Subgroups — C₂×#21422056 #28498419

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422056	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₉×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₇₂	C₄	1	1	yes	characteristic
6	C₂×C₄×D₂₃₆	—	2	1	no	—
7	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
8	C₄×D₄₇₂	—	2	1	no	—
9	C₄×D₄₇₂	—	2	1	no	—
10	C₄×D₄₇₂	C₄	1	2	yes	—
11	C₄×D₄₇₂	C₂²	1	2	yes	—
12	C₂×#14273661	C₄	1	1	yes	characteristic
13	C₂×#14273649	—	2	1	no	—
14	C₄×C₅₉⋊C₁₆	—	2	1	no	—
15	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	complemented
16	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#335	—	2	1	no	—
19	C₅₉×#335	C₂²	1	2	yes	complemented
20	C₂²×D₂₃₆	—	2	1	no	—
21	C₂²×C₄×D₅₉	—	4	1	no	—
22	C₂×D₄₇₂	—	2	1	no	—
23	C₂×D₄₇₂	—	2	2	no	—
24	C₂×D₄₇₂	C₈	1	4	yes	complemented
25	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
26	C₄×D₂₃₆	—	2	2	no	—
27	C₄×D₂₃₆	—	2	2	no	—
28	C₄×D₂₃₆	—	4	1	no	—
29	C₄×D₂₃₆	—	4	1	no	—
30	C₂×C₄⋊Dic₅₉	—	2	1	no	—
31	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
32	C₂×C₅₉⋊C₁₆	—	2	4	no	—
33	C₂×C₅₉⋊C₁₆	—	2	2	no	—
34	#14273661	—	2	1	no	—
35	#14273661	C₂×C₄	1	2	yes	—
36	#14273649	—	2	2	no	—
37	#14273649	—	2	2	no	—
38	C₂²×C₈×C₅₉	—	4	1	no	—
39	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
42	C₄⋊C₈×C₅₉	—	2	2	no	—
43	C₄⋊C₈×C₅₉	—	4	1	no	—
44	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	C₂³×D₅₉	—	4	1	no	—
49	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
52	C₂²×Dic₅₉	—	4	1	no	—
53	C₂×D₂₃₆	—	2	4	no	—
54	C₂×D₂₃₆	—	2	2	no	—
55	C₂×D₂₃₆	—	4	1	no	—
56	C₂×D₂₃₆	—	4	2	no	—
57	C₂×C₄×D₅₉	—	4	2	no	—
58	C₂×C₄×D₅₉	—	4	2	no	—
59	C₂×C₄×D₅₉	—	8	2	no	—
60	C₂×C₄×D₅₉	—	8	2	no	—
61	C₂×C₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	—	4	2	no	—
63	C₂×C₈×C₅₉	—	8	2	no	—
64	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
68	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₅₉	D₈	1	1	yes	characteristic
72	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
73	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
74	(C₂×D₅₉)⋊C₄	—	4	2	no	—
75	(C₂×D₅₉)⋊C₄	—	4	2	no	—
76	C₄⋊Dic₅₉	—	2	2	no	—
77	C₄⋊Dic₅₉	—	4	1	no	—
78	D₄₇₂	—	2	4	no	—
79	D₄₇₂	—	2	2	no	—
80	D₄₇₂	C₂×C₈	1	4	yes	complemented
81	C₅₉⋊C₁₆	—	2	4	no	—
82	C₂³×C₅₉	#102	1	1	yes	characteristic
83	C₂²×D₅₉	—	4	4	no	—
84	C₂²×D₅₉	—	4	2	no	—
85	C₂²×D₅₉	—	8	2	no	—
86	C₂²×D₅₉	—	8	2	no	—
87	C₂×C₄×C₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
89	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
92	C₂×C₄×C₅₉	#104	1	2	yes	—
93	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
94	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
95	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×Dic₅₉	—	4	2	no	—
99	C₂×Dic₅₉	—	8	2	no	—
100	D₂₃₆	—	2	4	no	—
101	D₂₃₆	—	4	4	no	—
102	D₂₃₆	—	4	2	no	—
103	C₄×D₅₉	—	8	4	no	—
104	C₄×D₅₉	—	8	2	no	—
105	C₄×D₅₉	—	8	2	no	—
106	C₈×C₅₉	—	2	1	no	—
107	C₈×C₅₉	—	2	1	no	—
