Subgroups — C₂²×#127×D₅₉ #29495374

Order 15104.

24030 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×D₅₉	C₂	1	6	yes	maximal, complemented
2	C₂²×#14273724	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14273723	C₂	1	6	yes	maximal, complemented
4	C₂×#127×D₅₉	C₂	1	48	yes	maximal, direct product factor
5	C₂²×C₄²×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#127×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₉	C₂²	1	4	yes	—
8	C₂³×D₂₃₆	C₂²	1	6	yes	complemented
9	C₂³×C₂²⋊D₅₉	C₂²	1	12	yes	complemented
10	C₂²×D₄×D₅₉	—	2	96	no	—
11	C₂²×D₄×D₅₉	C₂²	1	144	yes	complemented
12	C₂³×C₄×D₅₉	C₂²	1	3	yes	—
13	C₂×#14273724	C₂²	1	24	yes	complemented
14	C₂×#14273723	C₂²	1	144	yes	complemented
15	#127×D₅₉	C₂²	1	256	yes	direct product factor
16	C₂²×C₄×Dic₅₉	C₂²	1	3	yes	complemented
17	C₂×C₄²×D₅₉	C₂²	1	24	yes	complemented
18	C₂³×D₄×C₅₉	C₂²	1	6	yes	complemented
19	C₂×#127×C₅₉	C₂²	1	24	yes	complemented
20	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic, complemented
21	C₂⁴×D₅₉	—	2	96	no	—
22	C₂⁴×D₅₉	C₂³	1	48	yes	—
23	C₂⁴×D₅₉	C₂³	1	4	yes	—
24	C₂⁴×D₅₉	C₂³	1	1	yes	characteristic
25	C₂²×D₂₃₆	—	2	48	no	—
26	C₂²×D₂₃₆	C₂³	1	72	yes	complemented
27	C₂²×C₂²⋊D₅₉	—	2	96	no	—
28	C₂²×C₂²⋊D₅₉	C₂³	1	144	yes	complemented
29	C₂×D₄×D₅₉	—	2	1152	no	—
30	C₂×D₄×D₅₉	C₂³	1	384	yes	complemented
31	C₂³×Dic₅₉	C₂³	1	3	yes	—
32	C₂²×C₄×D₅₉	D₄	1	48	yes	complemented
33	C₂²×C₄×D₅₉	C₂³	1	36	yes	—
34	#14273724	C₂³	1	64	yes	complemented
35	#14273723	C₂³	1	384	yes	complemented
36	C₂×C₄×Dic₅₉	C₂³	1	36	yes	complemented
37	C₄²×D₅₉	C₂³	1	64	yes	complemented
38	C₂⁵×C₅₉	C₂³	1	4	yes	—
39	C₂²×D₄×C₅₉	—	2	48	no	—
40	C₂²×D₄×C₅₉	C₂³	1	72	yes	complemented
41	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
42	#127×C₅₉	C₂³	1	64	yes	complemented
43	C₂×C₄²×C₅₉	C₂³	1	12	yes	complemented
44	C₂⁴×C₅₉	—	2	48	no	—
45	C₂⁴×C₅₉	C₂⁴	1	24	yes	—
46	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
47	C₂³×D₅₉	—	2	48	no	—
48	C₂³×D₅₉	—	2	576	no	—
49	C₂³×D₅₉	—	4	256	no	—
50	C₂³×D₅₉	C₂×D₄	1	24	yes	—
51	C₂³×D₅₉	C₂⁴	1	64	yes	—
52	C₂³×D₅₉	C₂⁴	1	24	yes	—
53	C₂³×D₅₉	C₂⁴	1	6	yes	—
54	C₂×D₄×C₅₉	—	2	288	no	—
55	C₂×D₄×C₅₉	C₂⁴	1	96	yes	complemented
56	C₂²×C₄×C₅₉	C₂×D₄	1	24	yes	complemented
57	C₂²×C₄×C₅₉	C₂⁴	1	18	yes	—
58	C₂×C₂²⋊D₅₉	—	2	576	no	—
59	C₂×C₂²⋊D₅₉	C₂⁴	1	192	yes	complemented
