Subgroups — C₂×#13070458 #14907007

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×#10622753	C₂	1	2	yes	maximal, complemented
2	#13070458	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10622101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10622199	C₂	1	2	yes	maximal, complemented
5	C₂×#10622099	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₁×#443	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₁×#135	C₄	1	1	yes	characteristic
8	C₂×C₁₁×#131	C₂²	1	2	yes	—
9	C₁₁×#443	C₄	1	4	yes	—
10	C₁₁×#443	C₂²	1	4	yes	—
11	C₂×C₁₁×#108	C₄	1	1	yes	characteristic
12	C₂×C₁₁×#104	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	#10622753	—	2	4	no	—
15	#10622753	C₂²	1	8	yes	complemented
16	C₂×#66876	—	2	2	no	—
17	C₂×#66876	C₂²	1	2	yes	—
18	C₂²×C₁₁⋊C₁₆	—	2	2	no	—
19	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
21	#10622199	C₂²	1	8	yes	complemented
22	#10622101	C₂²	1	4	yes	complemented
23	#10622099	C₂²	1	4	yes	complemented
24	C₂×C₁₁×#41	C₂×C₄	1	2	yes	—
25	C₁₁×#135	C₈	1	4	yes	—
26	C₁₁×#135	C₂×C₄	1	4	yes	—
27	C₂×C₁₁×Q₁₆	C₈	1	1	yes	characteristic
28	C₂×C₁₁×SD₁₆	C₈	1	2	yes	—
29	C₂×C₁₁×D₈	C₈	1	1	yes	characteristic
30	C₁₁×#131	—	2	2	no	—
31	C₁₁×#131	—	2	2	no	—
32	C₁₁×#131	C₂×C₄	1	4	yes	—
33	C₁₁×#131	C₂³	1	4	yes	—
34	C₂×C₁₁×M₁₆	—	2	2	no	—
35	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₁	—	2	2	no	—
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
40	C₁₁×#108	C₂×C₄	1	4	yes	—
41	C₁₁×#104	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₁	C₂³	1	2	yes	—
44	#66876	—	2	2	no	—
45	#66876	—	2	8	no	—
46	#66876	C₂³	1	4	yes	—
47	C₂×C₁₁⋊C₁₆	—	2	2	no	—
48	C₂×C₁₁⋊C₁₆	—	2	1	no	—
49	C₂×C₁₁⋊C₁₆	—	2	1	no	—
50	C₂×C₁₁⋊C₁₆	—	2	8	no	—
51	C₂×C₁₁⋊C₁₆	D₄	1	2	yes	—
52	C₂×C₁₁⋊C₁₆	D₄	1	2	yes	—
53	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
54	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
55	C₂×#2171	—	11	1	no	maximal, Sylow
56	C₁₁×#41	—	2	2	no	—
57	C₁₁×#41	—	2	2	no	—
58	C₁₁×#41	C₂×C₈	1	4	yes	—
59	C₁₁×#41	C₂²×C₄	1	4	yes	—
60	C₂×Q₈×C₁₁	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₁₁	—	2	2	no	—
62	C₂×D₄×C₁₁	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₁	—	2	2	no	—
64	C₂²×C₄×C₁₁	—	2	1	no	—
65	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
66	C₁₁×Q₁₆	C₂×C₈	1	4	yes	—
67	C₁₁×SD₁₆	C₂×C₈	1	8	yes	—
68	C₁₁×D₈	C₂×C₈	1	4	yes	—
69	C₁₁×M₁₆	—	2	4	no	—
70	C₁₁×M₁₆	—	2	4	no	—
71	C₁₁×M₁₆	—	2	2	no	—
72	C₁₁×M₁₆	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₁₁	—	2	4	no	—
74	C₂×C₈×C₁₁	—	2	4	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
86	C₄²×C₁₁	—	2	1	no	—
87	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
88	C₁₁⋊C₁₆	—	2	4	no	—
89	C₁₁⋊C₁₆	—	2	2	no	—
90	C₁₁⋊C₁₆	C₂×D₄	1	4	yes	—
91	C₂×#504	—	11	2	no	—
92	C₂×#443	D₁₁	1	1	yes	characteristic
93	#2171	—	11	8	no	—
