Subgroups — C₂×#21426250 #28743301

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 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	#21426250	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837229	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837227	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837202	C₂	1	1	yes	maximal, characteristic
7	C₂×#16837202	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₃₆	—	2	1	no	—
11	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄×D₄₇₂	—	2	2	no	—
14	C₄×D₄₇₂	C₂²	1	4	yes	—
15	C₂×#14273661	C₂²	1	1	yes	characteristic
16	C₂×#14273649	—	2	1	no	—
17	#16837229	C₂²	1	4	yes	complemented
18	#16837227	C₂²	1	4	yes	complemented
19	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	complemented
20	#16837202	C₂²	1	4	yes	—
21	#16837202	C₂²	1	4	yes	—
22	C₂×#125×C₅₉	—	2	1	no	—
23	C₅₉×#499	—	2	2	no	—
24	C₅₉×#499	C₂²	1	4	yes	complemented
25	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#106×C₅₉	—	2	1	no	—
28	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₂²×D₂₃₆	—	2	1	no	—
30	C₂²×C₄×D₅₉	—	4	1	no	—
31	C₂×D₄₇₂	—	2	1	no	—
32	C₂×D₄₇₂	—	2	2	no	—
33	C₂×D₄₇₂	Q₈	1	4	yes	—
34	C₂×D₄₇₂	C₂³	1	2	yes	—
35	C₄×D₂₃₆	—	2	4	no	—
36	C₄×D₂₃₆	—	4	2	no	—
37	C₂×C₄⋊Dic₅₉	—	2	1	no	—
38	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
39	C₂×C₅₉⋊C₁₆	—	2	2	no	—
40	C₂×C₅₉⋊C₁₆	—	2	2	no	—
41	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
42	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
43	#14273661	—	2	1	no	—
44	#14273661	C₂³	1	2	yes	—
45	#14273649	—	2	4	no	—
46	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	#125×C₅₉	—	2	4	no	—
49	#125×C₅₉	—	4	2	no	—
50	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
53	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
54	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
55	#107×C₅₉	—	2	1	no	—
56	#107×C₅₉	—	2	1	no	—
57	#107×C₅₉	C₂³	1	2	yes	—
58	#107×C₅₉	C₂³	1	2	yes	—
59	#106×C₅₉	—	2	4	no	—
60	C₄×C₈×C₅₉	D₄	1	2	yes	—
61	C₄×C₈×C₅₉	C₂³	1	2	yes	—
62	C₂³×D₅₉	—	4	1	no	—
63	C₂²×C₄×C₅₉	—	4	1	no	—
64	C₂²×C₄×C₅₉	—	4	1	no	—
65	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₅₉	—	4	1	no	—
69	C₂×D₂₃₆	—	2	4	no	—
70	C₂×D₂₃₆	—	2	2	no	—
71	C₂×D₂₃₆	—	4	1	no	—
72	C₂×D₂₃₆	—	4	2	no	—
73	C₂×C₄×D₅₉	—	4	4	no	—
74	C₂×C₄×D₅₉	—	8	4	no	—
75	C₂×C₈×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₅₉	#41	1	2	yes	—
80	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₅₉	—	2	2	no	—
82	C₄⋊C₄×C₅₉	—	2	2	no	—
83	C₄⋊C₄×C₅₉	—	4	1	no	—
84	C₄⋊C₄×C₅₉	—	4	1	no	—
85	C₄⋊C₄×C₅₉	—	4	4	no	—
86	C₄⋊C₄×C₅₉	—	4	4	no	—
87	C₄²×C₅₉	D₈	1	2	yes	—
88	C₄²×C₅₉	C₂×D₄	1	2	yes	—
89	(C₂×D₅₉)⋊C₄	—	4	4	no	—
90	C₄⋊Dic₅₉	—	2	2	no	—
91	C₄⋊Dic₅₉	—	4	1	no	—
92	D₄₇₂	—	2	4	no	—
93	D₄₇₂	—	2	2	no	—
94	D₄₇₂	C₂×Q₈	1	4	yes	—
95	C₅₉⋊C₁₆	—	2	2	no	—
96	C₅₉⋊C₁₆	—	2	2	no	—
97	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
98	C₂²×D₅₉	—	4	2	no	—
99	C₂²×D₅₉	—	4	4	no	—
100	C₂²×D₅₉	—	8	2	no	—
101	C₂²×D₅₉	—	8	2	no	—
