Subgroups — #14668992

Order 2816.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622202	C₂	1	2	yes	maximal
2	C₂×#10622099	C₂	1	1	yes	maximal, characteristic
3	C₄×#66882	C₂	1	1	yes	maximal, characteristic
4	#13066707	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₁	—	2	1	no	—
8	C₄×C₁₁×Q₁₆	—	2	2	no	—
9	C₄×C₁₁×Q₁₆	C₄	1	4	yes	complemented
10	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×#103	—	2	1	no	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66882	C₂²	1	2	yes	—
14	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
15	C₄×C₄⋊Dic₁₁	—	2	1	no	—
16	#10622202	C₄	1	8	yes	complemented
17	#10622189	—	2	1	no	—
18	#10622099	—	2	2	no	—
19	C₂²×Q₈×C₁₁	—	2	1	no	—
20	C₂×C₁₁×Q₁₆	—	2	1	no	—
21	C₂×C₁₁×Q₁₆	—	2	2	no	—
22	C₂×C₁₁×Q₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
26	C₄×Q₈×C₁₁	—	2	4	no	—
27	C₄×Q₈×C₁₁	—	4	2	no	—
28	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₁×C₄⋊C₄	—	4	1	no	—
30	C₂×C₄²×C₁₁	—	4	1	no	—
31	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₁	—	2	2	no	—
33	C₂×C₄×Dic₁₁	—	4	1	no	—
34	C₁₁×#107	—	2	1	no	—
35	C₁₁×#107	C₂×C₄	1	2	yes	—
36	C₁₁×#103	—	2	4	no	—
37	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₁	D₄	1	2	yes	—
39	C₁₁⋊#95	—	4	1	no	—
40	#66882	—	2	2	no	—
41	#66882	C₂×C₄	1	4	yes	—
42	C₂×C₁₁⋊C₁₆	—	2	2	no	—
43	C₂×C₁₁⋊C₁₆	—	2	2	no	—
44	C₂×C₁₁⋊C₁₆	C₂×C₄	1	4	yes	—
45	#12638	—	11	1	no	maximal, Sylow
46	C₂×Q₈×C₁₁	—	2	2	no	—
47	C₂×Q₈×C₁₁	—	2	4	no	—
48	C₂×Q₈×C₁₁	—	4	1	no	—
49	C₂×Q₈×C₁₁	—	4	2	no	—
50	C₂²×C₄×C₁₁	—	4	1	no	—
51	C₂²×C₄×C₁₁	—	4	1	no	—
52	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₁₁	—	4	2	no	—
56	C₁₁×Q₁₆	—	2	2	no	—
57	C₁₁×Q₁₆	—	2	4	no	—
58	C₁₁×Q₁₆	C₄²	1	4	yes	complemented
59	C₂×C₈×C₁₁	—	2	2	no	—
60	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₁	#41	1	2	yes	—
65	C₁₁×C₄⋊C₄	—	2	2	no	—
66	C₁₁×C₄⋊C₄	—	4	4	no	—
67	C₁₁×C₄⋊C₄	—	4	1	no	—
68	C₄²×C₁₁	—	4	4	no	—
69	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₁₁	SD₁₆	1	2	yes	—
71	C₄⋊Dic₁₁	—	2	4	no	—
72	C₄⋊Dic₁₁	—	4	2	no	—
73	C₄×Dic₁₁	—	8	2	no	—
74	C₁₁⋊C₁₆	—	2	4	no	—
75	C₂×C₄×Q₁₆	D₁₁	1	1	yes	characteristic
76	C₂×#358	—	11	2	no	—
77	C₂×#346	—	11	1	no	—
78	C₄×#107	—	11	1	no	—
79	#1380	—	11	2	no	—
80	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
81	Q₈×C₁₁	—	2	4	no	—
82	Q₈×C₁₁	—	4	2	no	—
83	Q₈×C₁₁	—	4	4	no	—
84	C₂×C₄×C₁₁	—	2	1	no	—
85	C₂×C₄×C₁₁	—	2	2	no	—
86	C₂×C₄×C₁₁	—	2	1	no	—
87	C₂×C₄×C₁₁	—	4	2	no	—
88	C₂×C₄×C₁₁	—	4	4	no	—
89	C₂×C₄×C₁₁	—	4	2	no	—
90	C₂×C₄×C₁₁	—	8	1	no	—
91	C₂×C₄×C₁₁	—	8	1	no	—
92	C₂×C₄×C₁₁	#102	1	2	yes	—
93	C₂×C₄×C₁₁	#102	1	2	yes	—
94	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₁	#135	1	2	yes	—
98	C₂×Dic₁₁	—	4	4	no	—
99	C₂×Dic₁₁	—	8	2	no	—
100	C₂×Dic₁₁	—	8	2	no	—
101	C₈×C₁₁	—	2	2	no	—
102	C₈×C₁₁	C₂×C₄²	1	1	yes	characteristic
103	C₈×C₁₁	#117	1	1	yes	characteristic
104	C₈×C₁₁	C₄×D₄	1	2	yes	—
105	C₂²×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
106	C₂×C₄×Q₈	—	2	1	no	—
107	C₂²×C₁₆	—	11	2	no	—
108	C₄×Q₁₆	—	2	2	no	—
109	C₄×Q₁₆	Dic₁₁	1	4	yes	complemented
110	C₂×#107	—	11	2	no	—
111	C₂×#107	C₂×D₁₁	1	1	yes	characteristic
112	C₂×#103	—	2	1	no	—
113	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
114	C₄×C₄⋊C₄	—	22	1	no	—
115	#358	—	11	8	no	—
116	#346	—	22	2	no	—
117	#340	—	22	1	no	—
118	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
119	C₂²×C₁₁	C₄×D₈	1	2	yes	—
120	C₂²×C₁₁	#509	1	1	yes	characteristic
