Subgroups — C₂×#13067983 #14736104

Order 2816.

702 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×#10622707	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₂₂	C₂	1	1	yes	maximal, characteristic
3	#13067983	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622096	C₂	1	1	yes	maximal, characteristic
5	C₂×#10622185	C₂	1	2	yes	maximal
6	C₂×C₁₁×#494	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₁×Q₁₆	—	2	1	no	—
8	C₂²×C₁₁×Q₁₆	—	2	1	no	—
9	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₂	C₂²	1	1	yes	characteristic
11	C₁₁×#494	—	2	2	no	—
12	C₁₁×#494	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₁⋊C₈	—	2	1	no	—
15	C₂×C₈×Dic₁₁	—	2	1	no	—
16	#10622707	—	2	4	no	—
17	#10622707	C₂²	1	8	yes	—
18	C₂×#66874	—	2	2	no	—
19	C₂×#66872	—	2	2	no	—
20	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
22	C₈×Dic₂₂	—	2	2	no	—
23	C₈×Dic₂₂	C₂²	1	4	yes	—
24	#10622185	C₂²	1	8	yes	—
25	#10622096	C₂²	1	4	yes	—
26	C₂²×Q₈×C₁₁	—	2	2	no	—
27	C₂²×Dic₂₂	D₄	1	1	yes	characteristic
28	C₂×C₁₁×Q₁₆	—	2	4	no	—
29	C₂×C₁₁×Q₁₆	—	2	4	no	—
30	C₂×C₁₁×Q₁₆	—	4	4	no	—
31	C₂×C₁₁×Q₁₆	—	4	4	no	—
32	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
34	C₁₁×C₄⋊Q₈	—	2	2	no	—
35	C₁₁×C₄⋊Q₈	—	2	2	no	—
36	C₁₁×C₄⋊Q₈	—	2	2	no	—
37	C₁₁×C₄⋊Q₈	C₂³	1	4	yes	—
38	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
40	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
42	C₂×Dic₁₁⋊C₄	—	2	1	no	—
43	C₂×C₄×Dic₁₁	—	2	1	no	—
44	C₂²×C₁₁⋊C₈	—	2	2	no	—
45	C₂²×C₁₁⋊C₈	—	2	1	no	—
46	C₄×Dic₂₂	—	2	1	no	—
47	C₄×Dic₂₂	—	2	1	no	—
48	C₄×Dic₂₂	D₄	1	2	yes	—
49	C₄×Dic₂₂	C₂³	1	2	yes	—
50	C₄×C₈×C₁₁	D₄	1	2	yes	—
51	C₄×C₈×C₁₁	C₂³	1	2	yes	—
52	Dic₁₁⋊C₈	—	2	4	no	—
53	C₈×Dic₁₁	—	2	4	no	—
54	#66874	—	2	8	no	—
55	#66872	—	2	8	no	—
56	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
58	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
59	C₄⋊(C₁₁⋊C₈)	C₂³	1	4	yes	—
60	C₂×#1670	—	11	1	no	maximal, Sylow
61	C₂×Q₈×C₁₁	—	2	4	no	—
62	C₂×Q₈×C₁₁	—	4	2	no	—
63	C₂×Q₈×C₁₁	—	4	4	no	—
64	C₂×Q₈×C₁₁	—	4	4	no	—
65	C₂²×C₄×C₁₁	—	4	2	no	—
66	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
69	C₂²×Dic₁₁	—	2	1	no	—
70	C₂²×Dic₁₁	—	2	1	no	—
71	C₂×Dic₂₂	—	2	1	no	—
72	C₂×Dic₂₂	—	2	2	no	—
73	C₂×Dic₂₂	Q₁₆	1	4	yes	complemented
74	C₂×Dic₂₂	C₂×D₄	1	2	yes	—
75	C₁₁×Q₁₆	—	4	8	no	—
76	C₁₁×Q₁₆	—	4	8	no	—
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	SD₁₆	1	4	yes	—
79	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
81	C₁₁×C₄⋊C₄	—	2	4	no	—
82	C₁₁×C₄⋊C₄	—	2	4	no	—
83	C₁₁×C₄⋊C₄	—	4	2	no	—
84	C₁₁×C₄⋊C₄	—	4	2	no	—
85	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₁	#41	1	1	yes	characteristic
88	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
89	C₄⋊Dic₁₁	—	2	1	no	—
90	C₄⋊Dic₁₁	C₂×D₄	1	2	yes	—
91	Dic₁₁⋊C₄	—	2	2	no	—
92	Dic₁₁⋊C₄	—	2	2	no	—
93	C₄×Dic₁₁	—	2	2	no	—
94	C₄×Dic₁₁	—	2	2	no	—
95	C₂×C₁₁⋊C₈	—	2	4	no	—
96	C₂×C₁₁⋊C₈	—	2	2	no	—
97	C₂×C₁₁⋊C₈	—	4	4	no	—
98	C₂×C₁₁⋊C₈	—	4	2	no	—
