Subgroups — C₂×#13068145 #14736408

Order 2816.

618 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622295	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622463	C₂	1	2	yes	maximal
3	C₂×#10622292	C₂	1	1	yes	maximal, characteristic
4	#13068145	C₂	1	8	yes	maximal, direct product factor
5	C₂×#10622119	C₂	1	2	yes	maximal
6	C₂×C₁₁×#334	C₂	1	1	yes	maximal, characteristic
7	C₂×#66922	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₂	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×#66882	—	2	1	no	—
12	C₂×#66882	—	2	1	no	—
13	C₂×#66882	—	2	2	no	—
14	C₂×#66880	—	2	2	no	—
15	C₂×#66879	—	2	1	no	—
16	C₂×Dic₁₁⋊C₈	—	2	1	no	—
17	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
18	#10622463	—	2	4	no	—
19	#10622463	C₂²	1	8	yes	—
20	#10622295	—	2	2	no	—
21	#10622295	C₂²	1	4	yes	—
22	#10622292	—	2	2	no	—
23	#10622292	C₂²	1	4	yes	—
24	C₁₁×#334	C₂²	1	4	yes	—
25	#10622119	C₂²	1	8	yes	—
26	C₂²×Dic₂₂	—	2	2	no	—
27	C₂²×Dic₂₂	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	—	2	2	no	—
29	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₁	—	2	2	no	—
33	C₂×C₄⋊Dic₁₁	—	2	2	no	—
34	C₂×C₄⋊Dic₁₁	D₄	1	1	yes	characteristic
35	C₂×Dic₁₁⋊C₄	—	2	1	no	—
36	C₂×C₄×Dic₁₁	—	2	1	no	—
37	C₂²×C₁₁⋊C₈	—	2	1	no	—
38	#66922	—	2	2	no	—
39	#66922	—	2	2	no	—
40	#66922	—	2	2	no	—
41	#66922	C₂³	1	4	yes	—
42	C₄×Dic₂₂	—	2	1	no	—
43	C₄×Dic₂₂	—	2	1	no	—
44	C₄×Dic₂₂	D₄	1	2	yes	—
45	C₄×Dic₂₂	C₂³	1	2	yes	—
46	C₁₁×C₄⋊C₈	—	2	2	no	—
47	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
48	C₄×C₈×C₁₁	D₄	1	2	yes	—
49	C₄×C₈×C₁₁	C₂³	1	2	yes	—
50	#66882	—	2	4	no	—
51	#66882	—	2	4	no	—
52	#66882	—	2	8	no	—
53	#66880	—	2	8	no	—
54	#66879	—	2	4	no	—
55	Dic₁₁⋊C₈	—	2	4	no	—
56	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
58	C₂×#1696	—	11	1	no	maximal, Sylow
59	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
62	C₂²×Dic₁₁	—	2	1	no	—
63	C₂²×Dic₁₁	—	2	1	no	—
64	C₂²×Dic₁₁	—	4	2	no	—
65	C₂×Dic₂₂	—	2	1	no	—
66	C₂×Dic₂₂	—	2	4	no	—
67	C₂×Dic₂₂	—	2	4	no	—
68	C₂×Dic₂₂	—	4	2	no	—
69	C₂×Dic₂₂	—	8	4	no	—
70	C₂×Dic₂₂	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₁	—	2	2	no	—
72	C₂×C₈×C₁₁	—	2	4	no	—
73	C₂×C₈×C₁₁	—	4	4	no	—
74	C₂×C₈×C₁₁	Q₁₆	1	4	yes	—
75	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
77	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₁	#41	1	1	yes	characteristic
80	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
81	C₄⋊Dic₁₁	—	2	1	no	—
82	C₄⋊Dic₁₁	—	2	4	no	—
83	C₄⋊Dic₁₁	—	2	4	no	—
84	C₄⋊Dic₁₁	—	4	2	no	—
85	C₄⋊Dic₁₁	—	4	2	no	—
86	C₄⋊Dic₁₁	C₂×D₄	1	2	yes	—
87	Dic₁₁⋊C₄	—	2	2	no	—
88	Dic₁₁⋊C₄	—	2	2	no	—
89	C₄×Dic₁₁	—	2	2	no	—
90	C₄×Dic₁₁	—	2	2	no	—
91	C₂×C₁₁⋊C₈	—	2	2	no	—
92	C₂×C₁₁⋊C₈	—	4	2	no	—
93	C₂×#500	—	11	1	no	—
94	C₂×#477	—	11	2	no	—
95	C₂×#446	—	11	1	no	—
