Subgroups — C₂×#21423461 #28570924

Order 15104.

790 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837782	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837682	C₂	1	2	yes	maximal
3	#21423461	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837155	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837149	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₄×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₃₆	—	2	1	no	—
8	C₂²×Dic₂₃₆	—	2	1	no	—
9	C₂×#14273727	C₂²	1	2	yes	—
10	#16837782	—	2	2	no	—
11	#16837782	C₂²	1	4	yes	complemented
12	#16837682	—	2	4	no	—
13	#16837682	C₂²	1	8	yes	—
14	C₂×#14273660	—	2	2	no	—
15	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×#14273648	—	2	2	no	—
17	#16837155	C₂²	1	4	yes	complemented
18	#16837149	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₅₉	—	2	1	no	—
20	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₈×D₄×C₅₉	—	2	2	no	—
22	C₈×D₄×C₅₉	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₅₉	—	2	1	no	—
25	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₁₈	—	2	2	no	—
27	C₂×Dic₂₃₆	—	2	4	no	—
28	C₂×Dic₂₃₆	—	2	4	no	—
29	C₂×Dic₂₃₆	—	4	4	no	—
30	C₂×Dic₂₃₆	—	4	4	no	—
31	C₂²×C₅₉⋊C₈	—	2	2	no	—
32	#14273727	—	2	2	no	—
33	#14273727	—	2	2	no	—
34	#14273727	—	2	2	no	—
35	#14273727	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₅₉	—	2	2	no	—
37	C₂×C₄⋊Dic₅₉	—	2	2	no	—
38	#14273660	—	2	8	no	—
39	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
40	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
41	#14273648	—	2	8	no	—
42	C₂²×D₄×C₅₉	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₉	—	2	1	no	—
44	C₂²×C₈×C₅₉	—	2	1	no	—
45	C₂²×C₈×C₅₉	—	2	4	no	—
46	C₂²×C₈×C₅₉	—	4	4	no	—
47	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
49	C₄×D₄×C₅₉	—	2	1	no	—
50	C₄×D₄×C₅₉	—	2	1	no	—
51	C₄×D₄×C₅₉	D₄	1	2	yes	—
52	C₄×D₄×C₅₉	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	—	2	1	no	—
57	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
58	C₂²⋊C₈×C₅₉	—	2	4	no	—
59	C₄×C₈×C₅₉	D₄	1	2	yes	—
60	C₄×C₈×C₅₉	C₂³	1	2	yes	—
61	C₂⁴×C₅₉	—	2	1	no	—
62	C₂×D₄×C₅₉	—	2	1	no	—
63	C₂×D₄×C₅₉	—	2	2	no	—
64	C₂×D₄×C₅₉	Q₁₆	1	4	yes	complemented
65	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₅₉	—	2	1	no	—
67	C₂²×C₄×C₅₉	—	2	2	no	—
68	C₂²×C₄×C₅₉	—	2	2	no	—
69	C₂²×C₄×C₅₉	—	4	2	no	—
70	C₂²×C₄×C₅₉	—	4	2	no	—
71	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
74	C₂²×Dic₅₉	—	4	2	no	—
75	C₂×Dic₁₁₈	—	2	4	no	—
76	C₂×Dic₁₁₈	—	4	4	no	—
77	C₂×Dic₁₁₈	—	4	4	no	—
78	C₂×Dic₁₁₈	—	4	2	no	—
79	C₂×C₈×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	—	2	2	no	—
81	C₂×C₈×C₅₉	—	4	2	no	—
82	C₂×C₈×C₅₉	—	4	8	no	—
83	C₂×C₈×C₅₉	D₈	1	4	yes	—
84	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
86	C₄⋊C₄×C₅₉	—	2	1	no	—
87	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
88	C₂²⋊C₄×C₅₉	—	2	2	no	—
89	C₂²⋊C₄×C₅₉	—	2	2	no	—
90	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₉	#41	1	1	yes	characteristic
93	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
94	C₄⋊Dic₅₉	—	2	4	no	—
95	C₄⋊Dic₅₉	—	2	4	no	—
96	C₄⋊Dic₅₉	—	4	2	no	—
97	C₄⋊Dic₅₉	—	4	2	no	—
98	C₂×C₅₉⋊C₈	—	2	4	no	—
99	C₂×C₅₉⋊C₈	—	4	4	no	—
100	Dic₂₃₆	—	4	8	no	—
101	Dic₂₃₆	—	4	8	no	—
