Subgroups — #14665203

Order 2816.

330 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070415	C₂	1	1	yes	maximal, characteristic
2	#13067375	C₂	1	1	yes	maximal, characteristic
3	#13066671	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₁×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×C₄⋊Q₈	—	2	1	no	—
6	C₁₁×#500	—	2	1	no	—
7	C₁₁×#500	—	2	1	no	—
8	C₁₁×#500	C₄	1	2	yes	—
9	C₁₁×#500	C₂²	1	2	yes	—
10	C₂×C₁₁×#107	—	2	1	no	—
11	C₂×C₁₁×#107	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622537	—	2	1	no	—
14	C₂²⋊(C₁₁⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
16	C₄×C₁₁⋊C₁₆	—	2	1	no	—
17	#10622096	—	2	1	no	—
18	#10622096	C₂²	1	2	yes	—
19	C₂²×Q₈×C₁₁	—	4	1	no	—
20	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
22	C₁₁×C₄⋊Q₈	—	2	2	no	—
23	C₁₁×C₄⋊Q₈	—	2	2	no	—
24	C₁₁×C₄⋊Q₈	—	4	1	no	—
25	C₁₁×C₄⋊Q₈	—	4	1	no	—
26	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₁×C₄⋊C₄	—	4	1	no	—
28	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
29	C₂×#4318	—	4	1	no	—
30	C₁₁×#107	—	2	2	no	—
31	C₁₁×#107	—	2	2	no	—
32	C₁₁×#107	—	2	2	no	—
33	C₁₁×#107	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
38	C₂×C₁₁⋊C₁₆	—	2	2	no	—
39	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
40	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
41	#11971	—	11	1	no	maximal, Sylow
42	C₂×Q₈×C₁₁	—	4	2	no	—
43	C₂×Q₈×C₁₁	—	4	2	no	—
44	C₂×Q₈×C₁₁	—	8	2	no	—
45	C₂×Q₈×C₁₁	—	8	2	no	—
46	C₂²×C₄×C₁₁	—	4	2	no	—
47	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₁	—	2	2	no	—
51	C₂×C₈×C₁₁	—	2	2	no	—
52	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
56	C₁₁×C₄⋊C₄	—	2	4	no	—
57	C₁₁×C₄⋊C₄	—	4	2	no	—
58	C₁₁×C₄⋊C₄	—	4	2	no	—
59	C₁₁×C₄⋊C₄	—	4	2	no	—
60	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₁	D₈	1	1	yes	characteristic
62	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
63	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
64	#4318	—	8	2	no	—
65	C₂×C₁₁⋊C₈	—	4	2	no	—
66	C₁₁⋊C₁₆	—	4	2	no	—
67	C₂×#500	D₁₁	1	1	yes	characteristic
68	#2162	—	11	1	no	—
69	#1567	—	11	1	no	—
70	#1364	—	11	4	no	—
71	C₂³×C₁₁	#102	1	1	yes	characteristic
72	Q₈×C₁₁	—	8	4	no	—
73	Q₈×C₁₁	—	8	2	no	—
74	Q₈×C₁₁	—	8	2	no	—
75	C₂×C₄×C₁₁	—	2	2	no	—
76	C₂×C₄×C₁₁	—	2	1	no	—
77	C₂×C₄×C₁₁	—	2	1	no	—
78	C₂×C₄×C₁₁	—	4	4	no	—
79	C₂×C₄×C₁₁	—	8	4	no	—
80	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
81	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
82	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
83	C₂×C₄×C₁₁	D₁₆	1	2	yes	—
84	C₂×C₄×C₁₁	SD₃₂	1	2	yes	—
85	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₁₁	—	2	1	no	—
89	C₈×C₁₁	—	2	1	no	—
90	C₈×C₁₁	#101	1	2	yes	—
91	C₈×C₁₁	#137	1	2	yes	—
92	C₁₁⋊C₈	—	8	2	no	—
93	C₂×C₄⋊Q₈	—	2	1	no	—
94	#500	—	2	1	no	—
95	#500	—	2	1	no	—
96	#500	Dic₁₁	1	2	yes	—
97	#500	C₂×D₁₁	1	2	yes	—
98	C₂×#107	—	2	1	no	—
99	C₂×#107	Dic₁₁	1	2	yes	—
100	#423	—	22	1	no	—
101	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
102	C₄⋊C₁₆	—	11	2	no	—
103	C₂²⋊C₁₆	—	22	1	no	—
104	C₄×C₁₆	—	22	1	no	—
105	#335	—	11	2	no	—
106	#335	—	22	1	no	—
107	C₂²×C₁₁	—	2	1	no	—
108	C₂²×C₁₁	—	2	1	no	—
109	C₂²×C₁₁	#357	1	2	yes	—
110	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
