Subgroups — C₂³×D₄×#261 #16809628

Order 3328.

19998 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×#261	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄×#261	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊#261	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₂²⋊#261	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×#261	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁵×D₁₃	C₂²	1	2	yes	—
8	C₂⁴×#261	—	2	32	no	—
9	C₂⁴×#261	C₂²	1	1	yes	characteristic
10	C₂⁴×#261	C₂²	1	2	yes	—
11	C₂⁴×#261	C₂²	1	1	yes	characteristic
12	C₂⁴×#261	C₂²	1	56	yes	complemented
13	C₂³×D₄×C₁₃	C₄	1	1	yes	characteristic, complemented
14	C₂³×C₂²⋊D₁₃	C₄	1	2	yes	complemented
15	C₂²×D₄×D₁₃	C₄	1	28	yes	complemented
16	C₂²×D₄×D₁₃	C₂²	1	28	yes	—
17	C₂³×D₅₂	C₄	1	1	yes	characteristic, complemented
18	C₂³×C₄×D₁₃	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊#261	C₂²	1	56	yes	complemented
20	C₂×D₄×#261	C₂²	1	448	yes	direct product factor
21	C₂²×C₄⋊#261	C₂²	1	28	yes	complemented
22	C₂²×C₄×#261	C₂²	1	28	yes	complemented
23	C₂⁵×C₁₃	C₂×C₄	1	2	yes	complemented
24	C₂⁴×D₁₃	—	2	16	no	—
25	C₂⁴×D₁₃	—	2	16	no	—
26	C₂⁴×D₁₃	C₂×C₄	1	28	yes	complemented
27	C₂⁴×D₁₃	C₂×C₄	1	2	yes	complemented
28	C₂⁴×D₁₃	C₂³	1	28	yes	—
29	C₂⁴×D₁₃	C₂³	1	1	yes	characteristic
30	C₂³×#261	—	2	16	no	—
31	C₂³×#261	—	2	8	no	—
32	C₂³×#261	—	2	448	no	—
33	C₂³×#261	D₄	1	16	yes	direct product factor
34	C₂³×#261	C₂³	1	14	yes	—
35	C₂³×#261	C₂³	1	28	yes	—
36	C₂³×#261	C₂³	1	14	yes	—
37	C₂³×#261	C₂³	1	224	yes	complemented
38	C₂²×D₄×C₁₃	C₂×C₄	1	28	yes	complemented
39	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂²×C₂²⋊D₁₃	C₂×C₄	1	56	yes	complemented
41	C₂³×Dic₁₃	C₂×C₄	1	1	yes	characteristic, complemented
42	C₂×D₄×D₁₃	C₂×C₄	1	336	yes	complemented
43	C₂×D₄×D₁₃	C₂³	1	112	yes	—
44	C₂²×D₅₂	C₂×C₄	1	28	yes	complemented
45	C₂²×C₄×D₁₃	C₂×C₄	1	14	yes	complemented
46	C₂²×C₄×D₁₃	C₂³	1	14	yes	—
47	C₂×C₂²⋊#261	C₂³	1	224	yes	complemented
48	D₄×#261	C₂³	1	512	yes	direct product factor
49	C₂×C₄⋊#261	C₂³	1	112	yes	complemented
50	C₂×C₄×#261	C₂³	1	112	yes	complemented
51	C₂³×C₄×D₄	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	—	2	16	no	—
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂⁴×C₁₃	C₂²×C₄	1	28	yes	complemented
55	C₂³×D₁₃	—	2	16	no	—
56	C₂³×D₁₃	—	2	224	no	—
57	C₂³×D₁₃	—	2	112	no	—
58	C₂³×D₁₃	—	2	112	no	—
59	C₂³×D₁₃	C₂²×C₄	1	7	yes	—
