Subgroups — #14640689

Order 2816.

282 conjugacy classes of subgroups.

The subgroup lattice has 196 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	C₂×#10622096	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₁×#107	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁×#107	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₄×C₁₁×C₄⋊C₄	—	2	1	no	—
7	C₂²×C₁₁⋊C₁₆	—	2	1	no	—
8	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
9	C₁₁×#340	—	2	1	no	—
10	C₄×C₁₁⋊C₁₆	—	2	1	no	—
11	C₄×C₁₁⋊C₁₆	C₄	1	2	yes	complemented
12	#10622096	—	2	1	no	—
13	#10622096	C₄	1	2	yes	complemented
14	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
16	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
17	C₂×C₄²×C₁₁	—	4	1	no	—
18	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₁₁⋊C₈	—	4	1	no	—
20	C₁₁×#107	—	2	2	no	—
21	C₁₁×#107	C₈	1	4	yes	complemented
22	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
26	C₁₁×#95	—	4	1	no	—
27	C₂×C₁₁⋊C₁₆	—	2	4	no	—
28	C₂×C₁₁⋊C₁₆	—	2	2	no	—
29	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
30	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
31	#7171	—	11	1	no	maximal, Sylow
32	C₂²×C₄×C₁₁	—	4	2	no	—
33	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₁	—	2	2	no	—
37	C₂×C₈×C₁₁	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₁₁	M₁₆	1	2	yes	—
43	C₁₁×C₄⋊C₄	—	2	4	no	—
44	C₁₁×C₄⋊C₄	—	4	2	no	—
45	C₄²×C₁₁	—	8	1	no	—
46	C₄²×C₁₁	—	8	1	no	—
47	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
50	C₄²×C₁₁	Q₁₆	1	1	yes	characteristic
51	C₂×C₁₁⋊C₈	—	4	2	no	—
52	C₂×C₁₁⋊C₈	—	8	1	no	—
53	C₂×C₁₁⋊C₈	—	8	1	no	—
54	C₁₁⋊C₁₆	—	2	4	no	—
55	C₂×C₄×C₁₆	—	11	1	no	—
56	C₄×#107	D₁₁	1	1	yes	characteristic
57	C₂×#335	—	11	1	no	—
58	C₂³×C₁₁	#102	1	1	yes	characteristic
59	C₂×C₄×C₁₁	—	2	2	no	—
60	C₂×C₄×C₁₁	—	4	4	no	—
61	C₂×C₄×C₁₁	—	8	2	no	—
62	C₂×C₄×C₁₁	—	8	2	no	—
63	C₂×C₄×C₁₁	#95	1	1	yes	characteristic
64	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
66	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
67	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
68	C₂×C₄×C₁₁	#104	1	2	yes	—
69	C₂×C₄×C₁₁	#106	1	1	yes	characteristic
70	C₂×C₄×C₁₁	#107	1	1	yes	characteristic
71	C₂×C₄×C₁₁	D₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₁	SD₃₂	1	1	yes	characteristic
73	C₂×C₄×C₁₁	SD₃₂	1	1	yes	characteristic
74	C₂×C₄×C₁₁	Q₃₂	1	1	yes	characteristic
75	C₈×C₁₁	—	2	1	no	—
76	C₈×C₁₁	—	2	1	no	—
77	C₈×C₁₁	C₄×C₈	1	1	yes	characteristic
78	C₈×C₁₁	#97	1	1	yes	characteristic
79	C₈×C₁₁	C₄⋊C₈	1	2	yes	—
80	C₁₁⋊C₈	—	8	2	no	—
81	C₂²×C₁₆	—	22	1	no	—
82	C₂×#107	Dic₁₁	1	2	yes	—
83	C₂×C₄⋊C₈	—	22	1	no	—
84	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
85	C₄×C₄⋊C₄	—	2	1	no	—
86	C₄×C₁₆	—	11	2	no	—
87	C₄×C₁₆	—	22	1	no	—
88	#340	—	2	1	no	—
89	#335	—	11	2	no	—
90	#335	—	22	1	no	—
91	C₂²×C₁₁	D₄⋊C₈	1	2	yes	—
92	C₂²×C₁₁	#340	1	1	yes	characteristic
93	C₂²×C₁₁	#357	1	1	yes	characteristic
94	C₂²×C₁₁	#358	1	1	yes	characteristic
95	C₂²×C₁₁	#359	1	2	yes	—
96	C₄×C₁₁	—	2	1	no	—
97	C₄×C₁₁	—	2	1	no	—
98	C₄×C₁₁	—	8	4	no	—
99	C₄×C₁₁	Q₈⋊C₈	1	2	yes	—
100	C₄×C₁₁	#336	1	1	yes	characteristic
101	C₄×C₁₁	#339	1	1	yes	characteristic
102	C₄×C₁₁	#357	1	1	yes	characteristic
103	C₄×C₁₁	#358	1	1	yes	characteristic
104	C₄×C₁₁	#366	1	1	yes	characteristic
105	C₄×C₁₁	#367	1	1	yes	characteristic
106	C₂²×C₈	—	44	1	no	—
107	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	—	4	1	no	—
