Subgroups — C₂×#21425852 #28741668

Order 15104.

798 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273667	C₂	1	2	yes	maximal
2	#21425852	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	—	2	8	no	—
6	C₂×#14273667	C₂²	1	8	yes	—
7	C₂²×C₅₉⋊C₁₆	C₂²	1	4	yes	—
8	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	16	yes	complemented
9	C₂³×C₈×C₅₉	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₅₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₉	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₅₉	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	4	yes	—
16	#14273667	—	2	16	no	—
17	C₂×C₅₉⋊C₁₆	—	2	8	no	—
18	C₂×C₅₉⋊C₁₆	C₂³	1	8	yes	—
19	C₂³×C₄×C₅₉	C₈	1	2	yes	—
20	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₉	—	2	4	no	—
22	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₅₉	D₄	1	4	yes	—
27	C₂²×C₈×C₅₉	Q₈	1	4	yes	—
28	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
29	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	C₈	1	1	yes	characteristic
31	C₂×C₄²×C₅₉	C₈	1	4	yes	—
32	C₂×C₄²×C₅₉	C₈	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₅₉	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₅₉	C₂³	1	4	yes	—
38	C₂⁴×C₅₉	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	—	2	8	no	—
40	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₅₉	C₂×C₈	1	8	yes	—
43	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅₉	M₁₆	1	4	yes	—
47	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	—	2	8	no	—
50	C₂×C₈×C₅₉	—	2	4	no	—
51	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
52	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
61	C₄²×C₅₉	C₂×C₈	1	4	yes	—
62	C₄²×C₅₉	C₂×C₈	1	4	yes	—
63	C₄²×C₅₉	C₂×C₈	1	4	yes	—
64	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
65	C₅₉⋊C₁₆	—	2	8	no	—
66	C₂³×C₅₉	—	2	4	no	—
67	C₂³×C₅₉	C₄⋊C₈	1	4	yes	—
68	C₂³×C₅₉	C₂²×C₈	1	2	yes	—
69	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	4	no	—
71	C₂×C₄×C₅₉	—	2	16	no	—
72	C₂×C₄×C₅₉	C₄⋊C₈	1	8	yes	—
73	C₂×C₄×C₅₉	C₄⋊C₈	1	4	yes	—
74	C₂×C₄×C₅₉	M₃₂	1	8	yes	complemented
75	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₅₉	C₂²×C₈	1	4	yes	—
78	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂²×C₈	1	4	yes	—
81	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
82	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
84	C₈×C₅₉	—	2	4	no	—
85	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
86	C₈×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
87	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
89	C₂×#2150	—	59	1	no	maximal, Sylow
90	C₂²×C₅₉	—	2	4	no	—
91	C₂²×C₅₉	—	2	8	no	—
92	C₂²×C₅₉	C₂×C₄⋊C₈	1	4	yes	—
93	C₂²×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
94	C₂²×C₅₉	C₂×M₃₂	1	4	yes	—
95	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₅₉	—	2	4	no	—
97	C₄×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
98	C₄×C₅₉	C₂×C₄⋊C₈	1	4	yes	—
99	C₄×C₅₉	C₂×M₃₂	1	8	yes	complemented
100	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
102	C₂²×M₃₂	—	59	2	no	—
103	C₂²×C₄×C₈	D₅₉	1	1	yes	characteristic
104	#2150	—	59	8	no	—
105	C₂×C₄⋊C₁₆	—	59	4	no	—
106	C₂×C₅₉	—	2	4	no	—
107	C₂×C₅₉	#2150	1	4	yes	complemented
108	C₂×C₅₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
109	C₂×C₅₉	C₂²×M₃₂	1	2	yes	—
110	C₂³×C₈	Dic₅₉	1	1	yes	characteristic
111	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
112	C₂²×C₄²	Dic₅₉	1	1	yes	characteristic
113	C₂×M₃₂	—	59	8	no	—
114	C₂×M₃₂	—	118	8	no	—
115	C₂²×C₁₆	—	59	4	no	—
116	C₂×C₄×C₈	Dic₅₉	1	2	yes	—
117	C₂×C₄×C₈	Dic₅₉	1	4	yes	—
118	C₂×C₄×C₈	C₂×D₅₉	1	4	yes	—
119	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
120	C₄⋊C₁₆	—	59	16	no	—
121	C₅₉	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	C₅₉⋊C₈	1	2	yes	—
123	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
124	C₂²×C₈	—	2	4	no	—
125	C₂²×C₈	Dic₁₁₈	1	4	yes	—
126	C₂²×C₈	D₂₃₆	1	4	yes	—
127	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
128	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
