Subgroups — #14672375

Order 2816.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 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	1	yes	maximal, characteristic
2	#13068776	C₂	1	1	yes	maximal, characteristic
3	#13066707	C₂	1	4	yes	maximal
4	C₂×C₁₁×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×C₄⋊Q₈	—	2	1	no	—
6	C₁₁×#500	—	2	2	no	—
7	C₁₁×#500	C₄	1	4	yes	complemented
8	C₂×C₁₁×#107	—	2	1	no	—
9	C₂×C₁₁×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×#66882	—	2	1	no	—
12	C₂×#66882	C₂²	1	2	yes	—
13	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
14	#10622548	—	2	1	no	—
15	C₄×C₁₁⋊C₁₆	—	2	2	no	—
16	C₂²×Q₈×C₁₁	—	4	1	no	—
17	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
19	C₁₁×C₄⋊Q₈	—	2	4	no	—
20	C₁₁×C₄⋊Q₈	—	4	2	no	—
21	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
22	C₂×C₁₁×C₄⋊C₄	—	4	1	no	—
23	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₁	—	2	2	no	—
25	C₂×C₄⋊Dic₁₁	—	4	1	no	—
26	C₂×C₄⋊Dic₁₁	—	4	1	no	—
27	C₁₁×#107	—	2	4	no	—
28	C₁₁×#107	—	2	2	no	—
29	C₁₁×#107	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₁	D₄	1	2	yes	—
32	#66882	—	2	4	no	—
33	#66882	—	4	2	no	—
34	C₂×C₁₁⋊C₁₆	—	2	2	no	—
35	C₂×C₁₁⋊C₁₆	—	2	2	no	—
36	C₂×C₁₁⋊C₁₆	Q₈	1	4	yes	complemented
37	#13282	—	11	1	no	maximal, Sylow
38	C₂×Q₈×C₁₁	—	4	4	no	—
39	C₂×Q₈×C₁₁	—	8	2	no	—
40	C₂×Q₈×C₁₁	—	8	2	no	—
41	C₂²×C₄×C₁₁	—	4	2	no	—
42	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₁	—	4	2	no	—
46	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
48	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
49	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₁₁	#41	1	2	yes	—
53	C₁₁×C₄⋊C₄	—	2	4	no	—
54	C₁₁×C₄⋊C₄	—	4	4	no	—
55	C₁₁×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₁₁	SD₁₆	1	2	yes	—
58	C₄⋊Dic₁₁	—	4	2	no	—
59	C₄⋊Dic₁₁	—	4	2	no	—
60	C₄⋊Dic₁₁	—	8	2	no	—
61	C₄⋊Dic₁₁	—	8	2	no	—
62	C₁₁⋊C₁₆	—	2	4	no	—
63	C₂×#500	D₁₁	1	1	yes	characteristic
64	C₂×C₄×C₁₆	—	11	1	no	—
65	#1845	—	11	1	no	—
66	#1380	—	11	4	no	—
67	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₁₁	—	8	8	no	—
69	C₂×C₄×C₁₁	—	4	4	no	—
70	C₂×C₄×C₁₁	—	8	2	no	—
71	C₂×C₄×C₁₁	—	8	2	no	—
72	C₂×C₄×C₁₁	#102	1	2	yes	—
73	C₂×C₄×C₁₁	#102	1	2	yes	—
74	C₂×C₄×C₁₁	D₁₆	1	2	yes	—
75	C₂×C₄×C₁₁	SD₃₂	1	2	yes	—
76	C₂×C₄×C₁₁	SD₃₂	1	2	yes	—
77	C₂×C₄×C₁₁	Q₃₂	1	2	yes	—
78	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×Dic₁₁	—	8	2	no	—
85	C₂×Dic₁₁	—	8	2	no	—
86	C₂×Dic₁₁	—	8	2	no	—
87	C₈×C₁₁	#103	1	4	yes	—
88	C₈×C₁₁	C₄×Q₈	1	2	yes	—
89	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄⋊Q₈	—	2	1	no	—
91	C₂²×C₁₆	—	11	2	no	—
92	#500	—	2	2	no	—
93	#500	Dic₁₁	1	4	yes	complemented
94	C₂×#107	—	2	1	no	—
95	C₂×#107	—	11	2	no	—
96	C₂×#107	—	22	1	no	—
97	C₂×#107	C₂×D₁₁	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
99	#384	—	22	1	no	—
100	C₄×C₁₆	—	22	2	no	—
101	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
102	C₂²×C₁₁	#486	1	1	yes	characteristic
103	C₂²×C₁₁	#500	1	1	yes	characteristic
104	C₂²×C₁₁	C₂×D₁₆	1	1	yes	characteristic
105	C₂²×C₁₁	C₂×SD₃₂	1	1	yes	characteristic
106	C₂²×C₁₁	C₂×SD₃₂	1	1	yes	characteristic
107	C₂²×C₁₁	C₂×Q₃₂	1	1	yes	characteristic
108	C₄×C₁₁	—	8	4	no	—
109	C₄×C₁₁	#357	1	4	yes	—
110	C₄×C₁₁	#358	1	4	yes	—
111	C₄×C₁₁	#391	1	1	yes	characteristic
112	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
113	C₄×C₁₁	#487	1	1	yes	characteristic
114	C₄×C₁₁	#498	1	1	yes	characteristic
115	Dic₁₁	—	16	2	no	—
116	C₂²×Q₈	—	4	1	no	—
117	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
119	C₄⋊Q₈	—	2	4	no	—
