Subgroups — #14906962

Order 2816.

690 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×#10622753	C₂	1	2	yes	maximal, complemented
2	#13070448	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₁⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	#13070372	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070230	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₁×#2931	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₁×#131	C₂²	1	2	yes	—
9	C₁₁×#532	C₄	1	1	yes	characteristic
10	C₁₁×#436	C₄	1	4	yes	—
11	C₈×D₄×C₁₁	C₂²	1	4	yes	—
12	C₁₁×#423	C₄	1	1	yes	characteristic
13	C₁₁×#408	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	#10622753	—	2	8	no	—
16	C₂×#66876	C₂²	1	2	yes	—
17	C₂×#66876	C₂²	1	4	yes	—
18	C₂²×C₁₁⋊C₁₆	C₂²	1	4	yes	—
19	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	8	yes	—
22	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
23	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
24	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
25	C₂×C₁₁×#41	C₂×C₄	1	2	yes	—
26	C₁₁×#131	—	2	4	no	—
27	C₁₁×#131	—	2	4	no	—
28	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
33	C₁₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
34	C₁₁×#127	C₈	1	1	yes	characteristic
35	C₁₁×#124	C₈	1	2	yes	—
36	C₁₁×#121	C₈	1	4	yes	—
37	C₄×D₄×C₁₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
39	C₁₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
40	C₁₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₁	C₂³	1	2	yes	—
43	#66876	—	2	4	no	—
44	#66876	—	2	8	no	—
45	C₂×C₁₁⋊C₁₆	—	2	2	no	—
46	C₂×C₁₁⋊C₁₆	—	2	8	no	—
47	C₂×C₁₁⋊C₁₆	D₄	1	4	yes	complemented
48	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
49	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
50	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
52	#33434	—	11	1	no	maximal, Sylow
53	C₁₁×#41	—	2	4	no	—
54	C₁₁×#41	—	2	4	no	—
55	C₂×Q₈×C₁₁	C₂×C₈	1	2	yes	—
56	C₂×D₄×C₁₁	C₂×C₈	1	4	yes	—
57	C₂×D₄×C₁₁	C₂×C₈	1	4	yes	—
58	C₂×D₄×C₁₁	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₁₁×M₁₆	—	2	8	no	—
63	C₁₁×M₁₆	—	2	4	no	—
64	C₂×C₈×C₁₁	—	2	4	no	—
65	C₂×C₈×C₁₁	—	2	4	no	—
66	C₂×C₈×C₁₁	—	2	2	no	—
67	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₁	#41	1	2	yes	—
72	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
73	C₁₁×C₄⋊C₄	C₂×C₈	1	4	yes	—
74	C₁₁×C₂²⋊C₄	C₂×C₈	1	8	yes	—
75	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
78	C₁₁⋊C₁₆	—	2	4	no	—
79	C₁₁⋊C₁₆	—	2	2	no	—
80	C₁₁⋊C₁₆	C₂×D₄	1	2	yes	—
81	C₁₁⋊C₁₆	C₂×D₄	1	2	yes	—
82	C₂×#504	—	11	2	no	—
83	#2931	D₁₁	1	1	yes	characteristic
84	#2169	—	11	4	no	—
85	D₄×C₁₆	—	11	4	no	—
86	#2150	—	11	1	no	—
87	#2113	—	11	2	no	—
88	C₂×C₄×C₁₆	—	11	1	no	—
89	C₂³×C₁₁	C₂²×C₈	1	4	yes	—
90	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
91	Q₈×C₁₁	—	2	4	no	—
92	D₄×C₁₁	—	2	8	no	—
93	D₄×C₁₁	—	2	4	no	—
94	D₄×C₁₁	—	2	8	no	—
95	C₂×C₄×C₁₁	—	2	4	no	—
96	C₂×C₄×C₁₁	—	2	4	no	—
97	C₂×C₄×C₁₁	—	2	4	no	—
98	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₁₁	#131	1	2	yes	—
104	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
105	C₈×C₁₁	—	2	4	no	—
106	C₈×C₁₁	—	2	2	no	—
107	C₈×C₁₁	C₄×D₄	1	2	yes	—
108	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
110	C₂×#131	C₂×D₁₁	1	2	yes	—
111	#532	Dic₁₁	1	1	yes	characteristic
112	#504	—	22	8	no	—
113	C₂×M₃₂	—	11	2	no	—
114	C₂×M₃₂	—	11	4	no	—
115	C₂²×C₁₆	—	11	1	no	—
116	C₂²×C₁₆	—	11	1	no	—
117	C₂²×C₁₆	—	11	4	no	—
118	#436	Dic₁₁	1	4	yes	—
119	C₈×D₄	C₂×D₁₁	1	4	yes	—
120	#423	Dic₁₁	1	1	yes	characteristic
121	#408	Dic₁₁	1	2	yes	—
122	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
123	C₄⋊C₁₆	—	11	4	no	—
124	C₂²⋊C₁₆	—	11	8	no	—
125	C₄×C₁₆	—	11	2	no	—
126	C₄×C₁₆	—	11	2	no	—
127	C₂²×C₁₁	—	2	2	no	—
128	C₂²×C₁₁	—	2	8	no	—
129	C₂²×C₁₁	—	2	4	no	—
130	C₂²×C₁₁	C₈×D₄	1	2	yes	—
131	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
