Subgroups — C₂×C₈×#4318 #14719347

Order 2816.

1046 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622538	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#4318	C₂	1	1	yes	maximal, characteristic
3	C₈×#4318	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622092	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₁₁⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₁	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
9	C₂²×#4318	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₁	C₄	1	4	yes	—
11	C₂×C₄×C₈×C₁₁	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊(C₁₁⋊C₈)	C₄	1	2	yes	—
16	#10622538	C₄	1	4	yes	—
17	#10622538	C₂²	1	4	yes	—
18	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	2	yes	—
19	C₄×#4318	C₄	1	4	yes	—
20	C₄×#4318	C₂²	1	4	yes	—
21	C₂×#66867	C₄	1	2	yes	—
22	C₂×C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₁₁⋊C₈	C₂²	1	2	yes	—
24	#10622092	C₂²	1	8	yes	complemented
25	C₈×C₁₁⋊C₈	C₂²	1	8	yes	complemented
26	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	—	2	8	no	—
30	C₂²×C₈×C₁₁	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
41	C₂×#4318	C₈	1	16	yes	direct product factor
42	C₂×#4318	C₂×C₄	1	8	yes	—
43	C₂²×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
44	C₂²×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
45	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₁	C₂×C₄	1	8	yes	—
47	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₁	C₂³	1	2	yes	—
49	C₄×C₈×C₁₁	C₂³	1	2	yes	—
50	C₂²⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
51	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
52	#66867	C₂×C₄	1	8	yes	—
53	C₄×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
54	C₄×C₁₁⋊C₈	C₂×C₄	1	4	yes	—
55	C₄×C₁₁⋊C₈	C₂³	1	4	yes	—
56	C₄×C₁₁⋊C₈	C₂³	1	4	yes	—
57	C₂×C₈×M₁₆	—	11	1	no	maximal, Sylow
58	C₂⁴×C₁₁	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	—	2	4	no	—
60	C₂²×C₄×C₁₁	—	2	4	no	—
61	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
64	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
65	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
66	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂×C₈	1	8	yes	—
68	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₁	—	2	16	no	—
75	C₂×C₈×C₁₁	—	2	4	no	—
76	C₂×C₈×C₁₁	M₁₆	1	8	yes	complemented
77	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
79	C₄²×C₁₁	C₄²	1	2	yes	—
80	C₄²×C₁₁	C₄²	1	2	yes	—
81	C₄²×C₁₁	C₄²	1	2	yes	—
82	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
85	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
87	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
89	#4318	C₂×C₈	1	32	yes	direct product factor
90	C₂×C₁₁⋊C₈	—	2	8	no	—
91	C₂×C₁₁⋊C₈	C₂×C₈	1	16	yes	complemented
92	C₂×C₁₁⋊C₈	C₂²×C₄	1	8	yes	—
93	C₂×C₁₁⋊C₈	C₂²×C₄	1	8	yes	—
94	C₂×#431	—	11	1	no	—
95	C₂×C₄×M₁₆	—	11	1	no	—
96	C₂²×C₄×C₈	D₁₁	1	1	yes	characteristic
97	C₈×M₁₆	—	11	8	no	—
98	C₂×#322	—	11	2	no	—
99	C₂×C₈²	—	11	2	no	—
100	C₂³×C₁₁	—	2	2	no	—
101	C₂³×C₁₁	—	2	2	no	—
102	C₂³×C₁₁	C₄×C₈	1	4	yes	—
103	C₂³×C₁₁	C₂×C₄²	1	2	yes	—
104	C₂³×C₁₁	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₁	—	2	2	no	—
106	C₂×C₄×C₁₁	—	2	2	no	—
107	C₂×C₄×C₁₁	—	2	4	no	—
108	C₂×C₄×C₁₁	—	2	4	no	—
