Subgroups — #14646714

Order 2816.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068612	C₂	1	2	yes	maximal
2	#13067375	C₂	1	1	yes	maximal, characteristic
3	#13067372	C₂	1	1	yes	maximal, characteristic
4	#13066491	C₂	1	2	yes	maximal
5	C₂×C₁₁×#446	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₁₁×#446	—	2	1	no	—
8	C₁₁×#446	—	2	1	no	—
9	C₁₁×#446	—	2	1	no	—
10	C₁₁×#446	—	2	1	no	—
11	C₂×C₁₁×C₄⋊C₈	C₄	1	2	yes	—
12	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×#97	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×#66900	—	2	2	no	—
16	#10622537	C₂²	1	2	yes	—
17	#10622537	C₂²	1	2	yes	—
18	#10622168	—	2	2	no	—
19	#10622096	—	2	1	no	—
20	#10622096	—	2	1	no	—
21	#10622095	—	2	1	no	—
22	#10622095	—	2	1	no	—
23	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Q₈×C₁₁	—	2	1	no	—
28	C₄×Q₈×C₁₁	—	2	1	no	—
29	C₄×Q₈×C₁₁	—	2	1	no	—
30	C₄×Q₈×C₁₁	—	2	1	no	—
31	C₂×C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
32	C₂×C₁₁×C₄⋊C₄	Q₈	1	2	yes	—
33	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
35	C₂×#4318	—	2	2	no	—
36	C₂×#4318	—	2	2	no	—
37	C₁₁×C₄⋊C₈	—	2	2	no	—
38	C₁₁×C₄⋊C₈	—	2	2	no	—
39	C₁₁×C₄⋊C₈	—	2	1	no	—
40	C₁₁×C₄⋊C₈	—	2	1	no	—
41	C₁₁×#97	—	2	2	no	—
42	C₁₁×#97	—	2	2	no	—
43	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
47	#66900	—	4	4	no	—
48	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
49	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
50	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
51	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
52	#8619	—	11	1	no	maximal, Sylow
53	C₂×Q₈×C₁₁	—	2	1	no	—
54	C₂×Q₈×C₁₁	—	2	1	no	—
55	C₂×Q₈×C₁₁	—	2	2	no	—
56	C₂×Q₈×C₁₁	—	2	2	no	—
57	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₁₁	—	2	2	no	—
63	C₂×C₈×C₁₁	—	2	2	no	—
64	C₂×C₈×C₁₁	—	4	1	no	—
65	C₂×C₈×C₁₁	—	4	2	no	—
66	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₁₁×C₄⋊C₄	—	2	1	no	—
71	C₁₁×C₄⋊C₄	—	2	1	no	—
72	C₁₁×C₄⋊C₄	—	2	2	no	—
73	C₁₁×C₄⋊C₄	—	2	2	no	—
74	C₄²×C₁₁	—	2	2	no	—
75	C₄²×C₁₁	—	2	2	no	—
76	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₁	#41	1	1	yes	characteristic
79	C₄²×C₁₁	#41	1	1	yes	characteristic
80	#4318	—	4	2	no	—
81	#4318	—	4	2	no	—
82	#4318	—	4	2	no	—
83	#4318	—	4	2	no	—
84	C₂×C₁₁⋊C₈	—	4	2	no	—
85	C₂×C₁₁⋊C₈	—	4	2	no	—
86	C₂×#446	D₁₁	1	1	yes	characteristic
87	#1801	—	11	2	no	—
88	#1567	—	11	1	no	—
89	#1566	—	11	1	no	—
90	#1292	—	11	2	no	—
91	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	Q₈×C₁₁	—	4	2	no	—
93	Q₈×C₁₁	—	4	1	no	—
94	Q₈×C₁₁	—	4	1	no	—
95	C₂×C₄×C₁₁	—	2	1	no	—
96	C₂×C₄×C₁₁	—	2	1	no	—
97	C₂×C₄×C₁₁	—	2	2	no	—
98	C₂×C₄×C₁₁	—	2	2	no	—
99	C₂×C₄×C₁₁	—	2	4	no	—
100	C₂×C₄×C₁₁	—	4	1	no	—
101	C₂×C₄×C₁₁	—	4	2	no	—
102	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
104	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×C₄×C₁₁	#137	1	2	yes	—
110	C₈×C₁₁	—	2	1	no	—
111	C₈×C₁₁	—	2	1	no	—
112	C₈×C₁₁	—	2	1	no	—
113	C₈×C₁₁	—	4	2	no	—
114	C₈×C₁₁	#101	1	2	yes	complemented
115	C₁₁⋊C₈	—	8	2	no	—
116	C₁₁⋊C₈	—	8	2	no	—
117	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
118	#446	—	2	1	no	—
119	#446	—	2	1	no	—
120	#446	—	2	1	no	—
121	#446	—	2	1	no	—
122	#423	—	11	2	no	—
123	#423	—	11	2	no	—
124	C₂×C₄⋊C₈	Dic₁₁	1	2	yes	—
125	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
126	C₂×#101	—	22	2	no	—
127	C₂×#97	Dic₁₁	1	2	yes	—
128	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
129	#343	—	22	2	no	—
130	#335	—	22	1	no	—
131	#335	—	22	1	no	—
132	#334	—	22	1	no	—
133	#334	—	22	1	no	—
134	C₂²×C₁₁	—	2	1	no	—
