Subgroups — C₂×#13070422 #14906847

Order 2816.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622302	C₂	1	1	yes	maximal, characteristic
2	#13070422	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₁₁)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×(C₂×D₁₁)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#10622140	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#4313	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₁₁	C₂²	1	1	yes	characteristic
11	C₂×#66925	C₄	1	1	yes	characteristic
12	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×#66879	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×#66867	C₄	1	1	yes	characteristic
20	#10622302	C₄	1	4	yes	—
21	#10622302	C₂²	1	4	yes	—
22	C₁₁×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	—
24	(C₂×D₁₁)⋊C₁₆	C₂²	1	4	yes	—
25	#10622140	C₂²	1	4	yes	complemented
26	C₁₆×Dic₁₁	C₂²	1	4	yes	complemented
27	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	complemented
28	C₂²×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂×(C₂×D₁₁)⋊C₄	C₈	1	1	yes	characteristic
33	C₂×(C₂×D₁₁)⋊C₄	C₈	1	1	yes	characteristic
34	C₂×Dic₁₁⋊C₄	C₈	1	1	yes	characteristic
35	C₂×Dic₁₁⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×#4313	—	2	4	no	—
40	C₂×#4313	C₂×C₄	1	4	yes	—
41	C₂×C₈×D₁₁	—	2	2	no	—
42	C₂×C₈×D₁₁	—	2	2	no	—
43	C₂×C₈×D₁₁	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₁	C₂³	1	2	yes	—
45	#66925	C₈	1	4	yes	—
46	#66925	C₂×C₄	1	4	yes	—
47	C₂×C₁₁×C₁₆	—	2	2	no	—
48	C₂×C₁₁×C₁₆	—	2	2	no	—
49	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
51	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₁	C₂³	1	2	yes	—
53	#66879	C₂×C₄	1	4	yes	—
54	C₈×Dic₁₁	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₁	C₂³	1	2	yes	—
56	C₂×C₁₁⋊C₁₆	—	2	2	no	—
57	C₂×C₁₁⋊C₁₆	—	2	2	no	—
58	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
60	#66867	C₂×C₄	1	4	yes	—
61	C₂×#2164	—	11	1	no	maximal, Sylow
62	C₂³×D₁₁	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
68	C₂²×Dic₁₁	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₁₁	—	2	2	no	—
70	C₂×C₄×D₁₁	—	2	2	no	—
71	C₂×C₄×D₁₁	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₁₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₁	—	2	2	no	—
74	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₁	#41	1	2	yes	—
77	C₂×C₈×C₁₁	#41	1	2	yes	—
78	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
79	C₄²×C₁₁	C₂×C₈	1	2	yes	—
80	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
81	(C₂×D₁₁)⋊C₄	C₂×C₈	1	4	yes	—
82	(C₂×D₁₁)⋊C₄	C₂×C₈	1	4	yes	—
83	Dic₁₁⋊C₄	C₂×C₈	1	4	yes	—
84	Dic₁₁⋊C₄	C₂×C₈	1	4	yes	—
85	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
86	C₄×Dic₁₁	C₂²×C₄	1	2	yes	—
87	C₂×C₁₁⋊C₈	—	2	2	no	—
88	C₂×C₁₁⋊C₈	M₁₆	1	4	yes	—
89	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₁₁⋊C₈	C₂²×C₄	1	2	yes	—
91	#4313	—	2	8	no	—
92	C₈×D₁₁	—	2	4	no	—
93	C₈×D₁₁	—	2	2	no	—
94	C₈×D₁₁	—	2	2	no	—
95	C₁₁×C₁₆	—	2	2	no	—
96	C₁₁×C₁₆	—	2	2	no	—
97	C₁₁⋊C₁₆	—	2	2	no	—
98	C₁₁⋊C₁₆	—	2	2	no	—
99	C₂×#405	—	11	1	no	—
100	#2164	—	11	8	no	—
101	C₂×C₄⋊C₁₆	—	11	1	no	—
102	C₂×C₄⋊C₁₆	D₁₁	1	1	yes	characteristic, complemented
103	C₂×C₂²⋊C₁₆	—	11	1	no	—
