Subgroups — C₂×#21423580 #28571289

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837779	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837602	C₂	1	2	yes	maximal, complemented
3	#21423580	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837153	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837145	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11745671	—	2	2	no	—
8	C₂²×D₄₇₂	—	2	1	no	—
9	C₂²×D₄₇₂	—	2	1	no	—
10	C₂×#14273724	C₂²	1	2	yes	—
11	#16837779	—	2	2	no	—
12	#16837779	C₂²	1	4	yes	—
13	#16837602	—	2	4	no	—
14	#16837602	C₂²	1	8	yes	complemented
15	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×#14273647	—	2	2	no	—
17	#16837153	C₂²	1	4	yes	complemented
18	#16837145	C₂²	1	8	yes	complemented
19	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉×#446	—	2	2	no	—
21	C₅₉×#446	C₂²	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
23	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×#97×C₅₉	—	2	1	no	—
25	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂²×D₂₃₆	—	2	2	no	—
27	C₂²×D₂₃₆	—	2	2	no	—
28	C₂²×D₂₃₆	—	2	2	no	—
29	C₂×#11745671	—	2	8	no	—
30	C₂×#11745671	—	4	8	no	—
31	C₂×D₄₇₂	—	2	4	no	—
32	C₂×D₄₇₂	—	2	4	no	—
33	C₂×D₄₇₂	—	4	4	no	—
34	C₂×D₄₇₂	—	4	4	no	—
35	C₂²×C₅₉⋊C₈	—	2	2	no	—
36	#14273724	—	2	2	no	—
37	#14273724	—	2	2	no	—
38	#14273724	—	2	2	no	—
39	#14273724	C₂³	1	4	yes	—
40	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
41	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
42	#14273647	—	2	8	no	—
43	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₉	—	2	1	no	—
45	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₄×Q₈×C₅₉	—	2	1	no	—
48	C₄×Q₈×C₅₉	—	2	1	no	—
49	C₄×Q₈×C₅₉	D₄	1	2	yes	—
50	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₉	—	2	1	no	—
54	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₉	—	2	4	no	—
56	C₄⋊C₈×C₅₉	—	2	1	no	—
57	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
58	#97×C₅₉	—	2	4	no	—
59	C₄×C₈×C₅₉	D₄	1	2	yes	—
60	C₄×C₈×C₅₉	C₂³	1	2	yes	—
61	C₂³×D₅₉	—	4	2	no	—
62	C₂×Q₈×C₅₉	—	2	1	no	—
63	C₂×Q₈×C₅₉	—	2	4	no	—
64	C₂×Q₈×C₅₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₉	—	2	1	no	—
66	C₂²×C₄×C₅₉	—	2	1	no	—
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
70	C₂×D₂₃₆	—	2	4	no	—
71	C₂×D₂₃₆	—	2	4	no	—
72	C₂×D₂₃₆	—	2	4	no	—
73	C₂×D₂₃₆	—	4	4	no	—
74	C₂×D₂₃₆	—	4	4	no	—
75	C₂×D₂₃₆	—	4	4	no	—
76	C₂×D₂₃₆	—	4	4	no	—
77	C₂×D₂₃₆	—	4	2	no	—
78	C₂×D₂₃₆	—	4	2	no	—
79	C₂×D₂₃₆	—	4	2	no	—
80	C₂×D₂₃₆	—	8	4	no	—
81	C₂×C₈×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	4	2	no	—
84	C₂×C₈×C₅₉	SD₁₆	1	4	yes	complemented
85	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₅₉	—	2	2	no	—
88	C₄⋊C₄×C₅₉	—	2	1	no	—
89	C₄⋊C₄×C₅₉	—	2	2	no	—
90	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
91	C₄²×C₅₉	—	2	2	no	—
92	C₄²×C₅₉	—	2	2	no	—
93	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅₉	#41	1	1	yes	characteristic
96	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
97	#11745671	—	4	16	no	—
98	C₂×C₅₉⋊C₈	—	2	4	no	—
99	C₂×C₅₉⋊C₈	—	4	4	no	—
100	D₄₇₂	—	4	8	no	—
101	D₄₇₂	—	4	8	no	—
102	C₂³×C₅₉	#121	1	1	yes	characteristic
