Subgroups — C₂²×#10622825 #14912033

Order 2816.

8670 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₁₁	C₂	1	2	yes	maximal, complemented
2	C₂³×C₂²⋊Dic₁₁	C₂	1	2	yes	maximal
3	C₂×#10622825	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₁⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂⁵×C₄×C₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂⁶×C₁₁	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₁₁	—	2	16	no	—
8	C₂⁴×C₄×C₁₁	C₄	1	16	yes	complemented
9	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₁	C₂²	1	12	yes	—
11	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₁₁	C₂²	1	4	yes	—
13	C₂²×C₂²⋊Dic₁₁	—	2	16	no	—
14	C₂²×C₂²⋊Dic₁₁	C₂²	1	24	yes	—
15	C₂²×C₄⋊Dic₁₁	—	2	16	no	—
16	C₂²×C₄⋊Dic₁₁	C₂²	1	24	yes	complemented
17	#10622825	C₂²	1	64	yes	direct product factor
18	C₂×C₁₁⋊#95	C₂²	1	24	yes	complemented
19	C₂⁵×C₁₁	—	2	8	no	—
20	C₂⁵×C₁₁	—	2	8	no	—
21	C₂⁵×C₁₁	C₂×C₄	1	8	yes	—
22	C₂⁵×C₁₁	D₄	1	8	yes	—
23	C₂⁵×C₁₁	Q₈	1	8	yes	—
24	C₂⁵×C₁₁	C₂³	1	6	yes	—
25	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₁₁	—	2	8	no	—
27	C₂³×C₄×C₁₁	—	2	128	no	—
28	C₂³×C₄×C₁₁	—	2	96	no	—
29	C₂³×C₄×C₁₁	C₂×C₄	1	8	yes	—
30	C₂³×C₄×C₁₁	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₁₁	C₂×C₄	1	96	yes	complemented
32	C₂³×C₄×C₁₁	D₄	1	16	yes	—
33	C₂³×C₄×C₁₁	C₂³	1	6	yes	—
34	C₂³×C₄×C₁₁	C₂³	1	16	yes	—
35	C₂³×C₄×C₁₁	C₂³	1	6	yes	—
36	C₂³×Dic₁₁	—	2	16	no	—
37	C₂³×Dic₁₁	—	2	16	no	—
38	C₂³×Dic₁₁	—	2	16	no	—
39	C₂³×Dic₁₁	C₂³	1	24	yes	—
40	C₂×C₂²⋊Dic₁₁	—	2	96	no	—
41	C₂×C₂²⋊Dic₁₁	C₂³	1	32	yes	—
42	C₂×C₄⋊Dic₁₁	—	2	96	no	—
43	C₂×C₄⋊Dic₁₁	C₂³	1	32	yes	complemented
44	C₁₁⋊#95	C₂³	1	32	yes	complemented
45	C₂²×#380	—	11	1	no	maximal, Sylow
46	C₂⁴×C₁₁	—	2	32	no	—
47	C₂⁴×C₁₁	—	2	32	no	—
48	C₂⁴×C₁₁	—	2	64	no	—
49	C₂⁴×C₁₁	—	2	64	no	—
50	C₂⁴×C₁₁	—	2	24	no	—
51	C₂⁴×C₁₁	—	2	24	no	—
52	C₂⁴×C₁₁	C₄⋊C₄	1	64	yes	complemented
53	C₂⁴×C₁₁	C₂²×C₄	1	24	yes	—
54	C₂⁴×C₁₁	C₂²×C₄	1	4	yes	—
55	C₂⁴×C₁₁	C₂×D₄	1	24	yes	—
56	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
57	C₂⁴×C₁₁	C₂×D₄	1	4	yes	—
58	C₂⁴×C₁₁	C₂×Q₈	1	24	yes	—
