Subgroups — C₂×#10622197 #13068359

Order 1408.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#66979	C₂	1	1	yes	maximal, characteristic
3	C₂²×#4318	C₂	1	1	yes	maximal, characteristic
4	#10622197	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₄×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×C₁₁×M₁₆	—	2	1	no	—
7	C₂×C₁₁×M₁₆	—	2	1	no	—
8	C₂×C₁₁×M₁₆	—	2	1	no	—
9	C₂×C₁₁×M₁₆	C₄	1	2	yes	complemented
10	C₂×C₁₁×M₁₆	C₄	1	2	yes	complemented
11	C₂×C₁₁×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₁	—	2	1	no	—
13	C₂×C₂²⋊Dic₁₁	C₄	1	2	yes	—
14	#66979	—	2	1	no	—
15	#66979	—	2	1	no	—
16	#66979	C₄	1	2	yes	—
17	#66979	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₁	—	2	1	no	—
19	C₂×C₄×Dic₁₁	—	2	1	no	—
20	C₂×#4318	—	2	1	no	—
21	C₂×#4318	—	2	1	no	—
22	C₂×#4318	—	2	1	no	—
23	C₂×#4318	—	2	1	no	—
24	C₂×#4318	C₄	1	2	yes	complemented
25	C₂×#4318	C₂²	1	2	yes	—
26	C₂²×C₁₁⋊C₈	C₄	1	2	yes	complemented
27	C₂⁴×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₁	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₁	Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₁	—	2	2	no	—
43	C₁₁×M₁₆	—	2	2	no	—
44	C₁₁×M₁₆	—	2	2	no	—
45	C₁₁×M₁₆	—	2	2	no	—
46	C₁₁×M₁₆	C₂×C₄	1	4	yes	complemented
47	C₂×C₈×C₁₁	—	2	2	no	—
48	C₂×C₈×C₁₁	—	2	2	no	—
49	C₂×C₈×C₁₁	—	2	2	no	—
50	C₂²⋊Dic₁₁	—	2	2	no	—
51	C₂²⋊Dic₁₁	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₁₁	—	2	2	no	—
53	C₄⋊Dic₁₁	—	2	2	no	—
54	C₄×Dic₁₁	—	2	2	no	—
55	C₄×Dic₁₁	—	2	2	no	—
56	#4318	—	2	2	no	—
57	#4318	—	2	2	no	—
58	#4318	—	2	2	no	—
59	#4318	—	2	2	no	—
60	C₂×C₁₁⋊C₈	—	2	2	no	—
61	C₂×C₁₁⋊C₈	—	2	2	no	—
62	C₂×C₁₁⋊C₈	C₂×C₄	1	4	yes	complemented
63	C₂×#344	—	11	1	no	maximal, Sylow
64	C₂³×C₁₁	—	4	2	no	—
65	C₂³×C₁₁	C₄²	1	1	yes	characteristic
66	C₂³×C₁₁	C₄²	1	1	yes	characteristic
67	C₂³×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂³×C₁₁	C₄⋊C₄	1	1	yes	characteristic
70	C₂³×C₁₁	C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₁₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₁	C₄²	1	1	yes	characteristic
73	C₂×C₄×C₁₁	C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
100	C₂×Dic₁₁	—	2	4	no	—
101	C₂×Dic₁₁	—	4	4	no	—
102	C₈×C₁₁	—	2	4	no	—
103	C₁₁⋊C₈	—	2	4	no	—
104	C₂²×M₁₆	—	11	1	no	—
105	C₂²×M₁₆	D₁₁	1	1	yes	characteristic
106	C₂×#117	—	11	1	no	—
107	#344	—	11	4	no	—
108	C₂²×C₁₁	—	2	2	no	—
109	C₂²×C₁₁	—	2	2	no	—
110	C₂²×C₁₁	—	2	2	no	—
111	C₂²×C₁₁	—	4	1	no	—
112	C₂²×C₁₁	—	4	1	no	—
113	C₂²×C₁₁	—	4	1	no	—
114	C₂²×C₁₁	—	4	1	no	—
115	C₂²×C₁₁	#95	1	1	yes	characteristic
116	C₂²×C₁₁	#95	1	1	yes	characteristic
117	C₂²×C₁₁	#95	1	1	yes	characteristic
118	C₂²×C₁₁	#95	1	1	yes	characteristic
119	C₂²×C₁₁	C₂×C₄²	1	1	yes	characteristic
120	C₂²×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₁₁	#95	1	1	yes	characteristic
123	C₄×C₁₁	#95	1	1	yes	characteristic
124	C₄×C₁₁	#95	1	1	yes	characteristic
125	C₄×C₁₁	#95	1	1	yes	characteristic
126	C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
130	Dic₁₁	—	4	4	no	—
131	C₂³×C₄	C₂×D₁₁	1	1	yes	characteristic
132	C₂×M₁₆	—	2	1	no	—
133	C₂×M₁₆	—	2	1	no	—
134	C₂×M₁₆	—	2	1	no	—
135	C₂×M₁₆	—	11	2	no	—
136	C₂×M₁₆	—	11	2	no	—
137	C₂×M₁₆	—	22	1	no	—
138	C₂×M₁₆	—	22	1	no	—
139	C₂×M₁₆	—	22	1	no	—
140	C₂×M₁₆	—	22	1	no	—
141	C₂×M₁₆	Dic₁₁	1	2	yes	complemented
142	C₂×M₁₆	Dic₁₁	1	2	yes	complemented
143	C₂×M₁₆	C₂×D₁₁	1	2	yes	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	—	11	2	no	—
146	#117	—	11	2	no	—
147	#117	—	11	2	no	—
148	#117	—	22	1	no	—
149	#117	—	22	1	no	—
150	C₂×C₄⋊C₄	—	22	1	no	—
151	C₂×C₂²⋊C₄	—	11	2	no	—
152	C₂×C₄²	—	22	1	no	—
153	C₂×C₁₁	—	2	1	no	—
154	C₂×C₁₁	—	2	1	no	—
155	C₂×C₁₁	—	2	1	no	—
156	C₂×C₁₁	—	2	1	no	—
157	C₂×C₁₁	—	2	1	no	—
158	C₂×C₁₁	—	2	1	no	—
159	C₂×C₁₁	#344	1	2	yes	complemented
160	C₂×C₁₁	C₂×#95	1	1	yes	characteristic
161	C₂⁴	C₄×D₁₁	1	1	yes	characteristic
162	C₂²×C₄	—	22	2	no	—
163	C₂²×C₄	Dic₂₂	1	1	yes	characteristic
164	C₂²×C₄	Dic₂₂	1	1	yes	characteristic
165	C₂²×C₄	C₄×D₁₁	1	1	yes	characteristic
166	C₂²×C₄	C₄×D₁₁	1	1	yes	characteristic
167	C₂²×C₄	C₄×D₁₁	1	1	yes	characteristic
168	C₂²×C₄	D₄₄	1	1	yes	characteristic
