	#1728	1	1	yes	characteristic
151	C₂×C₁₁	C₂×#340	1	1	yes	characteristic
152	C₂×C₁₁	C₂×#341	1	1	yes	characteristic
153	C₂⁴	C₄⋊Dic₁₁	1	1	yes	characteristic
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	#4314	1	4	yes	—
157	C₂²×C₄	D₈₈	1	2	yes	—
158	C₂²×C₄	Dic₄₄	1	2	yes	—
159	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
160	C₂²×C₄	Dic₁₁⋊C₄	1	4	yes	—
161	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
164	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	4	yes	—
165	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
166	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
168	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
169	M₁₆	—	44	16	no	—
170	C₂×C₈	—	2	16	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	22	8	no	—
174	C₂×C₈	—	44	4	no	—
175	C₂×C₈	—	44	4	no	—
176	C₂×C₈	#4318	1	8	yes	complemented
177	C₄²	C₄×Dic₁₁	1	2	yes	—
178	C₄²	Dic₁₁⋊C₄	1	4	yes	—
179	C₄²	(C₂×D₁₁)⋊C₄	1	4	yes	—
180	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
181	C₄²	C₂×C₄×D₁₁	1	2	yes	—
182	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
183	C₁₁	#7874	1	1	yes	characteristic, complemented, Sylow
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	#66881	1	2	yes	—
187	C₂³	#66882	1	2	yes	—
188	C₂³	C₁₁⋊#95	1	2	yes	—
189	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	#66880	1	8	yes	—
196	C₂×C₄	#66881	1	2	yes	—
197	C₂×C₄	#66882	1	2	yes	—
198	C₂×C₄	#66883	1	4	yes	—
199	C₂×C₄	#66885	1	8	yes	—
200	C₂×C₄	C₁₁⋊#95	1	2	yes	—
201	C₂×C₄	C₁₁⋊#95	1	2	yes	—
202	C₂×C₄	C₁₁⋊#95	1	2	yes	—
203	C₂×C₄	C₂×#4314	1	2	yes	—
204	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
205	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
206	C₂×C₄	C₂×#4318	1	4	yes	—
207	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
208	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	2	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
211	C₈	—	2	8	no	—
212	C₈	—	44	8	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	#10622189	1	4	yes	—
218	C₂²	#10622537	1	2	yes	—
219	C₂²	C₂×#66881	1	1	yes	characteristic
220	C₂²	C₂×#66882	1	1	yes	characteristic
221	C₂²	C₂×#66883	1	2	yes	—
222	C₂²	C₂×C₁₁⋊#95	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	#10622189	1	4	yes	—
226	C₄	#10622190	1	4	yes	—
227	C₄	C₄×#4318	1	2	yes	—
228	C₄	C₂×#66880	1	2	yes	—
229	C₄	C₂×#66885	1	2	yes	—
230	C₄	#10622820	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	#13067372	1	2	yes	—
233	C₂	#13067375	1	2	yes	—
234	C₂	#13068310	1	1	yes	characteristic
235	C₂	C₂×#10622189	1	1	yes	characteristic
236	C₂	C₂×#10622190	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄₄
commutator	C₂×C₄×C₁₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂²×C₄×C₈×C₁₁	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂²×C₄×C₈	D₁₁
hypocenter	C₁₁	#7874
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13066451	C₂	semidirect
55. C₂×C₈×C₁₁	M₁₆	semidirect
176. C₂×C₈	#4318	semidirect
183. C₁₁	#7874	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622537	2
#13066451	2
C₂²×C₄×C₈×C₁₁	2
#7874	11