94	C₂×#364	—	11	1	no	—
95	C₂×#350	—	11	2	no	—
96	C₂×#346	—	11	1	no	—
97	C₂³×C₁₁	—	2	2	no	—
98	C₂³×C₁₁	#131	1	1	yes	characteristic
99	Q₈×C₁₁	—	2	2	no	—
100	Q₈×C₁₁	C₂²×C₈	1	4	yes	—
101	D₄×C₁₁	—	2	4	no	—
102	D₄×C₁₁	—	2	4	no	—
103	D₄×C₁₁	—	2	2	no	—
104	D₄×C₁₁	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₁₁	—	2	2	no	—
106	C₂×C₄×C₁₁	—	2	4	no	—
107	C₂×C₄×C₁₁	—	2	4	no	—
108	C₂×C₄×C₁₁	—	2	2	no	—
109	C₂×C₄×C₁₁	—	4	2	no	—
110	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₁	#131	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₁₁	—	2	4	no	—
117	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
118	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
119	C₈×C₁₁	C₂²×C₈	1	2	yes	—
120	C₈×C₁₁	C₂²×C₈	1	2	yes	—
121	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
123	C₂×#135	Dic₁₁	1	1	yes	characteristic
124	C₂×#131	C₂×D₁₁	1	2	yes	—
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	—	11	1	no	—
130	C₂²×C₁₆	—	11	1	no	—
131	C₂²×C₁₆	—	22	2	no	—
132	#443	Dic₁₁	1	4	yes	—
133	#443	C₂×D₁₁	1	4	yes	—
134	C₂×#108	Dic₁₁	1	1	yes	characteristic
135	C₂×#104	Dic₁₁	1	2	yes	—
136	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
137	#364	—	11	4	no	—
138	#350	—	11	8	no	—
139	#346	—	11	4	no	—
140	C₂²×C₁₁	—	2	2	no	—
141	C₂²×C₁₁	—	2	4	no	—
142	C₂²×C₁₁	—	4	4	no	—
143	C₂²×C₁₁	C₈×D₄	1	1	yes	characteristic
144	C₂²×C₁₁	C₈×D₄	1	1	yes	characteristic
145	C₂²×C₁₁	C₂×#131	1	1	yes	characteristic
146	C₄×C₁₁	—	2	4	no	—
147	C₄×C₁₁	—	4	2	no	—
148	C₄×C₁₁	C₈×D₄	1	1	yes	characteristic
149	C₄×C₁₁	C₈×D₄	1	1	yes	characteristic
150	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
152	C₂×#41	C₂×Dic₁₁	1	2	yes	—
153	#135	C₁₁⋊C₈	1	4	yes	—
154	#135	C₂×Dic₁₁	1	4	yes	—
155	C₂×Q₁₆	C₁₁⋊C₈	1	1	yes	characteristic
156	C₂×SD₁₆	C₁₁⋊C₈	1	2	yes	—
157	C₂×D₈	C₁₁⋊C₈	1	1	yes	characteristic
158	#131	—	2	2	no	—
159	#131	—	2	2	no	—
160	#131	C₂×Dic₁₁	1	4	yes	—
161	#131	C₂²×D₁₁	1	4	yes	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
168	M₃₂	—	11	4	no	—
169	M₃₂	—	22	8	no	—
170	M₃₂	—	22	2	no	—
171	C₂×C₁₆	—	11	2	no	—
172	C₂×C₁₆	—	11	2	no	—
173	C₂×C₁₆	—	11	2	no	—
174	C₂×C₁₆	—	11	2	no	—
175	C₂×C₁₆	—	22	8	no	—
176	C₂×C₁₆	—	22	2	no	—
177	C₂×C₁₆	—	22	1	no	—
178	C₂×C₁₆	—	22	1	no	—
179	#108	C₂×Dic₁₁	1	4	yes	—
180	#104	C₂×Dic₁₁	1	8	yes	—
181	C₄×C₈	C₂×Dic₁₁	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	—	4	4	no	—
186	C₂×C₁₁	#2171	1	2	yes	complemented
187	C₂×C₁₁	C₂×C₈×D₄	1	1	yes	characteristic
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	C₂×C₁₁⋊C₈	1	4	yes	—
191	#41	C₂²×Dic₁₁	1	4	yes	—
192	C₂×Q₈	C₂×C₁₁⋊C₈	1	2	yes	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	C₂×C₁₁⋊C₈	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
198	Q₁₆	C₂×C₁₁⋊C₈	1	4	yes	—
199	SD₁₆	C₂×C₁₁⋊C₈	1	8	yes	—