103	C₂²×D₅₉	—	4	4	no	—
104	C₂²×D₅₉	—	8	4	no	—
105	C₂²×D₅₉	—	8	4	no	—
106	C₂²×D₅₉	—	8	4	no	—
107	Q₈×C₅₉	—	2	4	no	—
108	Q₈×C₅₉	—	4	2	no	—
109	C₂×C₄×C₅₉	—	2	2	no	—
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	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
120	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
123	D₂₃₆	—	4	4	no	—
124	D₂₃₆	—	4	4	no	—
125	D₂₃₆	—	4	4	no	—
126	D₂₃₆	—	4	4	no	—
127	D₂₃₆	—	8	4	no	—
128	D₂₃₆	—	8	4	no	—
129	D₂₃₆	—	8	4	no	—
130	D₂₃₆	—	8	4	no	—
131	C₈×C₅₉	—	2	2	no	—
132	C₈×C₅₉	—	4	2	no	—
133	C₈×C₅₉	C₂×SD₁₆	1	4	yes	complemented
134	C₅₉⋊C₈	—	4	4	no	—
135	C₂×#1688	—	59	1	no	maximal, Sylow
136	C₂²×C₅₉	#460	1	2	yes	—
137	C₂²×C₅₉	#469	1	2	yes	—
138	C₂²×C₅₉	#472	1	2	yes	—
139	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
140	C₂×D₅₉	—	8	4	no	—
141	C₂×D₅₉	—	8	4	no	—
142	C₂×D₅₉	—	8	4	no	—
143	C₂×D₅₉	—	16	8	no	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	—	4	2	no	—
147	C₄×C₅₉	—	4	2	no	—
148	C₄×C₅₉	#460	1	2	yes	—
149	C₄×C₅₉	#469	1	2	yes	—
150	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
151	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
153	C₂×#493	—	59	1	no	—
154	C₂×#460	—	59	2	no	—
155	C₂×#446	D₅₉	1	1	yes	characteristic, complemented
156	#1688	—	59	8	no	—
157	C₂×#334	—	59	1	no	—
158	C₂×#331	—	59	2	no	—
159	C₂×C₅₉	#1688	1	4	yes	complemented
160	C₂×C₅₉	C₂×#460	1	1	yes	characteristic
161	C₂×C₅₉	C₂×#469	1	1	yes	characteristic
162	C₂×C₅₉	C₂×#472	1	1	yes	characteristic
163	D₅₉	—	16	4	no	—
164	C₂²×SD₁₆	—	118	2	no	—
165	C₂²×D₈	—	118	1	no	—
166	C₂²×D₈	—	118	1	no	—
167	C₂×#127	—	59	2	no	—
168	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
169	#493	—	59	4	no	—
170	#493	—	118	2	no	—
171	#460	—	59	8	no	—
172	#460	—	118	4	no	—
173	#446	—	2	2	no	—
174	#446	C₂×D₅₉	1	4	yes	complemented
175	C₂×C₄⋊C₈	—	2	1	no	—
176	C₂×C₄⋊C₈	—	59	2	no	—
177	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
178	C₂×#102	—	118	2	no	—
179	C₂×#97	—	2	1	no	—
180	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
181	#334	—	59	4	no	—
182	#331	—	59	8	no	—
183	C₅₉	C₂×#1688	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₈	D₂₃₆	1	1	yes	characteristic
185	C₂²×D₄	—	118	2	no	—
186	C₂²×D₄	—	118	2	no	—
187	C₂²×D₄	—	118	2	no	—
188	C₂×SD₁₆	—	118	8	no	—
189	C₂×SD₁₆	—	236	8	no	—
190	C₂×D₈	—	118	4	no	—
191	C₂×D₈	—	118	4	no	—
192	C₂×D₈	—	236	4	no	—
193	C₂×D₈	—	236	4	no	—
194	C₂²×C₈	—	2	1	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	#127	—	59	4	no	—
199	#127	—	118	2	no	—
200	#127	—	118	2	no	—
201	#127	—	118	2	no	—
202	C₄×Q₈	—	2	1	no	—
203	C₄×Q₈	—	2	1	no	—
204	C₄×Q₈	D₂₃₆	1	2	yes	—
205	C₄×Q₈	C₂²×D₅₉	1	2	yes	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
210	C₄⋊C₈	—	2	4	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	59	4	no	—
213	C₄⋊C₈	—	118	2	no	—
214	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
215	#102	—	118	8	no	—
216	#97	—	2	4	no	—
217	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
218	C₄×C₈	C₂²×D₅₉	1	2	yes	—
219	C₂⁴	—	236	2	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	C₂×D₂₃₆	1	2	yes	—
223	C₂×D₄	—	118	4	no	—
224	C₂×D₄	—	118	4	no	—
