Subgroups — C₂×#21425985 #28742120

Order 15104.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837843	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837525	C₂	1	1	yes	maximal, characteristic
3	#21425985	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837224	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837218	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837211	C₂	1	1	yes	maximal, characteristic
7	C₂×#16837180	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₉×#441	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₂²	1	1	yes	characteristic
10	C₂×#14273733	—	2	1	no	—
11	#16837843	C₄	1	4	yes	complemented
12	#16837843	C₂²	1	4	yes	—
13	C₂×#14273673	C₄	1	2	yes	complemented
14	C₂×#14273670	C₄	1	2	yes	complemented
15	C₂×#14273667	C₂²	1	1	yes	characteristic
16	C₂×#14273667	C₂²	1	1	yes	characteristic
17	#16837525	—	2	2	no	—
18	#16837525	C₂²	1	4	yes	—
19	C₂×#14273663	C₂²	1	1	yes	characteristic
20	C₂×#14273655	—	2	1	no	—
21	#16837224	C₂²	1	4	yes	complemented
22	#16837218	C₂²	1	4	yes	complemented
23	#16837211	C₂²	1	4	yes	—
24	#16837180	C₂²	1	4	yes	complemented
25	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₅₉	—	2	1	no	—
27	C₅₉×#441	—	2	2	no	—
28	C₅₉×#441	C₂²	1	4	yes	complemented
29	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×#103×C₅₉	—	2	1	no	—
31	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×#11745685	—	2	1	no	—
33	#14273749	—	2	1	no	—
34	#14273749	—	2	1	no	—
35	#14273749	C₂×C₄	1	2	yes	—
36	#14273749	C₂³	1	2	yes	—
37	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
38	C₂×#11745660	—	2	1	no	—
39	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
40	#14273733	—	2	4	no	—
41	#14273733	—	4	2	no	—
42	C₂×#11745664	—	2	1	no	—
43	C₂×#11745659	—	4	1	no	—
44	C₂×C₈×D₅₉	—	4	1	no	—
45	#14273673	C₂×C₄	1	8	yes	complemented
46	#14273670	C₂×C₄	1	8	yes	complemented
47	#14273667	—	2	1	no	—
48	#14273667	C₂×C₄	1	2	yes	—
49	#14273667	C₂³	1	2	yes	—
50	#14273667	C₂³	1	2	yes	—
51	C₂×C₅₉⋊C₁₆	—	2	1	no	—
52	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
53	#14273663	—	2	1	no	—
54	#14273663	C₂³	1	2	yes	—
55	#14273655	—	2	4	no	—
56	C₂²×Q₈×C₅₉	—	2	1	no	—
57	C₂×Q₁₆×C₅₉	—	2	1	no	—
58	C₂×Q₁₆×C₅₉	—	2	1	no	—
59	C₂×Q₁₆×C₅₉	—	2	1	no	—
60	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
61	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
62	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
63	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₅₉	—	2	4	no	—
66	C₄×Q₈×C₅₉	—	4	2	no	—
67	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
68	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
69	C₂×C₄²×C₅₉	—	4	1	no	—
70	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
71	#106×C₅₉	—	2	1	no	—
72	#106×C₅₉	C₂³	1	2	yes	—
73	#103×C₅₉	—	2	4	no	—
74	#97×C₅₉	D₄	1	2	yes	—
75	#97×C₅₉	C₂³	1	2	yes	—
76	C₂×Q₈×C₅₉	—	2	2	no	—
77	C₂×Q₈×C₅₉	—	2	2	no	—
78	C₂×Q₈×C₅₉	—	2	2	no	—
79	C₂×Q₈×C₅₉	—	4	1	no	—
80	C₂×Q₈×C₅₉	—	4	1	no	—
81	C₂×Q₈×C₅₉	—	4	1	no	—
82	C₂²×C₄×C₅₉	—	2	1	no	—
83	C₂²×C₄×C₅₉	—	4	1	no	—
84	C₂²×C₄×C₅₉	—	4	1	no	—
85	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₂²⋊D₅₉	—	4	1	no	—
87	#11745685	—	2	2	no	—
88	#11745685	—	2	2	no	—
89	#11745685	—	4	1	no	—
90	#11745685	—	4	1	no	—
91	C₂×D₂₃₆	—	2	1	no	—
92	C₂×C₄×D₅₉	—	4	1	no	—
93	C₂×Dic₁₁₈	—	2	1	no	—
94	Q₁₆×C₅₉	—	2	2	no	—
95	Q₁₆×C₅₉	—	2	2	no	—
96	Q₁₆×C₅₉	—	2	2	no	—
97	Q₁₆×C₅₉	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₅₉	—	2	1	no	—
