Subgroups — #14723071

Order 2816.

612 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077748	C₂	1	2	yes	maximal
2	#13067405	C₂	1	6	yes	maximal, complemented
3	#13067372	C₂	1	3	yes	maximal, complemented
4	#13067364	C₂	1	3	yes	maximal, complemented
5	C₈×C₁₁×#41	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₁×#41	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₁	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₁₁	C₄	1	1	yes	characteristic
9	C₈×D₄×C₁₁	C₄	1	3	yes	—
10	C₈×D₄×C₁₁	C₄	1	3	yes	—
11	C₁₁×#431	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
13	Q₈×C₁₁⋊C₈	—	2	2	no	—
14	#10622625	—	2	6	no	—
15	#10622538	C₂²	1	6	yes	—
16	#10622537	C₂²	1	6	yes	—
17	#10622096	C₂²	1	6	yes	—
18	#10622095	C₂²	1	1	yes	characteristic
19	#10622095	C₂²	1	3	yes	—
20	#10622095	C₂²	1	6	yes	—
21	C₂×C₁₁×#41	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₁₁	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₁₁	C₂×C₄	1	3	yes	—
24	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₁	D₄	1	1	yes	characteristic
26	C₄×D₄×C₁₁	C₂×C₄	1	3	yes	—
27	C₄×D₄×C₁₁	D₄	1	3	yes	—
28	C₁₁×#117	Q₈	1	3	yes	—
29	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
30	C₂×#4318	—	2	6	no	—
31	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
32	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₁₁×C₂²⋊C₈	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
37	C₄×C₈×C₁₁	C₂³	1	3	yes	—
38	C₂²⋊(C₁₁⋊C₈)	—	2	6	no	—
39	C₄⋊(C₁₁⋊C₈)	—	2	6	no	—
40	C₄⋊(C₁₁⋊C₈)	C₂³	1	6	yes	—
41	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
42	C₄×C₁₁⋊C₈	—	2	6	no	—
43	#17835	—	11	1	no	maximal, Sylow
44	C₁₁×#41	—	2	4	no	—
45	C₂×Q₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×D₄×C₁₁	C₄⋊C₄	1	3	yes	—
47	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₁₁	C₂×Q₈	1	3	yes	—
50	C₂×C₈×C₁₁	—	2	6	no	—
51	C₂×C₈×C₁₁	—	4	3	no	—
52	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
57	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
58	C₁₁×C₂²⋊C₄	C₄⋊C₄	1	6	yes	—
59	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
60	C₄²×C₁₁	C₂×D₄	1	3	yes	—
61	C₄²×C₁₁	C₂×Q₈	1	3	yes	—
62	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
63	#4318	—	8	6	no	—
64	C₂×C₁₁⋊C₈	—	2	2	no	—
65	C₂×C₁₁⋊C₈	—	2	6	no	—
66	#2991	—	11	2	no	—
67	C₈×#41	D₁₁	1	1	yes	characteristic
68	#1574	—	11	6	no	—
69	#1566	—	11	3	no	—
70	#1564	—	11	3	no	—
71	C₂³×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
72	Q₈×C₁₁	#108	1	4	yes	—
73	D₄×C₁₁	—	2	6	no	—
74	C₂×C₄×C₁₁	—	2	3	no	—
75	C₂×C₄×C₁₁	—	2	3	no	—
76	C₂×C₄×C₁₁	—	2	6	no	—
77	C₂×C₄×C₁₁	—	2	6	no	—
78	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
79	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
80	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
81	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
84	C₈×C₁₁	—	2	3	no	—
85	C₈×C₁₁	—	2	1	no	—
86	C₈×C₁₁	—	2	3	no	—
87	C₈×C₁₁	#131	1	2	yes	—
88	C₁₁⋊C₈	—	4	2	no	—
89	C₁₁⋊C₈	—	4	6	no	—
90	C₄×#41	C₂×D₁₁	1	1	yes	characteristic
91	C₈×Q₈	—	22	2	no	—
92	C₈×Q₈	Dic₁₁	1	1	yes	characteristic
93	C₈×Q₈	Dic₁₁	1	1	yes	characteristic
94	#436	—	22	6	no	—
95	C₈×D₄	Dic₁₁	1	3	yes	—
96	C₈×D₄	Dic₁₁	1	3	yes	—
97	#431	—	11	6	no	—
98	#431	C₂×D₁₁	1	3	yes	—
99	#423	—	11	6	no	—
100	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
101	#335	—	11	6	no	—
102	#334	—	11	1	no	—
103	#334	—	11	3	no	—
104	#334	—	11	6	no	—
105	C₂²×C₁₁	—	2	3	no	—
106	C₂²×C₁₁	—	2	3	no	—
107	C₂²×C₁₁	—	2	3	no	—
108	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₁₁	—	2	3	no	—
