Subgroups — C₂×#13070459 #14907008

Order 2816.

654 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622514	C₂	1	2	yes	maximal, complemented
2	C₂×#10622313	C₂	1	1	yes	maximal, characteristic
3	#13070459	C₂	1	8	yes	maximal, direct product factor
4	C₂×#10622101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10622146	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#66957	C₄	1	1	yes	characteristic
8	C₂×#66954	C₂²	1	2	yes	—
9	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×#66883	C₄	1	1	yes	characteristic
13	C₂×#66876	C₂²	1	2	yes	—
14	C₂×#66869	C₄	1	2	yes	—
15	#10622514	—	2	4	no	—
16	#10622514	C₂²	1	8	yes	complemented
17	C₂×#66862	—	2	2	no	—
18	C₂×C₁₆×D₁₁	—	2	2	no	—
19	#10622313	C₄	1	4	yes	—
20	#10622313	C₂²	1	4	yes	—
21	C₁₁×#346	C₂²	1	4	yes	complemented
22	#10622146	C₂²	1	8	yes	complemented
23	#10622101	C₂²	1	4	yes	complemented
24	C₂×#4339	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂×#4318	C₂×C₄	1	2	yes	—
29	C₂×Dic₄₄	C₈	1	1	yes	characteristic
30	#66957	C₈	1	4	yes	—
31	#66957	C₂×C₄	1	4	yes	—
32	C₂×D₈₈	C₈	1	1	yes	characteristic
33	C₂×#4314	C₈	1	2	yes	—
34	#66954	—	2	2	no	—
35	#66954	—	2	2	no	—
36	#66954	C₂×C₄	1	4	yes	—
37	#66954	C₂³	1	4	yes	—
38	C₂×#4313	—	2	2	no	—
39	C₂×C₈×D₁₁	—	2	2	no	—
40	C₂×C₁₁×C₁₆	—	2	1	no	—
41	C₂×C₁₁×C₁₆	—	2	1	no	—
42	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
43	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
44	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
45	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
46	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₁₁	C₂³	1	2	yes	—
48	#66883	C₂×C₄	1	4	yes	—
49	#66876	—	2	2	no	—
50	#66876	C₂³	1	4	yes	—
51	C₂×C₁₁⋊C₁₆	—	2	2	no	—
52	#66869	C₂×C₄	1	8	yes	—
53	#66862	—	2	8	no	—
54	C₁₆×D₁₁	—	2	8	no	—
55	C₂×#2171	—	11	1	no	maximal, Sylow
56	C₂²×C₄×C₁₁	—	2	1	no	—
57	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₁₁	—	2	2	no	—
59	#4339	—	2	2	no	—
60	#4339	—	2	2	no	—
61	#4339	C₂×C₈	1	4	yes	—
62	#4339	C₂²×C₄	1	4	yes	—
63	C₂×D₄₄	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₁₁	—	2	2	no	—
65	C₂×Dic₂₂	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
74	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
75	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
76	C₄²×C₁₁	—	2	1	no	—
77	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
78	#4318	—	2	2	no	—
79	#4318	C₂²×C₄	1	4	yes	—
80	C₂×C₁₁⋊C₈	—	2	2	no	—
81	Dic₄₄	C₂×C₈	1	4	yes	—
82	D₈₈	C₂×C₈	1	4	yes	—
83	#4314	C₂×C₈	1	8	yes	—
84	#4313	—	2	4	no	—
85	#4313	—	2	4	no	—
86	C₈×D₁₁	—	2	4	no	—
87	C₈×D₁₁	—	2	4	no	—
88	C₁₁×C₁₆	—	2	2	no	—
89	C₁₁×C₁₆	C₂×D₄	1	4	yes	—
90	C₁₁⋊C₁₆	—	2	4	no	—
91	C₂×#504	—	11	2	no	—
92	C₂×#443	—	11	1	no	—
93	#2171	—	11	8	no	—
94	C₂×#364	—	11	1	no	—
95	C₂×#350	—	11	2	no	—
96	C₂×#346	D₁₁	1	1	yes	characteristic, complemented
97	C₂³×C₁₁	#131	1	1	yes	characteristic