132	C₄×C₁₁	—	2	2	no	—
133	C₄×C₁₁	—	2	4	no	—
134	C₄×C₁₁	C₈×D₄	1	2	yes	—
135	C₄×C₁₁	#504	1	4	yes	complemented
136	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
138	C₂×#41	C₂×Dic₁₁	1	2	yes	—
139	#131	—	2	4	no	—
140	#131	—	2	4	no	—
141	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
142	C₂×M₁₆	C₂×Dic₁₁	1	4	yes	—
143	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₁₁	1	4	yes	—
145	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
146	C₄⋊Q₈	C₁₁⋊C₈	1	1	yes	characteristic
147	#127	C₁₁⋊C₈	1	1	yes	characteristic
148	#124	C₁₁⋊C₈	1	2	yes	—
149	#121	C₁₁⋊C₈	1	4	yes	—
150	C₄×D₄	C₂×Dic₁₁	1	4	yes	—
151	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
152	M₃₂	—	22	8	no	—
153	M₃₂	—	22	4	no	—
154	C₂×C₁₆	—	11	4	no	—
155	C₂×C₁₆	—	11	1	no	—
156	C₂×C₁₆	—	11	1	no	—
157	C₂×C₁₆	—	11	2	no	—
158	C₂×C₁₆	—	11	4	no	—
159	C₂×C₁₆	—	22	8	no	—
160	C₂×C₁₆	—	22	2	no	—
161	C₄⋊C₈	C₂×Dic₁₁	1	4	yes	—
162	C₂²⋊C₈	C₂×Dic₁₁	1	8	yes	—
163	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
164	C₄×C₈	C₂²×D₁₁	1	2	yes	—
165	C₂×C₁₁	—	2	2	no	—
166	C₂×C₁₁	—	4	4	no	—
167	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
168	C₂×C₁₁	C₂×#504	1	2	yes	—
169	#41	—	2	4	no	—
170	#41	—	2	4	no	—
171	C₂×Q₈	C₂×C₁₁⋊C₈	1	2	yes	—
172	C₂×D₄	C₂×C₁₁⋊C₈	1	4	yes	—
173	C₂×D₄	C₂×C₁₁⋊C₈	1	4	yes	—
174	C₂×D₄	C₂×C₁₁⋊C₈	1	2	yes	—
175	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₁₁	1	4	yes	—
177	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
178	M₁₆	—	2	4	no	—
179	M₁₆	—	2	8	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	D₄×D₁₁	1	2	yes	—
184	C₂×C₈	#4341	1	2	yes	—
185	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
186	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
187	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
188	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
189	C₄⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
190	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	8	yes	—
191	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
192	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
193	C₄²	C₂²×Dic₁₁	1	2	yes	—
194	C₁₆	—	11	2	no	—
195	C₁₆	—	11	2	no	—
196	C₁₆	—	22	4	no	—
197	C₁₆	—	22	2	no	—
198	C₁₁	#33434	1	1	yes	characteristic, complemented, Sylow
199	C₂³	C₂²×C₁₁⋊C₈	1	4	yes	—
200	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	D₄	—	2	8	no	—
203	D₄	—	2	8	no	—
204	D₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
209	C₂×C₄	C₂²×C₁₁⋊C₈	1	4	yes	—
210	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
211	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
212	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
213	C₂×C₄	#67001	1	2	yes	—
214	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	2	2	no	—
217	C₈	D₄×Dic₁₁	1	2	yes	—
218	C₈	C₂×D₄×D₁₁	1	1	yes	characteristic
219	C₈	C₂×#4341	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	D₄×C₁₁⋊C₈	1	2	yes	—
224	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	D₄×C₁₁⋊C₈	1	2	yes	—
228	C₄	#10622753	1	4	yes	complemented
229	C₄	C₂×D₄×Dic₁₁	1	1	yes	characteristic
230	C₄	C₂×#67001	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	4	4	no	—
233	C₂	C₂×D₄×C₁₁⋊C₈	1	1	yes	characteristic
234	C₂	C₂×#10622753	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2931	C₂
prime radical	C₁₁×#532	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2931	D₁₁
hypocenter	C₁₁	#33434
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. C₂×#10622753	C₂	semidirect
2. #13070448	C₂	semidirect
3. D₄×C₁₁⋊C₁₆	C₂	semidirect
4. #13070372	C₂	semidirect
5. #13070230	C₂	semidirect
6. C₂×C₄×C₁₁⋊C₁₆	C₂	semidirect
47. C₂×C₁₁⋊C₁₆	D₄	semidirect
135. C₄×C₁₁	#504	semidirect
198. C₁₁	#33434	semidirect
228. C₄	#10622753	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622753	2
#13070448	2
D₄×C₁₁⋊C₁₆	2
#13070372	2
#13070230	2
C₂×C₄×C₁₁⋊C₁₆	2
C₁₁×#2931	2
#33434	11