60	C₂³×D₁₃	C₂²×C₄	1	168	yes	complemented
61	C₂³×D₁₃	C₂²×C₄	1	28	yes	complemented
62	C₂³×D₁₃	C₂×D₄	1	8	yes	—
63	C₂³×D₁₃	#41	1	8	yes	—
64	C₂³×D₁₃	C₂⁴	1	56	yes	—
65	C₂³×D₁₃	C₂⁴	1	7	yes	—
66	C₂²×#261	—	2	112	no	—
67	C₂²×#261	—	2	56	no	—
68	C₂²×#261	—	2	896	no	—
69	C₂²×#261	C₂×D₄	1	112	yes	direct product factor
70	C₂²×#261	C₂⁴	1	28	yes	—
71	C₂²×#261	C₂⁴	1	56	yes	—
72	C₂²×#261	C₂⁴	1	28	yes	—
73	C₂²×#261	C₂⁴	1	128	yes	complemented
74	C₂×D₄×C₁₃	C₂²×C₄	1	112	yes	complemented
75	C₂²×C₄×C₁₃	C₂²×C₄	1	14	yes	complemented
76	C₂×C₂²⋊D₁₃	C₂²×C₄	1	224	yes	complemented
77	C₂²×Dic₁₃	C₂²×C₄	1	14	yes	complemented
78	D₄×D₁₃	C₂²×C₄	1	448	yes	complemented
79	D₄×D₁₃	C₂⁴	1	64	yes	—
80	C₂×D₅₂	C₂²×C₄	1	112	yes	complemented
81	C₂×C₄×D₁₃	C₂²×C₄	1	84	yes	complemented
82	C₂×C₄×D₁₃	C₂⁴	1	28	yes	—
83	C₂²⋊#261	C₂⁴	1	128	yes	complemented
84	C₄⋊#261	C₂⁴	1	64	yes	complemented
85	C₄×#261	C₂⁴	1	64	yes	complemented
86	C₂⁴×D₄	—	13	1	no	—
87	C₂⁵×C₄	—	13	2	no	—
88	C₂²×C₄×D₄	—	13	56	no	—
89	C₂³×C₄⋊C₄	—	13	1	no	—
90	C₂³×C₂²⋊C₄	—	13	2	no	—
91	C₂³×C₄²	—	13	1	no	—
92	C₂³×C₁₃	—	2	112	no	—
93	C₂³×C₁₃	C₄×D₄	1	8	yes	complemented
94	C₂³×C₁₃	C₂³×C₄	1	7	yes	—
95	C₂³×C₁₃	C₂³×C₄	1	56	yes	complemented
96	C₂²×D₁₃	—	2	112	no	—
97	C₂²×D₁₃	—	2	672	no	—
98	C₂²×D₁₃	—	2	112	no	—
99	C₂²×D₁₃	—	2	112	no	—
100	C₂²×D₁₃	C₄×D₄	1	56	yes	complemented
101	C₂²×D₁₃	C₂³×C₄	1	21	yes	—
102	C₂²×D₁₃	C₂³×C₄	1	112	yes	complemented
103	C₂²×D₁₃	C₂³×C₄	1	56	yes	complemented
104	C₂²×D₁₃	C₂²×D₄	1	28	yes	—
105	C₂²×D₁₃	C₂×#41	1	28	yes	—
106	C₂²×D₁₃	C₂⁵	1	16	yes	—
107	C₂²×D₁₃	C₂⁵	1	7	yes	—
108	C₂×#261	—	2	112	no	—
109	C₂×#261	—	2	56	no	—
110	C₂×#261	—	2	256	no	—
111	C₂×#261	C₂²×D₄	1	112	yes	direct product factor
112	C₂×#261	C₂⁵	1	8	yes	—
113	C₂×#261	C₂⁵	1	16	yes	—
114	C₂×#261	C₂⁵	1	8	yes	—
115	D₄×C₁₃	C₂³×C₄	1	64	yes	complemented
116	C₂×C₄×C₁₃	C₂³×C₄	1	28	yes	complemented
117	C₂²⋊D₁₃	C₂³×C₄	1	128	yes	complemented
118	C₂×Dic₁₃	C₂³×C₄	1	28	yes	complemented
119	D₅₂	C₂³×C₄	1	64	yes	complemented
120	C₄×D₁₃	C₂³×C₄	1	56	yes	complemented
121	C₄×D₁₃	C₂⁵	1	8	yes	—
122	C₂⁶	—	13	2	no	—
123	C₂³×D₄	—	13	2	no	—
124	C₂³×D₄	—	13	28	no	—
125	C₂³×D₄	—	13	28	no	—
126	C₂³×D₄	—	13	1	no	—
127	C₂³×D₄	#261	1	1	yes	characteristic, direct product factor
128	C₂⁴×C₄	—	13	1	no	—
129	C₂⁴×C₄	—	13	1	no	—
130	C₂⁴×C₄	—	13	2	no	—
131	C₂⁴×C₄	—	13	56	no	—
132	C₂⁴×C₄	—	13	1	no	—
133	C₂⁴×C₄	—	26	32	no	—
134	C₂×C₄×D₄	—	13	448	no	—