110	C₄×C₁₁	—	2	3	no	—
111	C₄×C₁₁	—	2	1	no	—
112	C₄×C₁₁	C₂×#108	1	6	yes	—
113	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
114	C₄×C₁₁	#568	1	1	yes	characteristic
115	C₄×C₁₁	#577	1	2	yes	—
116	C₂×#41	Dic₂₂	1	1	yes	characteristic
117	C₂×M₁₆	—	22	6	no	—
118	C₂²×C₈	C₂×Dic₁₁	1	3	yes	—
119	C₂²×C₈	C₂×Dic₁₁	1	3	yes	—
120	C₄×Q₈	D₄₄	1	1	yes	characteristic
121	C₄×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
122	C₄×D₄	D₄₄	1	3	yes	—
123	C₄×D₄	C₂×Dic₁₁	1	3	yes	—
124	#117	Dic₂₂	1	3	yes	—
125	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
126	C₄⋊C₈	—	11	6	no	—
127	C₄⋊C₈	—	11	2	no	—
128	C₄⋊C₈	—	22	6	no	—
129	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
130	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
131	C₂²⋊C₈	—	22	6	no	—
132	C₂²⋊C₈	C₂×Dic₁₁	1	6	yes	—
133	C₄×C₈	—	22	6	no	—
134	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
135	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
136	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
137	C₄×C₈	C₂²×D₁₁	1	3	yes	—
138	C₂×C₁₁	—	4	3	no	—
139	C₂×C₁₁	#2906	1	1	yes	characteristic
140	C₂×C₁₁	#2912	1	1	yes	characteristic
141	C₂×C₁₁	C₂²×#108	1	1	yes	characteristic
142	#41	—	2	4	no	—
143	C₂×Q₈	C₄⋊Dic₁₁	1	1	yes	characteristic
144	C₂×D₄	C₄⋊Dic₁₁	1	3	yes	—
145	C₂²×C₄	C₂×Dic₂₂	1	3	yes	—
146	C₂²×C₄	C₂×D₄₄	1	3	yes	—
147	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
148	M₁₆	—	88	6	no	—
149	C₂×C₈	—	2	6	no	—
150	C₂×C₈	—	4	3	no	—
151	C₂×C₈	—	22	2	no	—
152	C₂×C₈	—	22	6	no	—
153	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
154	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
155	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
156	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
157	C₄⋊C₄	C₄⋊Dic₁₁	1	3	yes	—
158	C₄⋊C₄	C₄⋊Dic₁₁	1	3	yes	—
159	C₂²⋊C₄	C₄⋊Dic₁₁	1	6	yes	—
160	C₄²	C₂×Dic₂₂	1	3	yes	—
161	C₄²	C₂×D₄₄	1	3	yes	—
162	C₄²	C₂²×Dic₁₁	1	3	yes	—
163	C₄²	C₂³×D₁₁	1	1	yes	characteristic
164	C₁₁	#17835	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₂×C₄⋊Dic₁₁	1	3	yes	—
166	Q₈	#66883	1	4	yes	—
167	D₄	—	2	6	no	—
168	C₂×C₄	—	2	6	no	—
169	C₂×C₄	—	2	6	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	2	3	no	—
172	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
175	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
176	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
177	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
178	C₈	—	2	3	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	3	no	—
181	C₈	—	44	6	no	—
182	C₈	—	44	2	no	—
183	C₈	#67001	1	2	yes	—
184	C₂²	—	2	3	no	—
185	C₂²	—	2	3	no	—
186	C₂²	—	2	3	no	—
187	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
188	C₄	—	2	3	no	—
189	C₄	—	2	3	no	—
190	C₄	—	2	1	no	—
191	C₄	C₂×#66883	1	6	yes	—
192	C₄	#10623282	1	2	yes	—
193	C₄	C₂×#67001	1	1	yes	characteristic
194	C₄	#10623350	1	1	yes	characteristic
195	C₂	—	4	3	no	—
196	C₂	#13077184	1	1	yes	characteristic
197	C₂	#13077211	1	1	yes	characteristic
198	C₂	C₂²×#66883	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₈×C₁₁×#41	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂×C₁₁×#41	Q₈
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₈×#41	D₁₁
hypocenter	C₁₁	#17835
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #13067405	C₂	semidirect
3. #13067372	C₂	semidirect
4. #13067364	C₂	semidirect
164. C₁₁	#17835	semidirect

Maximal subgroups

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

subgroup	index
#13077748	2
#13067405	2
#13067372	2
#13067364	2
C₈×C₁₁×#41	2
#17835	11