102	C₂×C₄×C₅₉	—	2	2	no	—
103	C₂×C₄×C₅₉	—	2	2	no	—
104	C₂×C₄×C₅₉	—	4	2	no	—
105	C₂×C₄×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	—	8	2	no	—
107	C₂×C₄×C₅₉	—	8	2	no	—
108	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#137	1	2	yes	—
113	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₅₉	—	4	2	no	—
115	C₂×Dic₅₉	—	8	2	no	—
116	D₂₃₆	—	2	4	no	—
117	D₂₃₆	—	4	2	no	—
118	D₂₃₆	—	4	4	no	—
119	C₄×D₅₉	—	8	8	no	—
120	C₈×C₅₉	—	2	2	no	—
121	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
122	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
124	C₂×#2228	—	59	1	no	maximal, Sylow
125	C₂²×C₅₉	#474	1	2	yes	—
126	C₂²×C₅₉	#508	1	2	yes	—
127	C₂²×C₅₉	#509	1	2	yes	—
128	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
129	C₂×D₅₉	—	4	4	no	—
130	C₂×D₅₉	—	8	2	no	—
131	C₂×D₅₉	—	8	2	no	—
132	C₂×D₅₉	—	8	8	no	—
133	C₄×C₅₉	—	2	2	no	—
134	C₄×C₅₉	—	2	2	no	—
135	C₄×C₅₉	—	8	2	no	—
136	C₄×C₅₉	—	8	2	no	—
137	C₄×C₅₉	#474	1	2	yes	—
138	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
140	Dic₅₉	—	8	2	no	—
141	C₂×#499	D₅₉	1	1	yes	characteristic, complemented
142	C₂×C₄×D₈	—	59	1	no	—
143	#2228	—	59	8	no	—
144	C₂×#367	—	59	1	no	—
145	C₂×#366	—	59	1	no	—
146	C₂×C₄⋊C₁₆	—	59	1	no	—
147	C₂×#357	—	59	1	no	—
148	C₂×#357	—	59	1	no	—
149	C₂×C₅₉	#2228	1	4	yes	complemented
150	C₂×C₅₉	C₂×#474	1	1	yes	characteristic
151	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
152	C₂×C₅₉	C₂×#509	1	1	yes	characteristic
153	D₅₉	—	8	4	no	—
154	C₂²×D₈	—	59	1	no	—
155	C₂×#125	—	2	1	no	—
156	C₂×C₄×D₄	—	118	1	no	—
157	C₂²×C₁₆	—	59	1	no	—
158	C₂²×C₁₆	—	59	1	no	—
159	#499	—	2	2	no	—
160	#499	C₂×D₅₉	1	4	yes	complemented
161	C₄×D₈	—	59	4	no	—
162	C₄×D₈	—	118	2	no	—
163	C₂×#107	—	59	1	no	—
164	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
165	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
166	C₂×#106	—	2	1	no	—
167	C₂×#102	—	118	1	no	—
168	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
169	#367	—	59	4	no	—
170	#366	—	59	4	no	—
171	C₄⋊C₁₆	—	59	4	no	—
172	#357	—	59	4	no	—
173	#357	—	59	4	no	—
174	C₅₉	C₂×#2228	1	1	yes	characteristic, complemented, Sylow
175	C₂²×D₄	—	118	1	no	—
176	C₂³×C₄	—	236	1	no	—
177	C₂×D₈	—	59	4	no	—
178	C₂×D₈	—	59	2	no	—
179	C₂×D₈	—	118	2	no	—
180	C₂×D₈	—	118	1	no	—
181	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
183	#125	—	2	4	no	—
184	#125	—	4	2	no	—
185	C₄×D₄	—	118	4	no	—
186	C₄×D₄	—	236	2	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	4	1	no	—
190	C₂×C₄⋊C₄	—	4	1	no	—
191	C₂×C₄⋊C₄	—	118	1	no	—
192	C₂×C₂²⋊C₄	—	236	1	no	—
193	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
194	C₂×C₁₆	—	59	2	no	—
195	C₂×C₁₆	—	59	2	no	—
196	C₂×C₁₆	—	118	2	no	—
197	C₂×C₁₆	—	118	2	no	—
198	#107	—	2	1	no	—
199	#107	—	2	1	no	—
200	#107	—	59	2	no	—
201	#107	—	118	1	no	—
202	#107	C₂²×D₅₉	1	2	yes	—
203	#107	C₂²×D₅₉	1	2	yes	—
204	#106	—	2	4	no	—
205	#102	—	118	4	no	—
206	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
207	C₄×C₈	C₂²×D₅₉	1	2	yes	—
208	C₂⁴	—	236	1	no	—