135	C₂²×C₄⋊C₄	—	13	28	no	—
136	C₂²×C₂²⋊C₄	—	13	56	no	—
137	C₂²×C₄²	—	13	28	no	—
138	C₂²×C₁₃	—	2	112	no	—
139	C₂²×C₁₃	C₂×C₄×D₄	1	28	yes	complemented
140	C₂²×C₁₃	C₂⁴×C₄	1	7	yes	—
141	C₂²×C₁₃	C₂⁴×C₄	1	16	yes	complemented
142	C₂×D₁₃	—	2	112	no	—
143	C₂×D₁₃	—	2	16	no	—
144	C₂×D₁₃	—	2	16	no	—
145	C₂×D₁₃	—	2	224	no	—
146	C₂×D₁₃	C₂×C₄×D₄	1	84	yes	complemented
147	C₂×D₁₃	C₂⁴×C₄	1	7	yes	—
148	C₂×D₁₃	C₂⁴×C₄	1	16	yes	complemented
149	C₂×D₁₃	C₂³×D₄	1	14	yes	—
150	C₂×D₁₃	C₂²×#41	1	14	yes	—
151	C₂×D₁₃	C₂⁶	1	1	yes	characteristic
152	#261	—	2	16	no	—
153	#261	—	2	8	no	—
154	#261	C₂³×D₄	1	16	yes	direct product factor
155	C₄×C₁₃	C₂⁴×C₄	1	8	yes	complemented
156	Dic₁₃	C₂⁴×C₄	1	8	yes	complemented
157	C₂⁵	—	13	1	no	—
158	C₂⁵	—	13	2	no	—
159	C₂⁵	—	13	28	no	—
160	C₂⁵	—	13	28	no	—
161	C₂⁵	—	26	16	no	—
162	C₂⁵	—	26	16	no	—
163	C₂⁵	C₂×#261	1	2	yes	complemented
164	C₂²×D₄	—	13	56	no	—
165	C₂²×D₄	—	13	112	no	—
166	C₂²×D₄	—	13	336	no	—
167	C₂²×D₄	—	13	28	no	—
168	C₂²×D₄	C₂×#261	1	28	yes	direct product factor
169	C₂³×C₄	—	13	14	no	—
170	C₂³×C₄	—	13	16	no	—
171	C₂³×C₄	—	13	1	no	—
172	C₂³×C₄	—	13	14	no	—
173	C₂³×C₄	—	13	14	no	—
174	C₂³×C₄	—	13	14	no	—
175	C₂³×C₄	—	13	28	no	—
176	C₂³×C₄	—	13	224	no	—
177	C₂³×C₄	—	26	448	no	—
178	C₂³×C₄	—	26	16	no	—
179	C₂³×C₄	—	26	8	no	—
180	C₂³×C₄	C₂×#261	1	1	yes	characteristic, complemented
181	C₄×D₄	—	13	512	no	—
182	C₂×C₄⋊C₄	—	13	112	no	—
183	C₂×C₂²⋊C₄	—	13	224	no	—
184	C₂×C₄²	—	13	112	no	—
185	C₂×C₁₃	—	2	16	no	—
186	C₂×C₁₃	C₂²×C₄×D₄	1	14	yes	complemented
187	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
188	D₁₃	—	2	16	no	—
189	D₁₃	C₂²×C₄×D₄	1	14	yes	complemented
190	D₁₃	C₂⁴×D₄	1	1	yes	characteristic
191	D₁₃	C₂³×#41	1	1	yes	characteristic
192	C₂⁴	—	2	16	no	—
193	C₂⁴	—	13	8	no	—
194	C₂⁴	—	13	8	no	—
195	C₂⁴	—	13	7	no	—
196	C₂⁴	—	13	7	no	—
197	C₂⁴	—	13	28	no	—
198	C₂⁴	—	13	56	no	—
199	C₂⁴	—	13	168	no	—
200	C₂⁴	—	26	112	no	—
201	C₂⁴	—	26	112	no	—
202	C₂⁴	—	26	224	no	—
203	C₂⁴	—	26	16	no	—
204	C₂⁴	C₂²×#261	1	1	yes	characteristic
205	C₂⁴	C₂²×#261	1	28	yes	complemented
206	C₂×D₄	—	13	224	no	—
207	C₂×D₄	—	13	64	no	—
208	C₂×D₄	—	13	448	no	—
209	C₂×D₄	—	13	112	no	—
210	C₂×D₄	C₂²×#261	1	112	yes	direct product factor
211	C₂²×C₄	—	13	28	no	—
212	C₂²×C₄	—	13	112	no	—
213	C₂²×C₄	—	13	14	no	—
214	C₂²×C₄	—	13	28	no	—
215	C₂²×C₄	—	13	84	no	—
216	C₂²×C₄	—	13	28	no	—
217	C₂²×C₄	—	13	56	no	—