99	C₂×#494	D₁₁	1	1	yes	characteristic
100	C₂×#475	—	11	2	no	—
101	C₂×C₈×Q₈	—	11	1	no	—
102	#1670	—	11	8	no	—
103	C₂×#335	—	11	1	no	—
104	C₂×#332	—	11	2	no	—
105	C₂³×C₁₁	#121	1	1	yes	characteristic
106	Q₈×C₁₁	—	4	4	no	—
107	Q₈×C₁₁	—	4	4	no	—
108	Q₈×C₁₁	—	8	4	no	—
109	C₂×C₄×C₁₁	—	2	2	no	—
110	C₂×C₄×C₁₁	—	4	4	no	—
111	C₂×C₄×C₁₁	—	8	4	no	—
112	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
117	C₂×C₄×C₁₁	#135	1	2	yes	—
118	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
122	C₂×Dic₁₁	—	2	2	no	—
123	C₂×Dic₁₁	—	2	4	no	—
124	C₂×Dic₁₁	—	2	2	no	—
125	C₂×Dic₁₁	—	4	2	no	—
126	Dic₂₂	—	2	2	no	—
127	Dic₂₂	—	2	4	no	—
128	Dic₂₂	C₂×Q₁₆	1	4	yes	complemented
129	C₈×C₁₁	—	2	2	no	—
130	C₈×C₁₁	C₂×SD₁₆	1	4	yes	—
131	C₁₁⋊C₈	—	4	4	no	—
132	C₁₁⋊C₈	—	4	2	no	—
133	C₂²×Q₁₆	—	2	1	no	—
134	C₂²×Q₁₆	—	2	1	no	—
135	C₂×C₄⋊Q₈	C₂×D₁₁	1	2	yes	—
136	C₂×C₄×Q₈	—	11	1	no	—
137	#494	—	2	2	no	—
138	#494	C₂×D₁₁	1	4	yes	—
139	#475	—	11	8	no	—
140	#475	—	22	4	no	—
141	C₈×Q₈	—	11	4	no	—
142	C₈×Q₈	—	22	2	no	—
143	C₂×#106	—	22	2	no	—
144	C₂×C₄⋊C₈	—	11	2	no	—
145	C₂×C₄⋊C₈	—	11	1	no	—
146	C₂×C₄⋊C₈	—	22	1	no	—
147	C₂×#103	—	22	2	no	—
148	C₂×C₄×C₈	—	22	1	no	—
149	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
150	#335	—	11	4	no	—
151	#332	—	11	8	no	—
152	C₂²×C₁₁	#460	1	2	yes	—
153	C₂²×C₁₁	#467	1	2	yes	—
154	C₂²×C₁₁	#473	1	2	yes	—
155	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
156	C₄×C₁₁	—	2	1	no	—
157	C₄×C₁₁	—	2	1	no	—
158	C₄×C₁₁	—	8	4	no	—
159	C₄×C₁₁	#460	1	2	yes	—
160	C₄×C₁₁	#467	1	2	yes	—
161	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
162	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
163	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
164	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	Dic₁₁	—	2	4	no	—
166	Dic₁₁	—	4	2	no	—
167	C₂²×Q₈	—	2	2	no	—
168	C₂²×Q₈	—	11	1	no	—
169	C₂×Q₁₆	—	2	4	no	—
170	C₂×Q₁₆	—	2	4	no	—
171	C₂×Q₁₆	—	4	4	no	—
172	C₂×Q₁₆	—	4	4	no	—
173	C₂²×C₈	—	22	2	no	—
174	C₂²×C₈	—	22	1	no	—
175	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
176	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	C₂²×D₁₁	1	4	yes	—
181	C₄×Q₈	—	11	2	no	—
182	C₄×Q₈	—	11	2	no	—
183	C₄×Q₈	—	22	1	no	—
184	C₄×Q₈	—	22	1	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	11	1	no	—
188	C₂×C₄⋊C₄	—	22	1	no	—
189	C₂×C₄²	—	22	1	no	—
190	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
191	#106	—	22	8	no	—
192	C₄⋊C₈	—	11	2	no	—
193	C₄⋊C₈	—	11	4	no	—
194	C₄⋊C₈	—	22	4	no	—
195	C₄⋊C₈	—	22	2	no	—
196	C₄⋊C₈	—	22	1	no	—
197	#103	—	22	8	no	—
198	C₄×C₈	—	22	4	no	—
199	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
200	C₄×C₈	C₂²×D₁₁	1	2	yes	—
201	C₂×C₁₁	#1670	1	4	yes	complemented
202	C₂×C₁₁	C₂×#460	1	1	yes	characteristic
203	C₂×C₁₁	C₂×#467	1	1	yes	characteristic
204	C₂×C₁₁	C₂×#473	1	1	yes	characteristic
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	4	no	—
208	C₂×Q₈	—	4	4	no	—
209	C₂×Q₈	—	11	4	no	—
210	C₂×Q₈	—	11	2	no	—