108	C₈×C₅₉	—	8	2	no	—
109	C₈×C₅₉	C₂²⋊C₈	1	2	yes	—
110	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
111	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
112	C₂×#1331	—	59	1	no	maximal, Sylow
113	C₂²×C₅₉	D₄⋊C₈	1	2	yes	—
114	C₂²×C₅₉	#357	1	2	yes	—
115	C₂²×C₅₉	#359	1	2	yes	—
116	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
117	C₂×D₅₉	—	4	4	no	—
118	C₂×D₅₉	—	8	8	no	—
119	C₂×D₅₉	—	8	2	no	—
120	C₂×D₅₉	—	8	2	no	—
121	C₄×C₅₉	—	2	1	no	—
122	C₄×C₅₉	—	2	1	no	—
123	C₄×C₅₉	D₄⋊C₈	1	2	yes	—
124	C₄×C₅₉	#357	1	2	yes	—
125	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
127	C₄×C₅₉	C₂×D₁₆	1	1	yes	characteristic
128	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
129	Dic₅₉	—	8	2	no	—
130	C₂×C₄×D₈	—	59	1	no	—
131	C₂×C₄×C₁₆	—	59	1	no	—
132	C₂×#335	D₅₉	1	1	yes	characteristic, complemented
133	#1331	—	59	4	no	—
134	C₂×C₅₉	#1331	1	4	yes	complemented
135	C₂×C₅₉	C₂×D₄⋊C₈	1	1	yes	characteristic
136	C₂×C₅₉	C₂×#357	1	1	yes	characteristic
137	C₂×C₅₉	C₂×#359	1	1	yes	characteristic
138	D₅₉	—	8	4	no	—
139	C₂²×D₈	—	59	1	no	—
140	C₂×C₄×D₄	—	118	1	no	—
141	C₂²×C₁₆	—	118	1	no	—
142	C₄×D₈	—	59	2	no	—
143	C₄×D₈	—	59	2	no	—
144	C₄×D₈	—	118	1	no	—
145	C₄×D₈	—	118	1	no	—
146	C₂×#107	—	59	1	no	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×#102	—	118	1	no	—
149	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
150	C₄×C₁₆	—	59	2	no	—
151	C₄×C₁₆	—	118	1	no	—
152	#335	—	2	1	no	—
153	#335	C₂×D₅₉	1	2	yes	complemented
154	C₅₉	C₂×#1331	1	1	yes	characteristic, complemented, Sylow
155	C₂²×D₄	—	118	1	no	—
156	C₂³×C₄	—	236	1	no	—
157	C₂×D₈	—	59	4	no	—
158	C₂×D₈	—	59	2	no	—
159	C₂×D₈	—	118	2	no	—
160	C₂×D₈	—	118	1	no	—
161	C₂²×C₈	—	4	1	no	—
162	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
163	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
164	C₄×D₄	—	118	2	no	—
165	C₄×D₄	—	118	2	no	—
166	C₄×D₄	—	236	1	no	—
167	C₄×D₄	—	236	1	no	—
168	C₂×C₄⋊C₄	—	118	1	no	—
169	C₂×C₂²⋊C₄	—	236	1	no	—
170	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
171	C₂×C₁₆	—	118	4	no	—
172	C₂×C₁₆	—	118	2	no	—
173	#107	—	59	2	no	—
174	#107	—	118	1	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	4	1	no	—
177	#102	—	118	2	no	—
178	#102	—	118	2	no	—
179	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
180	C₄×C₈	D₂₃₆	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
182	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
183	C₂⁴	—	236	1	no	—
184	C₂×D₄	—	118	4	no	—
185	C₂×D₄	—	118	2	no	—
186	C₂×D₄	—	236	1	no	—
187	C₂×D₄	—	236	2	no	—
188	C₂²×C₄	—	472	2	no	—
189	C₂²×C₄	—	472	2	no	—
190	C₂²×C₄	—	236	1	no	—
191	C₂²×C₄	—	236	2	no	—
192	C₂²×C₄	—	236	2	no	—
193	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	#11745669	1	1	yes	characteristic
195	C₂²×C₄	#11745671	1	1	yes	characteristic
196	D₈	—	59	4	no	—
197	D₈	—	118	4	no	—
198	D₈	—	118	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	8	2	no	—
