Subgroups — #16809638

Order 3328.

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)×#261	C₂	1	32	yes	maximal, complemented
2	#13131816	C₂	1	12	yes	maximal, complemented
3	C₂×#11719623	C₂	1	18	yes	maximal, complemented
4	C₂×D₁₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×H(5,𝔽₂)	C₄	1	1	yes	characteristic, complemented
6	D₁₃×H(5,𝔽₂)	C₄	1	16	yes	complemented
7	D₁₃×H(5,𝔽₂)	C₂²	1	16	yes	—
8	C₂×#88256	C₄	1	6	yes	complemented
9	C₂×#41×D₁₃	C₂²	1	6	yes	—
10	C₂×#88251	C₄	1	9	yes	complemented
11	C₂²×D₄×D₁₃	C₂²	1	9	yes	—
12	C₂²×C₂²⋊#261	C₂²	1	24	yes	complemented
13	#11719631	C₂²	1	96	yes	complemented
14	#41×#261	C₂²	1	96	yes	complemented
15	#11719627	C₂²	1	8	yes	complemented
16	#11719623	C₂²	1	72	yes	complemented
17	#11719623	C₂²	1	144	yes	complemented
18	C₂×D₄×#261	C₂²	1	144	yes	complemented
19	C₂×#88237	C₂²	1	36	yes	complemented
20	C₂⁴×D₁₃	C₂³	1	6	yes	—
21	C₂³×#261	C₂³	1	48	yes	complemented
22	C₁₃×H(5,𝔽₂)	C₂×C₄	1	16	yes	complemented
23	C₂×C₁₃×#41	C₂×C₄	1	6	yes	complemented
24	C₂²×D₄×C₁₃	C₂×C₄	1	9	yes	complemented
25	C₂²×C₂²⋊D₁₃	C₂×C₄	1	18	yes	complemented
26	#88256	C₂×C₄	1	96	yes	complemented
27	#41×D₁₃	C₂×C₄	1	48	yes	complemented
28	#41×D₁₃	C₂³	1	48	yes	—
29	C₂×#6131	C₂×C₄	1	6	yes	complemented
30	C₂×Q₈×D₁₃	C₂³	1	2	yes	—
31	#88251	C₂×C₄	1	144	yes	complemented
32	C₂×#6129	C₂×C₄	1	18	yes	complemented
33	C₂×D₄×D₁₃	C₂×C₄	1	36	yes	complemented
34	C₂×D₄×D₁₃	C₂×C₄	1	72	yes	complemented
35	C₂×D₄×D₁₃	C₂³	1	72	yes	—
36	C₂×D₄×D₁₃	C₂³	1	18	yes	—
37	C₂×#6127	C₂×C₄	1	18	yes	complemented
38	C₂²×D₅₂	C₂×C₄	1	9	yes	complemented
39	C₂²×C₄×D₁₃	C₂³	1	9	yes	—
40	C₂×C₂²⋊#261	C₂³	1	72	yes	—
41	C₂×C₂²⋊#261	C₂³	1	144	yes	complemented
42	Q₈×#261	C₂³	1	64	yes	complemented
43	D₄×#261	C₂³	1	576	yes	complemented
44	#88237	C₂³	1	48	yes	—
45	#88237	C₂³	1	144	yes	complemented
46	C₂×C₄⋊#261	C₂³	1	72	yes	complemented
47	C₂×C₄×#261	C₂³	1	72	yes	complemented
48	#62652	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂²×C₄	1	6	yes	complemented
50	C₂³×D₁₃	—	8	12	no	—
51	C₂³×D₁₃	C₂²×C₄	1	18	yes	complemented
52	C₂³×D₁₃	C₂²×C₄	1	24	yes	complemented
