Subgroups — C₂³×D₄×#6157 #28330801

Order 13568.

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×#6157	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄×#6157	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₂²⋊#6157	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×#6157	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×#6157	—	2	32	no	—
8	C₂⁴×#6157	C₂²	1	1	yes	characteristic
9	C₂⁴×#6157	C₂²	1	2	yes	—
10	C₂⁴×#6157	C₂²	1	1	yes	characteristic
11	C₂⁴×#6157	C₂²	1	56	yes	complemented
12	C₂×D₄×#6157	C₂²	1	448	yes	direct product factor
13	C₂²×C₄⋊#6157	C₂²	1	28	yes	complemented
14	C₂²×C₂²⋊#6157	C₂²	1	56	yes	complemented
15	C₂²×C₄×#6157	C₂²	1	28	yes	complemented
16	C₂⁵×D₅₃	C₂²	1	2	yes	—
17	C₂³×D₂₁₂	C₄	1	1	yes	characteristic, complemented
18	C₂³×C₂²⋊D₅₃	C₄	1	2	yes	complemented
19	C₂²×D₄×D₅₃	C₄	1	28	yes	complemented
20	C₂²×D₄×D₅₃	C₂²	1	28	yes	—
21	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
22	C₂³×D₄×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂³×#6157	—	2	16	no	—
24	C₂³×#6157	—	2	8	no	—
25	C₂³×#6157	—	2	448	no	—
26	C₂³×#6157	D₄	1	16	yes	direct product factor
27	C₂³×#6157	C₂³	1	14	yes	—
28	C₂³×#6157	C₂³	1	28	yes	—
29	C₂³×#6157	C₂³	1	14	yes	—
30	C₂³×#6157	C₂³	1	224	yes	complemented
31	D₄×#6157	C₂³	1	512	yes	direct product factor
32	C₂×C₄⋊#6157	C₂³	1	112	yes	complemented
33	C₂×C₂²⋊#6157	C₂³	1	224	yes	complemented
34	C₂×C₄×#6157	C₂³	1	112	yes	complemented
35	C₂⁴×D₅₃	—	2	16	no	—
36	C₂⁴×D₅₃	—	2	16	no	—
37	C₂⁴×D₅₃	C₂×C₄	1	28	yes	complemented
38	C₂⁴×D₅₃	C₂×C₄	1	2	yes	complemented
39	C₂⁴×D₅₃	C₂³	1	28	yes	—
40	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
41	C₂²×D₂₁₂	C₂×C₄	1	28	yes	complemented
42	C₂²×C₂²⋊D₅₃	C₂×C₄	1	56	yes	complemented
43	C₂×D₄×D₅₃	C₂×C₄	1	336	yes	complemented
44	C₂×D₄×D₅₃	C₂³	1	112	yes	—
45	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂²×C₄×D₅₃	C₂×C₄	1	14	yes	complemented
47	C₂²×C₄×D₅₃	C₂³	1	14	yes	—
48	C₂⁵×C₅₃	C₂×C₄	1	2	yes	complemented
49	C₂²×D₄×C₅₃	C₂×C₄	1	28	yes	complemented
50	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂⁴×C₅₃	—	2	16	no	—
52	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂⁴×C₅₃	C₂²×C₄	1	28	yes	complemented
54	C₂³×D₅₃	—	2	16	no	—
55	C₂³×D₅₃	—	2	224	no	—
56	C₂³×D₅₃	—	2	112	no	—
57	C₂³×D₅₃	—	2	112	no	—
58	C₂³×D₅₃	C₂²×C₄	1	7	yes	—
59	C₂³×D₅₃	C₂²×C₄	1	168	yes	complemented
60	C₂³×D₅₃	C₂²×C₄	1	28	yes	complemented
61	C₂³×D₅₃	C₂×D₄	1	8	yes	—