102	C₂³×C₅₉	—	2	2	no	—
103	C₂³×C₅₉	—	2	4	no	—
104	C₂³×C₅₉	—	4	2	no	—
105	C₂³×C₅₉	—	4	2	no	—
106	C₂³×C₅₉	#121	1	1	yes	characteristic
107	D₄×C₅₉	—	2	2	no	—
108	D₄×C₅₉	—	2	4	no	—
109	D₄×C₅₉	C₂×Q₁₆	1	4	yes	complemented
110	C₂×C₄×C₅₉	—	2	2	no	—
111	C₂×C₄×C₅₉	—	2	2	no	—
112	C₂×C₄×C₅₉	—	4	2	no	—
113	C₂×C₄×C₅₉	—	4	2	no	—
114	C₂×C₄×C₅₉	—	4	2	no	—
115	C₂×C₄×C₅₉	—	4	4	no	—
116	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
121	C₂×C₄×C₅₉	#135	1	2	yes	—
122	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₅₉	—	4	4	no	—
127	C₂×Dic₅₉	—	8	4	no	—
128	Dic₁₁₈	—	4	4	no	—
129	Dic₁₁₈	—	4	4	no	—
130	Dic₁₁₈	—	8	4	no	—
131	C₈×C₅₉	—	2	2	no	—
132	C₈×C₅₉	—	4	2	no	—
133	C₈×C₅₉	C₂×D₈	1	4	yes	—
134	C₅₉⋊C₈	—	4	4	no	—
135	C₂×#1669	—	59	1	no	maximal, Sylow
136	C₂²×C₅₉	—	2	4	no	—
137	C₂²×C₅₉	—	4	8	no	—
138	C₂²×C₅₉	—	4	2	no	—
139	C₂²×C₅₉	—	4	2	no	—
140	C₂²×C₅₉	#459	1	2	yes	—
141	C₂²×C₅₉	#467	1	2	yes	—
142	C₂²×C₅₉	#473	1	2	yes	—
143	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	—	4	2	no	—
147	C₄×C₅₉	#459	1	2	yes	—
148	C₄×C₅₉	#467	1	2	yes	—
149	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
151	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
152	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	Dic₅₉	—	8	4	no	—
154	C₂×#494	—	59	1	no	—
155	C₂×#474	—	59	2	no	—
156	C₂×C₈×D₄	D₅₉	1	1	yes	characteristic
157	#1669	—	59	8	no	—
158	C₂×#335	—	59	1	no	—
159	C₂×#332	—	59	2	no	—
160	C₂×C₅₉	—	4	4	no	—
161	C₂×C₅₉	#1669	1	4	yes	complemented
162	C₂×C₅₉	C₂×#459	1	1	yes	characteristic
163	C₂×C₅₉	C₂×#467	1	1	yes	characteristic
164	C₂×C₅₉	C₂×#473	1	1	yes	characteristic
165	C₂²×Q₁₆	—	118	1	no	—
166	C₂²×Q₁₆	—	118	1	no	—
167	C₂³×C₈	—	2	1	no	—
168	C₂×C₄⋊Q₈	—	59	2	no	—
169	C₂×C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
170	#494	—	59	4	no	—
171	#494	—	118	2	no	—
172	#474	—	59	8	no	—
173	#474	—	118	4	no	—
174	C₈×D₄	—	2	2	no	—
175	C₈×D₄	C₂×D₅₉	1	4	yes	—
176	C₂×#107	—	118	2	no	—
177	C₂×C₄⋊C₈	—	59	2	no	—
178	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
179	C₂×#102	—	118	2	no	—
180	C₂×C₂²⋊C₈	—	2	1	no	—
181	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
182	#335	—	59	4	no	—
183	#332	—	59	8	no	—
184	C₅₉	C₂×#1669	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	118	2	no	—
186	C₂²×D₄	D₂₃₆	1	1	yes	characteristic
187	C₂³×C₄	—	2	1	no	—
188	C₂×Q₁₆	—	118	4	no	—
189	C₂×Q₁₆	—	118	4	no	—
190	C₂×Q₁₆	—	236	4	no	—
191	C₂×Q₁₆	—	236	4	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	4	4	no	—
195	C₂²×C₈	—	118	2	no	—
196	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	59	4	no	—
199	C₄⋊Q₈	—	118	2	no	—
200	C₄⋊Q₈	—	118	2	no	—
201	C₄⋊Q₈	—	118	2	no	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	D₂₃₆	1	2	yes	—
205	C₄×D₄	C₂²×D₅₉	1	2	yes	—
206	C₂×C₄⋊C₄	—	118	2	no	—
207	C₂×C₄⋊C₄	—	118	2	no	—
208	C₂×C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
211	#107	—	118	8	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	59	4	no	—
214	C₄⋊C₈	—	118	2	no	—
215	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
216	#102	—	118	8	no	—
217	C₂²⋊C₈	—	2	4	no	—
218	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
219	C₄×C₈	C₂²×D₅₉	1	2	yes	—
220	C₂⁴	—	2	1	no	—
221	C₂×Q₈	—	118	4	no	—