53	C₂³×D₁₃	C₂⁴	1	9	yes	—
54	C₂³×D₁₃	C₂⁴	1	24	yes	—
55	C₂²×#261	—	8	48	no	—
56	C₂²×#261	C₂⁴	1	144	yes	—
57	C₁₃×#41	C₂²×C₄	1	48	yes	complemented
58	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	complemented
59	C₂×D₄×C₁₃	C₂²×C₄	1	18	yes	complemented
60	C₂×D₄×C₁₃	C₂²×C₄	1	72	yes	complemented
61	C₂²×C₄×C₁₃	C₂²×C₄	1	9	yes	complemented
62	C₂×C₂²⋊D₁₃	C₂²×C₄	1	36	yes	complemented
63	C₂×C₂²⋊D₁₃	C₂²×C₄	1	144	yes	complemented
64	C₂²×Dic₁₃	C₂²×C₄	1	9	yes	complemented
65	#6131	C₂²×C₄	1	48	yes	complemented
66	Q₈×D₁₃	C₂²×C₄	1	8	yes	complemented
67	Q₈×D₁₃	C₂⁴	1	8	yes	—
68	#6129	C₂²×C₄	1	144	yes	complemented
69	D₄×D₁₃	C₂²×C₄	1	72	yes	complemented
70	D₄×D₁₃	C₂²×C₄	1	288	yes	complemented
71	D₄×D₁₃	C₂⁴	1	72	yes	—
72	#6127	C₂²×C₄	1	144	yes	complemented
73	C₂×D₅₂	C₂²×C₄	1	18	yes	complemented
74	C₂×D₅₂	C₂²×C₄	1	72	yes	complemented
75	C₂×C₄×D₁₃	C₂²×C₄	1	18	yes	complemented
76	C₂×C₄×D₁₃	C₂²×C₄	1	36	yes	complemented
77	C₂×C₄×D₁₃	C₂⁴	1	36	yes	—
78	C₂×C₄×D₁₃	C₂⁴	1	6	yes	—
79	C₂×Dic₂₆	C₂²×C₄	1	6	yes	complemented
80	C₂²⋊#261	C₂⁴	1	16	yes	—
81	C₂²⋊#261	C₂⁴	1	144	yes	—
82	C₄⋊#261	C₂⁴	1	96	yes	—
83	C₄×#261	C₂⁴	1	96	yes	—
84	C₂²×H(5,𝔽₂)	—	13	1	no	—
85	C₄×H(5,𝔽₂)	—	13	32	no	—
86	#3428	—	13	12	no	—
87	C₂×#518	—	13	18	no	—
88	C₂³×C₁₃	—	8	6	no	—
89	C₂³×C₁₃	C₂³×C₄	1	9	yes	—
90	C₂³×C₁₃	C₂³×C₄	1	24	yes	—
91	C₂²×D₁₃	—	4	48	no	—
92	C₂²×D₁₃	—	4	18	no	—
93	C₂²×D₁₃	—	8	24	no	—
94	C₂²×D₁₃	—	8	18	no	—
95	C₂²×D₁₃	C₂³×C₄	1	9	yes	—
96	C₂²×D₁₃	C₂³×C₄	1	18	yes	—
97	C₂²×D₁₃	C₂³×C₄	1	72	yes	—
98	C₂²×D₁₃	C₂⁵	1	1	yes	characteristic
99	C₂²×D₁₃	C₂⁵	1	18	yes	—
100	C₂×#261	—	4	144	no	—
101	C₂×#261	C₂⁵	1	32	yes	—
102	Q₈×C₁₃	C₂³×C₄	1	8	yes	—
103	D₄×C₁₃	C₂³×C₄	1	72	yes	—
104	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	36	yes	—
106	C₂²⋊D₁₃	C₂³×C₄	1	144	yes	—
107	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
108	C₂×Dic₁₃	C₂³×C₄	1	36	yes	—
109	D₅₂	C₂³×C₄	1	72	yes	—
110	C₄×D₁₃	C₂³×C₄	1	12	yes	—
111	C₄×D₁₃	C₂³×C₄	1	72	yes	—
112	C₄×D₁₃	C₂⁵	1	12	yes	—
113	Dic₂₆	C₂³×C₄	1	24	yes	—
114	C₂×H(5,𝔽₂)	—	13	16	no	—
115	C₂×H(5,𝔽₂)	—	13	16	no	—