111	C₄×C₁₁	—	2	1	no	—
112	C₄×C₁₁	—	2	1	no	—
113	C₄×C₁₁	—	8	4	no	—
114	C₄×C₁₁	#357	1	2	yes	—
115	C₄×C₁₁	#362	1	2	yes	—
116	C₄×C₁₁	C₂×#101	1	1	yes	characteristic
117	C₄×C₁₁	#419	1	1	yes	characteristic
118	C₄×C₁₁	C₂×D₁₆	1	1	yes	characteristic
119	C₄×C₁₁	C₂×SD₃₂	1	1	yes	characteristic
120	C₂²×Q₈	—	4	1	no	—
121	C₂×M₁₆	—	44	1	no	—
122	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	4	1	no	—
127	C₄⋊Q₈	—	4	1	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
131	C₂×C₁₆	—	22	2	no	—
132	#107	—	2	2	no	—
133	#107	—	2	2	no	—
134	#107	—	2	2	no	—
135	#107	C₂×Dic₁₁	1	4	yes	—
136	C₄⋊C₈	—	22	2	no	—
137	C₄⋊C₈	—	44	1	no	—
138	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
141	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
142	C₂×C₁₁	—	2	2	no	—
143	C₂×C₁₁	#1546	1	1	yes	characteristic
144	C₂×C₁₁	C₂×#357	1	1	yes	characteristic
145	C₂×C₁₁	C₂×#362	1	1	yes	characteristic
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	8	2	no	—
149	C₂×Q₈	—	8	2	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	#4323	1	1	yes	characteristic
152	C₂²×C₄	#4324	1	1	yes	characteristic
153	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
154	M₁₆	—	88	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	44	2	no	—
158	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
159	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
160	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
161	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄²	#4323	1	1	yes	characteristic
167	C₄²	#4324	1	1	yes	characteristic
168	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
169	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
170	C₁₆	—	44	2	no	—
171	C₁₁	#11971	1	1	yes	characteristic, complemented, Sylow
172	C₂³	#66896	1	1	yes	characteristic
173	Q₈	—	8	2	no	—
174	Q₈	—	8	2	no	—
175	Q₈	—	8	4	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	#66890	1	2	yes	—
182	C₂×C₄	#66891	1	2	yes	—
183	C₂×C₄	#66896	1	1	yes	characteristic
184	C₂×C₄	#66896	1	1	yes	characteristic
185	C₂×C₄	#66896	1	1	yes	characteristic
186	C₂×C₄	C₂×#4323	1	1	yes	characteristic
187	C₂×C₄	C₂×#4324	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	88	2	no	—
192	C₈	#66900	1	2	yes	—
193	C₈	#66995	1	2	yes	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#10622200	1	2	yes	—
197	C₂²	C₂×#66896	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	#10622200	1	2	yes	—
202	C₄	#10622203	1	2	yes	—
203	C₄	C₂×#66890	1	1	yes	characteristic
204	C₄	C₂×#66891	1	1	yes	characteristic
205	C₄	#10622838	1	1	yes	characteristic
206	C₄	C₂×#66900	1	1	yes	characteristic
207	C₂	—	2	2	no	—
208	C₂	#13067288	1	1	yes	characteristic
209	C₂	C₂×#10622200	1	1	yes	characteristic
210	C₂	C₂×#10622203	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#66896
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#500	C₂
socle	C₂²×C₁₁	C₂×#102
prime radical	C₂³×C₁₁	#102
cosocle	C₄×C₈×C₁₁	C₂³
hypercenter	C₂×#500	D₁₁
hypocenter	C₁₁	#11971
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #13066671	C₂	semidirect
171. C₁₁	#11971	semidirect

Maximal subgroups

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

subgroup	index
#13070415	2
#13067375	2
#13066671	2
C₂×C₁₁×#500	2
#11971	11