96	#1696	—	11	8	no	—
97	C₂×#334	D₁₁	1	1	yes	characteristic
98	C₂×#333	—	11	2	no	—
99	C₂³×C₁₁	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₁	—	2	2	no	—
101	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
102	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₁₁	#137	1	2	yes	—
106	C₂×C₄×C₁₁	#137	1	4	yes	—
107	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₁₁	—	2	2	no	—
111	C₂×Dic₁₁	—	2	2	no	—
112	C₂×Dic₁₁	—	4	2	no	—
113	C₂×Dic₁₁	—	4	2	no	—
114	C₂×Dic₁₁	—	4	4	no	—
115	C₂×Dic₁₁	—	8	4	no	—
116	Dic₂₂	—	2	4	no	—
117	Dic₂₂	—	4	2	no	—
118	Dic₂₂	—	8	4	no	—
119	Dic₂₂	—	8	4	no	—
120	C₈×C₁₁	—	2	2	no	—
121	C₈×C₁₁	—	4	4	no	—
122	C₈×C₁₁	C₂×Q₁₆	1	4	yes	—
123	C₁₁⋊C₈	—	4	2	no	—
124	C₂×C₄⋊Q₈	—	11	2	no	—
125	C₂×C₄×Q₈	—	11	1	no	—
126	#500	—	11	4	no	—
127	#500	—	22	2	no	—
128	#477	—	11	8	no	—
129	#477	—	22	4	no	—
130	#446	—	11	4	no	—
131	#446	—	22	2	no	—
132	C₂×#107	—	22	2	no	—
133	C₂×#107	—	22	1	no	—
134	C₂×#107	—	22	1	no	—
135	C₂×C₄⋊C₈	—	11	1	no	—
136	C₂×C₄⋊C₈	—	22	1	no	—
137	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
138	C₂×#103	—	22	2	no	—
139	C₂×#97	—	22	1	no	—
140	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
141	#334	C₂×D₁₁	1	4	yes	—
142	#333	—	11	8	no	—
143	C₂²×C₁₁	#462	1	2	yes	—
144	C₂²×C₁₁	#470	1	2	yes	—
145	C₂²×C₁₁	#472	1	2	yes	—
146	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
147	C₄×C₁₁	—	2	1	no	—
148	C₄×C₁₁	—	2	1	no	—
149	C₄×C₁₁	#462	1	2	yes	—
150	C₄×C₁₁	#470	1	2	yes	—
151	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
152	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
153	C₄×C₁₁	C₂×#137	1	2	yes	—
154	Dic₁₁	—	4	2	no	—
155	Dic₁₁	—	4	2	no	—
156	Dic₁₁	—	8	4	no	—
157	C₂²×Q₈	—	11	1	no	—
158	C₂²×Q₈	—	22	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	22	1	no	—
161	C₂²×C₈	D₄₄	1	1	yes	characteristic
162	C₂²×C₈	D₄₄	1	1	yes	characteristic
163	C₄⋊Q₈	—	11	4	no	—
164	C₄⋊Q₈	—	22	2	no	—
165	C₄⋊Q₈	—	22	2	no	—
166	C₄⋊Q₈	—	22	2	no	—
167	C₄×Q₈	—	11	2	no	—
168	C₄×Q₈	—	11	2	no	—
169	C₄×Q₈	—	22	1	no	—
170	C₄×Q₈	—	22	1	no	—
171	C₂×C₄⋊C₄	—	11	1	no	—
172	C₂×C₄⋊C₄	—	22	1	no	—
173	C₂×C₄⋊C₄	—	22	2	no	—
174	C₂×C₄⋊C₄	—	22	2	no	—
175	C₂×C₄²	—	22	1	no	—
176	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
177	#107	—	22	8	no	—
178	#107	—	22	4	no	—
179	#107	—	22	4	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	11	2	no	—
182	C₄⋊C₈	—	22	4	no	—
183	C₄⋊C₈	—	22	1	no	—
184	C₄⋊C₈	C₂²×D₁₁	1	4	yes	—
185	#103	—	22	8	no	—
186	#97	—	22	4	no	—
187	C₄×C₈	D₄₄	1	2	yes	—
188	C₄×C₈	C₂²×D₁₁	1	2	yes	—
189	C₂×C₁₁	#1696	1	4	yes	complemented
190	C₂×C₁₁	C₂×#462	1	1	yes	characteristic
191	C₂×C₁₁	C₂×#470	1	1	yes	characteristic
192	C₂×C₁₁	C₂×#472	1	1	yes	characteristic
193	C₂×Q₈	—	11	2	no	—
194	C₂×Q₈	—	22	1	no	—
195	C₂×Q₈	—	22	4	no	—
196	C₂×Q₈	—	22	4	no	—