116	C₂×H(5,𝔽₂)	—	13	6	no	—
117	C₂×H(5,𝔽₂)	—	13	9	no	—
118	C₂×H(5,𝔽₂)	#261	1	1	yes	characteristic, complemented
119	C₂²×#41	—	13	6	no	—
120	C₂³×D₄	—	13	9	no	—
121	#520	—	13	96	no	—
122	#519	—	13	8	no	—
123	#518	—	13	72	no	—
124	#518	—	13	144	no	—
125	C₄×#41	—	13	96	no	—
126	C₂×C₄×D₄	—	13	144	no	—
127	C₂×#117	—	13	36	no	—
128	C₂²×C₂²⋊C₄	—	13	24	no	—
129	C₂²×C₁₃	—	4	9	no	—
130	C₂²×C₁₃	—	4	24	no	—
131	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
132	C₂²×C₁₃	C₂⁴×C₄	1	18	yes	—
133	C₂×D₁₃	—	2	36	no	—
134	C₂×D₁₃	—	2	2	no	—
135	C₂×D₁₃	—	4	18	no	—
136	C₂×D₁₃	—	4	9	no	—
137	C₂×D₁₃	—	4	72	no	—
138	C₂×D₁₃	C₂⁴×C₄	1	1	yes	characteristic
139	C₂×D₁₃	C₂⁴×C₄	1	18	yes	—
140	C₂×D₁₃	C₂⁶	1	1	yes	characteristic
141	#261	—	2	32	no	—
142	C₄×C₁₃	C₂⁴×C₄	1	12	yes	—
143	Dic₁₃	C₂⁴×C₄	1	12	yes	—
144	C₂⁵	—	13	6	no	—
145	H(5,𝔽₂)	—	13	96	no	—
146	H(5,𝔽₂)	—	13	144	no	—
147	H(5,𝔽₂)	C₂×#261	1	16	yes	complemented
148	C₂×#41	—	13	48	no	—
149	C₂×#41	—	13	48	no	—
150	C₂×#41	—	13	6	no	—
151	C₂×#41	—	13	18	no	—
152	C₂×#41	—	13	18	no	—
153	C₂×#41	C₂×#261	1	6	yes	complemented
154	C₂²×Q₈	—	13	2	no	—
155	C₂²×D₄	—	13	18	no	—
156	C₂²×D₄	—	13	72	no	—
157	C₂²×D₄	—	13	18	no	—
158	C₂²×D₄	—	13	36	no	—
159	C₂²×D₄	—	13	72	no	—
160	C₂²×D₄	—	13	9	no	—
161	C₂²×D₄	C₂×#261	1	9	yes	complemented
162	C₂³×C₄	—	13	9	no	—
163	C₂³×C₄	—	13	48	no	—
164	C₄×Q₈	—	13	64	no	—
165	C₄×D₄	—	13	576	no	—
166	#117	—	13	48	no	—
167	#117	—	13	144	no	—
168	C₂×C₄⋊C₄	—	13	72	no	—
169	C₂×C₂²⋊C₄	—	13	72	no	—
170	C₂×C₂²⋊C₄	—	13	144	no	—
171	C₂×C₄²	—	13	72	no	—
172	C₂×C₁₃	—	2	1	no	—
173	C₂×C₁₃	—	2	18	no	—
174	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
175	D₁₃	—	2	1	no	—
176	D₁₃	—	2	18	no	—
177	D₁₃	H(7,𝔽₂)	1	2	yes	—
178	C₂⁴	—	13	9	no	—
179	C₂⁴	—	13	24	no	—
180	C₂⁴	—	13	24	no	—
181	C₂⁴	—	13	18	no	—
182	C₂⁴	—	104	12	no	—
183	C₂⁴	C₂²×#261	1	6	yes	complemented
184	#41	—	13	48	no	—
185	#41	—	13	144	no	—
186	#41	—	13	144	no	—
187	#41	C₂²×#261	1	48	yes	complemented
188	C₂×Q₈	—	13	8	no	—
189	C₂×Q₈	—	13	8	no	—
190	C₂×Q₈	—	13	6	no	—
191	C₂×Q₈	C₂²×#261	1	2	yes	complemented
192	C₂×D₄	—	13	36	no	—
