Subgroups — #28330811

Order 13568.

11674 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×#6157	C₂	1	32	yes	maximal, complemented
2	#20291695	C₂	1	12	yes	maximal, complemented
3	C₂×#16811690	C₂	1	18	yes	maximal, complemented
4	C₂×H(5,𝔽₂)×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811692	C₂²	1	96	yes	complemented
6	#16811691	C₂²	1	8	yes	complemented
7	#16811690	C₂²	1	72	yes	complemented
8	#16811690	C₂²	1	144	yes	complemented
9	#41×#6157	C₂²	1	96	yes	complemented
10	C₂×D₄×#6157	C₂²	1	144	yes	complemented
11	C₂×#13133296	C₂²	1	36	yes	complemented
12	C₂²×C₂²⋊#6157	C₂²	1	24	yes	complemented
13	C₂×#13133263	C₄	1	6	yes	complemented
14	C₂×#13133262	C₄	1	9	yes	complemented
15	H(5,𝔽₂)×D₅₃	C₄	1	16	yes	complemented
16	H(5,𝔽₂)×D₅₃	C₂²	1	16	yes	—
17	C₂×#41×D₅₃	C₂²	1	6	yes	—
18	C₂²×D₄×D₅₃	C₂²	1	9	yes	—
19	C₂×H(5,𝔽₂)×C₅₃	C₄	1	1	yes	characteristic, complemented
20	C₂³×#6157	C₂³	1	48	yes	complemented
21	Q₈×#6157	C₂³	1	64	yes	complemented
22	D₄×#6157	C₂³	1	576	yes	complemented
23	#13133296	C₂³	1	48	yes	—
24	#13133296	C₂³	1	144	yes	complemented
25	C₂×C₄⋊#6157	C₂³	1	72	yes	complemented
26	C₂×C₂²⋊#6157	C₂³	1	72	yes	—
27	C₂×C₂²⋊#6157	C₂³	1	144	yes	complemented
28	C₂×C₄×#6157	C₂³	1	72	yes	complemented
29	C₂⁴×D₅₃	C₂³	1	6	yes	—
30	#13133263	C₂×C₄	1	96	yes	complemented
31	#13133262	C₂×C₄	1	144	yes	complemented
32	C₂×#11719949	C₂×C₄	1	6	yes	complemented
33	C₂×#11719945	C₂×C₄	1	18	yes	complemented
34	C₂×#11719947	C₂×C₄	1	18	yes	complemented
35	#41×D₅₃	C₂×C₄	1	48	yes	complemented
36	#41×D₅₃	C₂³	1	48	yes	—
37	C₂×Q₈×D₅₃	C₂³	1	2	yes	—
38	C₂²×D₂₁₂	C₂×C₄	1	9	yes	complemented
39	C₂²×C₂²⋊D₅₃	C₂×C₄	1	18	yes	complemented
40	C₂×D₄×D₅₃	C₂×C₄	1	72	yes	complemented
41	C₂×D₄×D₅₃	C₂×C₄	1	36	yes	complemented
42	C₂×D₄×D₅₃	C₂³	1	72	yes	—
43	C₂×D₄×D₅₃	C₂³	1	18	yes	—
44	C₂²×C₄×D₅₃	C₂³	1	9	yes	—
45	H(5,𝔽₂)×C₅₃	C₂×C₄	1	16	yes	complemented
46	C₂×#41×C₅₃	C₂×C₄	1	6	yes	complemented
47	C₂²×D₄×C₅₃	C₂×C₄	1	9	yes	complemented
48	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	complemented
49	C₂³×D₅₃	—	8	12	no	—
50	C₂³×D₅₃	C₂²×C₄	1	24	yes	complemented
51	C₂³×D₅₃	C₂²×C₄	1	18	yes	complemented