209	C₂×D₄	—	118	4	no	—
210	C₂×D₄	—	118	2	no	—
211	C₂×D₄	—	236	1	no	—
212	C₂×D₄	—	236	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	472	4	no	—
216	C₂²×C₄	—	236	1	no	—
217	C₂²×C₄	—	236	4	no	—
218	C₂²×C₄	#11745669	1	1	yes	characteristic
219	C₂²×C₄	#11745669	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
221	D₈	—	59	4	no	—
222	D₈	—	118	4	no	—
223	D₈	—	118	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	Q₈×D₅₉	1	2	yes	—
226	C₂×C₈	#11745689	1	2	yes	—
227	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
228	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₄⋊C₄	—	118	2	no	—
237	C₄⋊C₄	—	236	1	no	—
238	C₂²⋊C₄	—	236	4	no	—
239	C₄²	#11745669	1	2	yes	—
240	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
241	C₁₆	—	118	2	no	—
242	C₁₆	—	118	2	no	—
243	C₂³	—	472	2	no	—
244	C₂³	—	472	2	no	—
245	C₂³	—	236	2	no	—
246	C₂³	—	236	4	no	—
247	C₂³	C₂×#11745669	1	1	yes	characteristic
248	D₄	—	118	4	no	—
249	D₄	—	236	4	no	—
250	D₄	—	236	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	472	8	no	—
258	C₂×C₄	—	472	2	no	—
259	C₂×C₄	—	236	2	no	—
260	C₂×C₄	#14273703	1	2	yes	—
261	C₂×C₄	C₂×#11745669	1	2	yes	—
262	C₂×C₄	C₂×#11745669	1	2	yes	—
263	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
264	C₂×C₄	#14273758	1	2	yes	—
265	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	#14273703	1	2	yes	—
268	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
269	C₈	C₂×#11745689	1	1	yes	characteristic
270	C₂²	—	472	2	no	—
271	C₂²	—	472	2	no	—
272	C₂²	—	472	8	no	—
273	C₂²	—	236	4	no	—
274	C₂²	#16837679	1	2	yes	—
275	C₂²	#16837832	1	2	yes	—
276	C₂²	#16837837	1	2	yes	—
277	C₂²	C₂²×#11745669	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	472	2	no	—
283	C₄	#16837679	1	2	yes	—
284	C₄	C₂×#14273703	1	1	yes	characteristic
285	C₄	C₂×#14273758	1	1	yes	characteristic
286	C₂	—	472	4	no	—
287	C₂	#21426250	1	4	yes	direct product factor
288	C₂	C₂×#16837679	1	1	yes	characteristic
289	C₂	C₂×#16837832	1	1	yes	characteristic
290	C₂	C₂×#16837837	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#499	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂²×D₄₇₂	C₂²
hypercenter	C₂×#499	D₅₉
hypocenter	C₅₉	C₂×#2228
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #21426250	C₂	direct (N × H)
3. C₂×#16837229	C₂	semidirect
4. C₂×#16837227	C₂	semidirect
5. C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
8. C₂×C₅₉×#499	C₂	semidirect
17. #16837229	C₂²	semidirect
18. #16837227	C₂²	semidirect
19. C₄⋊(C₅₉⋊C₁₆)	C₂²	semidirect
24. C₅₉×#499	C₂²	semidirect
141. C₂×#499	D₅₉	semidirect
149. C₂×C₅₉	#2228	semidirect
160. #499	C₂×D₅₉	semidirect
174. C₅₉	C₂×#2228	semidirect
287. C₂	#21426250	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
#21426250	2
C₂×#16837229	2
C₂×#16837227	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×#16837202	2
C₂×C₅₉×#499	2
C₂×#2228	59

Subgroups — C2×#21426250 #28743301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426250 #28743301