193	C₂×D₄	—	13	144	no	—
194	C₂×D₄	—	13	72	no	—
195	C₂×D₄	—	13	72	no	—
196	C₂×D₄	—	13	288	no	—
197	C₂×D₄	—	13	18	no	—
198	C₂×D₄	—	13	72	no	—
199	C₂×D₄	C₂²×#261	1	18	yes	complemented
200	C₂×D₄	C₂²×#261	1	72	yes	complemented
201	C₂²×C₄	—	13	9	no	—
202	C₂²×C₄	—	13	36	no	—
203	C₂²×C₄	—	13	6	no	—
204	C₂²×C₄	—	13	18	no	—
205	C₂²×C₄	—	13	36	no	—
206	C₂²×C₄	—	13	144	no	—
207	C₂²×C₄	—	104	48	no	—
208	C₂²×C₄	C₂²×#261	1	9	yes	complemented
209	C₄⋊C₄	—	13	96	no	—
210	C₂²⋊C₄	—	13	16	no	—
211	C₂²⋊C₄	—	13	144	no	—
212	C₄²	—	13	96	no	—
213	C₁₃	#62652	1	1	yes	characteristic, complemented, Sylow
214	C₂³	—	8	6	no	—
215	C₂³	—	13	1	no	—
216	C₂³	—	13	18	no	—
217	C₂³	—	13	18	no	—
218	C₂³	—	13	9	no	—
219	C₂³	—	13	72	no	—
220	C₂³	—	52	48	no	—
221	C₂³	—	52	18	no	—
222	C₂³	—	104	18	no	—
223	C₂³	—	104	24	no	—
224	C₂³	C₂³×#261	1	9	yes	—
225	C₂³	C₂³×#261	1	24	yes	—
226	Q₈	—	13	24	no	—
227	Q₈	C₂³×#261	1	8	yes	—
228	D₄	—	13	144	no	—
229	D₄	—	13	72	no	—
230	D₄	C₂³×#261	1	72	yes	—
231	C₂×C₄	—	13	6	no	—
232	C₂×C₄	—	13	36	no	—
233	C₂×C₄	—	13	12	no	—
234	C₂×C₄	—	13	12	no	—
235	C₂×C₄	—	13	72	no	—
236	C₂×C₄	—	13	32	no	—
237	C₂×C₄	—	52	144	no	—
238	C₂×C₄	C₂³×#261	1	6	yes	—
239	C₂×C₄	C₂³×#261	1	36	yes	—
240	C₂²	—	4	9	no	—
241	C₂²	—	4	24	no	—
242	C₂²	—	13	1	no	—
243	C₂²	—	13	1	no	—
244	C₂²	—	13	18	no	—
245	C₂²	—	26	36	no	—
246	C₂²	—	26	2	no	—
247	C₂²	—	52	72	no	—
248	C₂²	—	52	9	no	—
249	C₂²	—	52	18	no	—
250	C₂²	C₂⁴×#261	1	1	yes	characteristic
251	C₂²	C₂⁴×#261	1	18	yes	—
252	C₄	—	13	12	no	—
253	C₄	—	26	32	no	—
254	C₄	C₂⁴×#261	1	12	yes	—
255	C₂	—	2	1	no	—
256	C₂	—	2	18	no	—
257	C₂	—	13	2	no	—
258	C₂	—	26	1	no	—
259	C₂	—	26	18	no	—
260	C₂	C₂⁵×#261	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×#261
commutator, socle	C₂×C₁₃	C₂⁵×C₄
Fitting	C₂×C₁₃×H(5,𝔽₂)	C₄
prime radical	C₂×D₁₃×H(5,𝔽₂)	C₂
cosocle	C₂×D₁₃	C₂⁶
hypercenter	C₂×H(5,𝔽₂)	#261
hypocenter	C₁₃	#62652
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
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)×#261	C₂	semidirect
2. #13131816	C₂	semidirect
3. C₂×#11719623	C₂	semidirect