52	C₂³×D₅₃	C₂⁴	1	9	yes	—
53	C₂³×D₅₃	C₂⁴	1	24	yes	—
54	C₂²×#6157	—	8	48	no	—
55	C₂²×#6157	C₂⁴	1	144	yes	—
56	#41×C₅₃	C₂²×C₄	1	48	yes	complemented
57	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	complemented
58	C₂×D₄×C₅₃	C₂²×C₄	1	18	yes	complemented
59	C₂×D₄×C₅₃	C₂²×C₄	1	72	yes	complemented
60	C₂²×C₄×C₅₃	C₂²×C₄	1	9	yes	complemented
61	C₂×C₂²⋊D₅₃	C₂²×C₄	1	36	yes	complemented
62	C₂×C₂²⋊D₅₃	C₂²×C₄	1	144	yes	complemented
63	C₂²×Dic₅₃	C₂²×C₄	1	9	yes	complemented
64	#11719949	C₂²×C₄	1	48	yes	complemented
65	Q₈×D₅₃	C₂²×C₄	1	8	yes	complemented
66	Q₈×D₅₃	C₂⁴	1	8	yes	—
67	#11719947	C₂²×C₄	1	144	yes	complemented
68	D₄×D₅₃	C₂²×C₄	1	72	yes	complemented
69	D₄×D₅₃	C₂²×C₄	1	288	yes	complemented
70	D₄×D₅₃	C₂⁴	1	72	yes	—
71	#11719945	C₂²×C₄	1	144	yes	complemented
72	C₂×D₂₁₂	C₂²×C₄	1	18	yes	complemented
73	C₂×D₂₁₂	C₂²×C₄	1	72	yes	complemented
74	C₂×C₄×D₅₃	C₂²×C₄	1	36	yes	complemented
75	C₂×C₄×D₅₃	C₂²×C₄	1	18	yes	complemented
76	C₂×C₄×D₅₃	C₂⁴	1	6	yes	—
77	C₂×C₄×D₅₃	C₂⁴	1	36	yes	—
78	C₂×Dic₁₀₆	C₂²×C₄	1	6	yes	complemented
79	C₂²⋊#6157	C₂⁴	1	16	yes	—
80	C₂²⋊#6157	C₂⁴	1	144	yes	—
81	C₄⋊#6157	C₂⁴	1	96	yes	—
82	C₄×#6157	C₂⁴	1	96	yes	—
83	C₂³×C₅₃	—	8	6	no	—
84	C₂³×C₅₃	C₂³×C₄	1	9	yes	—
85	C₂³×C₅₃	C₂³×C₄	1	24	yes	—
86	C₂²×D₅₃	—	4	48	no	—
87	C₂²×D₅₃	—	4	18	no	—
88	C₂²×D₅₃	—	8	24	no	—
89	C₂²×D₅₃	—	8	18	no	—
90	C₂²×D₅₃	C₂³×C₄	1	9	yes	—
91	C₂²×D₅₃	C₂³×C₄	1	18	yes	—
92	C₂²×D₅₃	C₂³×C₄	1	72	yes	—
93	C₂²×D₅₃	C₂⁵	1	1	yes	characteristic
94	C₂²×D₅₃	C₂⁵	1	18	yes	—
95	C₂×#6157	—	4	144	no	—
96	C₂×#6157	C₂⁵	1	32	yes	—
97	Q₈×C₅₃	C₂³×C₄	1	8	yes	—
98	D₄×C₅₃	C₂³×C₄	1	72	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
100	C₂×C₄×C₅₃	C₂³×C₄	1	36	yes	—
101	C₂²⋊D₅₃	C₂³×C₄	1	144	yes	—
102	C₂×Dic₅₃	C₂³×C₄	1	6	yes	—
103	C₂×Dic₅₃	C₂³×C₄	1	36	yes	—
104	D₂₁₂	C₂³×C₄	1	72	yes	—
105	C₄×D₅₃	C₂³×C₄	1	12	yes	—
106	C₄×D₅₃	C₂³×C₄	1	72	yes	—
107	C₄×D₅₃	C₂⁵	1	12	yes	—
108	Dic₁₀₆	C₂³×C₄	1	24	yes	—
109	#62652	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	4	9	no	—
111	C₂²×C₅₃	—	4	24	no	—
112	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
113	C₂²×C₅₃	C₂⁴×C₄	1	18	yes	—
114	C₂×D₅₃	—	2	36	no	—
115	C₂×D₅₃	—	2	2	no	—
