Subgroups — #14670240

Order 2816.

438 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#66876	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#13066694	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
9	C₄×C₁₁×C₂²⋊C₄	C₄	1	1	yes	characteristic
10	C₂×#66876	—	2	2	no	—
11	C₂×#66876	—	2	4	no	—
12	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
15	C₁₁×#336	C₄	1	2	yes	—
16	C₂²⋊(C₁₁⋊C₁₆)	—	2	2	no	—
17	C₄⋊(C₁₁⋊C₁₆)	—	2	4	no	—
18	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	—	2	2	no	—
20	C₂²×C₈×C₁₁	—	2	2	no	—
21	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	D₄	1	2	yes	—
24	C₂²×C₈×C₁₁	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
26	C₂×C₁₁×C₂²⋊C₄	C₈	1	2	yes	—
27	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
28	C₁₁×C₂²⋊C₈	—	2	2	no	—
29	C₄×C₈×C₁₁	—	2	2	no	—
30	C₄×C₈×C₁₁	—	2	2	no	—
31	C₁₁×#95	C₈	1	1	yes	characteristic
32	C₁₁×#95	C₈	1	1	yes	characteristic
33	#66876	—	2	8	no	—
34	C₂×C₁₁⋊C₁₆	—	2	2	no	—
35	C₂×C₁₁⋊C₁₆	—	2	2	no	—
36	C₂×C₁₁⋊C₁₆	—	2	2	no	—
37	C₂×C₁₁⋊C₁₆	—	2	1	no	—
38	C₂×C₁₁⋊C₁₆	—	2	1	no	—
39	C₂×C₁₁⋊C₁₆	—	2	2	no	—
40	C₂×C₁₁⋊C₁₆	D₄	1	4	yes	—
41	#12913	—	11	1	no	maximal, Sylow
42	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	—	2	2	no	—
44	C₂²×C₄×C₁₁	—	2	2	no	—
45	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
50	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₁	—	2	2	no	—
52	C₂×C₈×C₁₁	—	2	4	no	—
53	C₂×C₈×C₁₁	—	2	2	no	—
54	C₂×C₈×C₁₁	—	2	2	no	—
55	C₂×C₈×C₁₁	—	2	2	no	—
56	C₂×C₈×C₁₁	—	2	2	no	—
57	C₂×C₈×C₁₁	—	2	2	no	—
58	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₁₁	#41	1	2	yes	—
63	C₁₁×C₂²⋊C₄	—	2	4	no	—
64	C₄²×C₁₁	—	2	2	no	—
65	C₄²×C₁₁	—	2	2	no	—
66	C₁₁⋊C₁₆	—	2	4	no	—
67	C₁₁⋊C₁₆	—	4	1	no	—
68	C₁₁⋊C₁₆	—	4	1	no	—
69	C₂²×M₃₂	—	11	1	no	—
70	C₂×C₄⋊C₁₆	—	11	2	no	—
71	C₂×C₂²⋊C₁₆	—	11	1	no	—
72	C₈×C₂²⋊C₄	D₁₁	1	1	yes	characteristic
73	#1374	—	11	2	no	—
74	C₂³×C₁₁	—	2	2	no	—
75	C₂³×C₁₁	—	2	2	no	—
76	C₂³×C₁₁	—	2	1	no	—
77	C₂³×C₁₁	—	2	1	no	—
78	C₂³×C₁₁	C₄⋊C₈	1	2	yes	—
79	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₁	—	2	2	no	—
81	C₂×C₄×C₁₁	—	2	2	no	—
82	C₂×C₄×C₁₁	—	2	2	no	—
83	C₂×C₄×C₁₁	—	2	2	no	—
84	C₂×C₄×C₁₁	—	2	2	no	—
85	C₂×C₄×C₁₁	—	2	4	no	—
86	C₂×C₄×C₁₁	—	2	2	no	—
87	C₂×C₄×C₁₁	—	2	2	no	—
88	C₂×C₄×C₁₁	—	2	2	no	—
89	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
90	C₂×C₄×C₁₁	M₃₂	1	4	yes	complemented
91	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₁	#131	1	2	yes	—
95	C₈×C₁₁	—	2	2	no	—
96	C₈×C₁₁	—	4	2	no	—
97	C₈×C₁₁	#120	1	1	yes	characteristic
98	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
99	C₈×C₁₁	#123	1	1	yes	characteristic
100	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
101	C₂×M₃₂	—	22	4	no	—
102	C₂×M₃₂	—	22	2	no	—
103	C₂²×C₁₆	—	11	2	no	—
104	C₂²×C₁₆	—	11	1	no	—
105	C₂²×C₁₆	—	11	1	no	—
106	C₂×C₂²⋊C₈	Dic₁₁	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
108	C₄×C₂²⋊C₄	Dic₁₁	1	1	yes	characteristic
109	C₄⋊C₁₆	—	22	4	no	—
110	C₂²⋊C₁₆	—	22	2	no	—
111	#336	Dic₁₁	1	2	yes	—
112	C₂²×C₁₁	—	2	2	no	—
113	C₂²×C₁₁	—	2	2	no	—
114	C₂²×C₁₁	—	2	2	no	—
115	C₂²×C₁₁	—	4	2	no	—
116	C₂²×C₁₁	—	4	2	no	—
117	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂²×C₁₁	C₈×D₄	1	2	yes	—
119	C₂²×C₁₁	C₂×M₃₂	1	2	yes	—
120	C₂²×C₁₁	#504	1	1	yes	characteristic
