Subgroups — #14643166

Order 2816.

802 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#4318	C₂	1	2	yes	maximal
2	C₁₁⋊#1273	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₁×M₁₆	C₄	1	2	yes	—
5	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×#4318	C₄	1	4	yes	—
7	C₄×C₁₁×M₁₆	—	2	4	no	—
8	C₂×C₁₁×#97	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
10	C₄×#4318	—	2	4	no	—
11	C₄×#4318	—	2	4	no	—
12	C₂×#66867	C₄	1	4	yes	—
13	C₂×C₄×C₁₁⋊C₈	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₁	C₂×C₄	1	2	yes	—
15	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₁×M₁₆	—	2	4	no	—
17	C₂×C₁₁×M₁₆	—	2	8	no	—
18	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₁	D₄	1	2	yes	—
22	C₂×C₄²×C₁₁	D₄	1	4	yes	—
23	C₂×C₄²×C₁₁	Q₈	1	2	yes	—
24	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
25	C₂×#4318	—	2	8	no	—
26	C₂×#4318	—	2	16	no	—
27	C₂²×C₁₁⋊C₈	C₂×C₄	1	8	yes	—
28	C₁₁×#97	—	2	4	no	—
29	C₄×C₈×C₁₁	—	2	4	no	—
30	#66867	—	2	8	no	—
31	C₄×C₁₁⋊C₈	—	2	4	no	—
32	C₄×C₁₁⋊C₈	—	2	4	no	—
33	#7852	—	11	1	no	maximal, Sylow
34	C₂⁴×C₁₁	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	—	2	8	no	—
36	C₂²×C₄×C₁₁	—	2	4	no	—
37	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
39	C₂²×C₄×C₁₁	C₂²⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₁	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
45	C₁₁×M₁₆	—	2	16	no	—
46	C₂×C₈×C₁₁	—	2	4	no	—
47	C₂×C₈×C₁₁	—	2	4	no	—
48	C₂×C₈×C₁₁	M₁₆	1	8	yes	complemented
49	C₄²×C₁₁	C₂²⋊C₄	1	4	yes	—
50	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₁₁	C₄⋊C₄	1	4	yes	—
52	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
53	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₁	C₂×D₄	1	2	yes	—
55	C₄²×C₁₁	C₂×Q₈	1	1	yes	characteristic
56	#4318	—	2	32	no	—
57	C₂×C₁₁⋊C₈	—	2	8	no	—
58	C₂×C₁₁⋊C₈	—	2	8	no	—
59	C₂×C₁₁⋊C₈	M₁₆	1	16	yes	complemented
60	C₂×C₄×M₁₆	—	11	2	no	—
61	C₂×C₄×M₁₆	D₁₁	1	1	yes	characteristic
62	#1273	—	11	4	no	—
63	C₂³×C₁₁	—	2	4	no	—
64	C₂³×C₁₁	—	2	2	no	—
65	C₂³×C₁₁	#95	1	2	yes	—
66	C₂³×C₁₁	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₁₁	—	2	4	no	—
68	C₂×C₄×C₁₁	—	2	2	no	—
69	C₂×C₄×C₁₁	—	2	8	no	—
70	C₂×C₄×C₁₁	—	2	4	no	—
71	C₂×C₄×C₁₁	—	2	8	no	—
72	C₂×C₄×C₁₁	—	2	4	no	—
73	C₂×C₄×C₁₁	#95	1	2	yes	—
74	C₂×C₄×C₁₁	#95	1	4	yes	—
75	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	2	yes	—
76	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×M₁₆	1	8	yes	—
80	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
81	C₈×C₁₁	—	2	8	no	—
82	C₁₁⋊C₈	—	2	16	no	—
83	C₂²×M₁₆	—	11	4	no	—
84	C₂²×M₁₆	Dic₁₁	1	2	yes	—
85	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
86	C₄×M₁₆	—	2	4	no	—
87	C₄×M₁₆	—	22	4	no	—
88	C₄×M₁₆	—	22	4	no	—
89	C₂×#97	—	11	4	no	—
90	C₂×#97	Dic₁₁	1	2	yes	—
91	C₂×C₄×C₈	—	11	4	no	—
92	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
93	C₂²×C₁₁	—	2	4	no	—
94	C₂²×C₁₁	—	2	2	no	—
95	C₂²×C₁₁	—	4	4	no	—
96	C₂²×C₁₁	C₂×#95	1	1	yes	characteristic
97	C₂²×C₁₁	C₄×M₁₆	1	4	yes	—
98	C₂²×C₁₁	C₄×M₁₆	1	2	yes	—
99	C₄×C₁₁	—	2	4	no	—
100	C₄×C₁₁	—	2	2	no	—