211	C₂×Q₈	—	22	2	no	—
212	C₂×Q₈	—	22	1	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	22	1	no	—
215	C₂²×C₄	—	22	1	no	—
216	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
217	C₂²×C₄	#4341	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
219	Q₁₆	—	4	8	no	—
220	Q₁₆	—	4	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	22	4	no	—
223	C₂×C₈	—	22	2	no	—
224	C₂×C₈	—	44	4	no	—
225	C₂×C₈	—	44	2	no	—
226	C₂×C₈	#4324	1	4	yes	—
227	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
228	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	11	2	no	—
234	C₄⋊C₄	—	22	1	no	—
235	C₄⋊C₄	—	22	2	no	—
236	C₄⋊C₄	—	22	2	no	—
237	C₄²	—	22	2	no	—
238	C₄²	—	22	2	no	—
239	C₄²	D₄×D₁₁	1	1	yes	characteristic
240	C₄²	#4341	1	1	yes	characteristic
241	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
242	C₄²	C₂³×D₁₁	1	1	yes	characteristic
243	C₁₁	C₂×#1670	1	1	yes	characteristic, complemented, Sylow
244	C₂³	#66991	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	4	4	no	—
247	Q₈	—	8	4	no	—
248	Q₈	—	11	4	no	—
249	Q₈	—	22	2	no	—
250	Q₈	—	22	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	—	22	4	no	—
255	C₂×C₄	—	22	2	no	—
256	C₂×C₄	—	22	2	no	—
257	C₂×C₄	—	44	2	no	—
258	C₂×C₄	Q₁₆×D₁₁	1	2	yes	—
259	C₂×C₄	#66972	1	2	yes	—
260	C₂×C₄	#66985	1	2	yes	—
261	C₂×C₄	C₂×#4324	1	2	yes	—
262	C₂×C₄	#66991	1	1	yes	characteristic
263	C₂×C₄	#66991	1	1	yes	characteristic
264	C₂×C₄	#66991	1	1	yes	characteristic
265	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
266	C₂×C₄	C₂×#4341	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	44	4	no	—
270	C₈	—	44	2	no	—
271	C₈	C₂×#4324	1	4	yes	—
272	C₂²	#10622688	1	2	yes	—
273	C₂²	#10622801	1	2	yes	—
274	C₂²	#10622805	1	2	yes	—
275	C₂²	C₂×#66991	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	4	no	—
279	C₄	—	22	4	no	—
280	C₄	—	44	2	no	—
281	C₄	#10622688	1	2	yes	—
282	C₄	#10622801	1	2	yes	—
283	C₄	C₂×Q₁₆×D₁₁	1	1	yes	characteristic
284	C₄	C₂×#66972	1	1	yes	characteristic
285	C₄	C₂×#66985	1	1	yes	characteristic
286	C₄	C₂²×#4324	1	1	yes	characteristic
287	C₂	#13067983	1	4	yes	direct product factor
288	C₂	C₂×#10622688	1	1	yes	characteristic
289	C₂	C₂×#10622801	1	1	yes	characteristic
290	C₂	C₂×#10622805	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#66991
commutator, cosocle	C₄²×C₁₁	C₂⁴
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#494	C₂
socle, prime radical	C₂³×C₁₁	#121
hypercenter	C₂×#494	D₁₁
hypocenter	C₁₁	C₂×#1670
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #13067983	C₂	direct (N × H)
73. C₂×Dic₂₂	Q₁₆	semidirect
128. Dic₂₂	C₂×Q₁₆	semidirect
201. C₂×C₁₁	#1670	semidirect
243. C₁₁	C₂×#1670	semidirect
287. C₂	#13067983	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×#10622707	2
C₂×C₈×Dic₂₂	2
#13067983	2
C₂×#10622096	2
C₂×#10622185	2
C₂×C₁₁×#494	2
C₂×#1670	11

Subgroups — C2×#13067983 #14736104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13067983 #14736104