Subgroups — C₂²×#16838145 #29495562

Order 15104.

10246 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838145	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14273714	C₂	1	3	yes	maximal, complemented
3	C₂²×#14273713	C₂	1	6	yes	maximal, complemented
4	C₂²×#14273696	C₂	1	2	yes	maximal, complemented
5	C₂²×#14273695	C₂	1	3	yes	maximal, complemented
6	C₂²×#120×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₉	C₂²	1	1	yes	characteristic, complemented
8	C₂³×Dic₁₁₈	C₂²	1	3	yes	complemented
9	C₂³×C₂²⋊D₅₉	C₂²	1	6	yes	—
10	#16838145	C₂²	1	256	yes	direct product factor
11	C₂×#14273714	C₂²	1	72	yes	complemented
12	C₂×#14273713	C₂²	1	144	yes	complemented
13	C₂×#14273696	C₂²	1	48	yes	complemented
14	C₂×#14273695	C₂²	1	72	yes	complemented
15	C₂²×C₂²⋊Dic₅₉	C₂²	1	3	yes	complemented
16	C₂²×C₂²⋊Dic₅₉	C₂²	1	6	yes	complemented
17	C₂²×(C₂×D₅₉)⋊C₄	C₂²	1	6	yes	complemented
18	C₂²×C₄×Dic₅₉	C₂²	1	3	yes	complemented
19	C₂⁶×C₅₉	C₂²	1	1	yes	characteristic, complemented
20	C₂³×D₄×C₅₉	C₂²	1	3	yes	—
21	C₂×#120×C₅₉	C₂²	1	24	yes	complemented
22	C₂²×C₂²⋊C₄×C₅₉	C₂²	1	3	yes	complemented
23	C₂⁴×D₅₉	—	4	12	no	—
24	C₂⁴×D₅₉	C₂³	1	2	yes	—
25	C₂⁴×D₅₉	C₂³	1	12	yes	complemented
26	C₂²×Dic₁₁₈	—	4	12	no	—
27	C₂²×Dic₁₁₈	C₂³	1	36	yes	complemented
28	C₂²×C₂²⋊D₅₉	—	4	24	no	—
29	C₂²×C₂²⋊D₅₉	C₂³	1	72	yes	—
30	C₂³×Dic₅₉	C₂³	1	6	yes	—
31	#14273714	C₂³	1	192	yes	complemented
32	#14273713	C₂³	1	384	yes	complemented
33	#14273696	C₂³	1	128	yes	complemented
34	#14273695	C₂³	1	192	yes	complemented
35	C₂×C₂²⋊Dic₅₉	C₂³	1	72	yes	complemented
36	C₂×C₂²⋊Dic₅₉	C₂³	1	36	yes	complemented
37	C₂×(C₂×D₅₉)⋊C₄	C₂³	1	72	yes	complemented
38	C₂×C₄×Dic₅₉	C₂³	1	36	yes	complemented
39	C₂⁵×C₅₉	—	4	12	no	—
40	C₂⁵×C₅₉	C₂³	1	3	yes	—
41	C₂⁵×C₅₉	C₂³	1	1	yes	characteristic
42	C₂⁵×C₅₉	C₂³	1	12	yes	complemented
43	C₂²×D₄×C₅₉	—	4	12	no	—
44	C₂²×D₄×C₅₉	C₂³	1	36	yes	—
45	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
46	#120×C₅₉	C₂³	1	64	yes	complemented
47	C₂×C₂²⋊C₄×C₅₉	C₂³	1	36	yes	complemented
48	C₂⁴×C₅₉	—	2	12	no	—
49	C₂⁴×C₅₉	—	2	12	no	—
50	C₂⁴×C₅₉	—	2	24	no	—
51	C₂⁴×C₅₉	—	4	72	no	—
52	C₂⁴×C₅₉	—	4	64	no	—
53	C₂⁴×C₅₉	C₂⁴	1	18	yes	—
54	C₂⁴×C₅₉	C₂⁴	1	6	yes	—
55	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
56	C₂⁴×C₅₉	C₂⁴	1	16	yes	complemented
57	C₂³×D₅₉	—	2	24	no	—
58	C₂³×D₅₉	—	4	72	no	—
59	C₂³×D₅₉	—	4	64	no	—
60	C₂³×D₅₉	C₂⁴	1	12	yes	—