120	C₄⋊Q₈	—	4	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	4	1	no	—
123	C₂×C₄⋊C₄	—	22	2	no	—
124	C₂×C₄⋊C₄	—	44	1	no	—
125	C₂×C₄⋊C₄	—	44	1	no	—
126	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
127	C₂×C₁₆	—	11	4	no	—
128	C₂×C₁₆	—	22	2	no	—
129	C₂×C₁₆	—	22	2	no	—
130	#107	—	2	4	no	—
131	#107	—	2	2	no	—
132	#107	—	22	4	no	—
133	#107	—	44	2	no	—
134	#107	C₂×Dic₁₁	1	4	yes	—
135	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
136	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
137	C₂×C₁₁	#1990	1	1	yes	characteristic
138	C₂×C₁₁	C₂×#357	1	1	yes	characteristic
139	C₂×C₁₁	C₂×#358	1	1	yes	characteristic
140	C₂×C₁₁	#2240	1	1	yes	characteristic
141	C₂×C₁₁	#2241	1	1	yes	characteristic
142	C₂×C₁₁	#2252	1	1	yes	characteristic
143	C₂×C₁₁	#2254	1	1	yes	characteristic
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	8	2	no	—
146	C₂×Q₈	—	8	2	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	44	2	no	—
149	C₂²×C₄	#4323	1	1	yes	characteristic
150	C₂²×C₄	#4323	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
152	C₂×C₈	#4325	1	2	yes	—
153	C₂×C₈	#4326	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
155	C₂×C₈	Q₈×D₁₁	1	2	yes	—
156	C₂×C₈	#4343	1	2	yes	—
157	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
158	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	44	2	no	—
163	C₄⋊C₄	—	44	2	no	—
164	C₄⋊C₄	—	88	2	no	—
165	C₄⋊C₄	—	88	2	no	—
166	C₄²	#4324	1	2	yes	—
167	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
168	C₁₆	—	22	4	no	—
169	C₁₁	#13282	1	1	yes	characteristic, complemented, Sylow
170	C₂³	C₂×#4323	1	1	yes	characteristic
171	Q₈	—	8	8	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	88	2	no	—
176	C₂×C₄	—	88	2	no	—
177	C₂×C₄	—	88	2	no	—
178	C₂×C₄	#66890	1	2	yes	—
179	C₂×C₄	#66891	1	2	yes	—
180	C₂×C₄	#66892	1	2	yes	—
181	C₂×C₄	#66893	1	2	yes	—
182	C₂×C₄	#66896	1	2	yes	—
183	C₂×C₄	#66896	1	2	yes	—
184	C₂×C₄	C₂×#4324	1	1	yes	characteristic
185	C₂×C₄	C₂×#4325	1	1	yes	characteristic
186	C₂×C₄	C₂×#4326	1	1	yes	characteristic
187	C₂×C₄	#66997	1	1	yes	characteristic
188	C₂×C₄	#67000	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
190	C₈	#66899	1	4	yes	—
191	C₈	Q₈×Dic₁₁	1	2	yes	—
192	C₈	#66999	1	2	yes	—
193	C₂²	#10622691	1	1	yes	characteristic
194	C₂²	#10622701	1	1	yes	characteristic
195	C₂²	C₂×#66890	1	1	yes	characteristic
196	C₂²	C₂×#66891	1	1	yes	characteristic
197	C₂²	C₂×#66892	1	1	yes	characteristic
198	C₂²	C₂×#66893	1	1	yes	characteristic
199	C₂²	C₂×#66896	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	176	2	no	—
202	C₄	#10622200	1	4	yes	—
203	C₄	#10622202	1	4	yes	—
204	C₄	#10622692	1	1	yes	characteristic
205	C₄	#10622706	1	1	yes	characteristic
206	C₄	C₂×#66899	1	1	yes	characteristic
207	C₄	#10622843	1	1	yes	characteristic
208	C₂	#13069578	1	1	yes	characteristic
209	C₂	C₂×#10622200	1	1	yes	characteristic
210	C₂	C₂×#10622202	1	1	yes	characteristic
211	C₂	#13070841	1	1	yes	characteristic
212	C₂	#13070845	1	1	yes	characteristic
213	C₂	#13070889	1	1	yes	characteristic
214	C₂	#13070896	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#4323
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#500	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#500	D₁₁
hypocenter	C₁₁	#13282
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
7. C₁₁×#500	C₄	semidirect
36. C₂×C₁₁⋊C₁₆	Q₈	semidirect
93. #500	Dic₁₁	semidirect
169. C₁₁	#13282	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
#13068776	2
#13066707	2
C₂×C₁₁×#500	2
#13282	11