Subgroups — C₂×#21422877 #28557757

Order 15104.

670 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837455	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837455	C₂	1	1	yes	maximal, characteristic
3	#21422877	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837154	C₂	1	2	yes	maximal, complemented
5	C₂×#16837152	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#432	C₂	1	1	yes	maximal, characteristic
7	#16837455	—	2	2	no	—
8	#16837455	C₄	1	4	yes	—
9	#16837455	C₂²	1	4	yes	—
10	#16837455	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
12	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
14	C₂×C₂²⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
15	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
16	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	#16837154	C₂²	1	8	yes	complemented
19	#16837152	C₂²	1	8	yes	complemented
20	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#432	—	2	2	no	—
22	C₅₉×#432	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₅₉	—	2	1	no	—
25	C₂×#97×C₅₉	—	2	1	no	—
26	C₂²×C₅₉⋊C₈	—	2	2	no	—
27	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
30	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	8	yes	—
31	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
32	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
33	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	8	yes	—
34	C₄×C₅₉⋊C₈	—	2	4	no	—
35	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
38	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
39	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₉	Q₈	1	1	yes	characteristic
42	C₂²×C₈×C₅₉	—	2	2	no	—
43	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₉	D₄	1	2	yes	—
47	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
48	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
49	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
51	C₄⋊C₈×C₅₉	—	2	2	no	—
52	C₄⋊C₈×C₅₉	C₂³	1	4	yes	—
53	C₂²⋊C₈×C₅₉	—	2	4	no	—
54	#97×C₅₉	—	2	4	no	—
55	C₂⁴×C₅₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	—	2	4	no	—
57	C₂²×C₄×C₅₉	—	2	4	no	—
58	C₂²×C₄×C₅₉	—	4	2	no	—
59	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₅₉	—	2	4	no	—
72	C₂×C₈×C₅₉	—	4	4	no	—
73	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
75	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
76	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
78	C₄²×C₅₉	C₂×D₄	1	2	yes	—
79	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
81	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
83	C₂×C₅₉⋊C₈	—	2	4	no	—
84	C₂×C₅₉⋊C₈	—	2	4	no	—
85	C₂×C₅₉⋊C₈	—	2	4	no	—
86	C₂×C₅₉⋊C₈	—	4	4	no	—
87	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
89	C₂³×C₅₉	—	2	2	no	—
90	C₂³×C₅₉	—	2	2	no	—
91	C₂³×C₅₉	—	2	2	no	—
92	C₂³×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	2	2	no	—
98	C₂×C₄×C₅₉	—	2	4	no	—
99	C₂×C₄×C₅₉	—	2	4	no	—
100	C₂×C₄×C₅₉	—	2	4	no	—
101	C₂×C₄×C₅₉	—	2	4	no	—
102	C₂×C₄×C₅₉	—	4	2	no	—
103	C₂×C₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	#131	1	4	yes	—
111	C₂×C₄×C₅₉	#135	1	4	yes	—
112	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
115	C₈×C₅₉	—	4	4	no	—
116	C₅₉⋊C₈	—	2	4	no	—
117	C₅₉⋊C₈	—	2	4	no	—
118	C₅₉⋊C₈	—	4	4	no	—
119	C₂×#1572	—	59	1	no	maximal, Sylow
120	C₂²×C₅₉	—	2	2	no	—
121	C₂²×C₅₉	—	2	2	no	—
122	C₂²×C₅₉	—	2	2	no	—
123	C₂²×C₅₉	—	4	4	no	—
124	C₂²×C₅₉	#424	1	2	yes	—
125	C₂²×C₅₉	#427	1	2	yes	—
126	C₂²×C₅₉	#430	1	2	yes	—
127	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₅₉	—	2	2	no	—
129	C₄×C₅₉	—	2	2	no	—
130	C₄×C₅₉	—	2	2	no	—
131	C₄×C₅₉	—	2	1	no	—
132	C₄×C₅₉	—	2	1	no	—
133	C₄×C₅₉	#424	1	2	yes	—
134	C₄×C₅₉	#427	1	2	yes	—
135	C₄×C₅₉	C₂×#131	1	2	yes	—
136	C₄×C₅₉	C₂×#135	1	2	yes	—
137	C₂×#432	D₅₉	1	1	yes	characteristic
138	C₂×#431	—	59	1	no	—
139	C₂×#431	—	59	1	no	—
140	#1572	—	59	8	no	—
141	C₂×#335	—	59	2	no	—
142	C₂×#334	—	59	2	no	—
143	C₂×C₅₉	—	4	2	no	—
144	C₂×C₅₉	#1572	1	4	yes	complemented
145	C₂×C₅₉	C₂×#424	1	1	yes	characteristic
146	C₂×C₅₉	C₂×#427	1	1	yes	characteristic
147	C₂×C₅₉	C₂×#430	1	1	yes	characteristic
148	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