169	C₂²×C₄	D₄₄	1	1	yes	characteristic
170	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
171	C₂²×C₄	C₂×Dic₁₁	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
177	M₁₆	—	2	2	no	—
178	M₁₆	—	2	2	no	—
179	M₁₆	—	2	2	no	—
180	M₁₆	—	22	2	no	—
181	M₁₆	—	22	2	no	—
182	M₁₆	—	22	2	no	—
183	M₁₆	—	22	2	no	—
184	M₁₆	C₂×Dic₁₁	1	4	yes	complemented
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	11	4	no	—
189	C₂×C₈	—	22	2	no	—
190	C₂×C₈	—	22	2	no	—
191	C₄⋊C₄	—	22	2	no	—
192	C₄⋊C₄	—	22	2	no	—
193	C₂²⋊C₄	—	11	4	no	—
194	C₂²⋊C₄	—	22	2	no	—
195	C₄²	—	22	2	no	—
196	C₄²	—	22	2	no	—
197	C₁₁	C₂×#344	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	4	2	no	—
199	C₂³	C₄×Dic₁₁	1	1	yes	characteristic
200	C₂³	C₄×Dic₁₁	1	1	yes	characteristic
201	C₂³	Dic₁₁⋊C₄	1	1	yes	characteristic
202	C₂³	Dic₁₁⋊C₄	1	1	yes	characteristic
203	C₂³	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
204	C₂³	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
205	C₂³	C₂×C₄×D₁₁	1	1	yes	characteristic
206	C₂×C₄	—	22	4	no	—
207	C₂×C₄	—	44	4	no	—
208	C₂×C₄	C₄×Dic₁₁	1	1	yes	characteristic
209	C₂×C₄	C₄×Dic₁₁	1	1	yes	characteristic
210	C₂×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
211	C₂×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
212	C₂×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
213	C₂×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
214	C₂×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
215	C₂×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
216	C₂×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
217	C₂×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
218	C₂×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
219	C₂×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
220	C₂×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
221	C₂×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
224	C₂×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
225	C₂×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
227	C₂×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
228	C₂×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
230	C₂×C₄	C₂×Dic₂₂	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄₄	1	1	yes	characteristic
233	C₂×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	—	22	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	C₁₁⋊#95	1	1	yes	characteristic
246	C₂²	C₁₁⋊#95	1	1	yes	characteristic
247	C₂²	C₁₁⋊#95	1	1	yes	characteristic
248	C₂²	C₁₁⋊#95	1	1	yes	characteristic
249	C₂²	C₂×C₄×Dic₁₁	1	1	yes	characteristic
250	C₂²	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
251	C₂²	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
252	C₄	—	44	4	no	—
253	C₄	C₁₁⋊#95	1	1	yes	characteristic
254	C₄	C₁₁⋊#95	1	1	yes	characteristic
255	C₄	C₁₁⋊#95	1	1	yes	characteristic
256	C₄	C₁₁⋊#95	1	1	yes	characteristic
257	C₄	C₂×Dic₁₁⋊C₄	1	1	yes	characteristic
258	C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
259	C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
260	C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	#10622197	1	2	yes	direct product factor
268	C₂	C₂×C₁₁⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₁₁)⋊C₄
commutator	C₂²×C₁₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₁
Fitting	C₂²×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	#95
prime radical	C₂⁴×C₁₁	C₂×C₄
cosocle	C₂²×C₄×C₁₁	C₂³
hypercenter	C₂²×M₁₆	D₁₁
hypocenter	C₁₁	C₂×#344
LCS	G ▷ C₂²×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10622197	C₂	direct (N × H)
9. C₂×C₁₁×M₁₆	C₄	semidirect
10. C₂×C₁₁×M₁₆	C₄	semidirect
24. C₂×#4318	C₄	semidirect
26. C₂²×C₁₁⋊C₈	C₄	semidirect
46. C₁₁×M₁₆	C₂×C₄	semidirect
62. C₂×C₁₁⋊C₈	C₂×C₄	semidirect
141. C₂×M₁₆	Dic₁₁	semidirect
142. C₂×M₁₆	Dic₁₁	semidirect
159. C₂×C₁₁	#344	semidirect
184. M₁₆	C₂×Dic₁₁	semidirect
197. C₁₁	C₂×#344	semidirect
267. C₂	#10622197	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₁×M₁₆	2
C₂×#66979	2
C₂²×#4318	2
#10622197	2
C₂×#344	11

Subgroups — C2×#10622197 #13068359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10622197 #13068359