116	C₂×D₅₃	—	4	18	no	—
117	C₂×D₅₃	—	4	9	no	—
118	C₂×D₅₃	—	4	72	no	—
119	C₂×D₅₃	C₂⁴×C₄	1	1	yes	characteristic
120	C₂×D₅₃	C₂⁴×C₄	1	18	yes	—
121	C₂×D₅₃	C₂⁶	1	1	yes	characteristic
122	#6157	—	2	32	no	—
123	C₄×C₅₃	C₂⁴×C₄	1	12	yes	—
124	Dic₅₃	C₂⁴×C₄	1	12	yes	—
125	C₂²×H(5,𝔽₂)	—	53	1	no	—
126	C₄×H(5,𝔽₂)	—	53	32	no	—
127	#3428	—	53	12	no	—
128	C₂×#518	—	53	18	no	—
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	2	18	no	—
131	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
132	D₅₃	—	2	1	no	—
133	D₅₃	—	2	18	no	—
134	D₅₃	H(7,𝔽₂)	1	2	yes	—
135	C₂×H(5,𝔽₂)	—	53	16	no	—
136	C₂×H(5,𝔽₂)	—	53	16	no	—
137	C₂×H(5,𝔽₂)	—	53	6	no	—
138	C₂×H(5,𝔽₂)	—	53	9	no	—
139	C₂×H(5,𝔽₂)	#6157	1	1	yes	characteristic, complemented
140	C₂²×#41	—	53	6	no	—
141	C₂³×D₄	—	53	9	no	—
142	#520	—	53	96	no	—
143	#519	—	53	8	no	—
144	#518	—	53	72	no	—
145	#518	—	53	144	no	—
146	C₄×#41	—	53	96	no	—
147	C₂×C₄×D₄	—	53	144	no	—
148	C₂×#117	—	53	36	no	—
149	C₂²×C₂²⋊C₄	—	53	24	no	—
150	C₅₃	#62652	1	1	yes	characteristic, complemented, Sylow
151	C₂⁵	—	53	6	no	—
152	H(5,𝔽₂)	—	53	96	no	—
153	H(5,𝔽₂)	—	53	144	no	—
154	H(5,𝔽₂)	C₂×#6157	1	16	yes	complemented
155	C₂×#41	—	53	48	no	—
156	C₂×#41	—	53	48	no	—
157	C₂×#41	—	53	6	no	—
158	C₂×#41	—	53	18	no	—
159	C₂×#41	—	53	18	no	—
160	C₂×#41	C₂×#6157	1	6	yes	complemented
161	C₂²×Q₈	—	53	2	no	—
162	C₂²×D₄	—	53	18	no	—
163	C₂²×D₄	—	53	72	no	—
164	C₂²×D₄	—	53	18	no	—
165	C₂²×D₄	—	53	36	no	—
166	C₂²×D₄	—	53	72	no	—
167	C₂²×D₄	—	53	9	no	—
168	C₂²×D₄	C₂×#6157	1	9	yes	complemented
169	C₂³×C₄	—	53	9	no	—
170	C₂³×C₄	—	53	48	no	—
171	C₄×Q₈	—	53	64	no	—
172	C₄×D₄	—	53	576	no	—
173	#117	—	53	48	no	—
174	#117	—	53	144	no	—
175	C₂×C₄⋊C₄	—	53	72	no	—
176	C₂×C₂²⋊C₄	—	53	72	no	—
177	C₂×C₂²⋊C₄	—	53	144	no	—
178	C₂×C₄²	—	53	72	no	—
179	C₂⁴	—	53	24	no	—
180	C₂⁴	—	53	18	no	—
181	C₂⁴	—	53	9	no	—
182	C₂⁴	—	53	24	no	—
183	C₂⁴	—	424	12	no	—
184	C₂⁴	C₂²×#6157	1	6	yes	complemented
185	#41	—	53	48	no	—
186	#41	—	53	144	no	—
187	#41	—	53	144	no	—
188	#41	C₂²×#6157	1	48	yes	complemented
189	C₂×Q₈	—	53	8	no	—
190	C₂×Q₈	—	53	8	no	—
191	C₂×Q₈	—	53	6	no	—
192	C₂×Q₈	C₂²×#6157	1	2	yes	complemented