62	C₂³×D₅₃	#41	1	8	yes	—
63	C₂³×D₅₃	C₂⁴	1	56	yes	—
64	C₂³×D₅₃	C₂⁴	1	7	yes	—
65	C₂²×#6157	—	2	112	no	—
66	C₂²×#6157	—	2	56	no	—
67	C₂²×#6157	—	2	896	no	—
68	C₂²×#6157	C₂×D₄	1	112	yes	direct product factor
69	C₂²×#6157	C₂⁴	1	28	yes	—
70	C₂²×#6157	C₂⁴	1	56	yes	—
71	C₂²×#6157	C₂⁴	1	28	yes	—
72	C₂²×#6157	C₂⁴	1	128	yes	complemented
73	C₂×D₄×C₅₃	C₂²×C₄	1	112	yes	complemented
74	C₂²×C₄×C₅₃	C₂²×C₄	1	14	yes	complemented
75	C₂×C₂²⋊D₅₃	C₂²×C₄	1	224	yes	complemented
76	C₂²×Dic₅₃	C₂²×C₄	1	14	yes	complemented
77	D₄×D₅₃	C₂²×C₄	1	448	yes	complemented
78	D₄×D₅₃	C₂⁴	1	64	yes	—
79	C₂×D₂₁₂	C₂²×C₄	1	112	yes	complemented
80	C₂×C₄×D₅₃	C₂²×C₄	1	84	yes	complemented
81	C₂×C₄×D₅₃	C₂⁴	1	28	yes	—
82	C₂²⋊#6157	C₂⁴	1	128	yes	complemented
83	C₄⋊#6157	C₂⁴	1	64	yes	complemented
84	C₄×#6157	C₂⁴	1	64	yes	complemented
85	C₂³×C₅₃	—	2	112	no	—
86	C₂³×C₅₃	C₄×D₄	1	8	yes	complemented
87	C₂³×C₅₃	C₂³×C₄	1	7	yes	—
88	C₂³×C₅₃	C₂³×C₄	1	56	yes	complemented
89	C₂²×D₅₃	—	2	112	no	—
90	C₂²×D₅₃	—	2	672	no	—
91	C₂²×D₅₃	—	2	112	no	—
92	C₂²×D₅₃	—	2	112	no	—
93	C₂²×D₅₃	C₄×D₄	1	56	yes	complemented
94	C₂²×D₅₃	C₂³×C₄	1	21	yes	—
95	C₂²×D₅₃	C₂³×C₄	1	112	yes	complemented
96	C₂²×D₅₃	C₂³×C₄	1	56	yes	complemented
97	C₂²×D₅₃	C₂²×D₄	1	28	yes	—
98	C₂²×D₅₃	C₂×#41	1	28	yes	—
99	C₂²×D₅₃	C₂⁵	1	16	yes	—
100	C₂²×D₅₃	C₂⁵	1	7	yes	—
101	C₂×#6157	—	2	112	no	—
102	C₂×#6157	—	2	56	no	—
103	C₂×#6157	—	2	256	no	—
104	C₂×#6157	C₂²×D₄	1	112	yes	direct product factor
105	C₂×#6157	C₂⁵	1	8	yes	—
106	C₂×#6157	C₂⁵	1	16	yes	—
107	C₂×#6157	C₂⁵	1	8	yes	—
108	D₄×C₅₃	C₂³×C₄	1	64	yes	complemented
109	C₂×C₄×C₅₃	C₂³×C₄	1	28	yes	complemented
110	C₂²⋊D₅₃	C₂³×C₄	1	128	yes	complemented
111	C₂×Dic₅₃	C₂³×C₄	1	28	yes	complemented
112	D₂₁₂	C₂³×C₄	1	64	yes	complemented
113	C₄×D₅₃	C₂³×C₄	1	56	yes	complemented
114	C₄×D₅₃	C₂⁵	1	8	yes	—
115	C₂³×C₄×D₄	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	112	no	—
117	C₂²×C₅₃	C₂×C₄×D₄	1	28	yes	complemented
118	C₂²×C₅₃	C₂⁴×C₄	1	7	yes	—
119	C₂²×C₅₃	C₂⁴×C₄	1	16	yes	complemented
120	C₂×D₅₃	—	2	112	no	—
121	C₂×D₅₃	—	2	16	no	—
122	C₂×D₅₃	—	2	16	no	—
123	C₂×D₅₃	—	2	224	no	—
124	C₂×D₅₃	C₂×C₄×D₄	1	84	yes	complemented
125	C₂×D₅₃	C₂⁴×C₄	1	7	yes	—
126	C₂×D₅₃	C₂⁴×C₄	1	16	yes	complemented
127	C₂×D₅₃	C₂³×D₄	1	14	yes	—
128	C₂×D₅₃	C₂²×#41	1	14	yes	—
129	C₂×D₅₃	C₂⁶	1	1	yes	characteristic
130	#6157	—	2	16	no	—
131	#6157	—	2	8	no	—
132	#6157	C₂³×D₄	1	16	yes	direct product factor
133	C₄×C₅₃	C₂⁴×C₄	1	8	yes	complemented