218	C₂²×C₄	—	13	128	no	—
219	C₂²×C₄	—	26	896	no	—
220	C₂²×C₄	—	26	112	no	—
221	C₂²×C₄	—	26	56	no	—
222	C₂²×C₄	C₂²×#261	1	14	yes	complemented
223	C₄⋊C₄	—	13	64	no	—
224	C₂²⋊C₄	—	13	128	no	—
225	C₄²	—	13	64	no	—
226	C₁₃	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	2	112	no	—
228	C₂³	—	13	28	no	—
229	C₂³	—	13	28	no	—
230	C₂³	—	13	56	no	—
231	C₂³	—	13	7	no	—
232	C₂³	—	13	21	no	—
233	C₂³	—	13	56	no	—
234	C₂³	—	13	16	no	—
235	C₂³	—	13	112	no	—
236	C₂³	—	26	112	no	—
237	C₂³	—	26	672	no	—
238	C₂³	—	26	112	no	—
239	C₂³	—	26	112	no	—
240	C₂³	D₄×#261	1	8	yes	direct product factor
241	C₂³	C₂³×#261	1	7	yes	—
242	C₂³	C₂³×#261	1	56	yes	complemented
243	D₄	—	13	128	no	—
244	D₄	—	13	64	no	—
245	D₄	C₂³×#261	1	64	yes	direct product factor
246	C₂×C₄	—	13	8	no	—
247	C₂×C₄	—	13	112	no	—
248	C₂×C₄	—	13	28	no	—
249	C₂×C₄	—	13	8	no	—
250	C₂×C₄	—	13	56	no	—
251	C₂×C₄	—	13	8	no	—
252	C₂×C₄	—	13	16	no	—
253	C₂×C₄	—	26	256	no	—
254	C₂×C₄	—	26	112	no	—
255	C₂×C₄	—	26	56	no	—
256	C₂×C₄	C₂³×#261	1	28	yes	complemented
257	C₂²	—	2	112	no	—
258	C₂²	—	13	84	no	—
259	C₂²	—	13	14	no	—
260	C₂²	—	13	14	no	—
261	C₂²	—	13	7	no	—
262	C₂²	—	13	1	no	—
263	C₂²	—	13	16	no	—
264	C₂²	—	26	16	no	—
265	C₂²	—	26	16	no	—
266	C₂²	—	26	224	no	—
267	C₂²	—	26	112	no	—
268	C₂²	C₂×D₄×#261	1	28	yes	direct product factor
269	C₂²	C₂⁴×#261	1	7	yes	—
270	C₂²	C₂⁴×#261	1	16	yes	complemented
271	C₄	—	13	16	no	—
272	C₄	—	13	8	no	—
273	C₄	—	26	8	no	—
274	C₄	—	26	16	no	—
275	C₄	C₂⁴×#261	1	8	yes	complemented
276	C₂	—	2	16	no	—
277	C₂	—	13	1	no	—
278	C₂	—	13	1	no	—
279	C₂	—	13	14	no	—
280	C₂	—	26	16	no	—
281	C₂	C₂²×D₄×#261	1	14	yes	direct product factor
282	C₂	C₂⁵×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×#261
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#261
Fitting	C₂³×D₄×C₁₃	C₄
socle	C₂⁴×C₁₃	C₂²×C₄
prime radical	C₂³×D₄×D₁₃	C₂
cosocle	C₂×D₁₃	C₂⁶
hypercenter	C₂³×D₄	#261
hypocenter	C₁₃	C₂³×C₄×D₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×#261	C₂	semidirect
2. C₂²×D₄×#261	C₂	direct (N × H)
3. C₂³×C₄⋊#261	C₂	semidirect
4. C₂³×C₂²⋊#261	C₂	semidirect
5. C₂³×C₄×#261	C₂	semidirect
12. C₂⁴×#261	C₂²	semidirect
13. C₂³×D₄×C₁₃	C₄	semidirect
14. C₂³×C₂²⋊D₁₃	C₄	semidirect
15. C₂²×D₄×D₁₃	C₄	semidirect
17. C₂³×D₅₂	C₄	semidirect
19. C₂²×C₂²⋊#261	C₂²	semidirect
20. C₂×D₄×#261	C₂²	direct (N × H)
21. C₂²×C₄⋊#261	C₂²	semidirect
22. C₂²×C₄×#261	C₂²	semidirect