61	C₂³×D₅₉	C₂⁴	1	16	yes	complemented
62	C₂×D₄×C₅₉	—	4	72	no	—
63	C₂×D₄×C₅₉	C₂⁴	1	48	yes	—
64	C₂²×C₄×C₅₉	—	2	12	no	—
65	C₂²×C₄×C₅₉	C₂⁴	1	18	yes	—
66	C₂×C₂²⋊D₅₉	—	4	144	no	—
67	C₂×C₂²⋊D₅₉	C₂⁴	1	96	yes	—
68	C₂²×Dic₅₉	—	2	24	no	—
69	C₂²×Dic₅₉	C₂⁴	1	36	yes	—
70	C₂×Dic₁₁₈	—	4	72	no	—
71	C₂×Dic₁₁₈	C₂⁴	1	48	yes	complemented
72	C₂²⋊C₄×C₅₉	C₂⁴	1	48	yes	complemented
73	C₂²⋊Dic₅₉	C₂⁴	1	48	yes	complemented
74	C₂²⋊Dic₅₉	C₂⁴	1	96	yes	complemented
75	(C₂×D₅₉)⋊C₄	C₂⁴	1	96	yes	complemented
76	C₄×Dic₅₉	C₂⁴	1	48	yes	complemented
77	C₂³×C₅₉	—	2	72	no	—
78	C₂³×C₅₉	—	2	36	no	—
79	C₂³×C₅₉	—	2	36	no	—
80	C₂³×C₅₉	—	4	48	no	—
81	C₂³×C₅₉	—	4	192	no	—
82	C₂³×C₅₉	—	4	16	no	—
83	C₂³×C₅₉	—	4	48	no	—
84	C₂³×C₅₉	H(5,𝔽₂)	1	12	yes	—
85	C₂³×C₅₉	C₂⁵	1	4	yes	—
86	C₂³×C₅₉	C₂⁵	1	12	yes	—
87	C₂³×C₅₉	C₂⁵	1	3	yes	—
88	C₂²×D₅₉	—	2	72	no	—
89	C₂²×D₅₉	—	4	48	no	—
90	C₂²×D₅₉	—	4	192	no	—
91	C₂²×D₅₉	—	4	32	no	—
92	C₂²×D₅₉	C₂⁵	1	8	yes	—
93	D₄×C₅₉	—	4	48	no	—
94	C₂×C₄×C₅₉	—	2	36	no	—
95	C₂×C₄×C₅₉	C₂⁵	1	12	yes	—
96	C₂²⋊D₅₉	—	4	96	no	—
97	C₂×Dic₅₉	—	2	72	no	—
98	C₂×Dic₅₉	C₂⁵	1	24	yes	—
99	Dic₁₁₈	—	4	48	no	—
100	C₂²×H(3,𝔽₄)	—	59	1	no	maximal, Sylow
101	C₂²×C₅₉	—	2	24	no	—
102	C₂²×C₅₉	—	2	12	no	—
103	C₂²×C₅₉	—	2	12	no	—
104	C₂²×C₅₉	—	4	64	no	—
105	C₂²×C₅₉	—	4	24	no	—
106	C₂²×C₅₉	—	4	72	no	—
107	C₂²×C₅₉	H(3,𝔽₄)	1	16	yes	complemented
108	C₂²×C₅₉	C₂×H(5,𝔽₂)	1	18	yes	—
109	C₂²×C₅₉	C₂⁶	1	1	yes	characteristic
110	C₂×D₅₉	—	2	24	no	—
111	C₂×D₅₉	—	4	64	no	—
112	C₂×D₅₉	—	4	48	no	—
113	C₄×C₅₉	—	2	12	no	—
114	Dic₅₉	—	2	24	no	—
115	C₂×H(3,𝔽₄)	—	59	48	no	—
116	C₂²×#124	—	59	3	no	—
117	C₂²×#124	—	59	6	no	—
118	C₂²×#120	—	59	2	no	—
119	C₂²×#120	—	59	3	no	—
120	C₂²×#120	D₅₉	1	1	yes	characteristic, complemented
121	C₂×C₅₉	—	4	4	no	—
122	C₂×C₅₉	—	4	12	no	—
123	C₂×C₅₉	C₂×H(3,𝔽₄)	1	12	yes	complemented
124	C₂×C₅₉	C₂²×H(5,𝔽₂)	1	3	yes	—
125	D₅₉	—	4	8	no	—
126	C₂⁶	—	59	1	no	—
127	C₂⁶	C₂×D₅₉	1	1	yes	characteristic, complemented
128	C₂³×Q₈	—	59	3	no	—
129	C₂³×D₄	—	59	6	no	—
130	C₂³×D₄	C₂×D₅₉	1	3	yes	—
131	H(3,𝔽₄)	—	59	256	no	—
132	C₂×#124	—	59	72	no	—
133	C₂×#124	—	59	144	no	—
134	C₂×#120	—	59	48	no	—
135	C₂×#120	—	59	72	no	—
136	C₂×#120	C₂×D₅₉	1	24	yes	complemented
137	C₂²×C₂²⋊C₄	—	59	3	no	—
138	C₂²×C₂²⋊C₄	—	59	6	no	—
139	C₂²×C₂²⋊C₄	—	59	6	no	—
140	C₂²×C₂²⋊C₄	C₂×D₅₉	1	3	yes	complemented
141	C₂²×C₄²	—	59	3	no	—
142	C₅₉	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
143	C₂⁵	—	4	12	no	—