134	Dic₅₃	C₂⁴×C₄	1	8	yes	complemented
135	C₂⁴×D₄	—	53	1	no	—
136	C₂⁵×C₄	—	53	2	no	—
137	C₂²×C₄×D₄	—	53	56	no	—
138	C₂³×C₄⋊C₄	—	53	1	no	—
139	C₂³×C₂²⋊C₄	—	53	2	no	—
140	C₂³×C₄²	—	53	1	no	—
141	C₂×C₅₃	—	2	16	no	—
142	C₂×C₅₃	C₂²×C₄×D₄	1	14	yes	complemented
143	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
144	D₅₃	—	2	16	no	—
145	D₅₃	C₂²×C₄×D₄	1	14	yes	complemented
146	D₅₃	C₂⁴×D₄	1	1	yes	characteristic
147	D₅₃	C₂³×#41	1	1	yes	characteristic
148	C₂⁶	—	53	2	no	—
149	C₂³×D₄	—	53	2	no	—
150	C₂³×D₄	—	53	28	no	—
151	C₂³×D₄	—	53	28	no	—
152	C₂³×D₄	—	53	1	no	—
153	C₂³×D₄	#6157	1	1	yes	characteristic, direct product factor
154	C₂⁴×C₄	—	53	1	no	—
155	C₂⁴×C₄	—	53	1	no	—
156	C₂⁴×C₄	—	53	2	no	—
157	C₂⁴×C₄	—	53	56	no	—
158	C₂⁴×C₄	—	53	1	no	—
159	C₂⁴×C₄	—	106	32	no	—
160	C₂×C₄×D₄	—	53	448	no	—
161	C₂²×C₄⋊C₄	—	53	28	no	—
162	C₂²×C₂²⋊C₄	—	53	56	no	—
163	C₂²×C₄²	—	53	28	no	—
164	C₅₃	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
165	C₂⁵	—	53	1	no	—
166	C₂⁵	—	53	2	no	—
167	C₂⁵	—	53	28	no	—
168	C₂⁵	—	53	28	no	—
169	C₂⁵	—	106	16	no	—
170	C₂⁵	—	106	16	no	—
171	C₂⁵	C₂×#6157	1	2	yes	complemented
172	C₂²×D₄	—	53	56	no	—
173	C₂²×D₄	—	53	112	no	—
174	C₂²×D₄	—	53	336	no	—
175	C₂²×D₄	—	53	28	no	—
176	C₂²×D₄	C₂×#6157	1	28	yes	direct product factor
177	C₂³×C₄	—	53	14	no	—
178	C₂³×C₄	—	53	16	no	—
179	C₂³×C₄	—	53	1	no	—
180	C₂³×C₄	—	53	14	no	—
181	C₂³×C₄	—	53	14	no	—
182	C₂³×C₄	—	53	14	no	—
183	C₂³×C₄	—	53	28	no	—
184	C₂³×C₄	—	53	224	no	—
185	C₂³×C₄	—	106	448	no	—
186	C₂³×C₄	—	106	16	no	—
187	C₂³×C₄	—	106	8	no	—
188	C₂³×C₄	C₂×#6157	1	1	yes	characteristic, complemented
189	C₄×D₄	—	53	512	no	—
190	C₂×C₄⋊C₄	—	53	112	no	—
191	C₂×C₂²⋊C₄	—	53	224	no	—
192	C₂×C₄²	—	53	112	no	—
193	C₂⁴	—	2	16	no	—
194	C₂⁴	—	53	8	no	—
195	C₂⁴	—	53	8	no	—
196	C₂⁴	—	53	7	no	—
197	C₂⁴	—	53	7	no	—
198	C₂⁴	—	53	28	no	—
199	C₂⁴	—	53	168	no	—
200	C₂⁴	—	53	56	no	—
201	C₂⁴	—	106	112	no	—
202	C₂⁴	—	106	224	no	—
203	C₂⁴	—	106	112	no	—
204	C₂⁴	—	106	16	no	—
205	C₂⁴	C₂²×#6157	1	1	yes	characteristic
206	C₂⁴	C₂²×#6157	1	28	yes	complemented
207	C₂×D₄	—	53	224	no	—
208	C₂×D₄	—	53	448	no	—
209	C₂×D₄	—	53	64	no	—
210	C₂×D₄	—	53	112	no	—
211	C₂×D₄	C₂²×#6157	1	112	yes	direct product factor
212	C₂²×C₄	—	53	112	no	—
213	C₂²×C₄	—	53	28	no	—
214	C₂²×C₄	—	53	14	no	—
215	C₂²×C₄	—	53	84	no	—
216	C₂²×C₄	—	53	28	no	—
217	C₂²×C₄	—	53	28	no	—
218	C₂²×C₄	—	53	56	no	—
219	C₂²×C₄	—	53	128	no	—
