Subgroups — #30626732

Order 15616.

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)×#6923	C₂	1	32	yes	maximal, complemented
2	#21462799	C₂	1	12	yes	maximal, complemented
3	C₂×#16840829	C₂	1	18	yes	maximal, complemented
4	C₂×H(5,𝔽₂)×D₆₁	C₂	1	1	yes	maximal, characteristic
5	#16840831	C₂²	1	96	yes	complemented
6	#16840830	C₂²	1	8	yes	complemented
7	#16840829	C₂²	1	72	yes	complemented
8	#16840829	C₂²	1	144	yes	complemented
9	#41×#6923	C₂²	1	96	yes	complemented
10	C₂×D₄×#6923	C₂²	1	144	yes	complemented
11	C₂×#14519856	C₂²	1	36	yes	complemented
12	C₂²×C₂²⋊#6923	C₂²	1	24	yes	complemented
13	C₂×#14519823	C₄	1	6	yes	complemented
14	C₂×#14519822	C₄	1	9	yes	complemented
15	H(5,𝔽₂)×D₆₁	C₄	1	16	yes	complemented
16	H(5,𝔽₂)×D₆₁	C₂²	1	16	yes	—
17	C₂×#41×D₆₁	C₂²	1	6	yes	—
18	C₂²×D₄×D₆₁	C₂²	1	9	yes	—
19	C₂×H(5,𝔽₂)×C₆₁	C₄	1	1	yes	characteristic, complemented
20	C₂³×#6923	C₂³	1	48	yes	complemented
21	Q₈×#6923	C₂³	1	64	yes	complemented
22	D₄×#6923	C₂³	1	576	yes	complemented
23	#14519856	C₂³	1	48	yes	—
24	#14519856	C₂³	1	144	yes	complemented
25	C₂×C₄⋊#6923	C₂³	1	72	yes	complemented
26	C₂×C₂²⋊#6923	C₂³	1	72	yes	—
27	C₂×C₂²⋊#6923	C₂³	1	144	yes	complemented
28	C₂×C₄×#6923	C₂³	1	72	yes	complemented
29	C₂⁴×D₆₁	C₂³	1	6	yes	—
30	#14519823	C₂×C₄	1	96	yes	complemented
31	#14519822	C₂×C₄	1	144	yes	complemented
32	C₂×#11758226	C₂×C₄	1	6	yes	complemented
33	C₂×#11758222	C₂×C₄	1	18	yes	complemented
34	C₂×#11758224	C₂×C₄	1	18	yes	complemented
35	#41×D₆₁	C₂×C₄	1	48	yes	complemented
36	#41×D₆₁	C₂³	1	48	yes	—
37	C₂×Q₈×D₆₁	C₂³	1	2	yes	—
38	C₂²×D₂₄₄	C₂×C₄	1	9	yes	complemented
39	C₂²×C₂²⋊D₆₁	C₂×C₄	1	18	yes	complemented
40	C₂×D₄×D₆₁	C₂×C₄	1	72	yes	complemented
41	C₂×D₄×D₆₁	C₂×C₄	1	36	yes	complemented
42	C₂×D₄×D₆₁	C₂³	1	72	yes	—
43	C₂×D₄×D₆₁	C₂³	1	18	yes	—
44	C₂²×C₄×D₆₁	C₂³	1	9	yes	—
45	H(5,𝔽₂)×C₆₁	C₂×C₄	1	16	yes	complemented
46	C₂×#41×C₆₁	C₂×C₄	1	6	yes	complemented
47	C₂²×D₄×C₆₁	C₂×C₄	1	9	yes	complemented
48	C₂⁴×C₆₁	C₂²×C₄	1	6	yes	complemented
49	C₂³×D₆₁	—	8	12	no	—
50	C₂³×D₆₁	C₂²×C₄	1	24	yes	complemented
51	C₂³×D₆₁	C₂²×C₄	1	18	yes	complemented
52	C₂³×D₆₁	C₂⁴	1	9	yes	—
