Subgroups — C₂×#21425695 #28741326

Order 15104.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837357	C₂	1	1	yes	maximal, characteristic
3	#21425695	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837177	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₅₉	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11745664	C₄	1	1	yes	characteristic
8	C₂×#14273686	C₄	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₁₆	C₄	1	8	yes	complemented
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₅₉⋊C₁₆	C₂²	1	4	yes	—
13	#16837357	C₄	1	4	yes	—
14	#16837357	C₂²	1	4	yes	—
15	C₂×#14273637	C₄	1	2	yes	—
16	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
17	#16837177	C₂²	1	8	yes	complemented
18	C₁₆×Dic₅₉	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₅₉	C₄	1	8	yes	complemented
21	C₂×M₃₂×C₅₉	C₂²	1	4	yes	—
22	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
23	C₂×#11745664	C₈	1	8	yes	—
24	C₂×#11745664	C₂×C₄	1	4	yes	—
25	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
26	#14273686	C₈	1	4	yes	—
27	#14273686	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊Dic₅₉	C₈	1	2	yes	—
29	C₂×C₂²⋊Dic₅₉	C₈	1	2	yes	—
30	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
31	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₅₉⋊C₁₆	C₂×C₄	1	16	yes	complemented
34	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
35	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
36	#14273637	C₂×C₄	1	8	yes	—
37	C₈×Dic₅₉	C₂×C₄	1	4	yes	—
38	C₈×Dic₅₉	C₂³	1	4	yes	—
39	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
45	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
46	M₃₂×C₅₉	C₂×C₄	1	16	yes	complemented
47	C₂×C₁₆×C₅₉	C₂×C₄	1	8	yes	—
48	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
49	C₂⁴×C₅₉	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₄²	1	4	yes	—
51	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
56	C₂²×Dic₅₉	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₅₉	C₄²	1	4	yes	—
58	C₂×C₈×C₅₉	C₄²	1	8	yes	—
59	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
66	C₂²⋊Dic₅₉	C₂×C₈	1	8	yes	—
67	C₄⋊Dic₅₉	C₂×C₈	1	8	yes	—
68	C₄×Dic₅₉	C₂×C₈	1	4	yes	—
69	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
70	#11745664	C₂×C₈	1	16	yes	—
71	C₂×C₅₉⋊C₈	C₂×C₈	1	8	yes	—
72	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
73	C₁₆×C₅₉	C₂²×C₄	1	8	yes	—
74	C₅₉⋊C₁₆	C₂²×C₄	1	4	yes	—
75	C₅₉⋊C₁₆	C₂²×C₄	1	4	yes	—
76	C₂³×C₅₉	—	2	4	no	—
77	C₂³×C₅₉	C₄×C₈	1	4	yes	—
78	C₂³×C₅₉	C₂²×C₈	1	2	yes	—
79	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₄×C₈	1	8	yes	—
81	C₂×C₄×C₅₉	C₄×C₈	1	4	yes	—
82	C₂×C₄×C₅₉	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₅₉	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₅₉	—	2	8	no	—
90	C₂×Dic₅₉	C₂²×C₈	1	8	yes	—
91	C₈×C₅₉	C₂×C₄²	1	2	yes	—
92	C₈×C₅₉	C₂×C₄²	1	4	yes	—
93	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
94	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
95	C₅₉⋊C₈	C₂²×C₈	1	8	yes	—
96	C₂×#2108	—	59	1	no	maximal, Sylow
97	C₂²×C₅₉	—	2	8	no	—
98	C₂²×C₅₉	—	2	4	no	—
99	C₂²×C₅₉	C₂×C₄×C₈	1	4	yes	—
100	C₂²×C₅₉	C₂×C₄×C₈	1	2	yes	—
101	C₂²×C₅₉	#504	1	4	yes	—
102	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₅₉	C₂×C₄×C₈	1	2	yes	—
104	C₄×C₅₉	C₂×C₄×C₈	1	4	yes	—
105	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
106	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
107	Dic₅₉	—	2	8	no	—
108	C₂²×M₃₂	D₅₉	1	1	yes	characteristic
109	C₂³×C₁₆	—	59	1	no	—
110	C₂×#405	—	59	1	no	—
111	#2108	—	59	8	no	—
112	C₂×#346	—	59	2	no	—
113	C₂×C₄×C₁₆	—	59	2	no	—
114	C₂×C₅₉	—	2	4	no	—
115	C₂×C₅₉	#2108	1	4	yes	complemented
116	C₂×C₅₉	C₂²×C₄×C₈	1	1	yes	characteristic
117	C₂×C₅₉	C₂×#504	1	2	yes	—
118	C₂²×M₁₆	—	59	1	no	—
119	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
120	C₂×#117	—	59	1	no	—
121	C₂×M₃₂	Dic₅₉	1	8	yes	complemented
122	C₂×M₃₂	C₂×D₅₉	1	4	yes	—
123	C₂²×C₁₆	—	59	8	no	—
124	C₂²×C₁₆	—	59	4	no	—
125	C₂²×C₁₆	—	59	1	no	—
126	C₂²×C₁₆	—	59	1	no	—
127	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
128	#405	—	59	4	no	—
129	#405	—	59	4	no	—
130	C₂×#97	—	59	2	no	—
131	C₂×C₄×C₈	—	59	2	no	—