121	C₂²×C₁₁	#504	1	1	yes	characteristic
122	C₄×C₁₁	—	2	2	no	—
123	C₄×C₁₁	—	2	4	no	—
124	C₄×C₁₁	—	4	2	no	—
125	C₄×C₁₁	#385	1	1	yes	characteristic
126	C₄×C₁₁	#424	1	1	yes	characteristic
127	C₄×C₁₁	#435	1	2	yes	—
128	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	Dic₂₂	1	2	yes	—
132	C₂²×C₈	D₄₄	1	2	yes	—
133	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
134	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
136	C₂×C₂²⋊C₄	C₁₁⋊C₈	1	2	yes	—
137	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
138	M₃₂	—	22	8	no	—
139	C₂×C₁₆	—	11	4	no	—
140	C₂×C₁₆	—	22	2	no	—
141	C₂×C₁₆	—	22	2	no	—
142	C₂×C₁₆	—	22	1	no	—
143	C₂×C₁₆	—	22	1	no	—
144	C₂×C₁₆	—	22	2	no	—
145	C₂×C₁₆	—	22	2	no	—
146	C₂²⋊C₈	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	#95	C₁₁⋊C₈	1	1	yes	characteristic
150	#95	C₁₁⋊C₈	1	1	yes	characteristic
151	C₂×C₁₁	—	4	2	no	—
152	C₂×C₁₁	#1850	1	1	yes	characteristic
153	C₂×C₁₁	#2150	1	1	yes	characteristic
154	C₂×C₁₁	#2151	1	1	yes	characteristic
155	C₂×C₁₁	#2168	1	2	yes	—
156	C₂×C₁₁	#2169	1	2	yes	—
157	C₂⁴	C₂×C₁₁⋊C₈	1	1	yes	characteristic
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
161	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
163	C₂²×C₄	#4318	1	2	yes	—
164	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
165	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
174	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
176	C₂×C₈	D₄×D₁₁	1	2	yes	—
177	C₂×C₈	#4341	1	2	yes	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₁₆	—	22	4	no	—
182	C₁₆	—	44	1	no	—
183	C₁₆	—	44	1	no	—
184	C₁₁	#12913	1	1	yes	characteristic, complemented, Sylow
185	C₂³	—	2	1	no	—
186	C₂³	—	2	1	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
190	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
201	C₂×C₄	#66876	1	4	yes	complemented
202	C₂×C₄	C₂×#4318	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
204	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
205	C₂×C₄	#67001	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	4	2	no	—
208	C₈	#66931	1	2	yes	—
209	C₈	#66935	1	1	yes	characteristic
210	C₈	#66939	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
217	C₂²	D₄×C₁₁⋊C₈	1	2	yes	—
218	C₂²	C₂×#66876	1	2	yes	—
219	C₂²	#10622753	1	1	yes	characteristic
220	C₂²	#10622753	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	2	no	—
223	C₄	—	4	2	no	—
224	C₄	#10622563	1	1	yes	characteristic
225	C₄	#10622612	1	1	yes	characteristic
226	C₄	#10622624	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#13068796	1	1	yes	characteristic
229	C₂	#13070372	1	1	yes	characteristic
230	C₂	#13070375	1	1	yes	characteristic
231	C₂	#13070442	1	2	yes	—
232	C₂	#13070448	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator, socle	C₂³×C₁₁	C₂²×C₈
Fitting	C₈×C₁₁×C₂²⋊C₄	C₂
prime radical	C₂⁴×C₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₈×C₂²⋊C₄	D₁₁
hypocenter	C₁₁	#12913
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
4. #13066694	C₂	semidirect
90. C₂×C₄×C₁₁	M₃₂	semidirect
184. C₁₁	#12913	semidirect
201. C₂×C₄	#66876	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#66876	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×C₂²⋊(C₁₁⋊C₁₆)	2
#13066694	2
C₈×C₁₁×C₂²⋊C₄	2
#12913	11