53	C₂³×D₆₁	C₂⁴	1	24	yes	—
54	C₂²×#6923	—	8	48	no	—
55	C₂²×#6923	C₂⁴	1	144	yes	—
56	#41×C₆₁	C₂²×C₄	1	48	yes	complemented
57	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	complemented
58	C₂×D₄×C₆₁	C₂²×C₄	1	18	yes	complemented
59	C₂×D₄×C₆₁	C₂²×C₄	1	72	yes	complemented
60	C₂²×C₄×C₆₁	C₂²×C₄	1	9	yes	complemented
61	C₂×C₂²⋊D₆₁	C₂²×C₄	1	36	yes	complemented
62	C₂×C₂²⋊D₆₁	C₂²×C₄	1	144	yes	complemented
63	C₂²×Dic₆₁	C₂²×C₄	1	9	yes	complemented
64	#11758226	C₂²×C₄	1	48	yes	complemented
65	Q₈×D₆₁	C₂²×C₄	1	8	yes	complemented
66	Q₈×D₆₁	C₂⁴	1	8	yes	—
67	#11758224	C₂²×C₄	1	144	yes	complemented
68	D₄×D₆₁	C₂²×C₄	1	72	yes	complemented
69	D₄×D₆₁	C₂²×C₄	1	288	yes	complemented
70	D₄×D₆₁	C₂⁴	1	72	yes	—
71	#11758222	C₂²×C₄	1	144	yes	complemented
72	C₂×D₂₄₄	C₂²×C₄	1	18	yes	complemented
73	C₂×D₂₄₄	C₂²×C₄	1	72	yes	complemented
74	C₂×C₄×D₆₁	C₂²×C₄	1	36	yes	complemented
75	C₂×C₄×D₆₁	C₂²×C₄	1	18	yes	complemented
76	C₂×C₄×D₆₁	C₂⁴	1	6	yes	—
77	C₂×C₄×D₆₁	C₂⁴	1	36	yes	—
78	C₂×Dic₁₂₂	C₂²×C₄	1	6	yes	complemented
79	C₂²⋊#6923	C₂⁴	1	16	yes	—
80	C₂²⋊#6923	C₂⁴	1	144	yes	—
81	C₄⋊#6923	C₂⁴	1	96	yes	—
82	C₄×#6923	C₂⁴	1	96	yes	—
83	C₂³×C₆₁	—	8	6	no	—
84	C₂³×C₆₁	C₂³×C₄	1	9	yes	—
85	C₂³×C₆₁	C₂³×C₄	1	24	yes	—
86	C₂²×D₆₁	—	4	48	no	—
87	C₂²×D₆₁	—	4	18	no	—
88	C₂²×D₆₁	—	8	24	no	—
89	C₂²×D₆₁	—	8	18	no	—
90	C₂²×D₆₁	C₂³×C₄	1	9	yes	—
91	C₂²×D₆₁	C₂³×C₄	1	18	yes	—
92	C₂²×D₆₁	C₂³×C₄	1	72	yes	—
93	C₂²×D₆₁	C₂⁵	1	1	yes	characteristic
94	C₂²×D₆₁	C₂⁵	1	18	yes	—
95	C₂×#6923	—	4	144	no	—
96	C₂×#6923	C₂⁵	1	32	yes	—
97	Q₈×C₆₁	C₂³×C₄	1	8	yes	—
98	D₄×C₆₁	C₂³×C₄	1	72	yes	—
99	C₂×C₄×C₆₁	C₂³×C₄	1	6	yes	—
100	C₂×C₄×C₆₁	C₂³×C₄	1	36	yes	—
101	C₂²⋊D₆₁	C₂³×C₄	1	144	yes	—
102	C₂×Dic₆₁	C₂³×C₄	1	6	yes	—
103	C₂×Dic₆₁	C₂³×C₄	1	36	yes	—
104	D₂₄₄	C₂³×C₄	1	72	yes	—
105	C₄×D₆₁	C₂³×C₄	1	12	yes	—
106	C₄×D₆₁	C₂³×C₄	1	72	yes	—
107	C₄×D₆₁	C₂⁵	1	12	yes	—
108	Dic₁₂₂	C₂³×C₄	1	24	yes	—
109	#62652	—	61	1	no	maximal, Sylow
110	C₂²×C₆₁	—	4	9	no	—
111	C₂²×C₆₁	—	4	24	no	—
112	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
113	C₂²×C₆₁	C₂⁴×C₄	1	18	yes	—
114	C₂×D₆₁	—	2	36	no	—
115	C₂×D₆₁	—	2	2	no	—
116	C₂×D₆₁	—	4	18	no	—