132	#346	—	59	8	no	—
133	C₄×C₁₆	—	59	8	no	—
134	C₅₉	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
136	C₂×M₁₆	—	59	8	no	—
137	C₂×M₁₆	—	59	4	no	—
138	C₂²×C₈	—	59	2	no	—
139	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
140	C₂²×C₈	C₄×D₅₉	1	2	yes	—
141	C₂²×C₈	C₄×D₅₉	1	4	yes	—
142	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
143	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
144	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
145	#117	—	59	4	no	—
146	#117	—	59	4	no	—
147	C₂×C₄⋊C₄	—	59	2	no	—
148	C₂×C₂²⋊C₄	—	59	2	no	—
149	C₂×C₄²	—	59	2	no	—
150	M₃₂	C₂×Dic₅₉	1	16	yes	complemented
151	C₂×C₁₆	—	59	4	no	—
152	C₂×C₁₆	—	59	4	no	—
153	C₂×C₁₆	—	59	16	no	—
154	C₂×C₁₆	—	59	2	no	—
155	C₂×C₁₆	—	59	2	no	—
156	C₂×C₁₆	C₂×Dic₅₉	1	8	yes	—
157	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
158	#97	—	59	8	no	—
159	C₄×C₈	—	59	4	no	—
160	C₄×C₈	—	59	4	no	—
161	C₂⁴	C₈×D₅₉	1	1	yes	characteristic
162	C₂²×C₄	—	59	4	no	—
163	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
164	C₂²×C₄	C₈×D₅₉	1	2	yes	—
165	C₂²×C₄	C₈×D₅₉	1	4	yes	—
166	C₂²×C₄	C₄×Dic₅₉	1	4	yes	—
167	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
169	M₁₆	—	59	16	no	—
170	C₂×C₈	—	59	8	no	—
171	C₂×C₈	—	59	4	no	—
172	C₂×C₈	C₄×Dic₅₉	1	4	yes	—
173	C₂×C₈	C₄×Dic₅₉	1	8	yes	—
174	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
176	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
177	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
178	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
179	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
180	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
181	C₄⋊C₄	—	59	8	no	—
182	C₂²⋊C₄	—	59	8	no	—
183	C₄²	—	59	4	no	—
184	C₄²	—	59	4	no	—
185	C₁₆	—	59	4	no	—
186	C₁₆	—	59	4	no	—
187	C₁₆	C₂²×Dic₅₉	1	8	yes	—
188	C₂³	—	2	4	no	—
189	C₂³	C₈×Dic₅₉	1	4	yes	—
190	C₂³	C₂×C₈×D₅₉	1	2	yes	—
191	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
192	C₂×C₄	—	59	8	no	—
193	C₂×C₄	—	118	8	no	—
194	C₂×C₄	C₈×Dic₅₉	1	8	yes	—
195	C₂×C₄	C₈×Dic₅₉	1	4	yes	—
196	C₂×C₄	C₂×C₄×Dic₅₉	1	4	yes	—
197	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
198	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
199	C₂×C₄	C₂×C₈×D₅₉	1	4	yes	—
200	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
201	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
202	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
203	C₈	—	59	8	no	—
204	C₈	C₂×C₄×Dic₅₉	1	2	yes	—
205	C₈	C₂×C₄×Dic₅₉	1	4	yes	—
206	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
207	C₈	C₂³×Dic₅₉	1	1	yes	characteristic
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	C₂×C₈×Dic₅₉	1	4	yes	—
211	C₂²	C₂×C₈×Dic₅₉	1	2	yes	—
212	C₂²	#16837820	1	4	yes	—
213	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
214	C₄	—	118	8	no	—
215	C₄	C₂×C₈×Dic₅₉	1	2	yes	—
216	C₄	C₂×C₈×Dic₅₉	1	4	yes	—
217	C₄	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
218	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#21425695	1	4	yes	direct product factor
221	C₂	C₂²×C₈×Dic₅₉	1	1	yes	characteristic
222	C₂	C₂×#16837820	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₉
commutator	C₂×C₅₉	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂²×M₃₂×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₈
prime radical	C₂⁴×C₅₉	C₂×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂²×M₃₂	D₅₉
hypocenter	C₅₉	C₂×#2108
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #21425695	C₂	direct (N × H)
4. C₂×#16837177	C₂	semidirect
5. C₂×C₁₆×Dic₅₉	C₂	semidirect
9. C₂²×C₅₉⋊C₁₆	C₄	semidirect
17. #16837177	C₂²	semidirect
18. C₁₆×Dic₅₉	C₂²	semidirect
20. C₂×M₃₂×C₅₉	C₄	semidirect
33. C₂×C₅₉⋊C₁₆	C₂×C₄	semidirect
46. M₃₂×C₅₉	C₂×C₄	semidirect
115. C₂×C₅₉	#2108	semidirect
121. C₂×M₃₂	Dic₅₉	semidirect
134. C₅₉	C₂×#2108	semidirect
150. M₃₂	C₂×Dic₅₉	semidirect
220. C₂	#21425695	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×C₅₉⋊C₁₆	2
C₂×#16837357	2
#21425695	2
C₂×#16837177	2
C₂×C₁₆×Dic₅₉	2
C₂²×M₃₂×C₅₉	2
C₂×#2108	59

Subgroups — C2×#21425695 #28741326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425695 #28741326