220	C₂²×C₄	—	106	896	no	—
221	C₂²×C₄	—	106	56	no	—
222	C₂²×C₄	—	106	112	no	—
223	C₂²×C₄	C₂²×#6157	1	14	yes	complemented
224	C₄⋊C₄	—	53	64	no	—
225	C₂²⋊C₄	—	53	128	no	—
226	C₄²	—	53	64	no	—
227	C₂³	—	2	112	no	—
228	C₂³	—	53	56	no	—
229	C₂³	—	53	28	no	—
230	C₂³	—	53	28	no	—
231	C₂³	—	53	21	no	—
232	C₂³	—	53	7	no	—
233	C₂³	—	53	56	no	—
234	C₂³	—	53	112	no	—
235	C₂³	—	53	16	no	—
236	C₂³	—	106	112	no	—
237	C₂³	—	106	112	no	—
238	C₂³	—	106	672	no	—
239	C₂³	—	106	112	no	—
240	C₂³	D₄×#6157	1	8	yes	direct product factor
241	C₂³	C₂³×#6157	1	7	yes	—
242	C₂³	C₂³×#6157	1	56	yes	complemented
243	D₄	—	53	128	no	—
244	D₄	—	53	64	no	—
245	D₄	C₂³×#6157	1	64	yes	direct product factor
246	C₂×C₄	—	53	112	no	—
247	C₂×C₄	—	53	8	no	—
248	C₂×C₄	—	53	28	no	—
249	C₂×C₄	—	53	56	no	—
250	C₂×C₄	—	53	8	no	—
251	C₂×C₄	—	53	8	no	—
252	C₂×C₄	—	53	16	no	—
253	C₂×C₄	—	106	256	no	—
254	C₂×C₄	—	106	56	no	—
255	C₂×C₄	—	106	112	no	—
256	C₂×C₄	C₂³×#6157	1	28	yes	complemented
257	C₂²	—	2	112	no	—
258	C₂²	—	53	14	no	—
259	C₂²	—	53	14	no	—
260	C₂²	—	53	84	no	—
261	C₂²	—	53	1	no	—
262	C₂²	—	53	7	no	—
263	C₂²	—	53	16	no	—
264	C₂²	—	106	16	no	—
265	C₂²	—	106	16	no	—
266	C₂²	—	106	224	no	—
267	C₂²	—	106	112	no	—
268	C₂²	C₂×D₄×#6157	1	28	yes	direct product factor
269	C₂²	C₂⁴×#6157	1	7	yes	—
270	C₂²	C₂⁴×#6157	1	16	yes	complemented
271	C₄	—	53	16	no	—
272	C₄	—	53	8	no	—
273	C₄	—	106	16	no	—
274	C₄	—	106	8	no	—
275	C₄	C₂⁴×#6157	1	8	yes	complemented
276	C₂	—	2	16	no	—
277	C₂	—	53	1	no	—
278	C₂	—	53	1	no	—
279	C₂	—	53	14	no	—
280	C₂	—	106	16	no	—
281	C₂	C₂²×D₄×#6157	1	14	yes	direct product factor
282	C₂	C₂⁵×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×#6157
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#6157
Fitting	C₂³×D₄×C₅₃	C₄
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂³×D₄×D₅₃	C₂
cosocle	C₂×D₅₃	C₂⁶
hypercenter	C₂³×D₄	#6157
hypocenter	C₅₃	C₂³×C₄×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂⁵×#6157	C₂	semidirect
2. C₂²×D₄×#6157	C₂	direct (N × H)
3. C₂³×C₄⋊#6157	C₂	semidirect
4. C₂³×C₂²⋊#6157	C₂	semidirect
5. C₂³×C₄×#6157	C₂	semidirect
11. C₂⁴×#6157	C₂²	semidirect
12. C₂×D₄×#6157	C₂²	direct (N × H)
13. C₂²×C₄⋊#6157	C₂²	semidirect
14. C₂²×C₂²⋊#6157	C₂²	semidirect
15. C₂²×C₄×#6157	C₂²	semidirect
17. C₂³×D₂₁₂	C₄	semidirect
18. C₂³×C₂²⋊D₅₃	C₄	semidirect
19. C₂²×D₄×D₅₃	C₄	semidirect
22. C₂³×D₄×C₅₃	C₄	semidirect
26. C₂³×#6157	D₄	direct (N × H)