59	C₂⁴×C₁₁	C₂×Q₈	1	4	yes	—
60	C₂⁴×C₁₁	#41	1	8	yes	—
61	C₂⁴×C₁₁	C₂⁴	1	4	yes	—
62	C₂⁴×C₁₁	C₂⁴	1	3	yes	—
63	C₂²×C₄×C₁₁	—	2	32	no	—
64	C₂²×C₄×C₁₁	—	2	24	no	—
65	C₂²×C₄×C₁₁	—	2	384	no	—
66	C₂²×C₄×C₁₁	—	2	64	no	—
67	C₂²×C₄×C₁₁	C₂²⋊C₄	1	64	yes	complemented
68	C₂²×C₄×C₁₁	C₂²×C₄	1	24	yes	—
69	C₂²×C₄×C₁₁	C₂²×C₄	1	24	yes	—
70	C₂²×C₄×C₁₁	C₂²×C₄	1	64	yes	complemented
71	C₂²×C₄×C₁₁	C₂×D₄	1	48	yes	—
72	C₂²×C₄×C₁₁	C₂⁴	1	4	yes	—
73	C₂²×C₄×C₁₁	C₂⁴	1	4	yes	—
74	C₂²×Dic₁₁	—	2	48	no	—
75	C₂²×Dic₁₁	—	2	48	no	—
76	C₂²×Dic₁₁	—	2	48	no	—
77	C₂²×Dic₁₁	—	4	64	no	—
78	C₂²×Dic₁₁	C₂⁴	1	16	yes	—
79	C₂²⋊Dic₁₁	—	2	64	no	—
80	C₄⋊Dic₁₁	—	2	64	no	—
81	C₂⁵×C₄	D₁₁	1	1	yes	characteristic
82	C₂³×C₄⋊C₄	—	11	2	no	—
83	C₂³×C₂²⋊C₄	—	11	2	no	—
84	C₂×#380	—	11	24	no	—
85	C₂²×#95	—	11	2	no	—
86	C₂³×C₁₁	—	2	48	no	—
87	C₂³×C₁₁	—	2	48	no	—
88	C₂³×C₁₁	—	2	96	no	—
89	C₂³×C₁₁	—	2	96	no	—
90	C₂³×C₁₁	—	2	8	no	—
91	C₂³×C₁₁	—	2	8	no	—
92	C₂³×C₁₁	—	2	256	no	—
93	C₂³×C₁₁	C₂×C₂²⋊C₄	1	16	yes	—
94	C₂³×C₁₁	C₂×C₄⋊C₄	1	16	yes	—
95	C₂³×C₁₁	C₂×C₄⋊C₄	1	96	yes	complemented
96	C₂³×C₁₁	#117	1	16	yes	—
97	C₂³×C₁₁	#121	1	32	yes	—
98	C₂³×C₁₁	C₂²⋊Q₈	1	32	yes	—
99	C₂³×C₁₁	C₂³×C₄	1	8	yes	—
100	C₂³×C₁₁	C₂³×C₄	1	6	yes	—
101	C₂³×C₁₁	C₂²×D₄	1	8	yes	—
102	C₂³×C₁₁	C₂²×D₄	1	6	yes	—
103	C₂³×C₁₁	C₂²×D₄	1	12	yes	—
104	C₂³×C₁₁	C₂²×Q₈	1	8	yes	—
105	C₂³×C₁₁	C₂²×Q₈	1	6	yes	—
106	C₂³×C₁₁	C₂×#41	1	12	yes	—
107	C₂³×C₁₁	C₂⁵	1	1	yes	characteristic
108	C₂×C₄×C₁₁	—	2	48	no	—
109	C₂×C₄×C₁₁	—	2	8	no	—
110	C₂×C₄×C₁₁	—	2	128	no	—
111	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	96	yes	complemented
112	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
113	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₁₁	C₂²×D₄	1	16	yes	—
115	C₂×Dic₁₁	—	2	16	no	—
116	C₂×Dic₁₁	—	2	16	no	—
117	C₂×Dic₁₁	—	2	16	no	—
118	C₂×Dic₁₁	—	4	96	no	—
119	C₂⁶	C₂×D₁₁	1	1	yes	characteristic
120	C₂⁴×C₄	—	2	16	no	—
121	C₂⁴×C₄	—	11	4	no	—
122	C₂⁴×C₄	Dic₁₁	1	16	yes	complemented
123	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
124	C₂⁴×C₄	C₂×D₁₁	1	12	yes	—
125	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
