Subgroups — #14671799

Order 2816.

284 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070382	C₂	1	2	yes	maximal
2	C₂×#10622101	C₂	1	1	yes	maximal, characteristic
3	C₄×#66876	C₂	1	1	yes	maximal, characteristic
4	#13066702	C₂	1	2	yes	maximal
5	C₁₁×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#108	C₄	1	1	yes	characteristic
7	C₁₁×#427	C₄	1	1	yes	characteristic
8	C₁₁×#405	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×#66876	C₂²	1	2	yes	—
11	C₂×#66876	C₂²	1	1	yes	characteristic
12	C₂×#66876	C₂²	1	1	yes	characteristic
13	C₁₁×#339	C₄	1	2	yes	—
14	#10622101	—	2	1	no	—
15	#10622101	—	2	1	no	—
16	#10622101	—	2	4	no	—
17	#10622099	—	2	1	no	—
18	C₄×C₁₁⋊C₁₆	—	2	1	no	—
19	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₁₁×#117	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₁₁×#108	C₈	1	2	yes	—
25	C₁₁×#108	C₈	1	2	yes	—
26	C₁₁×#107	—	2	1	no	—
27	C₁₁×#106	—	2	1	no	—
28	C₁₁×#97	—	2	2	no	—
29	C₄×C₈×C₁₁	—	2	2	no	—
30	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
34	#66876	—	2	1	no	—
35	#66876	—	2	1	no	—
36	#66876	—	2	2	no	—
37	#66876	—	2	2	no	—
38	#66876	D₄	1	2	yes	—
39	#66876	Q₈	1	2	yes	—
40	C₂×C₁₁⋊C₁₆	—	2	1	no	—
41	C₂×C₁₁⋊C₁₆	—	2	1	no	—
42	C₂×C₁₁⋊C₁₆	—	2	2	no	—
43	#13164	—	11	1	no	maximal, Sylow
44	C₂²×C₄×C₁₁	—	2	1	no	—
45	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
46	C₁₁×M₁₆	—	2	2	no	—
47	C₁₁×M₁₆	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₁₁	—	2	2	no	—
49	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₁₁	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₁	M₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
60	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
61	C₁₁×C₄⋊C₄	—	2	2	no	—
62	C₁₁×C₂²⋊C₄	—	2	2	no	—
63	C₄²×C₁₁	—	2	1	no	—
64	C₄²×C₁₁	—	2	2	no	—
65	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
67	C₁₁⋊C₁₆	—	2	2	no	—
68	C₁₁⋊C₁₆	—	2	2	no	—
69	C₁₁⋊C₁₆	—	4	2	no	—
70	#2153	—	11	2	no	—
71	C₂×#364	—	11	1	no	—
72	C₄×M₃₂	—	11	1	no	—
73	#1772	D₁₁	1	1	yes	characteristic
74	#1378	—	11	2	no	—
75	C₂³×C₁₁	#131	1	1	yes	characteristic
76	C₂×C₄×C₁₁	—	2	1	no	—
77	C₂×C₄×C₁₁	—	2	1	no	—
78	C₂×C₄×C₁₁	—	4	2	no	—
79	C₂×C₄×C₁₁	—	4	2	no	—
80	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₁	#131	1	1	yes	characteristic
86	C₈×C₁₁	—	2	4	no	—
87	C₈×C₁₁	C₄⋊C₈	1	1	yes	characteristic
88	C₈×C₁₁	C₄⋊C₈	1	1	yes	characteristic
89	C₈×C₁₁	C₄⋊C₈	1	1	yes	characteristic
90	C₈×C₁₁	C₄⋊C₈	1	1	yes	characteristic
91	C₈×C₁₁	#121	1	1	yes	characteristic
92	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
93	C₈×C₁₁	#125	1	1	yes	characteristic
94	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×M₃₂	—	11	1	no	—
96	C₂×M₃₂	—	11	1	no	—
97	C₂×M₃₂	—	11	2	no	—
98	C₂×#108	Dic₁₁	1	1	yes	characteristic
99	#427	Dic₁₁	1	1	yes	characteristic
100	#405	C₂×D₁₁	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
102	#364	—	22	4	no	—
103	#364	—	22	1	no	—
104	#364	—	22	1	no	—
105	#346	—	22	1	no	—
106	C₄×C₁₆	—	22	1	no	—
107	#339	Dic₁₁	1	2	yes	—
108	C₂²×C₁₁	—	4	1	no	—
109	C₂²×C₁₁	#424	1	1	yes	characteristic
110	C₂²×C₁₁	#436	1	1	yes	characteristic
111	C₂²×C₁₁	C₈×Q₈	1	1	yes	characteristic
112	C₄×C₁₁	—	2	2	no	—
113	C₄×C₁₁	—	4	1	no	—
114	C₄×C₁₁	—	8	2	no	—
115	C₄×C₁₁	#389	1	1	yes	characteristic
116	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₄×C₁₁	C₈×D₄	1	1	yes	characteristic