121	C₂²×C₁₁	#510	1	2	yes	—
122	C₂²×C₁₁	#510	1	1	yes	characteristic
123	C₄×C₁₁	—	2	2	no	—
124	C₄×C₁₁	—	2	2	no	—
125	C₄×C₁₁	—	4	4	no	—
126	C₄×C₁₁	—	8	2	no	—
127	C₄×C₁₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₁₁	#416	1	1	yes	characteristic
129	C₄×C₁₁	C₄×SD₁₆	1	2	yes	—
130	Dic₁₁	—	8	4	no	—
131	C₂²×Q₈	—	2	1	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	1	no	—
134	C₂×Q₁₆	C₄×D₁₁	1	4	yes	—
135	C₂×Q₁₆	C₂×Dic₁₁	1	2	yes	—
136	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	4	2	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	22	2	no	—
143	C₂×C₄²	—	4	1	no	—
144	C₂×C₄²	—	44	1	no	—
145	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
146	C₂×C₁₆	—	11	4	no	—
147	C₂×C₁₆	—	22	2	no	—
148	C₂×C₁₆	—	22	2	no	—
149	#107	—	2	1	no	—
150	#107	—	11	4	no	—
151	#107	—	22	2	no	—
152	#107	C₂×Dic₁₁	1	2	yes	—
153	#103	—	2	4	no	—
154	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
155	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
156	#95	—	44	1	no	—
157	C₂×C₁₁	C₄×#102	1	1	yes	characteristic
158	C₂×C₁₁	#2139	1	1	yes	characteristic
159	C₂×C₁₁	#2141	1	1	yes	characteristic
160	C₂×C₁₁	#2175	1	2	yes	—
161	C₂×C₁₁	#2176	1	2	yes	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	44	2	no	—
169	C₂²×C₄	#4323	1	1	yes	characteristic
170	C₂²×C₄	#4323	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
172	Q₁₆	—	2	2	no	—
173	Q₁₆	—	2	4	no	—
174	Q₁₆	C₄×Dic₁₁	1	4	yes	complemented
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
178	C₂×C₈	#4339	1	2	yes	—
179	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	22	4	no	—
185	C₄⋊C₄	—	44	2	no	—
186	C₄²	—	4	4	no	—
187	C₄²	—	88	2	no	—
188	C₄²	#4324	1	2	yes	—
189	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
190	C₁₆	—	22	4	no	—
191	C₁₁	#12638	1	1	yes	characteristic, complemented, Sylow
192	C₂³	C₂×#4323	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	4	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	44	4	no	—
205	C₂×C₄	—	88	2	no	—
206	C₂×C₄	—	88	2	no	—
207	C₂×C₄	#66896	1	2	yes	—
208	C₂×C₄	#66896	1	2	yes	—
209	C₂×C₄	C₄×C₂²⋊D₁₁	1	2	yes	—
210	C₂×C₄	C₂×#4324	1	1	yes	characteristic
211	C₂×C₄	#67003	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	C₂×C₄×Dic₁₁	1	1	yes	characteristic
215	C₈	#66979	1	1	yes	characteristic
216	C₈	C₄×C₂²⋊D₁₁	1	2	yes	—
217	C₂²	C₄×#4323	1	2	yes	—
218	C₂²	#10622792	1	2	yes	—
219	C₂²	#10622806	1	1	yes	characteristic
220	C₂²	#10622808	1	1	yes	characteristic
221	C₂²	C₂×#66896	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	2	no	—
226	C₄	—	88	4	no	—
227	C₄	C₄×#4324	1	2	yes	—
228	C₄	C₄×C₂²⋊Dic₁₁	1	1	yes	characteristic
229	C₄	#10622846	1	1	yes	characteristic
230	C₂	C₄×#66896	1	1	yes	characteristic
231	C₂	#13070328	1	1	yes	characteristic
232	C₂	#13070336	1	1	yes	characteristic
233	C₂	#13070478	1	2	yes	—
234	C₂	#13070485	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#4323
commutator	C₈×C₁₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₄×C₁₁×Q₁₆	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₁
hypocenter	C₁₁	#12638
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×C₁₁×Q₁₆	C₄	semidirect
16. #10622202	C₄	semidirect
58. C₁₁×Q₁₆	C₄²	semidirect
109. C₄×Q₁₆	Dic₁₁	semidirect
174. Q₁₆	C₄×Dic₁₁	semidirect
191. C₁₁	#12638	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622202	2
C₂×#10622099	2
C₄×#66882	2
#13066707	2
C₂×C₄×C₁₁×Q₁₆	2
#12638	11