104	C₂×C₂²⋊C₁₆	—	11	1	no	—
105	C₂×#346	—	11	1	no	—
106	C₂×C₄×C₁₆	—	11	1	no	—
107	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
108	C₂²×D₁₁	—	2	2	no	—
109	C₂²×D₁₁	—	2	2	no	—
110	C₂²×D₁₁	—	2	2	no	—
111	C₂²×D₁₁	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	—	2	2	no	—
114	C₂×C₄×C₁₁	#117	1	2	yes	—
115	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₁₁	—	2	2	no	—
121	C₂×Dic₁₁	—	2	2	no	—
122	C₂×Dic₁₁	—	2	2	no	—
123	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
124	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
125	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
126	C₄×D₁₁	—	2	2	no	—
127	C₄×D₁₁	—	2	2	no	—
128	C₄×D₁₁	—	2	4	no	—
129	C₈×C₁₁	—	2	2	no	—
130	C₈×C₁₁	#117	1	2	yes	—
131	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
132	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
133	C₁₁⋊C₈	—	2	2	no	—
134	C₁₁⋊C₈	C₂×M₁₆	1	4	yes	—
135	C₂²×M₁₆	—	11	1	no	—
136	C₂³×C₈	—	11	1	no	—
137	C₂×#117	—	11	1	no	—
138	C₂²×C₁₆	—	11	1	no	—
139	C₂²×C₁₆	—	11	1	no	—
140	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
141	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
142	#405	—	11	4	no	—
143	#405	—	11	4	no	—
144	C₂×#97	—	11	1	no	—
145	C₂×#97	—	11	1	no	—
146	C₂×C₄×C₈	—	11	1	no	—
147	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
148	C₄⋊C₁₆	—	11	4	no	—
149	C₄⋊C₁₆	C₂×D₁₁	1	4	yes	complemented
150	C₂²⋊C₁₆	—	11	4	no	—
151	C₂²⋊C₁₆	—	11	4	no	—
152	#346	—	11	4	no	—
153	C₄×C₁₆	—	11	4	no	—
154	C₂²×C₁₁	#431	1	2	yes	—
155	C₂²×C₁₁	#504	1	2	yes	—
156	C₂²×C₁₁	#504	1	2	yes	—
157	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
158	C₂×D₁₁	—	2	2	no	—
159	C₂×D₁₁	—	2	2	no	—
160	C₂×D₁₁	—	2	2	no	—
161	C₂×D₁₁	—	4	4	no	—
162	C₄×C₁₁	—	2	2	no	—
163	C₄×C₁₁	—	2	2	no	—
164	C₄×C₁₁	#431	1	2	yes	—
165	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
166	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
167	Dic₁₁	—	2	2	no	—
168	Dic₁₁	—	2	2	no	—
169	Dic₁₁	—	2	2	no	—
170	C₂³×C₄	—	11	1	no	—
171	C₂×M₁₆	—	11	4	no	—
172	C₂×M₁₆	—	22	4	no	—
173	C₂²×C₈	—	11	1	no	—
174	C₂²×C₈	—	11	1	no	—
175	C₂²×C₈	—	11	2	no	—
176	C₂²×C₈	—	11	2	no	—
177	C₂²×C₈	—	22	2	no	—
178	C₂²×C₈	—	22	2	no	—
179	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
181	#117	—	11	4	no	—
182	#117	—	11	4	no	—
183	C₂×C₄⋊C₄	—	11	1	no	—
184	C₂×C₄⋊C₄	—	11	1	no	—
185	C₂×C₂²⋊C₄	—	11	1	no	—
186	C₂×C₂²⋊C₄	—	11	1	no	—
187	C₂×C₄²	—	11	1	no	—
188	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	11	2	no	—
192	C₂×C₁₆	—	11	2	no	—
193	C₂×C₁₆	—	22	2	no	—
194	C₂×C₁₆	—	22	2	no	—
195	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
196	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
197	#97	—	11	4	no	—
198	#97	—	11	4	no	—
199	C₄×C₈	—	11	2	no	—
200	C₄×C₈	—	11	2	no	—
201	C₄×C₈	C₄×D₁₁	1	2	yes	—
202	C₄×C₈	C₂²×D₁₁	1	2	yes	—
203	C₂×C₁₁	#2164	1	4	yes	complemented
204	C₂×C₁₁	C₂×#431	1	1	yes	characteristic
205	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
206	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
207	D₁₁	—	4	2	no	—
208	C₂⁴	—	11	1	no	—
209	C₂²×C₄	—	11	1	no	—
210	C₂²×C₄	—	11	1	no	—
211	C₂²×C₄	—	11	1	no	—
212	C₂²×C₄	—	11	2	no	—
213	C₂²×C₄	—	11	2	no	—