222	C₂×Q₈	—	236	4	no	—
223	C₂×Q₈	—	236	4	no	—
224	C₂×Q₈	—	236	2	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	Dic₂₃₆	1	4	yes	complemented
228	C₂×D₄	C₂×D₂₃₆	1	2	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	236	2	no	—
235	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
236	C₂²×C₄	#11745685	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
238	Q₁₆	—	236	8	no	—
239	Q₁₆	—	236	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	8	no	—
244	C₂×C₈	—	118	4	no	—
245	C₂×C₈	—	236	4	no	—
246	C₂×C₈	#11745669	1	4	yes	—
247	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	118	4	no	—
251	C₄⋊C₄	—	118	4	no	—
252	C₄⋊C₄	—	236	2	no	—
253	C₄⋊C₄	—	236	2	no	—
254	C₄⋊C₄	C₂×D₂₃₆	1	2	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
258	C₄²	#11745685	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
260	C₄²	C₂³×D₅₉	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	#14273701	1	1	yes	characteristic
266	Q₈	—	472	4	no	—
267	Q₈	—	236	4	no	—
268	Q₈	—	236	4	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	D₄	C₂×Dic₂₃₆	1	4	yes	complemented
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	472	4	no	—
279	C₂×C₄	—	236	4	no	—
280	C₂×C₄	#14273701	1	1	yes	characteristic
281	C₂×C₄	#14273701	1	1	yes	characteristic
282	C₂×C₄	#14273701	1	1	yes	characteristic
283	C₂×C₄	C₂×#11745669	1	2	yes	—
284	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
285	C₂×C₄	#14273749	1	2	yes	—
286	C₂×C₄	#14273752	1	2	yes	—
287	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
289	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	236	4	no	—
293	C₈	C₂×#11745669	1	4	yes	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	#16837597	1	2	yes	—
299	C₂²	#16837643	1	2	yes	—
300	C₂²	#16837677	1	2	yes	—
301	C₂²	C₂×#14273701	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	472	4	no	—
306	C₄	#16837597	1	2	yes	—
307	C₄	#16837643	1	2	yes	—
308	C₄	C₂²×#11745669	1	1	yes	characteristic
309	C₄	C₂²×Dic₂₃₆	1	1	yes	characteristic
310	C₄	C₂×#14273749	1	1	yes	characteristic
311	C₄	C₂×#14273752	1	1	yes	characteristic
312	C₂	—	4	4	no	—
313	C₂	#21423461	1	4	yes	direct product factor
314	C₂	C₂×#16837597	1	1	yes	characteristic
315	C₂	C₂×#16837643	1	1	yes	characteristic
316	C₂	C₂×#16837677	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273701
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₈×D₄×C₅₉	C₂
socle	C₂³×C₅₉	#121
prime radical	C₂²×D₄×C₅₉	D₄
hypercenter	C₂×C₈×D₄	D₅₉
hypocenter	C₅₉	C₂×#1669
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16837782	C₂	semidirect
3. #21423461	C₂	direct (N × H)
4. C₂×#16837155	C₂	semidirect
5. C₂×#16837149	C₂	semidirect
11. #16837782	C₂²	semidirect
17. #16837155	C₂²	semidirect
18. #16837149	C₂²	semidirect
64. C₂×D₄×C₅₉	Q₁₆	semidirect
109. D₄×C₅₉	C₂×Q₁₆	semidirect
161. C₂×C₅₉	#1669	semidirect
184. C₅₉	C₂×#1669	semidirect
227. C₂×D₄	Dic₂₃₆	semidirect
271. D₄	C₂×Dic₂₃₆	semidirect
313. C₂	#21423461	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×#16837782	2
C₂×#16837682	2
#21423461	2
C₂×#16837155	2
C₂×#16837149	2
C₂×C₈×D₄×C₅₉	2
C₂×#1669	59

Subgroups — C2×#21423461 #28570924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21423461 #28570924