111	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
112	C₂×C₁₆	—	22	4	no	—
113	C₂×C₁₆	—	22	2	no	—
114	#107	—	2	2	no	—
115	#107	C₁₁⋊C₈	1	4	yes	complemented
116	C₄⋊C₈	—	22	2	no	—
117	C₄⋊C₈	—	44	1	no	—
118	C₄×C₈	Dic₂₂	1	1	yes	characteristic
119	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
120	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
121	C₄×C₈	D₄₄	1	1	yes	characteristic
122	#95	—	4	1	no	—
123	C₂×C₁₁	#1279	1	1	yes	characteristic
124	C₂×C₁₁	#1331	1	2	yes	—
125	C₂×C₁₁	#1332	1	2	yes	—
126	C₂×C₁₁	#1380	1	1	yes	characteristic
127	C₂×C₁₁	#1382	1	1	yes	characteristic
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	#4323	1	1	yes	characteristic
130	C₂²×C₄	#4324	1	1	yes	characteristic
131	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	44	2	no	—
134	C₂×C₈	—	88	1	no	—
135	C₂×C₈	—	88	1	no	—
136	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
137	C₂×C₈	#4318	1	2	yes	—
138	C₂×C₈	C₄×Dic₁₁	1	1	yes	characteristic
139	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
140	C₂×C₈	C₄⋊Dic₁₁	1	1	yes	characteristic
141	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄²	—	8	1	no	—
145	C₄²	—	8	1	no	—
146	C₄²	Dic₁₁⋊C₄	1	1	yes	characteristic
147	C₄²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
148	C₄²	#4325	1	1	yes	characteristic
149	C₄²	#4326	1	1	yes	characteristic
150	C₁₆	—	22	4	no	—
151	C₁₁	#7171	1	1	yes	characteristic, complemented, Sylow
152	C₂³	#66896	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	#66871	1	1	yes	characteristic
158	C₂×C₄	#66872	1	1	yes	characteristic
159	C₂×C₄	#66873	1	1	yes	characteristic
160	C₂×C₄	#66874	1	1	yes	characteristic
161	C₂×C₄	#66890	1	1	yes	characteristic
162	C₂×C₄	#66891	1	1	yes	characteristic
163	C₂×C₄	#66892	1	1	yes	characteristic
164	C₂×C₄	#66893	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	2	yes	—
166	C₂×C₄	C₁₁⋊#95	1	1	yes	characteristic
167	C₂×C₄	#66899	1	1	yes	characteristic
168	C₂×C₄	#66901	1	2	yes	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	88	2	no	—
172	C₈	C₄×C₁₁⋊C₈	1	1	yes	characteristic
173	C₈	#66867	1	1	yes	characteristic
174	C₈	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
175	C₂²	#10622167	1	1	yes	characteristic
176	C₂²	D₄⋊(C₁₁⋊C₈)	1	2	yes	—
177	C₂²	#10622200	1	1	yes	characteristic
178	C₂²	#10622202	1	1	yes	characteristic
179	C₂²	#10622205	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	4	no	—
183	C₄	#10622127	1	1	yes	characteristic
184	C₄	#10622128	1	1	yes	characteristic
185	C₄	#10622129	1	1	yes	characteristic
186	C₄	#10622130	1	1	yes	characteristic
187	C₄	#10622162	1	1	yes	characteristic
188	C₄	#10622170	1	1	yes	characteristic
189	C₄	Q₈⋊(C₁₁⋊C₈)	1	2	yes	—
190	C₂	#13066457	1	1	yes	characteristic
191	C₂	#13066577	1	2	yes	—
192	C₂	#13066580	1	2	yes	—
193	C₂	#13066707	1	1	yes	characteristic
194	C₂	#13066711	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#4323
commutator	C₈×C₁₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₁
Fitting	C₄×C₁₁×#107	C₂
socle, prime radical	C₂³×C₁₁	#102
cosocle	C₂×C₄×C₈×C₁₁	C₂²
hypercenter	C₄×#107	D₁₁
hypocenter	C₁₁	#7171
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. C₄×C₁₁⋊C₁₆	C₄	semidirect
13. #10622096	C₄	semidirect
21. C₁₁×#107	C₈	semidirect
115. #107	C₁₁⋊C₈	semidirect
151. C₁₁	#7171	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
C₂×#10622096	2
C₄×C₁₁×#107	2
#7171	11