Subgroups — #14723010

Order 2816.

702 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	#13077184	C₂	1	2	yes	maximal
2	#13067391	C₂	1	6	yes	maximal, complemented
3	#13067375	C₂	1	3	yes	maximal, complemented
4	C₂×#10622096	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	C₂×#67001	—	2	2	no	—
8	C₈×Q₈×C₁₁	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₁₁	C₄	1	1	yes	characteristic
10	C₈×D₄×C₁₁	C₄	1	3	yes	—
11	C₈×D₄×C₁₁	C₄	1	3	yes	—
12	C₁₁×#431	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
14	#10622612	—	2	6	no	—
15	#10622537	C₂²	1	6	yes	—
16	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	6	yes	—
17	#10622096	C₂²	1	1	yes	characteristic
18	#10622096	C₂²	1	3	yes	—
19	#10622096	C₂²	1	6	yes	—
20	#10622096	C₂²	1	6	yes	—
21	C₂×C₁₁×#41	D₄	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	Q₈	1	1	yes	characteristic
26	C₄×D₄×C₁₁	C₂×C₄	1	3	yes	—
27	C₄×D₄×C₁₁	Q₈	1	3	yes	—
28	C₁₁×#117	D₄	1	3	yes	—
29	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
30	#67001	—	4	8	no	—
31	C₂×#4318	—	2	6	no	—
32	C₂²×C₁₁⋊C₈	—	2	6	no	—
33	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
34	C₁₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
35	C₁₁×C₂²⋊C₈	C₂×C₄	1	6	yes	—
36	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₁₁	C₂×C₄	1	3	yes	—
38	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₁₁	C₂³	1	3	yes	—
40	C₄⋊(C₁₁⋊C₈)	—	2	6	no	—
41	C₄⋊(C₁₁⋊C₈)	—	2	6	no	—
42	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
43	C₄⋊(C₁₁⋊C₈)	C₂³	1	6	yes	—
44	#17828	—	11	1	no	maximal, Sylow
45	C₁₁×#41	D₈	1	4	yes	—
46	C₁₁×#41	Q₁₆	1	4	yes	—
47	C₂×Q₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₄×C₁₁	C₄⋊C₄	1	3	yes	—
49	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₁₁	C₂×Q₈	1	3	yes	—
52	C₂×C₈×C₁₁	—	2	6	no	—
53	C₂×C₈×C₁₁	—	4	3	no	—
54	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
58	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
59	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
60	C₁₁×C₂²⋊C₄	C₄⋊C₄	1	6	yes	—
61	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
62	C₄²×C₁₁	C₂×D₄	1	3	yes	—
63	C₄²×C₁₁	C₂×Q₈	1	3	yes	—
64	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
65	#4318	—	4	12	no	—
66	C₂×C₁₁⋊C₈	—	2	2	no	—
67	C₂×C₁₁⋊C₈	—	2	6	no	—
68	C₂×C₁₁⋊C₈	—	4	12	no	—
69	C₈×#41	D₁₁	1	1	yes	characteristic
70	#2906	—	11	2	no	—
71	#1571	—	11	6	no	—
72	#1567	—	11	3	no	—
73	C₂×#335	—	11	3	no	—
74	C₂³×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
75	Q₈×C₁₁	#107	1	4	yes	—
76	D₄×C₁₁	#107	1	12	yes	—
77	C₂×C₄×C₁₁	—	2	3	no	—
78	C₂×C₄×C₁₁	—	2	3	no	—
79	C₂×C₄×C₁₁	—	4	3	no	—
80	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
81	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
82	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
83	C₂×C₄×C₁₁	C₂×D₈	1	6	yes	—
84	C₂×C₄×C₁₁	C₂×Q₁₆	1	6	yes	—
85	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
88	C₈×C₁₁	—	2	3	no	—
89	C₈×C₁₁	—	2	1	no	—
90	C₈×C₁₁	—	2	3	no	—
91	C₈×C₁₁	#131	1	2	yes	complemented
92	C₁₁⋊C₈	—	4	2	no	—
93	C₁₁⋊C₈	—	4	6	no	—
94	C₂×#131	—	22	2	no	—
95	C₄×#41	C₂×D₁₁	1	1	yes	characteristic
96	C₈×Q₈	Dic₁₁	1	1	yes	characteristic
97	C₈×Q₈	Dic₁₁	1	1	yes	characteristic
98	C₈×D₄	Dic₁₁	1	3	yes	—
99	C₈×D₄	Dic₁₁	1	3	yes	—
100	#431	C₂×D₁₁	1	3	yes	—
101	#424	—	22	6	no	—
102	#423	—	11	6	no	—
103	C₂×C₄⋊C₈	—	11	6	no	—
104	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
105	#335	—	11	1	no	—
106	#335	—	11	3	no	—
107	#335	—	11	6	no	—
108	#335	—	11	6	no	—
109	C₂²×C₁₁	—	2	3	no	—
110	C₂²×C₁₁	—	2	3	no	—
111	C₂²×C₁₁	C₂×#107	1	6	yes	—