98	C₂²×D₁₁	—	2	2	no	—
99	C₂×C₄×C₁₁	—	2	2	no	—
100	C₂×C₄×C₁₁	—	4	2	no	—
101	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₁	#131	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
107	C₂²⋊D₁₁	—	2	4	no	—
108	C₂²⋊D₁₁	—	2	4	no	—
109	C₂×Dic₁₁	—	2	2	no	—
110	D₄₄	—	2	2	no	—
111	D₄₄	C₂²×C₈	1	4	yes	—
112	C₄×D₁₁	—	2	4	no	—
113	C₄×D₁₁	—	2	4	no	—
114	Dic₂₂	—	2	2	no	—
115	Dic₂₂	C₂²×C₈	1	4	yes	—
116	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
117	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
118	C₈×C₁₁	C₂²×C₈	1	2	yes	—
119	C₈×C₁₁	C₂²×C₈	1	2	yes	—
120	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
122	C₁₁⋊C₈	—	2	4	no	—
123	C₂×#135	—	11	1	no	—
124	C₂×#131	—	11	2	no	—
125	#504	—	11	8	no	—
126	#504	—	22	4	no	—
127	C₂×M₃₂	—	11	2	no	—
128	C₂×M₃₂	—	22	2	no	—
129	C₂²×C₁₆	—	22	2	no	—
130	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
131	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
132	#443	—	11	4	no	—
133	#443	—	11	4	no	—
134	C₂×#108	—	11	1	no	—
135	C₂×#104	—	11	2	no	—
136	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
137	#364	—	11	4	no	—
138	#350	—	11	8	no	—
139	#346	C₂×D₁₁	1	4	yes	complemented
140	C₂²×C₁₁	—	2	2	no	—
141	C₂²×C₁₁	C₈×D₄	1	1	yes	characteristic
142	C₂²×C₁₁	C₈×D₄	1	1	yes	characteristic
143	C₂²×C₁₁	C₂×#131	1	1	yes	characteristic
144	C₂×D₁₁	—	2	4	no	—
145	C₂×D₁₁	—	4	4	no	—
146	C₄×C₁₁	—	4	2	no	—
147	C₄×C₁₁	C₈×D₄	1	1	yes	characteristic
148	C₄×C₁₁	C₈×D₄	1	1	yes	characteristic
149	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
151	Dic₁₁	—	2	4	no	—
152	C₂×#41	—	11	2	no	—
153	#135	—	11	4	no	—
154	#135	—	11	4	no	—
155	C₂×Q₁₆	—	11	1	no	—
156	C₂×SD₁₆	—	11	2	no	—
157	C₂×D₈	—	11	1	no	—
158	#131	—	11	4	no	—
159	#131	—	11	4	no	—
160	#131	—	22	2	no	—
161	#131	—	22	2	no	—
162	C₂×M₁₆	—	11	2	no	—
163	C₂×M₁₆	—	22	2	no	—
164	C₂²×C₈	—	22	2	no	—
165	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
167	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
168	M₃₂	—	11	4	no	—
169	M₃₂	—	22	2	no	—
170	M₃₂	—	22	8	no	—
171	C₂×C₁₆	—	2	1	no	—
172	C₂×C₁₆	—	2	1	no	—
173	C₂×C₁₆	—	22	2	no	—
174	C₂×C₁₆	—	22	8	no	—
175	C₂×C₁₆	D₄₄	1	2	yes	—
176	C₂×C₁₆	D₄₄	1	2	yes	—
177	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
178	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
179	#108	—	11	4	no	—
180	#104	—	11	8	no	—
181	C₄×C₈	C₄×D₁₁	1	2	yes	—
182	C₄×C₈	C₂²×D₁₁	1	2	yes	—
183	C₂×C₁₁	—	2	1	no	—
184	C₂×C₁₁	—	2	1	no	—
185	C₂×C₁₁	#2171	1	2	yes	complemented
186	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
187	D₁₁	—	4	4	no	—
188	#41	—	11	4	no	—
189	#41	—	11	4	no	—
190	#41	—	22	2	no	—
191	#41	—	22	2	no	—
192	C₂×Q₈	—	11	2	no	—
193	C₂×D₄	—	11	2	no	—
194	C₂×D₄	—	22	2	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	22	2	no	—
197	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
198	Q₁₆	—	11	4	no	—
199	SD₁₆	—	11	8	no	—