101	C₄×C₁₁	#407	1	4	yes	—
102	C₄×C₁₁	#407	1	2	yes	—
103	C₄×C₁₁	#423	1	4	yes	—
104	C₄×C₁₁	#423	1	2	yes	—
105	C₂³×C₄	C₄×D₁₁	1	2	yes	—
106	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
107	C₂×M₁₆	—	2	4	no	—
108	C₂×M₁₆	—	2	8	no	—
109	C₂×M₁₆	—	22	16	no	—
110	C₂×M₁₆	—	22	8	no	—
111	C₂²×C₈	—	11	8	no	—
112	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
113	C₂×C₄²	Dic₂₂	1	2	yes	—
114	C₂×C₄²	C₄×D₁₁	1	2	yes	—
115	C₂×C₄²	D₄₄	1	2	yes	—
116	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
117	C₂×C₄²	C₂²⋊D₁₁	1	4	yes	—
118	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
119	#97	—	2	4	no	—
120	#97	—	22	8	no	—
121	C₄×C₈	—	2	4	no	—
122	C₄×C₈	—	22	4	no	—
123	C₄×C₈	—	22	4	no	—
124	C₂×C₁₁	—	4	2	no	—
125	C₂×C₁₁	#1455	1	4	yes	—
126	C₂×C₁₁	#1728	1	2	yes	—
127	C₂×C₁₁	#1728	1	1	yes	characteristic
128	C₂⁴	C₄×Dic₁₁	1	1	yes	characteristic
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
132	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
133	C₂²×C₄	Dic₁₁⋊C₄	1	4	yes	—
134	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
135	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	4	yes	—
136	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
137	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
139	M₁₆	—	2	16	no	—
140	M₁₆	—	22	32	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	11	16	no	—
144	C₂×C₈	—	22	8	no	—
145	C₂×C₈	—	22	8	no	—
146	C₂×C₈	#4318	1	8	yes	complemented
147	C₄²	Dic₁₁⋊C₄	1	4	yes	—
148	C₄²	C₄⋊Dic₁₁	1	2	yes	—
149	C₄²	(C₂×D₁₁)⋊C₄	1	4	yes	—
150	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
151	C₄²	C₂×Dic₂₂	1	1	yes	characteristic
152	C₄²	C₂×D₄₄	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
154	C₁₁	#7852	1	1	yes	characteristic, complemented, Sylow
155	C₂³	—	2	4	no	—
156	C₂³	—	2	2	no	—
157	C₂³	C₁₁⋊#95	1	2	yes	—
158	C₂³	C₂×C₄×Dic₁₁	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	C₁₁⋊#95	1	2	yes	—
166	C₂×C₄	C₁₁⋊#95	1	4	yes	—
167	C₂×C₄	M₁₆×D₁₁	1	8	yes	—
168	C₂×C₄	C₂×#4318	1	4	yes	—
169	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
171	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
173	C₈	—	2	8	no	—
174	C₈	—	22	16	no	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	4	4	no	—
178	C₂²	C₄×#4318	1	2	yes	—
179	C₂²	M₁₆×Dic₁₁	1	4	yes	—
180	C₂²	C₂×C₁₁⋊#95	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	#10622537	1	2	yes	—
184	C₄	#10622731	1	4	yes	—
185	C₄	#10622739	1	4	yes	—
186	C₄	#10622820	1	2	yes	—
187	C₂	—	4	2	no	—
188	C₂	#13066919	1	4	yes	—
189	C₂	#13068309	1	2	yes	—
190	C₂	#13068310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₁
commutator, socle	C₂³×C₁₁	C₂×C₄²
Fitting	C₂×C₄×C₁₁×M₁₆	C₂
prime radical	C₂⁴×C₁₁	C₄²
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₁
hypocenter	C₁₁	#7852
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₁₁⋊#1273	C₂	semidirect
48. C₂×C₈×C₁₁	M₁₆	semidirect
59. C₂×C₁₁⋊C₈	M₁₆	semidirect
146. C₂×C₈	#4318	semidirect
154. C₁₁	#7852	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#4318	2
C₁₁⋊#1273	2
C₂×C₄×C₁₁×M₁₆	2
#7852	11