193	C₂×D₄	—	53	36	no	—
194	C₂×D₄	—	53	144	no	—
195	C₂×D₄	—	53	72	no	—
196	C₂×D₄	—	53	288	no	—
197	C₂×D₄	—	53	72	no	—
198	C₂×D₄	—	53	18	no	—
199	C₂×D₄	—	53	72	no	—
200	C₂×D₄	C₂²×#6157	1	18	yes	complemented
201	C₂×D₄	C₂²×#6157	1	72	yes	complemented
202	C₂²×C₄	—	53	9	no	—
203	C₂²×C₄	—	53	18	no	—
204	C₂²×C₄	—	53	36	no	—
205	C₂²×C₄	—	53	36	no	—
206	C₂²×C₄	—	53	6	no	—
207	C₂²×C₄	—	53	144	no	—
208	C₂²×C₄	—	424	48	no	—
209	C₂²×C₄	C₂²×#6157	1	9	yes	complemented
210	C₄⋊C₄	—	53	96	no	—
211	C₂²⋊C₄	—	53	16	no	—
212	C₂²⋊C₄	—	53	144	no	—
213	C₄²	—	53	96	no	—
214	C₂³	—	8	6	no	—
215	C₂³	—	53	1	no	—
216	C₂³	—	53	18	no	—
217	C₂³	—	53	18	no	—
218	C₂³	—	53	9	no	—
219	C₂³	—	53	72	no	—
220	C₂³	—	424	18	no	—
221	C₂³	—	424	24	no	—
222	C₂³	—	212	48	no	—
223	C₂³	—	212	18	no	—
224	C₂³	C₂³×#6157	1	9	yes	—
225	C₂³	C₂³×#6157	1	24	yes	—
226	Q₈	—	53	24	no	—
227	Q₈	C₂³×#6157	1	8	yes	—
228	D₄	—	53	144	no	—
229	D₄	—	53	72	no	—
230	D₄	C₂³×#6157	1	72	yes	—
231	C₂×C₄	—	53	6	no	—
232	C₂×C₄	—	53	36	no	—
233	C₂×C₄	—	53	12	no	—
234	C₂×C₄	—	53	72	no	—
235	C₂×C₄	—	53	12	no	—
236	C₂×C₄	—	53	32	no	—
237	C₂×C₄	—	212	144	no	—
238	C₂×C₄	C₂³×#6157	1	6	yes	—
239	C₂×C₄	C₂³×#6157	1	36	yes	—
240	C₂²	—	4	9	no	—
241	C₂²	—	4	24	no	—
242	C₂²	—	53	1	no	—
243	C₂²	—	53	1	no	—
244	C₂²	—	53	18	no	—
245	C₂²	—	106	36	no	—
246	C₂²	—	106	2	no	—
247	C₂²	—	212	9	no	—
248	C₂²	—	212	18	no	—
249	C₂²	—	212	72	no	—
250	C₂²	C₂⁴×#6157	1	1	yes	characteristic
251	C₂²	C₂⁴×#6157	1	18	yes	—
252	C₄	—	53	12	no	—
253	C₄	—	106	32	no	—
254	C₄	C₂⁴×#6157	1	12	yes	—
255	C₂	—	2	1	no	—
256	C₂	—	2	18	no	—
257	C₂	—	53	2	no	—
258	C₂	—	106	1	no	—
259	C₂	—	106	18	no	—
260	C₂	C₂⁵×#6157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×#6157
commutator, socle	C₂×C₅₃	C₂⁵×C₄
Fitting	C₂×H(5,𝔽₂)×C₅₃	C₄
prime radical	C₂×H(5,𝔽₂)×D₅₃	C₂
cosocle	C₂×D₅₃	C₂⁶
hypercenter	C₂×H(5,𝔽₂)	#6157
hypocenter	C₅₃	#62652
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×#6157	C₂	semidirect
2. #20291695	C₂	semidirect
3. C₂×#16811690	C₂	semidirect
5. #16811692	C₂²	semidirect
6. #16811691	C₂²	semidirect
7. #16811690	C₂²	semidirect
8. #16811690	C₂²	semidirect
9. #41×#6157	C₂²	semidirect