109	C₂×C₄×C₁₁	—	2	4	no	—
110	C₂×C₄×C₁₁	—	2	4	no	—
111	C₂×C₄×C₁₁	C₄×C₈	1	8	yes	—
112	C₂×C₄×C₁₁	C₄×C₈	1	4	yes	—
113	C₂×C₄×C₁₁	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₁	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
120	C₂×C₄×C₁₁	C₂²×C₈	1	8	yes	—
121	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
122	C₂×C₄×C₁₁	#131	1	4	yes	—
123	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₁₁	—	2	4	no	—
127	C₈×C₁₁	C₂×M₁₆	1	8	yes	complemented
128	C₁₁⋊C₈	—	2	8	no	—
129	C₁₁⋊C₈	C₂²×C₈	1	16	yes	complemented
130	C₂²×M₁₆	—	11	2	no	—
131	C₂³×C₈	Dic₁₁	1	2	yes	—
132	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
133	#431	—	11	4	no	—
134	#431	—	11	4	no	—
135	C₂×C₄⋊C₈	—	11	2	no	—
136	C₂×C₂²⋊C₈	—	11	2	no	—
137	C₄×M₁₆	—	11	4	no	—
138	C₄×M₁₆	—	11	4	no	—
139	C₂×#97	—	11	2	no	—
140	C₂×C₄×C₈	—	11	2	no	—
141	C₂×C₄×C₈	—	11	2	no	—
142	C₂×C₄×C₈	Dic₁₁	1	2	yes	—
143	C₂×C₄×C₈	Dic₁₁	1	4	yes	—
144	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₁₁	1	4	yes	—
146	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
147	#322	—	11	8	no	—
148	C₈²	—	11	8	no	—
149	C₂²×C₁₁	—	2	2	no	—
150	C₂²×C₁₁	—	2	2	no	—
151	C₂²×C₁₁	—	2	8	no	—
152	C₂²×C₁₁	C₂×C₄×C₈	1	4	yes	—
153	C₂²×C₁₁	C₂×C₄×C₈	1	2	yes	—
154	C₂²×C₁₁	C₄×M₁₆	1	2	yes	—
155	C₂²×C₁₁	#405	1	2	yes	—
156	C₂²×C₁₁	C₂²×C₄²	1	1	yes	characteristic
157	C₄×C₁₁	—	2	2	no	—
158	C₄×C₁₁	—	2	2	no	—
159	C₄×C₁₁	C₂×C₄×C₈	1	2	yes	—
160	C₄×C₁₁	C₂×C₄×C₈	1	4	yes	—
161	C₄×C₁₁	C₄×M₁₆	1	2	yes	—
162	C₄×C₁₁	#405	1	2	yes	—
163	C₄×C₁₁	C₂³×C₈	1	2	yes	—
164	C₄×C₁₁	C₂²×M₁₆	1	2	yes	—
165	C₄×C₁₁	C₂×#131	1	2	yes	—
166	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
167	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
169	C₂×M₁₆	—	11	16	no	—
170	C₂×M₁₆	—	11	8	no	—
171	C₂²×C₈	—	2	8	no	—
172	C₂²×C₈	—	11	4	no	—
173	C₂²×C₈	—	11	4	no	—
174	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
175	C₂²×C₈	C₂×Dic₁₁	1	8	yes	—
176	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
177	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₁₁	1	2	yes	—
179	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
180	C₂×C₄²	C₄×D₁₁	1	2	yes	—
181	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
182	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
185	C₄⋊C₈	—	11	8	no	—
186	C₂²⋊C₈	—	11	8	no	—
187	#97	—	11	8	no	—
188	C₄×C₈	—	11	4	no	—
189	C₄×C₈	—	11	4	no	—
190	C₄×C₈	—	11	4	no	—
191	C₄×C₈	—	11	4	no	—
192	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
193	C₄×C₈	C₂×Dic₁₁	1	8	yes	—
194	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
195	C₄×C₈	C₂²×D₁₁	1	2	yes	—
196	C₄×C₈	C₂²×D₁₁	1	2	yes	—
197	C₂×C₁₁	—	2	4	no	—
198	C₂×C₁₁	C₈×M₁₆	1	4	yes	complemented
199	C₂×C₁₁	C₂²×C₄×C₈	1	1	yes	characteristic
200	C₂×C₁₁	C₂×C₄×M₁₆	1	1	yes	characteristic
201	C₂×C₁₁	C₂×#405	1	1	yes	characteristic
202	C₂⁴	C₄×Dic₁₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₈×D₁₁	1	8	yes	—
206	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
207	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
208	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
209	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
210	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