117	C₂×D₆₁	—	4	9	no	—
118	C₂×D₆₁	—	4	72	no	—
119	C₂×D₆₁	C₂⁴×C₄	1	1	yes	characteristic
120	C₂×D₆₁	C₂⁴×C₄	1	18	yes	—
121	C₂×D₆₁	C₂⁶	1	1	yes	characteristic
122	#6923	—	2	32	no	—
123	C₄×C₆₁	C₂⁴×C₄	1	12	yes	—
124	Dic₆₁	C₂⁴×C₄	1	12	yes	—
125	C₂²×H(5,𝔽₂)	—	61	1	no	—
126	C₄×H(5,𝔽₂)	—	61	32	no	—
127	#3428	—	61	12	no	—
128	C₂×#518	—	61	18	no	—
129	C₂×C₆₁	—	2	1	no	—
130	C₂×C₆₁	—	2	18	no	—
131	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
132	D₆₁	—	2	1	no	—
133	D₆₁	—	2	18	no	—
134	D₆₁	H(7,𝔽₂)	1	2	yes	—
135	C₂×H(5,𝔽₂)	—	61	16	no	—
136	C₂×H(5,𝔽₂)	—	61	16	no	—
137	C₂×H(5,𝔽₂)	—	61	6	no	—
138	C₂×H(5,𝔽₂)	—	61	9	no	—
139	C₂×H(5,𝔽₂)	#6923	1	1	yes	characteristic, complemented
140	C₂²×#41	—	61	6	no	—
141	C₂³×D₄	—	61	9	no	—
142	#520	—	61	96	no	—
143	#519	—	61	8	no	—
144	#518	—	61	72	no	—
145	#518	—	61	144	no	—
146	C₄×#41	—	61	96	no	—
147	C₂×C₄×D₄	—	61	144	no	—
148	C₂×#117	—	61	36	no	—
149	C₂²×C₂²⋊C₄	—	61	24	no	—
150	C₆₁	#62652	1	1	yes	characteristic, complemented, Sylow
151	C₂⁵	—	61	6	no	—
152	H(5,𝔽₂)	—	61	96	no	—
153	H(5,𝔽₂)	—	61	144	no	—
154	H(5,𝔽₂)	C₂×#6923	1	16	yes	complemented
155	C₂×#41	—	61	48	no	—
156	C₂×#41	—	61	48	no	—
157	C₂×#41	—	61	6	no	—
158	C₂×#41	—	61	18	no	—
159	C₂×#41	—	61	18	no	—
160	C₂×#41	C₂×#6923	1	6	yes	complemented
161	C₂²×Q₈	—	61	2	no	—
162	C₂²×D₄	—	61	18	no	—
163	C₂²×D₄	—	61	72	no	—
164	C₂²×D₄	—	61	36	no	—
165	C₂²×D₄	—	61	72	no	—
166	C₂²×D₄	—	61	18	no	—
167	C₂²×D₄	—	61	9	no	—
168	C₂²×D₄	C₂×#6923	1	9	yes	complemented
169	C₂³×C₄	—	61	9	no	—
170	C₂³×C₄	—	61	48	no	—
171	C₄×Q₈	—	61	64	no	—
172	C₄×D₄	—	61	576	no	—
173	#117	—	61	48	no	—
174	#117	—	61	144	no	—
175	C₂×C₄⋊C₄	—	61	72	no	—
176	C₂×C₂²⋊C₄	—	61	72	no	—
177	C₂×C₂²⋊C₄	—	61	144	no	—
178	C₂×C₄²	—	61	72	no	—
179	C₂⁴	—	61	24	no	—
180	C₂⁴	—	61	18	no	—
181	C₂⁴	—	61	9	no	—
182	C₂⁴	—	61	24	no	—
183	C₂⁴	—	488	12	no	—
184	C₂⁴	C₂²×#6923	1	6	yes	complemented
185	#41	—	61	48	no	—
186	#41	—	61	144	no	—
187	#41	—	61	144	no	—
188	#41	C₂²×#6923	1	48	yes	complemented
189	C₂×Q₈	—	61	8	no	—
190	C₂×Q₈	—	61	8	no	—
191	C₂×Q₈	—	61	6	no	—
192	C₂×Q₈	C₂²×#6923	1	2	yes	complemented
