Subgroups — C₂×#21422870 #28557749

Order 15104.

770 conjugacy classes of subgroups.

The subgroup lattice has 282 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×#16837426	C₂	1	1	yes	maximal, characteristic
3	#21422870	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837154	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11745664	—	2	1	no	—
7	#16837455	C₄	1	4	yes	—
8	#16837455	C₂²	1	4	yes	—
9	#16837426	—	2	2	no	—
10	#16837426	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
12	C₂×C₄⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
13	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₅₉⋊C₈)	C₄	1	2	yes	—
15	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
16	#16837154	C₂²	1	16	yes	complemented
17	C₂³×C₈×C₅₉	—	2	1	no	—
18	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
20	C₂×C₄⋊C₈×C₅₉	—	2	2	no	—
21	C₂×C₄⋊C₈×C₅₉	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
23	C₂×#11745664	—	2	4	no	—
24	C₂×#11745664	—	4	4	no	—
25	C₂²×C₅₉⋊C₈	—	2	2	no	—
26	C₂²×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
27	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
28	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
29	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	8	yes	—
30	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
31	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	8	yes	—
32	C₄×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
33	C₄×C₅₉⋊C₈	C₂³	1	4	yes	—
34	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₉	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	—	2	4	no	—
38	C₂²×C₈×C₅₉	—	2	2	no	—
39	C₂²×C₈×C₅₉	—	4	4	no	—
40	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₉	D₄	1	2	yes	—
44	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
46	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
48	C₄⋊C₈×C₅₉	—	2	4	no	—
49	C₄⋊C₈×C₅₉	—	2	2	no	—
50	C₄⋊C₈×C₅₉	C₂³	1	4	yes	—
51	C₂⁴×C₅₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	—	2	4	no	—
53	C₂²×C₄×C₅₉	—	4	2	no	—
54	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	D₈	1	4	yes	—
61	C₂²×C₄×C₅₉	Q₁₆	1	4	yes	—
62	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₅₉	—	2	4	no	—
69	C₂×C₈×C₅₉	—	4	4	no	—
70	C₂×C₈×C₅₉	—	4	8	no	—
71	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₅₉	C₂×D₄	1	2	yes	—
77	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
79	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
81	#11745664	—	4	8	no	—
82	C₂×C₅₉⋊C₈	—	2	8	no	—
83	C₂×C₅₉⋊C₈	—	2	4	no	—
84	C₂×C₅₉⋊C₈	—	4	4	no	—
85	C₂×C₅₉⋊C₈	C₂²×C₄	1	8	yes	—
86	C₂³×C₅₉	—	2	2	no	—
87	C₂³×C₅₉	—	2	2	no	—
88	C₂³×C₅₉	#107	1	4	yes	—
89	C₂³×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₉	—	2	2	no	—
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	4	no	—
95	C₂×C₄×C₅₉	—	2	4	no	—
96	C₂×C₄×C₅₉	—	4	2	no	—
97	C₂×C₄×C₅₉	—	4	2	no	—
98	C₂×C₄×C₅₉	#107	1	8	yes	—
99	C₂×C₄×C₅₉	#107	1	4	yes	—
100	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#131	1	4	yes	—
107	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
109	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
110	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
114	C₈×C₅₉	—	4	4	no	—
115	C₅₉⋊C₈	—	2	8	no	—
116	C₅₉⋊C₈	—	4	4	no	—
117	C₂×#1571	—	59	1	no	maximal, Sylow
118	C₂²×C₅₉	—	2	8	no	—
119	C₂²×C₅₉	—	2	2	no	—
120	C₂²×C₅₉	—	2	2	no	—
121	C₂²×C₅₉	#424	1	2	yes	—
122	C₂²×C₅₉	C₂×#107	1	4	yes	—
123	C₂²×C₅₉	C₂×#107	1	2	yes	—
124	C₂²×C₅₉	#430	1	2	yes	—
125	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₅₉	—	2	2	no	—
127	C₄×C₅₉	—	2	2	no	—
128	C₄×C₅₉	—	2	1	no	—
129	C₄×C₅₉	—	2	1	no	—
130	C₄×C₅₉	#424	1	2	yes	—
131	C₄×C₅₉	C₂×#107	1	2	yes	—
132	C₄×C₅₉	C₂×#107	1	4	yes	—
133	C₄×C₅₉	C₂×#131	1	2	yes	—
134	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
136	C₂×#431	—	59	1	no	—
137	C₂×#423	—	59	1	no	—
138	C₂²×C₄⋊C₈	D₅₉	1	1	yes	characteristic
139	#1571	—	59	8	no	—
140	C₂×#335	—	59	4	no	—
141	C₂×C₅₉	—	2	4	no	—
142	C₂×C₅₉	#1571	1	4	yes	complemented
143	C₂×C₅₉	C₂×#424	1	1	yes	characteristic
144	C₂×C₅₉	C₂²×#107	1	1	yes	characteristic
145	C₂×C₅₉	C₂×#430	1	1	yes	characteristic
146	C₂²×M₁₆	—	118	1	no	—
147	C₂³×C₈	—	2	1	no	—
148	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
149	#431	—	59	4	no	—
150	#431	—	59	4	no	—
151	#423	—	59	4	no	—
152	#423	—	118	2	no	—
