Subgroups — C₂×#21425904 #28741765

Order 15104.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837356	C₂	1	1	yes	maximal, characteristic
2	#21425904	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₅₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×(C₂×D₅₉)⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11745659	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₅₉	C₂²	1	1	yes	characteristic
9	C₂×#14273684	C₄	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
11	#16837356	C₄	1	4	yes	—
12	#16837356	C₂²	1	4	yes	—
13	C₂×#14273638	C₄	1	1	yes	characteristic
14	C₂×#14273637	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	Dic₅₉⋊C₁₆	C₂²	1	8	yes	complemented
17	(C₂×D₅₉)⋊C₁₆	C₂²	1	8	yes	—
18	C₄×C₅₉⋊C₁₆	C₂²	1	4	yes	complemented
19	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#346	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂×#11745659	—	2	4	no	—
24	C₂×#11745659	C₂×C₄	1	4	yes	—
25	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₅₉	—	2	4	no	—
28	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₅₉	C₂³	1	2	yes	—
30	#14273684	C₈	1	4	yes	—
31	#14273684	C₂×C₄	1	4	yes	—
32	C₂×Dic₅₉⋊C₄	C₈	1	2	yes	—
33	C₂×(C₂×D₅₉)⋊C₄	C₈	1	2	yes	—
34	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅₉⋊C₁₆	—	2	4	no	—
36	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
37	#14273638	C₂×C₄	1	4	yes	—
38	#14273637	C₂×C₄	1	4	yes	—
39	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
40	C₈×Dic₅₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₉	—	2	4	no	—
45	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
46	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₉	C₂³	1	2	yes	—
48	C₂³×D₅₉	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
52	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₅₉	—	2	2	no	—
54	C₂×C₄×D₅₉	—	2	2	no	—
55	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
58	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₅₉	#41	1	4	yes	—
61	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
62	C₄²×C₅₉	C₂×C₈	1	2	yes	—
63	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
64	(C₂×D₅₉)⋊C₄	C₂×C₈	1	8	yes	—
65	Dic₅₉⋊C₄	C₂×C₈	1	8	yes	—
66	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
67	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
68	C₂×C₅₉⋊C₈	—	2	2	no	—
69	C₂×C₅₉⋊C₈	—	2	2	no	—
70	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
72	#11745659	—	2	8	no	—
73	C₈×D₅₉	—	2	4	no	—
74	C₈×D₅₉	—	2	4	no	—
75	C₁₆×C₅₉	—	2	4	no	—
76	C₅₉⋊C₁₆	—	2	4	no	—
77	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
78	C₂²×D₅₉	—	2	2	no	—
79	C₂²×D₅₉	—	2	4	no	—
80	C₂²×D₅₉	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₅₉	—	2	4	no	—
82	C₂×C₄×C₅₉	#117	1	2	yes	—
83	C₂×C₄×C₅₉	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₅₉	—	2	2	no	—
88	C₂×Dic₅₉	—	2	4	no	—
89	C₂×Dic₅₉	C₂²×C₈	1	4	yes	—
90	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
91	C₄×D₅₉	—	2	2	no	—
92	C₄×D₅₉	—	2	2	no	—
93	C₄×D₅₉	—	2	4	no	—
94	C₈×C₅₉	#117	1	2	yes	—
95	C₈×C₅₉	C₂×M₁₆	1	4	yes	—
96	C₈×C₅₉	C₂×#41	1	2	yes	—
97	C₅₉⋊C₈	—	2	2	no	—
98	C₅₉⋊C₈	—	2	2	no	—
99	C₂×#2164	—	59	1	no	maximal, Sylow
100	C₂²×C₅₉	#431	1	2	yes	—
101	C₂²×C₅₉	#504	1	4	yes	—
102	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
103	C₂×D₅₉	—	2	4	no	—
104	C₂×D₅₉	—	2	2	no	—
105	C₂×D₅₉	—	4	4	no	—
106	C₄×C₅₉	—	2	4	no	—
107	C₄×C₅₉	#431	1	2	yes	—
108	C₄×C₅₉	C₂×#117	1	1	yes	characteristic
109	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
110	Dic₅₉	—	2	4	no	—
111	Dic₅₉	—	2	2	no	—
112	C₂×#405	—	59	1	no	—
113	#2164	—	59	8	no	—
114	C₂×C₄⋊C₁₆	—	59	2	no	—
115	C₂×C₂²⋊C₁₆	—	59	2	no	—
116	C₂×#346	D₅₉	1	1	yes	characteristic, complemented
117	C₂×C₄×C₁₆	—	59	1	no	—
118	C₂×C₅₉	#2164	1	4	yes	complemented
119	C₂×C₅₉	C₂×#431	1	1	yes	characteristic
120	C₂×C₅₉	C₂×#504	1	2	yes	—
121	D₅₉	—	4	2	no	—