60	C₂²×Dic₅₉	C₂×D₄	1	24	yes	complemented
61	C₂²×Dic₅₉	C₂⁴	1	18	yes	—
62	D₄×D₅₉	—	2	1536	no	—
63	C₂×D₂₃₆	—	2	288	no	—
64	C₂×D₂₃₆	C₂⁴	1	96	yes	complemented
65	C₂×C₄×D₅₉	C₂×D₄	1	288	yes	complemented
66	C₂×C₄×D₅₉	C₂⁴	1	48	yes	—
67	C₄²×C₅₉	C₂⁴	1	16	yes	complemented
68	C₄×Dic₅₉	C₂⁴	1	48	yes	complemented
69	C₂³×C₅₉	—	2	144	no	—
70	C₂³×C₅₉	—	4	64	no	—
71	C₂³×C₅₉	C₂²×D₄	1	12	yes	—
72	C₂³×C₅₉	C₂⁵	1	16	yes	—
73	C₂³×C₅₉	C₂⁵	1	3	yes	—
74	C₂²×D₅₉	—	2	144	no	—
75	C₂²×D₅₉	—	2	384	no	—
76	C₂²×D₅₉	—	4	64	no	—
77	C₂²×D₅₉	—	4	768	no	—
78	C₂²×D₅₉	#127	1	64	yes	direct product factor
79	C₂²×D₅₉	C₂²×D₄	1	72	yes	—
80	C₂²×D₅₉	C₂⁵	1	16	yes	—
81	C₂²×D₅₉	C₂⁵	1	4	yes	—
82	D₄×C₅₉	—	2	192	no	—
83	C₂×C₄×C₅₉	C₂²×D₄	1	72	yes	complemented
84	C₂×C₄×C₅₉	C₂⁵	1	12	yes	—
85	C₂²⋊D₅₉	—	2	384	no	—
86	C₂×Dic₅₉	C₂²×D₄	1	72	yes	complemented
87	C₂×Dic₅₉	C₂⁵	1	12	yes	—
88	D₂₃₆	—	2	192	no	—
89	C₄×D₅₉	C₂²×D₄	1	192	yes	complemented
90	C₂³×#127	—	59	1	no	maximal, Sylow
91	C₂²×C₅₉	—	2	48	no	—
92	C₂²×C₅₉	—	4	96	no	—
93	C₂²×C₅₉	C₂×#127	1	16	yes	complemented
94	C₂²×C₅₉	C₂³×D₄	1	18	yes	—
95	C₂²×C₅₉	C₂⁶	1	1	yes	characteristic
96	C₂×D₅₉	—	2	48	no	—
97	C₂×D₅₉	—	4	96	no	—
98	C₂×D₅₉	—	4	256	no	—
99	C₂×D₅₉	C₂×#127	1	96	yes	direct product factor
100	C₂×D₅₉	C₂³×D₄	1	24	yes	—
101	C₄×C₅₉	C₂³×D₄	1	24	yes	complemented
102	Dic₅₉	C₂³×D₄	1	24	yes	complemented
103	C₂⁴×D₄	—	59	6	no	—
104	C₂²×#127	—	59	1	no	—
105	C₂²×#127	—	59	6	no	—
106	C₂²×#127	—	59	48	no	—
107	C₂²×#127	D₅₉	1	1	yes	characteristic, direct product factor
108	C₂³×C₄²	—	59	1	no	—
109	C₂×C₅₉	—	4	16	no	—
110	C₂×C₅₉	C₂²×#127	1	12	yes	complemented
111	C₂×C₅₉	C₂⁴×D₄	1	3	yes	—
112	D₅₉	—	4	16	no	—
113	D₅₉	C₂²×#127	1	16	yes	direct product factor
114	C₂⁶	—	59	4	no	—
115	C₂³×D₄	—	59	12	no	—
116	C₂³×D₄	—	59	144	no	—
117	C₂³×D₄	—	59	6	no	—
118	C₂³×D₄	—	118	96	no	—
119	C₂³×D₄	C₂×D₅₉	1	6	yes	complemented
120	C₂⁴×C₄	—	59	3	no	—
121	C₂×#127	—	59	144	no	—
122	C₂×#127	—	59	256	no	—
123	C₂×#127	—	59	24	no	—
124	C₂×#127	C₂×D₅₉	1	24	yes	direct product factor
125	C₂²×C₄²	—	59	3	no	—
126	C₂²×C₄²	—	59	24	no	—
127	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic, complemented
128	C₅₉	C₂³×#127	1	1	yes	characteristic, complemented, Sylow
129	C₂⁵	—	59	1	no	—
130	C₂⁵	—	59	48	no	—
131	C₂⁵	—	59	4	no	—
132	C₂⁵	—	118	96	no	—