112	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₄×C₁₁	—	2	3	no	—
114	C₄×C₁₁	—	2	1	no	—
115	C₄×C₁₁	—	2	3	no	—
116	C₄×C₁₁	C₂×#107	1	6	yes	—
117	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
118	C₄×C₁₁	#568	1	1	yes	characteristic
119	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
121	C₂×#41	D₄₄	1	1	yes	characteristic
122	#131	—	44	8	no	—
123	C₂×M₁₆	—	22	6	no	—
124	C₂²×C₈	—	22	6	no	—
125	C₂²×C₈	C₂×Dic₁₁	1	3	yes	—
126	C₂²×C₈	C₂×Dic₁₁	1	3	yes	—
127	C₄×Q₈	Dic₂₂	1	1	yes	characteristic
128	C₄×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
129	C₄×D₄	Dic₂₂	1	3	yes	—
130	C₄×D₄	C₂×Dic₁₁	1	3	yes	—
131	#117	D₄₄	1	3	yes	—
132	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
133	C₄⋊C₈	—	11	2	no	—
134	C₄⋊C₈	—	11	6	no	—
135	C₄⋊C₈	—	22	6	no	—
136	C₄⋊C₈	—	22	6	no	—
137	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
138	C₄⋊C₈	C₂×Dic₁₁	1	3	yes	—
139	C₂²⋊C₈	C₂×Dic₁₁	1	6	yes	—
140	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
141	C₄×C₈	C₂×Dic₁₁	1	3	yes	—
142	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
143	C₄×C₈	C₂²×D₁₁	1	3	yes	—
144	C₂×C₁₁	—	2	6	no	—
145	C₂×C₁₁	#2906	1	1	yes	characteristic
146	C₂×C₁₁	C₂²×#107	1	1	yes	characteristic
147	C₂×C₁₁	#2916	1	1	yes	characteristic
148	#41	D₈₈	1	4	yes	—
149	#41	Dic₄₄	1	4	yes	—
150	C₂×Q₈	C₄⋊Dic₁₁	1	1	yes	characteristic
151	C₂×D₄	C₄⋊Dic₁₁	1	3	yes	—
152	C₂²×C₄	C₂×Dic₂₂	1	3	yes	—
153	C₂²×C₄	C₂×D₄₄	1	3	yes	—
154	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
155	M₁₆	—	44	12	no	—
156	C₂×C₈	—	2	6	no	—
157	C₂×C₈	—	4	3	no	—
158	C₂×C₈	—	22	2	no	—
159	C₂×C₈	—	22	6	no	—
160	C₂×C₈	—	44	12	no	—
161	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
162	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
163	C₂×C₈	C₂²×Dic₁₁	1	3	yes	—
164	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
165	C₄⋊C₄	C₄⋊Dic₁₁	1	3	yes	—
166	C₄⋊C₄	C₄⋊Dic₁₁	1	3	yes	—
167	C₂²⋊C₄	C₄⋊Dic₁₁	1	6	yes	—
168	C₄²	C₂×Dic₂₂	1	3	yes	—
169	C₄²	C₂×D₄₄	1	3	yes	—
170	C₄²	C₂²×Dic₁₁	1	3	yes	—
171	C₄²	C₂³×D₁₁	1	1	yes	characteristic
172	C₁₁	#17828	1	1	yes	characteristic, complemented, Sylow
173	C₂³	C₂×C₄⋊Dic₁₁	1	3	yes	—
174	Q₈	#66882	1	4	yes	—
175	D₄	#66882	1	12	yes	—
176	C₂×C₄	—	2	3	no	—
177	C₂×C₄	—	2	3	no	—
178	C₂×C₄	—	4	3	no	—
179	C₂×C₄	C₂×D₈₈	1	6	yes	—
180	C₂×C₄	C₂×Dic₄₄	1	6	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
184	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
185	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
186	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	3	no	—
190	C₈	—	44	2	no	—
191	C₈	—	44	6	no	—
192	C₈	#67001	1	2	yes	complemented
193	C₂²	—	2	3	no	—
194	C₂²	—	2	3	no	—
195	C₂²	C₂×#66882	1	6	yes	—
196	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	3	no	—
200	C₄	C₂×#66882	1	6	yes	—
201	C₄	C₂²×D₈₈	1	1	yes	characteristic
202	C₄	C₂²×Dic₄₄	1	1	yes	characteristic
203	C₄	C₂×#67001	1	1	yes	characteristic
204	C₄	#10623350	1	1	yes	characteristic
205	C₂	—	2	6	no	—
206	C₂	#13077184	1	1	yes	characteristic
207	C₂	C₂²×#66882	1	1	yes	characteristic
208	C₂	#13077227	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	D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₈×#41	D₁₁
hypocenter	C₁₁	#17828
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. #13067391	C₂	semidirect
3. #13067375	C₂	semidirect
4. C₂×#10622096	C₂	semidirect
91. C₈×C₁₁	#131	semidirect
172. C₁₁	#17828	semidirect
192. C₈	#67001	semidirect

Maximal subgroups

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

subgroup	index
#13077184	2
#13067391	2
#13067375	2
C₂×#10622096	2
C₈×C₁₁×#41	2
#17828	11