23. C₂⁵×C₁₃	C₂×C₄	semidirect
26. C₂⁴×D₁₃	C₂×C₄	semidirect
27. C₂⁴×D₁₃	C₂×C₄	semidirect
33. C₂³×#261	D₄	direct (N × H)
37. C₂³×#261	C₂³	semidirect
38. C₂²×D₄×C₁₃	C₂×C₄	semidirect
39. C₂³×C₄×C₁₃	C₂×C₄	semidirect
40. C₂²×C₂²⋊D₁₃	C₂×C₄	semidirect
41. C₂³×Dic₁₃	C₂×C₄	semidirect
42. C₂×D₄×D₁₃	C₂×C₄	semidirect
44. C₂²×D₅₂	C₂×C₄	semidirect
45. C₂²×C₄×D₁₃	C₂×C₄	semidirect
47. C₂×C₂²⋊#261	C₂³	semidirect
48. D₄×#261	C₂³	direct (N × H)
49. C₂×C₄⋊#261	C₂³	semidirect
50. C₂×C₄×#261	C₂³	semidirect
54. C₂⁴×C₁₃	C₂²×C₄	semidirect
60. C₂³×D₁₃	C₂²×C₄	semidirect
61. C₂³×D₁₃	C₂²×C₄	semidirect
69. C₂²×#261	C₂×D₄	direct (N × H)
73. C₂²×#261	C₂⁴	semidirect
74. C₂×D₄×C₁₃	C₂²×C₄	semidirect
75. C₂²×C₄×C₁₃	C₂²×C₄	semidirect
76. C₂×C₂²⋊D₁₃	C₂²×C₄	semidirect
77. C₂²×Dic₁₃	C₂²×C₄	semidirect
78. D₄×D₁₃	C₂²×C₄	semidirect
80. C₂×D₅₂	C₂²×C₄	semidirect
81. C₂×C₄×D₁₃	C₂²×C₄	semidirect
83. C₂²⋊#261	C₂⁴	semidirect
84. C₄⋊#261	C₂⁴	semidirect
85. C₄×#261	C₂⁴	semidirect
93. C₂³×C₁₃	C₄×D₄	semidirect
95. C₂³×C₁₃	C₂³×C₄	semidirect
100. C₂²×D₁₃	C₄×D₄	semidirect
102. C₂²×D₁₃	C₂³×C₄	semidirect
103. C₂²×D₁₃	C₂³×C₄	semidirect
111. C₂×#261	C₂²×D₄	direct (N × H)
115. D₄×C₁₃	C₂³×C₄	semidirect
116. C₂×C₄×C₁₃	C₂³×C₄	semidirect
117. C₂²⋊D₁₃	C₂³×C₄	semidirect
118. C₂×Dic₁₃	C₂³×C₄	semidirect
119. D₅₂	C₂³×C₄	semidirect
120. C₄×D₁₃	C₂³×C₄	semidirect
127. C₂³×D₄	#261	direct (N × H)
139. C₂²×C₁₃	C₂×C₄×D₄	semidirect
141. C₂²×C₁₃	C₂⁴×C₄	semidirect
146. C₂×D₁₃	C₂×C₄×D₄	semidirect
148. C₂×D₁₃	C₂⁴×C₄	semidirect
154. #261	C₂³×D₄	direct (N × H)
155. C₄×C₁₃	C₂⁴×C₄	semidirect
156. Dic₁₃	C₂⁴×C₄	semidirect
163. C₂⁵	C₂×#261	semidirect
168. C₂²×D₄	C₂×#261	direct (N × H)
180. C₂³×C₄	C₂×#261	semidirect
186. C₂×C₁₃	C₂²×C₄×D₄	semidirect
189. D₁₃	C₂²×C₄×D₄	semidirect
205. C₂⁴	C₂²×#261	semidirect
210. C₂×D₄	C₂²×#261	direct (N × H)
222. C₂²×C₄	C₂²×#261	semidirect
226. C₁₃	C₂³×C₄×D₄	semidirect
240. C₂³	D₄×#261	direct (N × H)
242. C₂³	C₂³×#261	semidirect
245. D₄	C₂³×#261	direct (N × H)
256. C₂×C₄	C₂³×#261	semidirect
268. C₂²	C₂×D₄×#261	direct (N × H)
270. C₂²	C₂⁴×#261	semidirect
275. C₄	C₂⁴×#261	semidirect
281. C₂	C₂²×D₄×#261	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×#261	2
C₂²×D₄×#261	2
C₂³×C₄⋊#261	2
C₂³×C₂²⋊#261	2
C₂³×C₄×#261	2
C₂³×D₄×D₁₃	2
C₂³×C₄×D₄	13

Subgroups — C23×D4×#261 #16809628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₄×#261 #16809628