5. C₂×C₁₃×H(5,𝔽₂)	C₄	semidirect
6. D₁₃×H(5,𝔽₂)	C₄	semidirect
8. C₂×#88256	C₄	semidirect
10. C₂×#88251	C₄	semidirect
12. C₂²×C₂²⋊#261	C₂²	semidirect
13. #11719631	C₂²	semidirect
14. #41×#261	C₂²	semidirect
15. #11719627	C₂²	semidirect
16. #11719623	C₂²	semidirect
17. #11719623	C₂²	semidirect
18. C₂×D₄×#261	C₂²	semidirect
19. C₂×#88237	C₂²	semidirect
21. C₂³×#261	C₂³	semidirect
22. C₁₃×H(5,𝔽₂)	C₂×C₄	semidirect
23. C₂×C₁₃×#41	C₂×C₄	semidirect
24. C₂²×D₄×C₁₃	C₂×C₄	semidirect
25. C₂²×C₂²⋊D₁₃	C₂×C₄	semidirect
26. #88256	C₂×C₄	semidirect
27. #41×D₁₃	C₂×C₄	semidirect
29. C₂×#6131	C₂×C₄	semidirect
31. #88251	C₂×C₄	semidirect
32. C₂×#6129	C₂×C₄	semidirect
33. C₂×D₄×D₁₃	C₂×C₄	semidirect
34. C₂×D₄×D₁₃	C₂×C₄	semidirect
37. C₂×#6127	C₂×C₄	semidirect
38. C₂²×D₅₂	C₂×C₄	semidirect
41. C₂×C₂²⋊#261	C₂³	semidirect
42. Q₈×#261	C₂³	semidirect
43. D₄×#261	C₂³	semidirect
45. #88237	C₂³	semidirect
46. C₂×C₄⋊#261	C₂³	semidirect
47. C₂×C₄×#261	C₂³	semidirect
49. C₂⁴×C₁₃	C₂²×C₄	semidirect
51. C₂³×D₁₃	C₂²×C₄	semidirect
52. C₂³×D₁₃	C₂²×C₄	semidirect
57. C₁₃×#41	C₂²×C₄	semidirect
58. C₂×Q₈×C₁₃	C₂²×C₄	semidirect
59. C₂×D₄×C₁₃	C₂²×C₄	semidirect
60. C₂×D₄×C₁₃	C₂²×C₄	semidirect
61. C₂²×C₄×C₁₃	C₂²×C₄	semidirect
62. C₂×C₂²⋊D₁₃	C₂²×C₄	semidirect
63. C₂×C₂²⋊D₁₃	C₂²×C₄	semidirect
64. C₂²×Dic₁₃	C₂²×C₄	semidirect
65. #6131	C₂²×C₄	semidirect
66. Q₈×D₁₃	C₂²×C₄	semidirect
68. #6129	C₂²×C₄	semidirect
69. D₄×D₁₃	C₂²×C₄	semidirect
70. D₄×D₁₃	C₂²×C₄	semidirect
72. #6127	C₂²×C₄	semidirect
73. C₂×D₅₂	C₂²×C₄	semidirect
74. C₂×D₅₂	C₂²×C₄	semidirect
75. C₂×C₄×D₁₃	C₂²×C₄	semidirect
76. C₂×C₄×D₁₃	C₂²×C₄	semidirect
79. C₂×Dic₂₆	C₂²×C₄	semidirect
118. C₂×H(5,𝔽₂)	#261	semidirect
147. H(5,𝔽₂)	C₂×#261	semidirect
153. C₂×#41	C₂×#261	semidirect
161. C₂²×D₄	C₂×#261	semidirect
183. C₂⁴	C₂²×#261	semidirect
187. #41	C₂²×#261	semidirect
191. C₂×Q₈	C₂²×#261	semidirect
199. C₂×D₄	C₂²×#261	semidirect
200. C₂×D₄	C₂²×#261	semidirect
208. C₂²×C₄	C₂²×#261	semidirect
213. C₁₃	#62652	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)×#261	2
#13131816	2
C₂×#11719623	2
C₂×D₁₃×H(5,𝔽₂)	2
#62652	13