126	C₂²×C₄⋊C₄	—	11	24	no	—
127	C₂²×C₄⋊C₄	—	22	16	no	—
128	C₂²×C₂²⋊C₄	—	11	24	no	—
129	C₂²×C₂²⋊C₄	—	22	16	no	—
130	#380	—	11	64	no	—
131	C₂×#95	—	11	24	no	—
132	C₂²×C₁₁	—	2	8	no	—
133	C₂²×C₁₁	—	2	8	no	—
134	C₂²×C₁₁	—	2	16	no	—
135	C₂²×C₁₁	—	2	16	no	—
136	C₂²×C₁₁	—	2	192	no	—
137	C₂²×C₁₁	#380	1	64	yes	complemented
138	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	12	yes	—
139	C₂²×C₁₁	C₂²×C₄⋊C₄	1	12	yes	—
140	C₂²×C₁₁	C₂²×C₄⋊C₄	1	16	yes	complemented
141	C₂²×C₁₁	C₂×#117	1	12	yes	—
142	C₂²×C₁₁	C₂×#121	1	24	yes	—
143	C₂²×C₁₁	C₂×C₂²⋊Q₈	1	24	yes	—
144	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
145	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
146	C₂²×C₁₁	C₂³×D₄	1	2	yes	—
147	C₂²×C₁₁	C₂³×Q₈	1	1	yes	characteristic
148	C₂²×C₁₁	C₂²×#41	1	2	yes	—
149	C₄×C₁₁	—	2	8	no	—
150	C₄×C₁₁	C₂²×C₂²⋊C₄	1	16	yes	complemented
151	Dic₁₁	—	4	16	no	—
152	C₂⁵	—	2	8	no	—
153	C₂⁵	—	2	8	no	—
154	C₂⁵	Dic₂₂	1	8	yes	—
155	C₂⁵	D₄₄	1	8	yes	—
156	C₂⁵	C₂×Dic₁₁	1	8	yes	—
157	C₂⁵	C₂²×D₁₁	1	6	yes	—
158	C₂⁵	C₂²×D₁₁	1	1	yes	characteristic
159	C₂³×C₄	—	2	8	no	—
160	C₂³×C₄	—	2	128	no	—
161	C₂³×C₄	—	2	96	no	—
162	C₂³×C₄	—	11	24	no	—
163	C₂³×C₄	—	22	16	no	—
164	C₂³×C₄	—	22	16	no	—
165	C₂³×C₄	—	22	16	no	—
166	C₂³×C₄	C₂×Dic₁₁	1	8	yes	—
167	C₂³×C₄	C₂×Dic₁₁	1	8	yes	—
168	C₂³×C₄	C₂×Dic₁₁	1	96	yes	complemented
169	C₂³×C₄	C₂²⋊D₁₁	1	16	yes	—
170	C₂³×C₄	C₂²×D₁₁	1	6	yes	—
171	C₂³×C₄	C₂²×D₁₁	1	16	yes	—
172	C₂³×C₄	C₂²×D₁₁	1	6	yes	—
173	C₂×C₄⋊C₄	—	11	32	no	—
174	C₂×C₄⋊C₄	—	22	96	no	—
175	C₂×C₂²⋊C₄	—	11	32	no	—
176	C₂×C₂²⋊C₄	—	22	96	no	—
177	#95	—	11	32	no	—
178	C₂×C₁₁	—	2	16	no	—
179	C₂×C₁₁	C₂×#380	1	24	yes	complemented
180	C₂×C₁₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₁₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₁₁	C₂²×#117	1	1	yes	characteristic
183	C₂×C₁₁	C₂²×#121	1	2	yes	—
184	C₂×C₁₁	C₂²×C₂²⋊Q₈	1	2	yes	—
185	C₂⁴	—	2	64	no	—
186	C₂⁴	—	2	32	no	—
187	C₂⁴	—	2	64	no	—
188	C₂⁴	—	2	32	no	—
189	C₂⁴	—	2	24	no	—
190	C₂⁴	—	2	24	no	—
191	C₂⁴	C₄⋊Dic₁₁	1	64	yes	complemented
192	C₂⁴	C₂×Dic₂₂	1	24	yes	—