193	C₂×D₄	—	61	36	no	—
194	C₂×D₄	—	61	144	no	—
195	C₂×D₄	—	61	72	no	—
196	C₂×D₄	—	61	72	no	—
197	C₂×D₄	—	61	288	no	—
198	C₂×D₄	—	61	18	no	—
199	C₂×D₄	—	61	72	no	—
200	C₂×D₄	C₂²×#6923	1	18	yes	complemented
201	C₂×D₄	C₂²×#6923	1	72	yes	complemented
202	C₂²×C₄	—	61	9	no	—
203	C₂²×C₄	—	61	36	no	—
204	C₂²×C₄	—	61	6	no	—
205	C₂²×C₄	—	61	18	no	—
206	C₂²×C₄	—	61	36	no	—
207	C₂²×C₄	—	61	144	no	—
208	C₂²×C₄	—	488	48	no	—
209	C₂²×C₄	C₂²×#6923	1	9	yes	complemented
210	C₄⋊C₄	—	61	96	no	—
211	C₂²⋊C₄	—	61	16	no	—
212	C₂²⋊C₄	—	61	144	no	—
213	C₄²	—	61	96	no	—
214	C₂³	—	8	6	no	—
215	C₂³	—	61	1	no	—
216	C₂³	—	61	18	no	—
217	C₂³	—	61	18	no	—
218	C₂³	—	61	9	no	—
219	C₂³	—	61	72	no	—
220	C₂³	—	488	18	no	—
221	C₂³	—	488	24	no	—
222	C₂³	—	244	48	no	—
223	C₂³	—	244	18	no	—
224	C₂³	C₂³×#6923	1	9	yes	—
225	C₂³	C₂³×#6923	1	24	yes	—
226	Q₈	—	61	24	no	—
227	Q₈	C₂³×#6923	1	8	yes	—
228	D₄	—	61	144	no	—
229	D₄	—	61	72	no	—
230	D₄	C₂³×#6923	1	72	yes	—
231	C₂×C₄	—	61	6	no	—
232	C₂×C₄	—	61	36	no	—
233	C₂×C₄	—	61	12	no	—
234	C₂×C₄	—	61	72	no	—
235	C₂×C₄	—	61	12	no	—
236	C₂×C₄	—	61	32	no	—
237	C₂×C₄	—	244	144	no	—
238	C₂×C₄	C₂³×#6923	1	6	yes	—
239	C₂×C₄	C₂³×#6923	1	36	yes	—
240	C₂²	—	4	9	no	—
241	C₂²	—	4	24	no	—
242	C₂²	—	61	1	no	—
243	C₂²	—	61	1	no	—
244	C₂²	—	61	18	no	—
245	C₂²	—	122	36	no	—
246	C₂²	—	122	2	no	—
247	C₂²	—	244	9	no	—
248	C₂²	—	244	18	no	—
249	C₂²	—	244	72	no	—
250	C₂²	C₂⁴×#6923	1	1	yes	characteristic
251	C₂²	C₂⁴×#6923	1	18	yes	—
252	C₄	—	61	12	no	—
253	C₄	—	122	32	no	—
254	C₄	C₂⁴×#6923	1	12	yes	—
255	C₂	—	2	1	no	—
256	C₂	—	2	18	no	—
257	C₂	—	61	2	no	—
258	C₂	—	122	1	no	—
259	C₂	—	122	18	no	—
260	C₂	C₂⁵×#6923	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁵×#6923
commutator, socle	C₂×C₆₁	C₂⁵×C₄
Fitting	C₂×H(5,𝔽₂)×C₆₁	C₄
prime radical	C₂×H(5,𝔽₂)×D₆₁	C₂
cosocle	C₂×D₆₁	C₂⁶
hypercenter	C₂×H(5,𝔽₂)	#6923
hypocenter	C₆₁	#62652
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
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)×#6923	C₂	semidirect
2. #21462799	C₂	semidirect
3. C₂×#16840829	C₂	semidirect
5. #16840831	C₂²	semidirect
6. #16840830	C₂²	semidirect