144	C₂⁵	—	59	2	no	—
145	C₂⁵	—	59	12	no	—
146	C₂⁵	—	236	12	no	—
147	C₂⁵	C₂²×D₅₉	1	1	yes	characteristic
148	C₂⁵	C₂²×D₅₉	1	3	yes	—
149	C₂⁵	C₂²×D₅₉	1	12	yes	complemented
150	C₂²×Q₈	—	59	36	no	—
151	C₂²×Q₈	—	236	12	no	—
152	C₂²×D₄	—	4	12	no	—
153	C₂²×D₄	—	59	72	no	—
154	C₂²×D₄	—	236	24	no	—
155	C₂²×D₄	C₂²×D₅₉	1	36	yes	—
156	C₂³×C₄	—	59	6	no	—
157	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
158	#124	—	59	192	no	—
159	#124	—	59	384	no	—
160	#120	—	59	128	no	—
161	#120	—	59	192	no	—
162	#120	C₂²×D₅₉	1	64	yes	complemented
163	C₂×C₂²⋊C₄	—	59	36	no	—
164	C₂×C₂²⋊C₄	—	59	72	no	—
165	C₂×C₂²⋊C₄	—	59	72	no	—
166	C₂×C₂²⋊C₄	C₂²×D₅₉	1	36	yes	complemented
167	C₂×C₄²	—	59	36	no	—
168	C₂⁴	—	2	12	no	—
169	C₂⁴	—	2	12	no	—
170	C₂⁴	—	2	24	no	—
171	C₂⁴	—	4	72	no	—
172	C₂⁴	—	4	64	no	—
173	C₂⁴	—	59	12	no	—
174	C₂⁴	—	59	16	no	—
175	C₂⁴	—	118	24	no	—
176	C₂⁴	—	236	72	no	—
177	C₂⁴	—	236	64	no	—
178	C₂⁴	C₂³×D₅₉	1	6	yes	—
179	C₂⁴	C₂³×D₅₉	1	18	yes	—
180	C₂⁴	C₂³×D₅₉	1	1	yes	characteristic
181	C₂⁴	C₂³×D₅₉	1	16	yes	complemented
182	C₂×Q₈	—	59	48	no	—
183	C₂×Q₈	—	236	72	no	—
184	C₂×D₄	—	4	72	no	—
185	C₂×D₄	—	59	96	no	—
186	C₂×D₄	—	236	144	no	—
187	C₂×D₄	C₂³×D₅₉	1	48	yes	—
188	C₂²×C₄	—	2	12	no	—
189	C₂²×C₄	—	59	36	no	—
190	C₂²×C₄	—	118	24	no	—
191	C₂²×C₄	C₂³×D₅₉	1	18	yes	—
192	C₂²⋊C₄	—	59	48	no	—
193	C₂²⋊C₄	—	59	96	no	—
194	C₂²⋊C₄	—	59	96	no	—
195	C₂²⋊C₄	C₂³×D₅₉	1	48	yes	complemented
196	C₄²	—	59	48	no	—
197	C₂³	—	2	36	no	—
198	C₂³	—	2	36	no	—
199	C₂³	—	2	72	no	—
200	C₂³	—	4	48	no	—
201	C₂³	—	4	192	no	—
202	C₂³	—	4	48	no	—
203	C₂³	—	4	16	no	—
204	C₂³	—	59	8	no	—
205	C₂³	—	118	72	no	—
206	C₂³	—	236	48	no	—
207	C₂³	—	236	192	no	—
208	C₂³	—	236	32	no	—
209	C₂³	#14273772	1	12	yes	—
210	C₂³	C₂⁴×D₅₉	1	4	yes	—
211	C₂³	C₂⁴×D₅₉	1	12	yes	—
212	C₂³	C₂⁴×D₅₉	1	3	yes	—
213	Q₈	—	236	48	no	—
214	D₄	—	4	48	no	—
215	D₄	—	236	96	no	—
216	C₂×C₄	—	2	36	no	—
217	C₂×C₄	—	59	24	no	—
218	C₂×C₄	—	118	72	no	—
219	C₂×C₄	C₂⁴×D₅₉	1	12	yes	—
220	C₂²	—	2	12	no	—
221	C₂²	—	2	12	no	—
222	C₂²	—	2	24	no	—
223	C₂²	—	4	64	no	—
224	C₂²	—	4	72	no	—
225	C₂²	—	4	24	no	—
226	C₂²	—	118	24	no	—
227	C₂²	—	236	64	no	—
228	C₂²	—	236	48	no	—
229	C₂²	#16838145	1	16	yes	direct product factor
230	C₂²	C₂×#14273772	1	18	yes	—
231	C₂²	C₂⁵×D₅₉	1	1	yes	characteristic
232	C₄	—	2	12	no	—
233	C₄	—	118	24	no	—
234	C₂	—	4	12	no	—
235	C₂	—	4	4	no	—
236	C₂	—	236	8	no	—