225	C₂×D₄	—	118	4	no	—
226	C₂×D₄	—	472	4	no	—
227	C₂×D₄	—	236	2	no	—
228	C₂×D₄	—	236	2	no	—
229	C₂×D₄	—	236	2	no	—
230	C₂×D₄	—	236	4	no	—
231	C₂×D₄	—	236	4	no	—
232	C₂×D₄	—	236	4	no	—
233	C₂×D₄	—	236	4	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
237	C₂²×C₄	#11745685	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
239	SD₁₆	—	236	16	no	—
240	D₈	—	236	8	no	—
241	D₈	—	236	8	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	118	4	no	—
246	C₂×C₈	—	236	4	no	—
247	C₂×C₈	#11745671	1	4	yes	complemented
248	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	C₂×D₂₃₆	1	2	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
257	C₄²	#11745685	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
259	C₄²	C₂³×D₅₉	1	1	yes	characteristic
260	C₂³	—	472	4	no	—
261	C₂³	—	472	4	no	—
262	C₂³	—	472	4	no	—
263	C₂³	—	236	4	no	—
264	C₂³	#14273701	1	1	yes	characteristic
265	Q₈	—	2	4	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	472	4	no	—
268	D₄	—	472	4	no	—
269	D₄	—	472	4	no	—
270	D₄	—	472	4	no	—
271	D₄	—	236	4	no	—
272	D₄	—	236	4	no	—
273	D₄	—	236	4	no	—
274	D₄	—	236	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	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×#11745671	1	2	yes	—
284	C₂×C₄	#14273752	1	2	yes	—
285	C₂×C₄	#14273755	1	4	yes	—
286	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
288	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	236	4	no	—
292	C₈	C₂×#11745671	1	4	yes	complemented
293	C₂²	—	944	8	no	—
294	C₂²	—	472	4	no	—
295	C₂²	—	472	4	no	—
296	C₂²	—	472	4	no	—
297	C₂²	#16837602	1	2	yes	—
298	C₂²	#16837655	1	2	yes	—
299	C₂²	#16837672	1	2	yes	—
300	C₂²	C₂×#14273701	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	#16837602	1	2	yes	—
306	C₄	#16837655	1	2	yes	—
307	C₄	C₂²×#11745671	1	1	yes	characteristic
308	C₄	C₂×#14273752	1	1	yes	characteristic
309	C₄	C₂×#14273755	1	2	yes	—
310	C₂	—	944	4	no	—
311	C₂	#21423580	1	4	yes	direct product factor
312	C₂	C₂×#16837602	1	1	yes	characteristic
313	C₂	C₂×#16837655	1	1	yes	characteristic
314	C₂	C₂×#16837672	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273701
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#446	C₂
socle	C₂³×C₅₉	#121
prime radical	C₂×#16837779	C₂
hypercenter	C₂×#446	D₅₉
hypocenter	C₅₉	C₂×#1688
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
2. C₂×#16837602	C₂	semidirect
3. #21423580	C₂	direct (N × H)
4. C₂×#16837153	C₂	semidirect
5. C₂×#16837145	C₂	semidirect
6. C₂×C₅₉×#446	C₂	semidirect
14. #16837602	C₂²	semidirect
17. #16837153	C₂²	semidirect
18. #16837145	C₂²	semidirect
21. C₅₉×#446	C₂²	semidirect
84. C₂×C₈×C₅₉	SD₁₆	semidirect
133. C₈×C₅₉	C₂×SD₁₆	semidirect
155. C₂×#446	D₅₉	semidirect
159. C₂×C₅₉	#1688	semidirect
174. #446	C₂×D₅₉	semidirect
183. C₅₉	C₂×#1688	semidirect
247. C₂×C₈	#11745671	semidirect
292. C₈	C₂×#11745671	semidirect
311. C₂	#21423580	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×#16837779	2
C₂×#16837602	2
#21423580	2
C₂×#16837153	2
C₂×#16837145	2
C₂×C₅₉×#446	2
C₂×#1688	59

Subgroups — C2×#21423580 #28571289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21423580 #28571289