202	C₂×C₈	C₈×D₅₉	1	2	yes	—
203	C₂×C₈	#11745659	1	2	yes	—
204	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
207	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
208	C₄⋊C₄	—	118	2	no	—
209	C₄⋊C₄	—	236	1	no	—
210	C₂²⋊C₄	—	236	2	no	—
211	C₂²⋊C₄	—	236	2	no	—
212	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
213	C₄²	#11745669	1	1	yes	characteristic
214	C₄²	#11745671	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
216	C₁₆	—	118	4	no	—
217	C₂³	—	472	2	no	—
218	C₂³	—	472	2	no	—
219	C₂³	—	236	4	no	—
220	C₂³	—	236	2	no	—
221	C₂³	#14273647	1	1	yes	characteristic
222	D₄	—	118	4	no	—
223	D₄	—	236	4	no	—
224	D₄	—	236	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	472	2	no	—
227	C₂×C₄	—	472	4	no	—
228	C₂×C₄	—	472	2	no	—
229	C₂×C₄	—	472	2	no	—
230	C₂×C₄	—	236	2	no	—
231	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
232	C₂×C₄	#14273647	1	1	yes	characteristic
233	C₂×C₄	#14273647	1	1	yes	characteristic
234	C₂×C₄	#14273647	1	1	yes	characteristic
235	C₂×C₄	#14273653	1	2	yes	—
236	C₂×C₄	#14273668	1	2	yes	—
237	C₂×C₄	#14273670	1	2	yes	—
238	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
240	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	8	2	no	—
244	C₈	(C₂×D₅₉)⋊C₈	1	2	yes	—
245	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
246	C₈	C₂×#11745659	1	1	yes	characteristic
247	C₂²	—	472	2	no	—
248	C₂²	—	472	2	no	—
249	C₂²	—	472	8	no	—
250	C₂²	—	236	4	no	—
251	C₂²	#16837129	1	2	yes	—
252	C₂²	#16837202	1	2	yes	—
253	C₂²	#16837208	1	2	yes	—
254	C₂²	C₂×#14273647	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	472	2	no	—
258	C₄	#16837129	1	2	yes	—
259	C₄	#16837202	1	2	yes	—
260	C₄	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
261	C₄	C₂×#14273653	1	1	yes	characteristic
262	C₄	C₂×#14273668	1	1	yes	characteristic
263	C₄	C₂×#14273670	1	1	yes	characteristic
264	C₂	—	472	4	no	—
265	C₂	#21422056	1	4	yes	direct product factor
266	C₂	C₂×#16837129	1	1	yes	characteristic
267	C₂	C₂×#16837202	1	1	yes	characteristic
268	C₂	C₂×#16837208	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11745669
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#335	C₂
socle	C₂³×C₅₉	#102
prime radical	C₂²×D₄₇₂	C₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×#335	D₅₉
hypocenter	C₅₉	C₂×#1331
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅₉⋊C₁₆	C₂	semidirect
3. #21422056	C₂	direct (N × H)
4. C₂×C₅₉×#335	C₂	semidirect
15. C₄×C₅₉⋊C₁₆	C₂²	semidirect
19. C₅₉×#335	C₂²	semidirect
24. C₂×D₄₇₂	C₈	semidirect
80. D₄₇₂	C₂×C₈	semidirect
132. C₂×#335	D₅₉	semidirect
134. C₂×C₅₉	#1331	semidirect
153. #335	C₂×D₅₉	semidirect
154. C₅₉	C₂×#1331	semidirect
265. C₂	#21422056	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×C₄×D₄₇₂	2
C₂×C₄×C₅₉⋊C₁₆	2
#21422056	2
C₂×C₅₉×#335	2
C₂×#1331	59

Subgroups — C2×#21422056 #28498419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422056 #28498419