149	#432	—	2	2	no	—
150	#432	C₂×D₅₉	1	4	yes	—
151	#431	—	59	4	no	—
152	#431	—	59	4	no	—
153	#431	—	59	4	no	—
154	#431	—	118	2	no	—
155	C₂×C₄⋊C₈	—	59	2	no	—
156	C₂×C₄⋊C₈	—	59	2	no	—
157	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
158	C₂×C₂²⋊C₈	—	2	1	no	—
159	C₂×C₂²⋊C₈	—	59	2	no	—
160	C₂×C₂²⋊C₈	—	118	1	no	—
161	C₂×#97	—	2	1	no	—
162	C₂×C₄×C₈	—	59	1	no	—
163	C₂×C₄×C₈	—	59	1	no	—
164	C₂×C₄×C₈	—	118	1	no	—
165	#335	—	59	8	no	—
166	#334	—	59	8	no	—
167	C₅₉	C₂×#1572	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	Dic₁₁₈	1	1	yes	characteristic
169	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
170	C₂³×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	59	2	no	—
173	C₂²×C₈	—	59	2	no	—
174	C₂²×C₈	—	118	2	no	—
175	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
176	C₂×C₄²	Dic₁₁₈	1	2	yes	—
177	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₅₉	1	2	yes	—
179	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
182	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	—	59	8	no	—
185	C₄⋊C₈	—	59	4	no	—
186	C₄⋊C₈	—	118	2	no	—
187	C₄⋊C₈	C₂²×D₅₉	1	4	yes	—
188	C₂²⋊C₈	—	2	4	no	—
189	C₂²⋊C₈	—	59	8	no	—
190	C₂²⋊C₈	—	118	4	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	—	59	2	no	—
193	C₄×C₈	—	59	2	no	—
194	C₄×C₈	—	59	2	no	—
195	C₄×C₈	—	59	2	no	—
196	C₄×C₈	—	118	4	no	—
197	C₂⁴	Dic₅₉⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
203	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
205	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
207	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
208	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
210	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
212	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	59	4	no	—
216	C₂×C₈	—	59	4	no	—
217	C₂×C₈	—	118	4	no	—
218	C₂×C₈	—	118	4	no	—
219	C₂×C₈	—	118	4	no	—
220	C₂×C₈	—	236	4	no	—
221	C₄²	Dic₅₉⋊C₄	1	2	yes	—
222	C₄²	Dic₅₉⋊C₄	1	2	yes	—
223	C₄²	Dic₅₉⋊C₄	1	2	yes	—
224	C₄²	C₂×Dic₁₁₈	1	2	yes	—
225	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
226	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
227	C₄²	C₂×C₄×D₅₉	1	2	yes	—
228	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
229	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
230	C₄²	C₂³×D₅₉	1	1	yes	characteristic
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₂×Dic₅₉⋊C₄	1	2	yes	—
235	C₂³	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
247	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
248	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
250	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
252	C₂×C₄	#14273735	1	4	yes	—
253	C₂×C₄	#14273747	1	4	yes	—
254	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
256	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
257	C₈	—	4	4	no	—
258	C₈	—	118	4	no	—
259	C₈	—	118	4	no	—
260	C₈	—	236	4	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	4	no	—
265	C₂²	#16837430	1	2	yes	—
266	C₂²	#16837438	1	2	yes	—
267	C₂²	#16837450	1	2	yes	—
268	C₂²	C₂²×Dic₅₉⋊C₄	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	#16837430	1	2	yes	—
275	C₄	#16837438	1	2	yes	—
276	C₄	C₂×#14273735	1	2	yes	—
277	C₄	C₂×#14273747	1	2	yes	—
278	C₂	—	4	2	no	—
279	C₂	#21422877	1	4	yes	direct product factor
280	C₂	C₂×#16837430	1	1	yes	characteristic
281	C₂	C₂×#16837438	1	1	yes	characteristic
282	C₂	C₂×#16837450	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#432	C₂
socle	C₂³×C₅₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₉	C₄⋊C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂×#432	D₅₉
hypocenter	C₅₉	C₂×#1572
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #21422877	C₂	direct (N × H)
4. C₂×#16837154	C₂	semidirect
5. C₂×#16837152	C₂	semidirect
18. #16837154	C₂²	semidirect
19. #16837152	C₂²	semidirect
144. C₂×C₅₉	#1572	semidirect
167. C₅₉	C₂×#1572	semidirect
279. C₂	#21422877	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×#16837455	2
#21422877	2
C₂×#16837154	2
C₂×#16837152	2
C₂×C₅₉×#432	2
C₂×#1572	59

Subgroups — C2×#21422877 #28557757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422877 #28557757