237	C₂	C₂×#16838145	1	12	yes	direct product factor
238	C₂	C₂²×#14273772	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₉
commutator, cosocle	C₂²×C₅₉	C₂⁶
Frattini	C₂²	C₂⁵×D₅₉
Fitting	C₂²×#120×C₅₉	C₂
socle	C₂⁴×C₅₉	C₂⁴
hypercenter	C₂²×#120	D₅₉
hypocenter	C₅₉	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#120

Complements

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

N	H ≅ G/N	type
1. C₂×#16838145	C₂	direct (N × H)
2. C₂²×#14273714	C₂	semidirect
3. C₂²×#14273713	C₂	semidirect
4. C₂²×#14273696	C₂	semidirect
5. C₂²×#14273695	C₂	semidirect
6. C₂²×#120×C₅₉	C₂	semidirect
7. C₂⁵×D₅₉	C₂²	semidirect
8. C₂³×Dic₁₁₈	C₂²	semidirect
10. #16838145	C₂²	direct (N × H)
11. C₂×#14273714	C₂²	semidirect
12. C₂×#14273713	C₂²	semidirect
13. C₂×#14273696	C₂²	semidirect
14. C₂×#14273695	C₂²	semidirect
15. C₂²×C₂²⋊Dic₅₉	C₂²	semidirect
16. C₂²×C₂²⋊Dic₅₉	C₂²	semidirect
17. C₂²×(C₂×D₅₉)⋊C₄	C₂²	semidirect
18. C₂²×C₄×Dic₅₉	C₂²	semidirect
19. C₂⁶×C₅₉	C₂²	semidirect
21. C₂×#120×C₅₉	C₂²	semidirect
22. C₂²×C₂²⋊C₄×C₅₉	C₂²	semidirect
25. C₂⁴×D₅₉	C₂³	semidirect
27. C₂²×Dic₁₁₈	C₂³	semidirect
31. #14273714	C₂³	semidirect
32. #14273713	C₂³	semidirect
33. #14273696	C₂³	semidirect
34. #14273695	C₂³	semidirect
35. C₂×C₂²⋊Dic₅₉	C₂³	semidirect
36. C₂×C₂²⋊Dic₅₉	C₂³	semidirect
37. C₂×(C₂×D₅₉)⋊C₄	C₂³	semidirect
38. C₂×C₄×Dic₅₉	C₂³	semidirect
42. C₂⁵×C₅₉	C₂³	semidirect
46. #120×C₅₉	C₂³	semidirect
47. C₂×C₂²⋊C₄×C₅₉	C₂³	semidirect
56. C₂⁴×C₅₉	C₂⁴	semidirect
61. C₂³×D₅₉	C₂⁴	semidirect
71. C₂×Dic₁₁₈	C₂⁴	semidirect
72. C₂²⋊C₄×C₅₉	C₂⁴	semidirect
73. C₂²⋊Dic₅₉	C₂⁴	semidirect
74. C₂²⋊Dic₅₉	C₂⁴	semidirect
75. (C₂×D₅₉)⋊C₄	C₂⁴	semidirect
76. C₄×Dic₅₉	C₂⁴	semidirect
107. C₂²×C₅₉	H(3,𝔽₄)	semidirect
120. C₂²×#120	D₅₉	semidirect
123. C₂×C₅₉	C₂×H(3,𝔽₄)	semidirect
127. C₂⁶	C₂×D₅₉	semidirect
136. C₂×#120	C₂×D₅₉	semidirect
140. C₂²×C₂²⋊C₄	C₂×D₅₉	semidirect
142. C₅₉	C₂²×H(3,𝔽₄)	semidirect
149. C₂⁵	C₂²×D₅₉	semidirect
162. #120	C₂²×D₅₉	semidirect
166. C₂×C₂²⋊C₄	C₂²×D₅₉	semidirect
181. C₂⁴	C₂³×D₅₉	semidirect
195. C₂²⋊C₄	C₂³×D₅₉	semidirect
229. C₂²	#16838145	direct (N × H)
237. C₂	C₂×#16838145	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16838145	2
C₂²×#14273714	2
C₂²×#14273713	2
C₂²×#14273696	2
C₂²×#14273695	2
C₂²×#120×C₅₉	2
C₂²×H(3,𝔽₄)	59

Subgroups — C22×#16838145 #29495562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16838145 #29495562