135	C₂²×C₁₁	—	2	1	no	—
136	C₂²×C₁₁	#380	1	1	yes	characteristic
137	C₂²×C₁₁	#428	1	2	yes	—
138	C₄×C₁₁	—	2	1	no	—
139	C₄×C₁₁	—	2	1	no	—
140	C₄×C₁₁	—	2	1	no	—
141	C₄×C₁₁	—	4	2	no	—
142	C₄×C₁₁	—	4	2	no	—
143	C₄×C₁₁	C₂×#101	1	1	yes	characteristic
144	C₄×C₁₁	#413	1	1	yes	characteristic
145	C₄×C₁₁	#468	1	2	yes	—
146	C₄×C₁₁	#469	1	1	yes	characteristic
147	C₄×C₁₁	#470	1	1	yes	characteristic
148	C₂²×Q₈	D₄₄	1	1	yes	characteristic
149	C₂×M₁₆	—	22	2	no	—
150	C₂×M₁₆	—	22	2	no	—
151	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
152	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
153	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	—	2	1	no	—
157	C₄×Q₈	—	2	1	no	—
158	C₂×C₄⋊C₄	Dic₂₂	1	2	yes	—
159	C₂×C₄⋊C₄	D₄₄	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
161	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	1	no	—
165	C₄⋊C₈	—	2	1	no	—
166	C₄⋊C₈	—	22	2	no	—
167	C₄⋊C₈	—	22	2	no	—
168	C₄⋊C₈	—	22	2	no	—
169	C₄⋊C₈	—	22	2	no	—
170	#101	—	44	4	no	—
171	#97	—	2	2	no	—
172	#97	—	2	2	no	—
173	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
176	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
177	C₂×C₁₁	—	2	2	no	—
178	C₂×C₁₁	#1801	1	1	yes	characteristic
179	C₂×C₁₁	#1809	1	1	yes	characteristic
180	C₂×C₁₁	#1813	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	2	1	no	—
185	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
186	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
187	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
189	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
190	M₁₆	—	44	2	no	—
191	M₁₆	—	44	2	no	—
192	M₁₆	—	44	2	no	—
193	M₁₆	—	44	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	44	2	no	—
199	C₂×C₈	—	44	2	no	—
200	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	#4339	1	1	yes	characteristic
211	C₄²	#4339	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
214	C₁₁	#8619	1	1	yes	characteristic, complemented, Sylow
215	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
216	Q₈	—	4	2	no	—
217	Q₈	—	4	1	no	—
218	Q₈	—	4	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	#66961	1	2	yes	—
227	C₂×C₄	#66962	1	2	yes	—
228	C₂×C₄	#66977	1	1	yes	characteristic
229	C₂×C₄	#66977	1	1	yes	characteristic
230	C₂×C₄	#66979	1	1	yes	characteristic
231	C₂×C₄	#66983	1	1	yes	characteristic
232	C₂×C₄	#66983	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	4	2	no	—
238	C₈	—	88	2	no	—
239	C₈	—	88	2	no	—
240	C₈	#66900	1	2	yes	complemented
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#10622734	1	2	yes	—
244	C₂²	#10622825	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	#10622747	1	2	yes	—
251	C₄	#10622748	1	1	yes	characteristic
252	C₄	#10622750	1	1	yes	characteristic
253	C₄	C₂×#66900	1	1	yes	characteristic
254	C₄	#10622850	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	#13068612	1	1	yes	characteristic
257	C₂	#13068649	1	1	yes	characteristic
258	C₂	#13068666	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10622825
commutator	C₂²×C₄×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂×C₁₁×#446	C₂
socle	C₂²×C₁₁	#380
prime radical	C₂³×C₁₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×#446	D₁₁
hypocenter	C₁₁	#8619
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
114. C₈×C₁₁	#101	semidirect
214. C₁₁	#8619	semidirect
240. C₈	#66900	semidirect

Maximal subgroups

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

subgroup	index
#13068612	2
#13067375	2
#13067372	2
#13066491	2
C₂×C₁₁×#446	2
#8619	11