211	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₁₁	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
218	M₁₆	—	11	32	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	16	no	—
221	C₂×C₈	—	11	16	no	—
222	C₂×C₈	—	11	8	no	—
223	C₂×C₈	—	11	8	no	—
224	C₂×C₈	—	22	8	no	—
225	C₂×C₈	#4318	1	8	yes	direct product factor
226	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
227	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
228	C₄²	C₄×Dic₁₁	1	2	yes	—
229	C₄²	C₄×Dic₁₁	1	2	yes	—
230	C₄²	C₄×Dic₁₁	1	2	yes	—
231	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
232	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
233	C₄²	C₂×C₄×D₁₁	1	2	yes	—
234	C₄²	C₂×C₄×D₁₁	1	2	yes	—
235	C₄²	C₂²×Dic₁₁	1	2	yes	—
236	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
237	C₄²	C₂³×D₁₁	1	1	yes	characteristic
238	C₁₁	C₂×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₈×Dic₁₁	1	4	yes	—
242	C₂³	C₂×C₄×Dic₁₁	1	2	yes	—
243	C₂³	C₂×C₄×Dic₁₁	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	C₈×Dic₁₁	1	8	yes	—
251	C₂×C₄	C₈×Dic₁₁	1	4	yes	—
252	C₂×C₄	C₂×C₈×D₁₁	1	4	yes	—
253	C₂×C₄	C₂×C₈×D₁₁	1	8	yes	—
254	C₂×C₄	#66954	1	4	yes	—
255	C₂×C₄	C₂×#4318	1	4	yes	—
256	C₂×C₄	C₂×C₄×Dic₁₁	1	2	yes	—
257	C₂×C₄	C₂×C₄×Dic₁₁	1	2	yes	—
258	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄×Dic₁₁	1	2	yes	—
261	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
264	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	11	16	no	—
267	C₈	—	22	8	no	—
268	C₈	C₂×#4318	1	8	yes	direct product factor
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	8	no	—
272	C₂²	C₄×#4318	1	2	yes	—
273	C₂²	C₂×C₈×Dic₁₁	1	4	yes	—
274	C₂²	C₂×C₈×Dic₁₁	1	2	yes	—
275	C₂²	#10622714	1	2	yes	—
276	C₂²	C₂²×C₄×Dic₁₁	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	C₄×#4318	1	2	yes	—
280	C₄	C₂×C₈×Dic₁₁	1	2	yes	—
281	C₄	C₂×C₈×Dic₁₁	1	4	yes	—
282	C₄	#10622714	1	2	yes	—
283	C₄	C₂²×C₈×D₁₁	1	2	yes	—
284	C₄	C₂×#66954	1	2	yes	—
285	C₄	C₂²×#4318	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	C₈×#4318	1	4	yes	direct product factor
288	C₂	C₂²×C₈×Dic₁₁	1	1	yes	characteristic
289	C₂	C₂×C₄×#4318	1	1	yes	characteristic
290	C₂	C₂×#10622714	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₁₁
commutator	C₂×C₁₁	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂²×C₄×C₈×C₁₁	C₂
socle	C₂³×C₁₁	C₂×C₄²
prime radical	C₂⁴×C₁₁	C₄²
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₁
hypocenter	C₁₁	C₂×C₈×M₁₆
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₈×#4318	C₂	direct (N × H)
4. C₂×#10622092	C₂	semidirect
5. C₂×C₈×C₁₁⋊C₈	C₂	semidirect
24. #10622092	C₂²	semidirect
25. C₈×C₁₁⋊C₈	C₂²	semidirect
41. C₂×#4318	C₈	direct (N × H)
76. C₂×C₈×C₁₁	M₁₆	semidirect
89. #4318	C₂×C₈	direct (N × H)
91. C₂×C₁₁⋊C₈	C₂×C₈	semidirect
127. C₈×C₁₁	C₂×M₁₆	semidirect
129. C₁₁⋊C₈	C₂²×C₈	semidirect
198. C₂×C₁₁	C₈×M₁₆	semidirect
225. C₂×C₈	#4318	direct (N × H)
238. C₁₁	C₂×C₈×M₁₆	semidirect
268. C₈	C₂×#4318	direct (N × H)
287. C₂	C₈×#4318	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#10622538	2
C₂×C₄×#4318	2
C₈×#4318	2
C₂×#10622092	2
C₂×C₈×C₁₁⋊C₈	2
C₂²×C₄×C₈×C₁₁	2
C₂×C₈×M₁₆	11

Subgroups — C2×C8×#4318 #14719347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×#4318 #14719347