153	C₂×C₄⋊C₈	—	2	2	no	—
154	C₂×C₄⋊C₈	—	2	1	no	—
155	C₂×C₄⋊C₈	—	59	2	no	—
156	C₂×C₄⋊C₈	—	59	2	no	—
157	C₂×C₄⋊C₈	—	118	1	no	—
158	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
159	C₂×C₄⋊C₈	C₂×D₅₉	1	4	yes	—
160	C₂×C₂²⋊C₈	—	59	2	no	—
161	C₂×C₄×C₈	—	59	2	no	—
162	#335	—	59	16	no	—
163	C₅₉	C₂×#1571	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	Dic₁₁₈	1	1	yes	characteristic
165	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
166	C₂³×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
167	C₂×M₁₆	—	118	4	no	—
168	C₂×M₁₆	—	236	4	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	4	4	no	—
172	C₂²×C₈	—	59	4	no	—
173	C₂²×C₈	—	118	2	no	—
174	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
175	C₂×C₄²	Dic₁₁₈	1	2	yes	—
176	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₅₉	1	2	yes	—
178	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
179	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
181	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	59	8	no	—
185	C₄⋊C₈	—	59	4	no	—
186	C₄⋊C₈	—	118	2	no	—
187	C₄⋊C₈	—	118	4	no	—
188	C₄⋊C₈	C₂²×D₅₉	1	4	yes	—
189	C₂²⋊C₈	—	59	8	no	—
190	C₄×C₈	—	59	4	no	—
191	C₄×C₈	—	59	4	no	—
192	C₂⁴	Dic₅₉⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
197	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
199	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	Dic₅₉⋊C₄	1	2	yes	—
201	C₂²×C₄	#11745669	1	4	yes	—
202	C₂²×C₄	#11745672	1	4	yes	—
203	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
205	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
206	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
209	M₁₆	—	236	8	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	4	no	—
212	C₂×C₈	—	4	8	no	—
213	C₂×C₈	—	59	8	no	—
214	C₂×C₈	—	118	8	no	—
215	C₂×C₈	—	118	4	no	—
216	C₂×C₈	—	236	4	no	—
217	C₄²	Dic₅₉⋊C₄	1	2	yes	—
218	C₄²	Dic₅₉⋊C₄	1	2	yes	—
219	C₄²	Dic₅₉⋊C₄	1	2	yes	—
220	C₄²	C₂×Dic₁₁₈	1	2	yes	—
221	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
222	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
223	C₄²	C₂×C₄×D₅₉	1	2	yes	—
224	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
225	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
226	C₄²	C₂³×D₅₉	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	#14273660	1	4	yes	—
230	C₂³	C₂×Dic₅₉⋊C₄	1	2	yes	—
231	C₂³	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	#14273660	1	8	yes	—
240	C₂×C₄	#14273660	1	4	yes	—
241	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
242	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
244	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
246	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
247	C₂×C₄	#14273733	1	4	yes	—
248	C₂×C₄	C₂×#11745669	1	2	yes	—
249	C₂×C₄	C₂×#11745669	1	4	yes	—
250	C₂×C₄	C₂×#11745672	1	4	yes	—
251	C₂×C₄	C₂×#11745672	1	2	yes	—
252	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
254	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
255	C₈	—	4	4	no	—
256	C₈	—	118	8	no	—
257	C₈	—	236	4	no	—
258	C₂²	—	2	8	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	#16837428	1	2	yes	—
262	C₂²	C₂×#14273660	1	4	yes	—
263	C₂²	C₂×#14273660	1	2	yes	—
264	C₂²	#16837450	1	2	yes	—
265	C₂²	C₂²×Dic₅₉⋊C₄	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	#16837428	1	2	yes	—
271	C₄	C₂×#14273660	1	4	yes	—
272	C₄	C₂×#14273660	1	2	yes	—
273	C₄	C₂×#14273733	1	2	yes	—
274	C₄	C₂²×#11745669	1	1	yes	characteristic
275	C₄	C₂²×#11745672	1	1	yes	characteristic
276	C₂	—	2	4	no	—
277	C₂	#21422870	1	4	yes	direct product factor
278	C₂	C₂×#16837428	1	1	yes	characteristic
279	C₂	C₂²×#14273660	1	1	yes	characteristic
280	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	C₂²×C₄⋊C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₉	C₄⋊C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₅₉
hypocenter	C₅₉	C₂×#1571
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #21422870	C₂	direct (N × H)
4. C₂×#16837154	C₂	semidirect
16. #16837154	C₂²	semidirect
142. C₂×C₅₉	#1571	semidirect
163. C₅₉	C₂×#1571	semidirect
277. C₂	#21422870	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
C₂×#16837426	2
#21422870	2
C₂×#16837154	2
C₂²×C₄⋊C₈×C₅₉	2
C₂×#1571	59

Subgroups — C2×#21422870 #28557749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422870 #28557749