133	C₂⁵	C₂²×D₅₉	1	4	yes	—
134	C₂²×D₄	—	2	48	no	—
135	C₂²×D₄	—	59	144	no	—
136	C₂²×D₄	—	59	384	no	—
137	C₂²×D₄	—	59	72	no	—
138	C₂²×D₄	—	118	1152	no	—
139	C₂²×D₄	—	118	96	no	—
140	C₂²×D₄	—	118	48	no	—
141	C₂²×D₄	C₂²×D₅₉	1	72	yes	complemented
142	C₂³×C₄	—	59	3	no	—
143	C₂³×C₄	—	59	48	no	—
144	C₂³×C₄	—	59	36	no	—
145	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
146	#127	—	59	384	no	—
147	#127	—	59	64	no	—
148	#127	C₂²×D₅₉	1	64	yes	direct product factor
149	C₂×C₄²	—	59	36	no	—
150	C₂×C₄²	—	59	64	no	—
151	C₂×C₄²	C₂²×D₅₉	1	12	yes	complemented
152	C₂⁴	—	2	48	no	—
153	C₂⁴	—	59	24	no	—
154	C₂⁴	—	59	6	no	—
155	C₂⁴	—	59	64	no	—
156	C₂⁴	—	59	24	no	—
157	C₂⁴	—	118	576	no	—
158	C₂⁴	—	118	48	no	—
159	C₂⁴	—	236	256	no	—
160	C₂⁴	C₂³×D₅₉	1	24	yes	—
161	C₂⁴	C₂³×D₅₉	1	1	yes	characteristic
162	C₂×D₄	—	2	288	no	—
163	C₂×D₄	—	59	192	no	—
164	C₂×D₄	—	59	96	no	—
165	C₂×D₄	—	118	1536	no	—
166	C₂×D₄	—	118	576	no	—
167	C₂×D₄	—	118	288	no	—
168	C₂×D₄	C₂³×D₅₉	1	96	yes	complemented
169	C₂²×C₄	—	59	24	no	—
170	C₂²×C₄	—	59	18	no	—
171	C₂²×C₄	—	59	288	no	—
172	C₂²×C₄	—	59	48	no	—
173	C₂²×C₄	D₄×D₅₉	1	24	yes	complemented
174	C₂²×C₄	C₂³×D₅₉	1	18	yes	—
175	C₄²	—	59	48	no	—
176	C₄²	C₂³×D₅₉	1	16	yes	complemented
177	C₂³	—	2	144	no	—
178	C₂³	—	4	64	no	—
179	C₂³	—	59	64	no	—
180	C₂³	—	59	72	no	—
181	C₂³	—	59	4	no	—
182	C₂³	—	59	16	no	—
183	C₂³	—	118	384	no	—
184	C₂³	—	118	144	no	—
185	C₂³	—	236	768	no	—
186	C₂³	—	236	64	no	—
187	C₂³	C₂×D₄×D₅₉	1	12	yes	—
188	C₂³	C₂⁴×D₅₉	1	16	yes	—
189	C₂³	C₂⁴×D₅₉	1	3	yes	—
190	D₄	—	2	192	no	—
191	D₄	—	118	384	no	—
192	D₄	—	118	192	no	—
193	C₂×C₄	—	59	72	no	—
194	C₂×C₄	—	59	12	no	—
195	C₂×C₄	—	59	192	no	—
196	C₂×C₄	C₂×D₄×D₅₉	1	72	yes	complemented
197	C₂×C₄	C₂⁴×D₅₉	1	12	yes	—
198	C₂²	—	2	48	no	—
199	C₂²	—	4	96	no	—
200	C₂²	—	59	24	no	—
201	C₂²	—	59	96	no	—
202	C₂²	—	118	48	no	—
203	C₂²	—	236	256	no	—
204	C₂²	—	236	96	no	—
205	C₂²	#127×D₅₉	1	16	yes	direct product factor
206	C₂²	C₂²×D₄×D₅₉	1	18	yes	—
207	C₂²	C₂⁵×D₅₉	1	1	yes	characteristic
208	C₄	—	59	24	no	—
209	C₄	C₂²×D₄×D₅₉	1	24	yes	complemented
210	C₂	—	4	16	no	—
211	C₂	—	59	16	no	—
212	C₂	—	236	16	no	—
213	C₂	C₂×#127×D₅₉	1	12	yes	direct product factor
214	C₂	C₂³×D₄×D₅₉	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₉
commutator, cosocle	C₂²×C₅₉	C₂⁶