193	C₂⁴	C₂×Dic₂₂	1	4	yes	—
194	C₂⁴	C₂×D₄₄	1	24	yes	—
195	C₂⁴	C₂×D₄₄	1	4	yes	—
196	C₂⁴	#4339	1	8	yes	—
197	C₂⁴	C₂²×Dic₁₁	1	24	yes	—
198	C₂⁴	C₂²×Dic₁₁	1	4	yes	—
199	C₂⁴	C₂×C₂²⋊D₁₁	1	8	yes	—
200	C₂⁴	C₂³×D₁₁	1	4	yes	—
201	C₂⁴	C₂³×D₁₁	1	3	yes	—
202	C₂²×C₄	—	2	32	no	—
203	C₂²×C₄	—	2	24	no	—
204	C₂²×C₄	—	2	384	no	—
205	C₂²×C₄	—	2	64	no	—
206	C₂²×C₄	—	11	16	no	—
207	C₂²×C₄	—	22	48	no	—
208	C₂²×C₄	—	22	48	no	—
209	C₂²×C₄	—	22	48	no	—
210	C₂²×C₄	—	44	64	no	—
211	C₂²×C₄	C₂²⋊Dic₁₁	1	64	yes	complemented
212	C₂²×C₄	C₂²×Dic₁₁	1	24	yes	—
213	C₂²×C₄	C₂²×Dic₁₁	1	24	yes	—
214	C₂²×C₄	C₂²×Dic₁₁	1	64	yes	complemented
215	C₂²×C₄	C₂×C₂²⋊D₁₁	1	48	yes	—
216	C₂²×C₄	C₂³×D₁₁	1	4	yes	—
217	C₂²×C₄	C₂³×D₁₁	1	4	yes	—
218	C₄⋊C₄	—	22	64	no	—
219	C₂²⋊C₄	—	22	64	no	—
220	C₁₁	C₂²×#380	1	1	yes	characteristic, complemented, Sylow
221	C₂³	—	2	96	no	—
222	C₂³	—	2	48	no	—
223	C₂³	—	2	96	no	—
224	C₂³	—	2	256	no	—
225	C₂³	—	2	48	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	2	8	no	—
228	C₂³	#66977	1	32	yes	—
229	C₂³	C₂×C₄⋊Dic₁₁	1	16	yes	—
230	C₂³	C₂×C₄⋊Dic₁₁	1	96	yes	complemented
231	C₂³	#66979	1	16	yes	—
232	C₂³	#66983	1	32	yes	—
233	C₂³	C₂×C₂²⋊Dic₁₁	1	16	yes	—
234	C₂³	C₂²×Dic₂₂	1	8	yes	—
235	C₂³	C₂²×Dic₂₂	1	6	yes	—
236	C₂³	C₂²×D₄₄	1	8	yes	—
237	C₂³	C₂²×D₄₄	1	6	yes	—
238	C₂³	C₂×#4339	1	12	yes	—
239	C₂³	C₂³×Dic₁₁	1	8	yes	—
240	C₂³	C₂³×Dic₁₁	1	6	yes	—
241	C₂³	C₂²×C₂²⋊D₁₁	1	12	yes	—
242	C₂³	C₂⁴×D₁₁	1	1	yes	characteristic
243	C₂×C₄	—	2	48	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	128	no	—
246	C₂×C₄	—	22	16	no	—
247	C₂×C₄	—	22	16	no	—
248	C₂×C₄	—	22	16	no	—
249	C₂×C₄	—	44	96	no	—
250	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	96	yes	complemented
251	C₂×C₄	C₂³×Dic₁₁	1	8	yes	—
252	C₂×C₄	C₂³×Dic₁₁	1	8	yes	—
253	C₂×C₄	C₂²×C₂²⋊D₁₁	1	16	yes	—
254	C₂²	—	2	192	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	16	no	—
258	C₂²	—	2	16	no	—
259	C₂²	#10622825	1	64	yes	direct product factor
260	C₂²	C₂×#66977	1	24	yes	—
261	C₂²	C₂²×C₄⋊Dic₁₁	1	12	yes	—
262	C₂²	C₂²×C₄⋊Dic₁₁	1	16	yes	complemented