200	D₈	C₂×C₁₁⋊C₈	1	4	yes	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	C₂²×Dic₁₁	1	4	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
209	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
210	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
211	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
212	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
213	C₂×C₈	#4341	1	1	yes	characteristic
214	C₂×C₈	#4341	1	1	yes	characteristic
215	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
216	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
217	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
218	C₄²	—	2	1	no	—
219	C₄²	C₂²×Dic₁₁	1	2	yes	—
220	C₁₆	—	11	4	no	—
221	C₁₆	—	22	4	no	—
222	C₁₆	—	22	2	no	—
223	C₁₁	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	2	2	no	—
225	C₂³	#67001	1	1	yes	characteristic
226	Q₈	—	2	2	no	—
227	Q₈	C₂²×C₁₁⋊C₈	1	4	yes	—
228	D₄	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	D₄	C₂²×C₁₁⋊C₈	1	4	yes	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
239	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
240	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
241	C₂×C₄	#67001	1	1	yes	characteristic
242	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	C₂²×C₁₁⋊C₈	1	2	yes	—
245	C₈	C₂²×C₁₁⋊C₈	1	2	yes	—
246	C₈	D₄×Dic₁₁	1	1	yes	characteristic
247	C₈	D₄×Dic₁₁	1	1	yes	characteristic
248	C₈	C₂×D₄×D₁₁	1	1	yes	characteristic
249	C₈	C₂×#4341	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	D₄×C₁₁⋊C₈	1	1	yes	characteristic
254	C₂²	D₄×C₁₁⋊C₈	1	1	yes	characteristic
255	C₂²	C₂×#67001	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	D₄×C₁₁⋊C₈	1	1	yes	characteristic
259	C₄	D₄×C₁₁⋊C₈	1	1	yes	characteristic
260	C₄	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
261	C₄	C₂×D₄×Dic₁₁	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	4	4	no	—
265	C₂	#13070458	1	2	yes	direct product factor
266	C₂	C₂×D₄×C₁₁⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₁
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×#443	C₂
socle	C₂²×C₁₁	C₈×D₄
prime radical	C₂×C₁₁×#135	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂×#443	D₁₁
hypocenter	C₁₁	C₂×#2171
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#10622753	C₂	semidirect
2. #13070458	C₂	direct (N × H)
3. C₂×#10622101	C₂	semidirect
4. C₂×#10622199	C₂	semidirect
5. C₂×#10622099	C₂	semidirect
15. #10622753	C₂²	semidirect
21. #10622199	C₂²	semidirect
22. #10622101	C₂²	semidirect
23. #10622099	C₂²	semidirect
186. C₂×C₁₁	#2171	semidirect
223. C₁₁	C₂×#2171	semidirect
265. C₂	#13070458	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×#10622753	2
#13070458	2
C₂×#10622101	2
C₂×#10622199	2
C₂×#10622099	2
C₂×C₁₁×#443	2
C₂×#2171	11

Subgroups — C2×#13070458 #14907007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13070458 #14907007