Frattini	C₂²	C₂⁵×D₅₉
Fitting	C₂²×#127×C₅₉	C₂
socle	C₂⁴×C₅₉	C₂⁴
hypercenter	C₂²×#127	D₅₉
hypocenter	C₅₉	C₂³×#127
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₅₉	C₂	semidirect
2. C₂²×#14273724	C₂	semidirect
3. C₂²×#14273723	C₂	semidirect
4. C₂×#127×D₅₉	C₂	direct (N × H)
5. C₂²×C₄²×D₅₉	C₂	semidirect
6. C₂²×#127×C₅₉	C₂	semidirect
8. C₂³×D₂₃₆	C₂²	semidirect
9. C₂³×C₂²⋊D₅₉	C₂²	semidirect
11. C₂²×D₄×D₅₉	C₂²	semidirect
13. C₂×#14273724	C₂²	semidirect
14. C₂×#14273723	C₂²	semidirect
15. #127×D₅₉	C₂²	direct (N × H)
16. C₂²×C₄×Dic₅₉	C₂²	semidirect
17. C₂×C₄²×D₅₉	C₂²	semidirect
18. C₂³×D₄×C₅₉	C₂²	semidirect
19. C₂×#127×C₅₉	C₂²	semidirect
20. C₂²×C₄²×C₅₉	C₂²	semidirect
26. C₂²×D₂₃₆	C₂³	semidirect
28. C₂²×C₂²⋊D₅₉	C₂³	semidirect
30. C₂×D₄×D₅₉	C₂³	semidirect
32. C₂²×C₄×D₅₉	D₄	semidirect
34. #14273724	C₂³	semidirect
35. #14273723	C₂³	semidirect
36. C₂×C₄×Dic₅₉	C₂³	semidirect
37. C₄²×D₅₉	C₂³	semidirect
40. C₂²×D₄×C₅₉	C₂³	semidirect
42. #127×C₅₉	C₂³	semidirect
43. C₂×C₄²×C₅₉	C₂³	semidirect
55. C₂×D₄×C₅₉	C₂⁴	semidirect
56. C₂²×C₄×C₅₉	C₂×D₄	semidirect
59. C₂×C₂²⋊D₅₉	C₂⁴	semidirect
60. C₂²×Dic₅₉	C₂×D₄	semidirect
64. C₂×D₂₃₆	C₂⁴	semidirect
65. C₂×C₄×D₅₉	C₂×D₄	semidirect
67. C₄²×C₅₉	C₂⁴	semidirect
68. C₄×Dic₅₉	C₂⁴	semidirect
78. C₂²×D₅₉	#127	direct (N × H)
83. C₂×C₄×C₅₉	C₂²×D₄	semidirect
86. C₂×Dic₅₉	C₂²×D₄	semidirect
89. C₄×D₅₉	C₂²×D₄	semidirect
93. C₂²×C₅₉	C₂×#127	semidirect
99. C₂×D₅₉	C₂×#127	direct (N × H)
101. C₄×C₅₉	C₂³×D₄	semidirect
102. Dic₅₉	C₂³×D₄	semidirect
107. C₂²×#127	D₅₉	direct (N × H)
110. C₂×C₅₉	C₂²×#127	semidirect
113. D₅₉	C₂²×#127	direct (N × H)
119. C₂³×D₄	C₂×D₅₉	semidirect
124. C₂×#127	C₂×D₅₉	direct (N × H)
127. C₂²×C₄²	C₂×D₅₉	semidirect
128. C₅₉	C₂³×#127	semidirect
141. C₂²×D₄	C₂²×D₅₉	semidirect
148. #127	C₂²×D₅₉	direct (N × H)
151. C₂×C₄²	C₂²×D₅₉	semidirect
168. C₂×D₄	C₂³×D₅₉	semidirect
173. C₂²×C₄	D₄×D₅₉	semidirect
176. C₄²	C₂³×D₅₉	semidirect
196. C₂×C₄	C₂×D₄×D₅₉	semidirect
205. C₂²	#127×D₅₉	direct (N × H)
209. C₄	C₂²×D₄×D₅₉	semidirect
213. C₂	C₂×#127×D₅₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×D₅₉	2
C₂²×#14273724	2
C₂²×#14273723	2
C₂×#127×D₅₉	2
C₂²×C₄²×D₅₉	2
C₂²×#127×C₅₉	2
C₂³×#127	59

Subgroups — C22×#127×D59 #29495374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#127×D₅₉ #29495374