10. C₂×D₄×#6157	C₂²	semidirect
11. C₂×#13133296	C₂²	semidirect
12. C₂²×C₂²⋊#6157	C₂²	semidirect
13. C₂×#13133263	C₄	semidirect
14. C₂×#13133262	C₄	semidirect
15. H(5,𝔽₂)×D₅₃	C₄	semidirect
19. C₂×H(5,𝔽₂)×C₅₃	C₄	semidirect
20. C₂³×#6157	C₂³	semidirect
21. Q₈×#6157	C₂³	semidirect
22. D₄×#6157	C₂³	semidirect
24. #13133296	C₂³	semidirect
25. C₂×C₄⋊#6157	C₂³	semidirect
27. C₂×C₂²⋊#6157	C₂³	semidirect
28. C₂×C₄×#6157	C₂³	semidirect
30. #13133263	C₂×C₄	semidirect
31. #13133262	C₂×C₄	semidirect
32. C₂×#11719949	C₂×C₄	semidirect
33. C₂×#11719945	C₂×C₄	semidirect
34. C₂×#11719947	C₂×C₄	semidirect
35. #41×D₅₃	C₂×C₄	semidirect
38. C₂²×D₂₁₂	C₂×C₄	semidirect
39. C₂²×C₂²⋊D₅₃	C₂×C₄	semidirect
40. C₂×D₄×D₅₃	C₂×C₄	semidirect
41. C₂×D₄×D₅₃	C₂×C₄	semidirect
45. H(5,𝔽₂)×C₅₃	C₂×C₄	semidirect
46. C₂×#41×C₅₃	C₂×C₄	semidirect
47. C₂²×D₄×C₅₃	C₂×C₄	semidirect
48. C₂⁴×C₅₃	C₂²×C₄	semidirect
50. C₂³×D₅₃	C₂²×C₄	semidirect
51. C₂³×D₅₃	C₂²×C₄	semidirect
56. #41×C₅₃	C₂²×C₄	semidirect
57. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
58. C₂×D₄×C₅₃	C₂²×C₄	semidirect
59. C₂×D₄×C₅₃	C₂²×C₄	semidirect
60. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
61. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
62. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
63. C₂²×Dic₅₃	C₂²×C₄	semidirect
64. #11719949	C₂²×C₄	semidirect
65. Q₈×D₅₃	C₂²×C₄	semidirect
67. #11719947	C₂²×C₄	semidirect
68. D₄×D₅₃	C₂²×C₄	semidirect
69. D₄×D₅₃	C₂²×C₄	semidirect
71. #11719945	C₂²×C₄	semidirect
72. C₂×D₂₁₂	C₂²×C₄	semidirect
73. C₂×D₂₁₂	C₂²×C₄	semidirect
74. C₂×C₄×D₅₃	C₂²×C₄	semidirect
75. C₂×C₄×D₅₃	C₂²×C₄	semidirect
78. C₂×Dic₁₀₆	C₂²×C₄	semidirect
139. C₂×H(5,𝔽₂)	#6157	semidirect
150. C₅₃	#62652	semidirect
154. H(5,𝔽₂)	C₂×#6157	semidirect
160. C₂×#41	C₂×#6157	semidirect
168. C₂²×D₄	C₂×#6157	semidirect
184. C₂⁴	C₂²×#6157	semidirect
188. #41	C₂²×#6157	semidirect
192. C₂×Q₈	C₂²×#6157	semidirect
200. C₂×D₄	C₂²×#6157	semidirect
201. C₂×D₄	C₂²×#6157	semidirect
209. C₂²×C₄	C₂²×#6157	semidirect

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×#6157	2
#20291695	2
C₂×#16811690	2
C₂×H(5,𝔽₂)×D₅₃	2
#62652	53