Subgroups — C₂×#21426002 #28742174

Order 15104.

542 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#21426002	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₁₈	C₄	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₁₈	C₄	1	4	yes	—
13	C₈×Dic₁₁₈	C₂²	1	4	yes	—
14	C₂×C₄⋊(C₅₉⋊C₈)	C₄	1	1	yes	characteristic
15	C₂×Dic₅₉⋊C₈	C₄	1	1	yes	characteristic
16	C₂×Dic₅₉⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
19	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
20	Dic₅₉⋊C₁₆	C₂²	1	4	yes	—
21	Dic₅₉⋊C₁₆	C₂²	1	4	yes	—
22	#16837177	C₂²	1	4	yes	—
23	C₁₆×Dic₅₉	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#346	C₂²	1	4	yes	—
28	C₂²×Dic₁₁₈	C₈	1	1	yes	characteristic
29	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₁₈	C₈	1	4	yes	—
32	C₄×Dic₁₁₈	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₅₉	C₈	1	1	yes	characteristic
34	C₂×Dic₅₉⋊C₄	C₈	1	1	yes	characteristic
35	C₂×Dic₅₉⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅₉⋊C₁₆	—	2	2	no	—
39	C₂×C₅₉⋊C₁₆	—	2	2	no	—
40	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
41	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
42	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
43	Dic₅₉⋊C₈	C₂×C₄	1	4	yes	—
44	Dic₅₉⋊C₈	C₂×C₄	1	4	yes	—
45	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
46	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
47	C₈×Dic₅₉	C₂³	1	2	yes	—
48	C₈×Dic₅₉	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₅₉	—	2	2	no	—
53	C₂×C₁₆×C₅₉	Q₈	1	4	yes	complemented
54	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
55	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
56	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₅₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
65	C₂×Dic₁₁₈	—	2	4	no	—
66	C₂×Dic₁₁₈	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₅₉	#41	1	2	yes	—
72	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
73	C₄²×C₅₉	C₂×C₈	1	2	yes	—
74	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
75	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
76	Dic₅₉⋊C₄	C₂×C₈	1	4	yes	—
77	Dic₅₉⋊C₄	C₂×C₈	1	4	yes	—
78	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
79	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
80	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
81	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
82	C₂×C₅₉⋊C₈	—	2	2	no	—
83	C₂×C₅₉⋊C₈	—	2	2	no	—
84	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
86	C₁₆×C₅₉	—	2	2	no	—
87	C₁₆×C₅₉	C₂×Q₈	1	4	yes	complemented
88	C₅₉⋊C₁₆	—	2	2	no	—
89	C₅₉⋊C₁₆	—	2	2	no	—
90	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₉	—	2	2	no	—
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#131	1	2	yes	—
98	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
99	C₂×Dic₅₉	—	2	2	no	—
100	C₂×Dic₅₉	—	2	2	no	—
101	C₂×Dic₅₉	—	2	2	no	—
102	C₂×Dic₅₉	M₃₂	1	4	yes	—
103	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
104	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
105	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
106	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
107	Dic₁₁₈	—	2	8	no	—
108	C₈×C₅₉	—	2	2	no	—
109	C₈×C₅₉	C₄×Q₈	1	2	yes	—
110	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
111	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
112	C₅₉⋊C₈	—	2	2	no	—
113	C₅₉⋊C₈	—	2	2	no	—
114	C₂×#2183	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	C₈×Q₈	1	2	yes	—
116	C₂²×C₅₉	C₂×M₃₂	1	2	yes	—
117	C₂²×C₅₉	#504	1	2	yes	—
118	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
119	C₄×C₅₉	—	2	2	no	—
120	C₄×C₅₉	—	2	2	no	—
121	C₄×C₅₉	C₈×Q₈	1	2	yes	—
122	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
124	Dic₅₉	—	2	2	no	—
125	Dic₅₉	—	2	2	no	—
126	Dic₅₉	—	2	2	no	—
127	Dic₅₉	C₂×M₃₂	1	4	yes	—
128	C₂×C₈×Q₈	—	59	1	no	—
129	#2183	—	59	8	no	—
130	C₂×C₄⋊C₁₆	—	59	1	no	—
131	C₂×C₄⋊C₁₆	—	59	1	no	—
132	C₂×C₄⋊C₁₆	—	59	1	no	—
133	C₂×#346	—	59	1	no	—
134	C₂×#346	D₅₉	1	1	yes	characteristic
135	C₂×C₄×C₁₆	—	59	1	no	—
136	C₂×C₅₉	#2183	1	4	yes	complemented
137	C₂×C₅₉	C₂×C₈×Q₈	1	1	yes	characteristic
138	C₂×C₅₉	C₂²×M₃₂	1	1	yes	characteristic
139	C₂×C₅₉	C₂×#504	1	1	yes	characteristic
140	C₂×C₄×Q₈	—	59	1	no	—
141	C₂²×C₁₆	—	59	1	no	—