263	C₂²	C₂×#66979	1	12	yes	—
264	C₂²	C₂×#66983	1	24	yes	—
265	C₂²	C₂²×C₂²⋊Dic₁₁	1	12	yes	—
266	C₂²	C₂³×Dic₂₂	1	1	yes	characteristic
267	C₂²	C₂³×D₄₄	1	1	yes	characteristic
268	C₂²	C₂²×#4339	1	2	yes	—
269	C₂²	C₂⁴×Dic₁₁	1	1	yes	characteristic
270	C₂²	C₂³×C₂²⋊D₁₁	1	2	yes	—
271	C₄	—	2	8	no	—
272	C₄	—	44	16	no	—
273	C₄	C₂²×C₂²⋊Dic₁₁	1	16	yes	complemented
274	C₂	—	2	16	no	—
275	C₂	C₂×#10622825	1	24	yes	direct product factor
276	C₂	C₂³×C₂²⋊Dic₁₁	1	1	yes	characteristic
277	C₂	C₂³×C₄⋊Dic₁₁	1	1	yes	characteristic
278	C₂	C₂²×#66979	1	1	yes	characteristic
279	C₂	C₂²×#66983	1	2	yes	—
280	C₂	C₂²×#66977	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₁₁
Fitting	C₂⁵×C₄×C₁₁	C₂
socle	C₂⁵×C₁₁	C₂³
prime radical	C₂⁶×C₁₁	C₂²
cosocle	C₂³×C₁₁	C₂⁵
hypercenter	C₂⁵×C₄	D₁₁
hypocenter	C₁₁	C₂²×#380
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄⋊Dic₁₁	C₂	semidirect
3. C₂×#10622825	C₂	direct (N × H)
4. C₂²×C₁₁⋊#95	C₂	semidirect
8. C₂⁴×C₄×C₁₁	C₄	semidirect
16. C₂²×C₄⋊Dic₁₁	C₂²	semidirect
17. #10622825	C₂²	direct (N × H)
18. C₂×C₁₁⋊#95	C₂²	semidirect
31. C₂³×C₄×C₁₁	C₂×C₄	semidirect
43. C₂×C₄⋊Dic₁₁	C₂³	semidirect
44. C₁₁⋊#95	C₂³	semidirect
52. C₂⁴×C₁₁	C₄⋊C₄	semidirect
67. C₂²×C₄×C₁₁	C₂²⋊C₄	semidirect
70. C₂²×C₄×C₁₁	C₂²×C₄	semidirect
95. C₂³×C₁₁	C₂×C₄⋊C₄	semidirect
111. C₂×C₄×C₁₁	C₂×C₂²⋊C₄	semidirect
122. C₂⁴×C₄	Dic₁₁	semidirect
137. C₂²×C₁₁	#380	semidirect
140. C₂²×C₁₁	C₂²×C₄⋊C₄	semidirect
150. C₄×C₁₁	C₂²×C₂²⋊C₄	semidirect
168. C₂³×C₄	C₂×Dic₁₁	semidirect
179. C₂×C₁₁	C₂×#380	semidirect
191. C₂⁴	C₄⋊Dic₁₁	semidirect
211. C₂²×C₄	C₂²⋊Dic₁₁	semidirect
214. C₂²×C₄	C₂²×Dic₁₁	semidirect
220. C₁₁	C₂²×#380	semidirect
230. C₂³	C₂×C₄⋊Dic₁₁	semidirect
250. C₂×C₄	C₂×C₂²⋊Dic₁₁	semidirect
259. C₂²	#10622825	direct (N × H)
262. C₂²	C₂²×C₄⋊Dic₁₁	semidirect
273. C₄	C₂²×C₂²⋊Dic₁₁	semidirect
275. C₂	C₂×#10622825	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₁₁	2
C₂³×C₂²⋊Dic₁₁	2
C₂×#10622825	2
C₂²×C₁₁⋊#95	2
C₂⁵×C₄×C₁₁	2
C₂²×#380	11

Subgroups — C22×#10622825 #14912033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10622825 #14912033