129	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
131	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
132	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
133	C₂×C₄²	C₅₉⋊C₈	1	1	yes	characteristic
134	C₂×C₄²	C₅₉⋊C₈	1	4	yes	—
135	C₂×C₄²	C₅₉⋊C₈	1	1	yes	characteristic
136	C₂×C₄²	C₂×Dic₅₉	1	4	yes	—
137	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
138	M₃₂	—	118	16	no	—
139	C₂×C₁₆	—	59	8	no	—
140	C₂×C₁₆	—	118	8	no	—
141	C₄×C₈	C₂×Dic₅₉	1	8	yes	—
142	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
143	C₄×C₈	C₂²×D₅₉	1	4	yes	—
144	C₂⁴	C₂×C₅₉⋊C₈	1	1	yes	characteristic
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
147	C₂²×C₄	C₂×C₅₉⋊C₈	1	8	yes	—
148	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
149	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
151	C₂²×C₄	#11745664	1	4	yes	—
152	C₂²×C₄	C₄⋊Dic₅₉	1	4	yes	—
153	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
154	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
158	C₂×C₈	C₄⋊Dic₅₉	1	8	yes	—
159	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
160	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
161	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
162	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
163	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
164	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
165	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
166	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
167	C₄²	C₂×C₅₉⋊C₈	1	4	yes	—
168	C₄²	C₂×C₅₉⋊C₈	1	4	yes	—
169	C₄²	C₂×C₅₉⋊C₈	1	4	yes	—
170	C₄²	C₂²×Dic₅₉	1	4	yes	—
171	C₁₆	—	118	8	no	—
172	C₂³	—	2	4	no	—
173	C₂³	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
174	C₂³	C₂²×C₅₉⋊C₈	1	2	yes	—
175	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	8	yes	—
179	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
180	C₂×C₄	#14273667	1	8	yes	complemented
181	C₂×C₄	C₂×C₄⋊Dic₅₉	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
183	C₂×C₄	C₂²×C₅₉⋊C₈	1	4	yes	—
184	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
185	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
186	C₂×C₄	C₂²×C₅₉⋊C₈	1	4	yes	—
187	C₂×C₄	C₂×#11745664	1	4	yes	—
188	C₂×C₄	C₂×#11745664	1	2	yes	—
189	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	C₂×C₄⋊Dic₅₉	1	4	yes	—
192	C₈	C₂×C₄⋊Dic₅₉	1	2	yes	—
193	C₈	C₂²×D₂₃₆	1	1	yes	characteristic
194	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
195	C₂²	—	2	4	no	—
196	C₂²	—	2	8	no	—
197	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	4	yes	—
198	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	2	yes	—
199	C₂²	C₂×#14273667	1	4	yes	—
200	C₂²	C₂³×C₅₉⋊C₈	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	2	yes	—
203	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	4	yes	—
204	C₄	C₂×#14273667	1	8	yes	complemented
205	C₄	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
206	C₄	C₂²×#11745664	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#21425852	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
210	C₂	C₂²×#14273667	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂²×C₄×C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₈
prime radical	C₂⁴×C₅₉	C₂×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₉
hypocenter	C₅₉	C₂×#2150
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #21425852	C₂	direct (N × H)
3. C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
8. C₄⋊(C₅₉⋊C₁₆)	C₂²	semidirect
74. C₂×C₄×C₅₉	M₃₂	semidirect
99. C₄×C₅₉	C₂×M₃₂	semidirect
107. C₂×C₅₉	#2150	semidirect
121. C₅₉	C₂×#2150	semidirect
180. C₂×C₄	#14273667	semidirect
204. C₄	C₂×#14273667	semidirect
208. C₂	#21425852	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14273667	2
#21425852	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂²×C₄×C₈×C₅₉	2
C₂×#2150	59

Subgroups — C2×#21425852 #28741668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425852 #28741668