7. #16840829	C₂²	semidirect
8. #16840829	C₂²	semidirect
9. #41×#6923	C₂²	semidirect
10. C₂×D₄×#6923	C₂²	semidirect
11. C₂×#14519856	C₂²	semidirect
12. C₂²×C₂²⋊#6923	C₂²	semidirect
13. C₂×#14519823	C₄	semidirect
14. C₂×#14519822	C₄	semidirect
15. H(5,𝔽₂)×D₆₁	C₄	semidirect
19. C₂×H(5,𝔽₂)×C₆₁	C₄	semidirect
20. C₂³×#6923	C₂³	semidirect
21. Q₈×#6923	C₂³	semidirect
22. D₄×#6923	C₂³	semidirect
24. #14519856	C₂³	semidirect
25. C₂×C₄⋊#6923	C₂³	semidirect
27. C₂×C₂²⋊#6923	C₂³	semidirect
28. C₂×C₄×#6923	C₂³	semidirect
30. #14519823	C₂×C₄	semidirect
31. #14519822	C₂×C₄	semidirect
32. C₂×#11758226	C₂×C₄	semidirect
33. C₂×#11758222	C₂×C₄	semidirect
34. C₂×#11758224	C₂×C₄	semidirect
35. #41×D₆₁	C₂×C₄	semidirect
38. C₂²×D₂₄₄	C₂×C₄	semidirect
39. C₂²×C₂²⋊D₆₁	C₂×C₄	semidirect
40. C₂×D₄×D₆₁	C₂×C₄	semidirect
41. C₂×D₄×D₆₁	C₂×C₄	semidirect
45. H(5,𝔽₂)×C₆₁	C₂×C₄	semidirect
46. C₂×#41×C₆₁	C₂×C₄	semidirect
47. C₂²×D₄×C₆₁	C₂×C₄	semidirect
48. C₂⁴×C₆₁	C₂²×C₄	semidirect
50. C₂³×D₆₁	C₂²×C₄	semidirect
51. C₂³×D₆₁	C₂²×C₄	semidirect
56. #41×C₆₁	C₂²×C₄	semidirect
57. C₂×Q₈×C₆₁	C₂²×C₄	semidirect
58. C₂×D₄×C₆₁	C₂²×C₄	semidirect
59. C₂×D₄×C₆₁	C₂²×C₄	semidirect
60. C₂²×C₄×C₆₁	C₂²×C₄	semidirect
61. C₂×C₂²⋊D₆₁	C₂²×C₄	semidirect
62. C₂×C₂²⋊D₆₁	C₂²×C₄	semidirect
63. C₂²×Dic₆₁	C₂²×C₄	semidirect
64. #11758226	C₂²×C₄	semidirect
65. Q₈×D₆₁	C₂²×C₄	semidirect
67. #11758224	C₂²×C₄	semidirect
68. D₄×D₆₁	C₂²×C₄	semidirect
69. D₄×D₆₁	C₂²×C₄	semidirect
71. #11758222	C₂²×C₄	semidirect
72. C₂×D₂₄₄	C₂²×C₄	semidirect
73. C₂×D₂₄₄	C₂²×C₄	semidirect
74. C₂×C₄×D₆₁	C₂²×C₄	semidirect
75. C₂×C₄×D₆₁	C₂²×C₄	semidirect
78. C₂×Dic₁₂₂	C₂²×C₄	semidirect
139. C₂×H(5,𝔽₂)	#6923	semidirect
150. C₆₁	#62652	semidirect
154. H(5,𝔽₂)	C₂×#6923	semidirect
160. C₂×#41	C₂×#6923	semidirect
168. C₂²×D₄	C₂×#6923	semidirect
184. C₂⁴	C₂²×#6923	semidirect
188. #41	C₂²×#6923	semidirect
192. C₂×Q₈	C₂²×#6923	semidirect
200. C₂×D₄	C₂²×#6923	semidirect
201. C₂×D₄	C₂²×#6923	semidirect
209. C₂²×C₄	C₂²×#6923	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)×#6923	2
#21462799	2
C₂×#16840829	2
C₂×H(5,𝔽₂)×D₆₁	2
#62652	61