142	C₂²×C₁₆	—	59	1	no	—
143	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
144	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
145	C₈×Q₈	—	59	4	no	—
146	C₈×Q₈	—	59	4	no	—
147	C₂×C₄⋊C₈	—	59	1	no	—
148	C₂×C₄⋊C₈	—	59	1	no	—
149	C₂×C₄⋊C₈	—	59	1	no	—
150	C₂×C₄×C₈	—	59	1	no	—
151	C₂×C₄×C₈	—	59	1	no	—
152	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
153	C₄⋊C₁₆	—	59	4	no	—
154	C₄⋊C₁₆	—	59	4	no	—
155	C₄⋊C₁₆	—	59	4	no	—
156	#346	—	59	4	no	—
157	#346	C₂×D₅₉	1	4	yes	—
158	C₄×C₁₆	—	59	4	no	—
159	C₅₉	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	59	1	no	—
161	C₂²×C₈	—	59	1	no	—
162	C₂²×C₈	—	59	1	no	—
163	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
165	C₄×Q₈	—	59	4	no	—
166	C₄×Q₈	—	59	4	no	—
167	C₂×C₄⋊C₄	—	59	1	no	—
168	C₂×C₄⋊C₄	—	59	1	no	—
169	C₂×C₄⋊C₄	—	59	1	no	—
170	C₂×C₄²	—	59	1	no	—
171	C₂×C₄²	—	59	1	no	—
172	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	59	2	no	—
175	C₂×C₁₆	—	59	2	no	—
176	C₂×C₁₆	—	118	2	no	—
177	C₂×C₁₆	—	118	2	no	—
178	C₂×C₁₆	Dic₁₁₈	1	4	yes	complemented
179	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
180	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
181	C₄⋊C₈	—	59	4	no	—
182	C₄⋊C₈	—	59	4	no	—
183	C₄⋊C₈	—	59	4	no	—
184	C₄×C₈	—	59	2	no	—
185	C₄×C₈	—	59	2	no	—
186	C₄×C₈	—	59	2	no	—
187	C₄×C₈	—	59	2	no	—
188	C₄×C₈	C₄×D₅₉	1	2	yes	—
189	C₄×C₈	C₂²×D₅₉	1	2	yes	—
190	C₂×Q₈	—	59	4	no	—
191	C₂×Q₈	—	118	4	no	—
192	C₂²×C₄	—	59	1	no	—
193	C₂²×C₄	—	59	1	no	—
194	C₂²×C₄	—	59	1	no	—
195	C₂²×C₄	—	59	1	no	—
196	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
197	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	59	2	no	—
201	C₂×C₈	—	59	2	no	—
202	C₂×C₈	—	118	2	no	—
203	C₂×C₈	—	118	2	no	—
204	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
205	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
207	C₂×C₈	#11745685	1	2	yes	—
208	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
209	C₄⋊C₄	—	59	4	no	—
210	C₄⋊C₄	—	59	4	no	—
211	C₄⋊C₄	—	59	4	no	—
212	C₄²	—	59	2	no	—
213	C₄²	—	59	2	no	—
214	C₄²	—	59	2	no	—
215	C₄²	—	59	2	no	—
216	C₄²	C₈×D₅₉	1	2	yes	—
217	C₄²	C₂×C₄×D₅₉	1	2	yes	—
218	C₁₆	—	2	2	no	—
219	C₁₆	—	118	2	no	—
220	C₁₆	—	118	2	no	—
221	C₁₆	C₂×Dic₁₁₈	1	4	yes	complemented
222	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
223	Q₈	—	118	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	59	2	no	—
227	C₂×C₄	—	59	4	no	—
228	C₂×C₄	—	59	2	no	—
229	C₂×C₄	—	59	2	no	—
230	C₂×C₄	—	59	2	no	—
231	C₂×C₄	—	118	2	no	—
232	C₂×C₄	—	118	2	no	—
233	C₂×C₄	—	118	2	no	—
234	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
235	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
236	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
237	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
238	C₂×C₄	#14273733	1	2	yes	—
239	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	118	2	no	—
242	C₈	—	118	2	no	—
243	C₈	C₄×Dic₁₁₈	1	2	yes	—
244	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
245	C₈	C₂×#11745685	1	1	yes	characteristic
246	C₂²	C₈×Dic₁₁₈	1	2	yes	—
247	C₂²	M₃₂×D₅₉	1	2	yes	—
248	C₂²	#16837820	1	2	yes	—
249	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	59	4	no	—
253	C₄	—	118	2	no	—
254	C₄	—	118	2	no	—
255	C₄	—	118	2	no	—
256	C₄	C₈×Dic₁₁₈	1	2	yes	—
257	C₄	C₂×C₄×Dic₁₁₈	1	1	yes	characteristic
258	C₄	C₂×#14273733	1	1	yes	characteristic
259	C₂	#21426002	1	4	yes	direct product factor
260	C₂	C₂×C₈×Dic₁₁₈	1	1	yes	characteristic
261	C₂	C₂×M₃₂×D₅₉	1	1	yes	characteristic
262	C₂	C₂×#16837820	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle, prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	C₂×#2183
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. #21426002	C₂	direct (N × H)
53. C₂×C₁₆×C₅₉	Q₈	semidirect
87. C₁₆×C₅₉	C₂×Q₈	semidirect
136. C₂×C₅₉	#2183	semidirect
159. C₅₉	C₂×#2183	semidirect
178. C₂×C₁₆	Dic₁₁₈	semidirect
221. C₁₆	C₂×Dic₁₁₈	semidirect
259. C₂	#21426002	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×C₈×Dic₁₁₈	2
#21426002	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×Dic₅₉⋊C₁₆	2
C₂×#16837177	2
C₂×C₁₆×Dic₅₉	2
C₂×C₅₉×#346	2
C₂×#2183	59

Subgroups — C2×#21426002 #28742174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426002 #28742174