122	C₂²×M₁₆	—	59	1	no	—
123	C₂³×C₈	—	59	1	no	—
124	C₂×#117	—	59	1	no	—
125	C₂²×C₁₆	—	59	2	no	—
126	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
127	#405	—	59	4	no	—
128	#405	—	59	4	no	—
129	C₂×#97	—	59	1	no	—
130	C₂×#97	—	59	1	no	—
131	C₂×C₄×C₈	—	59	1	no	—
132	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
133	C₄⋊C₁₆	—	59	8	no	—
134	C₂²⋊C₁₆	—	59	8	no	—
135	#346	C₂×D₅₉	1	4	yes	complemented
136	C₄×C₁₆	—	59	4	no	—
137	C₅₉	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	—	59	1	no	—
139	C₂×M₁₆	—	59	4	no	—
140	C₂×M₁₆	—	118	4	no	—
141	C₂²×C₈	—	59	1	no	—
142	C₂²×C₈	—	59	1	no	—
143	C₂²×C₈	—	59	2	no	—
144	C₂²×C₈	—	59	2	no	—
145	C₂²×C₈	—	118	4	no	—
146	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
148	#117	—	59	4	no	—
149	#117	—	59	4	no	—
150	C₂×C₄⋊C₄	—	59	2	no	—
151	C₂×C₂²⋊C₄	—	59	2	no	—
152	C₂×C₄²	—	59	1	no	—
153	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	—	59	4	no	—
156	C₂×C₁₆	—	118	4	no	—
157	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
158	#97	—	59	4	no	—
159	#97	—	59	4	no	—
160	C₄×C₈	—	59	2	no	—
161	C₄×C₈	—	59	2	no	—
162	C₄×C₈	C₄×D₅₉	1	2	yes	—
163	C₄×C₈	C₂²×D₅₉	1	2	yes	—
164	C₂⁴	—	59	1	no	—
165	C₂²×C₄	—	59	2	no	—
166	C₂²×C₄	—	59	1	no	—
167	C₂²×C₄	—	59	2	no	—
168	C₂²×C₄	—	59	2	no	—
169	C₂²×C₄	—	118	2	no	—
170	C₂²×C₄	—	118	2	no	—
171	C₂²×C₄	C₈×D₅₉	1	2	yes	—
172	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
173	M₁₆	—	118	8	no	—
174	C₂×C₈	—	59	2	no	—
175	C₂×C₈	—	59	2	no	—
176	C₂×C₈	—	118	2	no	—
177	C₂×C₈	—	118	2	no	—
178	C₂×C₈	—	118	4	no	—
179	C₂×C₈	—	118	4	no	—
180	C₂×C₈	#11745659	1	4	yes	—
181	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
183	C₂×C₈	#11745685	1	4	yes	—
184	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
185	C₄⋊C₄	—	59	8	no	—
186	C₂²⋊C₄	—	59	8	no	—
187	C₄²	—	59	2	no	—
188	C₄²	—	59	2	no	—
189	C₄²	C₈×D₅₉	1	2	yes	—
190	C₄²	C₂×C₄×D₅₉	1	2	yes	—
191	C₁₆	—	2	4	no	—
192	C₁₆	—	118	4	no	—
193	C₂³	—	59	2	no	—
194	C₂³	—	118	2	no	—
195	C₂³	—	118	4	no	—
196	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	59	4	no	—
199	C₂×C₄	—	59	2	no	—
200	C₂×C₄	—	118	2	no	—
201	C₂×C₄	—	118	4	no	—
202	C₂×C₄	—	118	2	no	—
203	C₂×C₄	—	118	4	no	—
204	C₂×C₄	—	118	2	no	—
205	C₂×C₄	#14273684	1	2	yes	—
206	C₂×C₄	C₂×C₈×D₅₉	1	4	yes	—
207	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
208	C₂×C₄	C₂×#11745659	1	2	yes	—
209	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
210	C₈	—	118	2	no	—
211	C₈	—	118	2	no	—
212	C₈	#14273684	1	2	yes	—
213	C₈	C₂×#11745659	1	4	yes	—
214	C₈	C₂×#11745685	1	2	yes	—
215	C₂²	—	118	4	no	—
216	C₂²	—	118	2	no	—
217	C₂²	—	236	4	no	—
218	C₂²	#16837453	1	2	yes	—
219	C₂²	#16837820	1	4	yes	—
220	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	118	4	no	—
223	C₄	—	118	2	no	—
224	C₄	#16837453	1	2	yes	—
225	C₄	C₂×#14273684	1	1	yes	characteristic
226	C₄	C₂²×#11745659	1	1	yes	characteristic
227	C₂	—	236	2	no	—
228	C₂	#21425904	1	4	yes	direct product factor
229	C₂	C₂×#16837453	1	1	yes	characteristic
230	C₂	C₂×#16837820	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle	C₂³×C₅₉	C₂²×C₈
prime radical	C₂³×D₅₉	C₂×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	C₂×#2164
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. #21425904	C₂	direct (N × H)
3. C₂×Dic₅₉⋊C₁₆	C₂	semidirect
5. C₂×C₄×C₅₉⋊C₁₆	C₂	semidirect
6. C₂×C₅₉×#346	C₂	semidirect
16. Dic₅₉⋊C₁₆	C₂²	semidirect
18. C₄×C₅₉⋊C₁₆	C₂²	semidirect
21. C₅₉×#346	C₂²	semidirect
116. C₂×#346	D₅₉	semidirect
118. C₂×C₅₉	#2164	semidirect
135. #346	C₂×D₅₉	semidirect
137. C₅₉	C₂×#2164	semidirect
228. C₂	#21425904	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×#16837356	2
#21425904	2
C₂×Dic₅₉⋊C₁₆	2
C₂×(C₂×D₅₉)⋊C₁₆	2
C₂×C₄×C₅₉⋊C₁₆	2
C₂×C₅₉×#346	2
C₂×#2164	59

Subgroups — C2×#21425904 #28741765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425904 #28741765