214	C₂²×C₄	—	22	2	no	—
215	C₂²×C₄	—	22	2	no	—
216	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
217	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
219	M₁₆	—	22	8	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	11	4	no	—
222	C₂×C₈	—	11	2	no	—
223	C₂×C₈	—	11	2	no	—
224	C₂×C₈	—	22	2	no	—
225	C₂×C₈	—	22	4	no	—
226	C₂×C₈	—	22	2	no	—
227	C₂×C₈	—	22	2	no	—
228	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
230	C₂×C₈	#4341	1	2	yes	—
231	C₂×C₈	#4343	1	2	yes	—
232	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
233	C₄⋊C₄	—	11	4	no	—
234	C₄⋊C₄	—	11	4	no	—
235	C₂²⋊C₄	—	11	4	no	—
236	C₂²⋊C₄	—	11	4	no	—
237	C₄²	—	11	2	no	—
238	C₄²	—	11	2	no	—
239	C₄²	C₈×D₁₁	1	2	yes	—
240	C₄²	C₂×C₄×D₁₁	1	2	yes	—
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	22	2	no	—
244	C₁₆	—	22	2	no	—
245	C₁₁	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	11	2	no	—
247	C₂³	—	22	2	no	—
248	C₂³	—	22	2	no	—
249	C₂³	—	22	2	no	—
250	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	11	2	no	—
254	C₂×C₄	—	11	2	no	—
255	C₂×C₄	—	11	2	no	—
256	C₂×C₄	—	22	2	no	—
257	C₂×C₄	—	22	2	no	—
258	C₂×C₄	—	22	2	no	—
259	C₂×C₄	—	22	2	no	—
260	C₂×C₄	—	22	2	no	—
261	C₂×C₄	—	22	4	no	—
262	C₂×C₄	#66945	1	2	yes	—
263	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
264	C₂×C₄	C₂×C₈×D₁₁	1	2	yes	—
265	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
266	C₂×C₄	M₁₆×D₁₁	1	2	yes	—
267	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	11	4	no	—
270	C₈	—	22	2	no	—
271	C₈	#66945	1	2	yes	—
272	C₈	C₂×#4341	1	1	yes	characteristic
273	C₈	C₂×#4343	1	1	yes	characteristic
274	C₂²	—	22	2	no	—
275	C₂²	—	22	2	no	—
276	C₂²	—	22	2	no	—
277	C₂²	—	44	4	no	—
278	C₂²	#10622446	1	2	yes	—
279	C₂²	#10622514	1	2	yes	—
280	C₂²	#10622520	1	2	yes	—
281	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	22	2	no	—
285	C₄	—	22	2	no	—
286	C₄	—	22	2	no	—
287	C₄	#10622446	1	2	yes	—
288	C₄	C₂×#66945	1	1	yes	characteristic
289	C₄	C₂×M₁₆×D₁₁	1	1	yes	characteristic
290	C₂	—	44	2	no	—
291	C₂	#13070422	1	4	yes	direct product factor
292	C₂	C₂×#10622446	1	1	yes	characteristic
293	C₂	C₂×#10622514	1	1	yes	characteristic
294	C₂	C₂×#10622520	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×C₄⋊C₁₆	C₂
socle	C₂³×C₁₁	C₂²×C₈
prime radical	C₂³×D₁₁	C₂×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₁₁
hypocenter	C₁₁	C₂×#2164
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #13070422	C₂	direct (N × H)
3. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
6. C₂×#10622140	C₂	semidirect
7. C₂×C₁₆×Dic₁₁	C₂	semidirect
8. C₂×C₁₁×C₄⋊C₁₆	C₂	semidirect
22. C₁₁×C₄⋊C₁₆	C₂²	semidirect
25. #10622140	C₂²	semidirect
26. C₁₆×Dic₁₁	C₂²	semidirect
27. C₄⋊(C₁₁⋊C₁₆)	C₂²	semidirect
102. C₂×C₄⋊C₁₆	D₁₁	semidirect
149. C₄⋊C₁₆	C₂×D₁₁	semidirect
203. C₂×C₁₁	#2164	semidirect
245. C₁₁	C₂×#2164	semidirect
291. C₂	#13070422	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×#10622302	2
#13070422	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×(C₂×D₁₁)⋊C₁₆	2
C₂×#10622140	2
C₂×C₁₆×Dic₁₁	2
C₂×C₁₁×C₄⋊C₁₆	2
C₂×#2164	11

Subgroups — C2×#13070422 #14906847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13070422 #14906847