118	C₄×C₁₁	#446	1	1	yes	characteristic
119	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
120	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
122	#117	C₂×Dic₁₁	1	2	yes	—
123	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
124	M₃₂	—	11	2	no	—
125	M₃₂	—	11	2	no	—
126	M₃₂	—	22	1	no	—
127	M₃₂	—	22	1	no	—
128	M₃₂	—	22	2	no	—
129	M₃₂	—	22	2	no	—
130	C₂×C₁₆	—	22	1	no	—
131	C₂×C₁₆	—	22	1	no	—
132	C₂×C₁₆	—	22	2	no	—
133	#108	C₁₁⋊C₈	1	2	yes	—
134	#108	C₁₁⋊C₈	1	2	yes	—
135	#107	—	2	1	no	—
136	#106	—	2	1	no	—
137	#97	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	Dic₂₂	1	1	yes	characteristic
140	C₄×C₈	Dic₂₂	1	1	yes	characteristic
141	C₄×C₈	D₄₄	1	1	yes	characteristic
142	C₄×C₈	D₄₄	1	1	yes	characteristic
143	C₂×C₁₁	—	2	1	no	—
144	C₂×C₁₁	—	2	1	no	—
145	C₂×C₁₁	—	2	1	no	—
146	C₂×C₁₁	#1939	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	C₂×C₁₁⋊C₈	1	4	yes	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
153	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
154	C₂×C₈	#4318	1	1	yes	characteristic
155	C₂×C₈	#4318	1	1	yes	characteristic
156	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
157	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
158	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
159	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
160	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
161	C₂×C₈	#4341	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
163	C₂×C₈	#4343	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	2	no	—
168	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
169	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
170	C₁₆	—	22	2	no	—
171	C₁₆	—	22	2	no	—
172	C₁₆	—	44	2	no	—
173	C₁₁	#13164	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#67001	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×#4318	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₁₁	1	1	yes	characteristic
184	C₂×C₄	#67001	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
187	C₈	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
188	C₈	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
189	C₈	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
190	C₈	#66941	1	1	yes	characteristic
191	C₈	#66943	1	1	yes	characteristic
192	C₈	#66948	1	1	yes	characteristic
193	C₈	#66950	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#10622612	1	1	yes	characteristic
196	C₂²	#10622625	1	1	yes	characteristic
197	C₂²	Q₈×C₁₁⋊C₈	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	8	2	no	—
201	C₄	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
202	C₄	#10622598	1	1	yes	characteristic
203	C₄	D₄×C₁₁⋊C₈	1	1	yes	characteristic
204	C₄	#10622637	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#13069335	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#66948
commutator	C₂×C₄×C₁₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1772	C₂
socle	C₂×C₁₁	#1939
prime radical	C₂³×C₁₁	#131
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1772	D₁₁
hypocenter	C₁₁	#13164
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
173. C₁₁	#13164	semidirect

Maximal subgroups

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

subgroup	index
#13070382	2
C₂×#10622101	2
C₄×#66876	2
#13066702	2
C₁₁×#1772	2
#13164	11