30. C₂³×#6157	C₂³	semidirect
31. D₄×#6157	C₂³	direct (N × H)
32. C₂×C₄⋊#6157	C₂³	semidirect
33. C₂×C₂²⋊#6157	C₂³	semidirect
34. C₂×C₄×#6157	C₂³	semidirect
37. C₂⁴×D₅₃	C₂×C₄	semidirect
38. C₂⁴×D₅₃	C₂×C₄	semidirect
41. C₂²×D₂₁₂	C₂×C₄	semidirect
42. C₂²×C₂²⋊D₅₃	C₂×C₄	semidirect
43. C₂×D₄×D₅₃	C₂×C₄	semidirect
45. C₂³×Dic₅₃	C₂×C₄	semidirect
46. C₂²×C₄×D₅₃	C₂×C₄	semidirect
48. C₂⁵×C₅₃	C₂×C₄	semidirect
49. C₂²×D₄×C₅₃	C₂×C₄	semidirect
50. C₂³×C₄×C₅₃	C₂×C₄	semidirect
53. C₂⁴×C₅₃	C₂²×C₄	semidirect
59. C₂³×D₅₃	C₂²×C₄	semidirect
60. C₂³×D₅₃	C₂²×C₄	semidirect
68. C₂²×#6157	C₂×D₄	direct (N × H)
72. C₂²×#6157	C₂⁴	semidirect
73. C₂×D₄×C₅₃	C₂²×C₄	semidirect
74. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
75. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
76. C₂²×Dic₅₃	C₂²×C₄	semidirect
77. D₄×D₅₃	C₂²×C₄	semidirect
79. C₂×D₂₁₂	C₂²×C₄	semidirect
80. C₂×C₄×D₅₃	C₂²×C₄	semidirect
82. C₂²⋊#6157	C₂⁴	semidirect
83. C₄⋊#6157	C₂⁴	semidirect
84. C₄×#6157	C₂⁴	semidirect
86. C₂³×C₅₃	C₄×D₄	semidirect
88. C₂³×C₅₃	C₂³×C₄	semidirect
93. C₂²×D₅₃	C₄×D₄	semidirect
95. C₂²×D₅₃	C₂³×C₄	semidirect
96. C₂²×D₅₃	C₂³×C₄	semidirect
104. C₂×#6157	C₂²×D₄	direct (N × H)
108. D₄×C₅₃	C₂³×C₄	semidirect
109. C₂×C₄×C₅₃	C₂³×C₄	semidirect
110. C₂²⋊D₅₃	C₂³×C₄	semidirect
111. C₂×Dic₅₃	C₂³×C₄	semidirect
112. D₂₁₂	C₂³×C₄	semidirect
113. C₄×D₅₃	C₂³×C₄	semidirect
117. C₂²×C₅₃	C₂×C₄×D₄	semidirect
119. C₂²×C₅₃	C₂⁴×C₄	semidirect
124. C₂×D₅₃	C₂×C₄×D₄	semidirect
126. C₂×D₅₃	C₂⁴×C₄	semidirect
132. #6157	C₂³×D₄	direct (N × H)
133. C₄×C₅₃	C₂⁴×C₄	semidirect
134. Dic₅₃	C₂⁴×C₄	semidirect
142. C₂×C₅₃	C₂²×C₄×D₄	semidirect
145. D₅₃	C₂²×C₄×D₄	semidirect
153. C₂³×D₄	#6157	direct (N × H)
164. C₅₃	C₂³×C₄×D₄	semidirect
171. C₂⁵	C₂×#6157	semidirect
176. C₂²×D₄	C₂×#6157	direct (N × H)
188. C₂³×C₄	C₂×#6157	semidirect
206. C₂⁴	C₂²×#6157	semidirect
211. C₂×D₄	C₂²×#6157	direct (N × H)
223. C₂²×C₄	C₂²×#6157	semidirect
240. C₂³	D₄×#6157	direct (N × H)
242. C₂³	C₂³×#6157	semidirect
245. D₄	C₂³×#6157	direct (N × H)
256. C₂×C₄	C₂³×#6157	semidirect
268. C₂²	C₂×D₄×#6157	direct (N × H)
270. C₂²	C₂⁴×#6157	semidirect
275. C₄	C₂⁴×#6157	semidirect
281. C₂	C₂²×D₄×#6157	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×#6157	2
C₂²×D₄×#6157	2
C₂³×C₄⋊#6157	2
C₂³×C₂²⋊#6157	2
C₂³×C₄×#6157	2
C₂³×D₄×D₅₃	2
C₂³×C₄×D₄	53

Subgroups — C23×D4×#6157 #28330801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₄×#6157 #28330801