197	C₂×Q₈	—	44	2	no	—
198	C₂×Q₈	—	88	4	no	—
199	C₂²×C₄	—	22	1	no	—
200	C₂²×C₄	—	22	1	no	—
201	C₂²×C₄	—	44	2	no	—
202	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
203	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
204	C₂²×C₄	#4343	1	1	yes	characteristic
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	22	2	no	—
209	C₂×C₈	—	44	2	no	—
210	C₂×C₈	Dic₄₄	1	4	yes	—
211	C₂×C₈	C₂×D₄₄	1	2	yes	—
212	C₂×C₈	C₂×D₄₄	1	2	yes	—
213	C₄⋊C₄	—	11	2	no	—
214	C₄⋊C₄	—	22	1	no	—
215	C₄⋊C₄	—	22	2	no	—
216	C₄⋊C₄	—	22	2	no	—
217	C₄⋊C₄	—	22	4	no	—
218	C₄⋊C₄	—	22	4	no	—
219	C₄⋊C₄	—	44	2	no	—
220	C₄⋊C₄	—	44	2	no	—
221	C₄²	—	22	2	no	—
222	C₄²	—	22	2	no	—
223	C₄²	C₂×D₄₄	1	1	yes	characteristic
224	C₄²	D₄×D₁₁	1	1	yes	characteristic
225	C₄²	#4343	1	1	yes	characteristic
226	C₄²	C₂³×D₁₁	1	1	yes	characteristic
227	C₁₁	C₂×#1696	1	1	yes	characteristic, complemented, Sylow
228	C₂³	#66948	1	1	yes	characteristic
229	Q₈	—	22	4	no	—
230	Q₈	—	44	2	no	—
231	Q₈	—	88	4	no	—
232	Q₈	—	88	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	22	2	no	—
235	C₂×C₄	—	22	2	no	—
236	C₂×C₄	—	44	2	no	—
237	C₂×C₄	—	44	2	no	—
238	C₂×C₄	—	44	4	no	—
239	C₂×C₄	—	88	4	no	—
240	C₂×C₄	#66948	1	1	yes	characteristic
241	C₂×C₄	#66948	1	1	yes	characteristic
242	C₂×C₄	#66948	1	1	yes	characteristic
243	C₂×C₄	C₂×Dic₄₄	1	2	yes	—
244	C₂×C₄	#66962	1	2	yes	—
245	C₂×C₄	#66968	1	4	yes	—
246	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
248	C₂×C₄	C₂×#4343	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	4	4	no	—
251	C₈	—	44	2	no	—
252	C₈	C₂×Dic₄₄	1	4	yes	—
253	C₂²	#10622462	1	2	yes	—
254	C₂²	#10622481	1	2	yes	—
255	C₂²	#10622510	1	2	yes	—
256	C₂²	C₂×#66948	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	44	2	no	—
260	C₄	—	44	2	no	—
261	C₄	—	88	4	no	—
262	C₄	#10622462	1	2	yes	—
263	C₄	#10622481	1	2	yes	—
264	C₄	C₂²×Dic₄₄	1	1	yes	characteristic
265	C₄	C₂×#66962	1	1	yes	characteristic
266	C₄	C₂×#66968	1	2	yes	—
267	C₂	#13068145	1	4	yes	direct product factor
268	C₂	C₂×#10622462	1	1	yes	characteristic
269	C₂	C₂×#10622481	1	1	yes	characteristic
270	C₂	C₂×#10622510	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#66948
commutator, cosocle	C₄²×C₁₁	C₂⁴
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂×C₁₁×#334	C₂
socle, prime radical	C₂³×C₁₁	#121
hypercenter	C₂×#334	D₁₁
hypocenter	C₁₁	C₂×#1696
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
4. #13068145	C₂	direct (N × H)
189. C₂×C₁₁	#1696	semidirect
227. C₁₁	C₂×#1696	semidirect
267. C₂	#13068145	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×#10622295	2
C₂×#10622463	2
C₂×#10622292	2
#13068145	2
C₂×#10622119	2
C₂×C₁₁×#334	2
C₂×#1696	11

Subgroups — C2×#13068145 #14736408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13068145 #14736408