200	D₈	—	11	4	no	—
201	M₁₆	—	11	4	no	—
202	M₁₆	—	22	2	no	—
203	M₁₆	—	22	4	no	—
204	M₁₆	—	22	4	no	—
205	C₂×C₈	—	22	2	no	—
206	C₂×C₈	—	22	4	no	—
207	C₂×C₈	—	22	4	no	—
208	C₂×C₈	C₈×D₁₁	1	1	yes	characteristic
209	C₂×C₈	C₈×D₁₁	1	1	yes	characteristic
210	C₂×C₈	C₈×D₁₁	1	2	yes	—
211	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
213	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
215	C₂×C₈	#4339	1	1	yes	characteristic
216	C₂×C₈	#4339	1	1	yes	characteristic
217	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
218	C₄²	—	2	1	no	—
219	C₄²	C₂×C₄×D₁₁	1	2	yes	—
220	C₁₆	—	2	2	no	—
221	C₁₆	—	22	4	no	—
222	C₁₆	C₂×D₄₄	1	4	yes	—
223	C₁₁	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	22	2	no	—
225	C₂³	#66954	1	1	yes	characteristic
226	Q₈	—	11	4	no	—
227	Q₈	—	22	2	no	—
228	D₄	—	11	4	no	—
229	D₄	—	22	4	no	—
230	D₄	—	22	4	no	—
231	D₄	—	22	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	22	2	no	—
235	C₂×C₄	—	22	4	no	—
236	C₂×C₄	—	22	4	no	—
237	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
239	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
241	C₂×C₄	#66954	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
243	C₈	—	22	4	no	—
244	C₈	C₄×D₄₄	1	1	yes	characteristic
245	C₈	C₄×D₄₄	1	1	yes	characteristic
246	C₈	C₂×C₈×D₁₁	1	2	yes	—
247	C₈	C₂×C₈×D₁₁	1	2	yes	—
248	C₈	C₂²×D₄₄	1	1	yes	characteristic
249	C₈	C₂×#4339	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	22	4	no	—
252	C₂²	—	44	4	no	—
253	C₂²	C₈×D₄₄	1	1	yes	characteristic
254	C₂²	C₈×D₄₄	1	1	yes	characteristic
255	C₂²	C₂×#66954	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	22	4	no	—
258	C₄	C₈×D₄₄	1	1	yes	characteristic
259	C₄	C₈×D₄₄	1	1	yes	characteristic
260	C₄	C₂×C₄×D₄₄	1	1	yes	characteristic
261	C₄	C₂²×C₈×D₁₁	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	44	4	no	—
265	C₂	#13070459	1	2	yes	direct product factor
266	C₂	C₂×C₈×D₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄₄
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×#346	C₂
socle	C₂²×C₁₁	C₈×D₄
prime radical	C₂×#66957	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂×#346	D₁₁
hypocenter	C₁₁	C₂×#2171
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#10622514	C₂	semidirect
3. #13070459	C₂	direct (N × H)
4. C₂×#10622101	C₂	semidirect
5. C₂×#10622146	C₂	semidirect
6. C₂×C₁₁×#346	C₂	semidirect
16. #10622514	C₂²	semidirect
21. C₁₁×#346	C₂²	semidirect
22. #10622146	C₂²	semidirect
23. #10622101	C₂²	semidirect
96. C₂×#346	D₁₁	semidirect
139. #346	C₂×D₁₁	semidirect
185. C₂×C₁₁	#2171	semidirect
223. C₁₁	C₂×#2171	semidirect
265. C₂	#13070459	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×#10622514	2
C₂×#10622313	2
#13070459	2
C₂×#10622101	2
C₂×#10622146	2
C₂×C₁₁×#346	2
C₂×#2171	11

Subgroups — C2×#13070459 #14907008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13070459 #14907008