99	C₂×C₈×C₅₉	—	2	1	no	—
100	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
105	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
106	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₅₉	—	2	2	no	—
108	C₄⋊C₄×C₅₉	—	4	4	no	—
109	C₄⋊C₄×C₅₉	—	4	1	no	—
110	C₄²×C₅₉	—	2	1	no	—
111	C₄²×C₅₉	—	4	4	no	—
112	C₄²×C₅₉	C₂×D₄	1	2	yes	—
113	#11745664	—	2	2	no	—
114	#11745664	—	4	1	no	—
115	C₂×C₅₉⋊C₈	—	4	1	no	—
116	Dic₂₃₆	—	2	1	no	—
117	Dic₂₃₆	C₂²×C₄	1	2	yes	—
118	D₄₇₂	—	2	1	no	—
119	D₄₇₂	C₂²×C₄	1	2	yes	—
120	#11745660	—	2	2	no	—
121	#11745660	—	2	2	no	—
122	#11745659	—	4	4	no	—
123	C₈×D₅₉	—	4	4	no	—
124	C₅₉⋊C₁₆	—	2	2	no	—
125	C₅₉⋊C₁₆	C₂²×C₄	1	4	yes	—
126	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²×D₅₉	—	4	1	no	—
128	Q₈×C₅₉	—	2	4	no	—
129	Q₈×C₅₉	—	4	2	no	—
130	Q₈×C₅₉	—	4	2	no	—
131	Q₈×C₅₉	—	4	2	no	—
132	C₂×C₄×C₅₉	—	2	2	no	—
133	C₂×C₄×C₅₉	—	4	2	no	—
134	C₂×C₄×C₅₉	—	4	2	no	—
135	C₂×C₄×C₅₉	—	4	2	no	—
136	C₂×C₄×C₅₉	—	4	2	no	—
137	C₂×C₄×C₅₉	—	4	2	no	—
138	C₂×C₄×C₅₉	—	8	2	no	—
139	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₅₉	—	4	2	no	—
146	C₂²⋊D₅₉	—	4	2	no	—
147	C₂×Dic₅₉	—	4	1	no	—
148	D₂₃₆	—	2	2	no	—
149	D₂₃₆	—	4	1	no	—
150	C₄×D₅₉	—	4	2	no	—
151	C₄×D₅₉	—	4	2	no	—
152	Dic₁₁₈	—	2	2	no	—
153	Dic₁₁₈	—	4	1	no	—
154	C₈×C₅₉	—	2	2	no	—
155	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
156	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
157	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
158	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
159	C₅₉⋊C₈	—	4	2	no	—
160	C₂×#2180	—	59	1	no	maximal, Sylow
161	C₂²×C₅₉	—	2	2	no	—
162	C₂²×C₅₉	#442	1	1	yes	characteristic
163	C₂²×C₅₉	#442	1	1	yes	characteristic
164	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×D₅₉	—	4	2	no	—
166	C₂×D₅₉	—	8	2	no	—
167	C₄×C₅₉	—	4	2	no	—
168	C₄×C₅₉	—	4	4	no	—
169	C₄×C₅₉	—	8	2	no	—
170	C₄×C₅₉	#442	1	1	yes	characteristic
171	C₄×C₅₉	#442	1	1	yes	characteristic
172	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	Dic₅₉	—	4	2	no	—
175	C₂×#510	—	59	1	no	—
176	C₂×#444	—	59	1	no	—
177	C₂×#441	D₅₉	1	1	yes	characteristic, complemented
178	#2180	—	59	8	no	—
179	C₂×#365	—	59	1	no	—
180	C₂×#362	—	59	1	no	—
181	C₂×#360	—	59	1	no	—
182	C₂×#347	—	59	1	no	—
183	C₂×C₅₉	—	2	1	no	—
184	C₂×C₅₉	—	2	1	no	—
185	C₂×C₅₉	#2180	1	2	yes	complemented
186	C₂×C₅₉	C₂×#442	1	1	yes	characteristic
187	D₅₉	—	8	2	no	—
188	C₂×#135	—	59	1	no	—
189	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
190	C₂×#131	—	118	1	no	—
191	C₂×C₄×Q₈	—	2	1	no	—
192	#510	—	59	4	no	—
193	#510	—	59	4	no	—
194	C₂×Q₃₂	—	59	2	no	—
195	C₂×SD₃₂	—	59	2	no	—
196	C₂×M₃₂	—	59	1	no	—
197	C₂×M₃₂	—	59	1	no	—
198	#444	—	59	4	no	—
199	#444	—	118	2	no	—
200	#441	—	2	2	no	—
201	#441	C₂×D₅₉	1	4	yes	complemented
202	C₂×#108	—	59	1	no	—
203	C₂×#106	C₂×D₅₉	1	1	yes	characteristic
204	C₂×#104	—	118	1	no	—
205	C₂×#103	—	2	1	no	—
206	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
207	#365	—	59	4	no	—
208	#362	—	59	4	no	—
209	#360	—	59	4	no	—
210	#347	—	59	4	no	—
211	C₅₉	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	—	118	1	no	—
213	C₂²×Q₈	—	2	1	no	—
214	#135	—	59	2	no	—
215	#135	—	59	2	no	—
216	#135	—	118	1	no	—
217	#135	—	118	1	no	—
218	C₂×Q₁₆	—	2	1	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	—	59	1	no	—
222	C₂×Q₁₆	C₄×D₅₉	1	2	yes	—
223	C₂×Q₁₆	C₄×D₅₉	1	2	yes	—
224	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
225	C₂×SD₁₆	—	118	1	no	—
226	C₂×D₈	—	59	1	no	—
227	#131	—	118	4	no	—
228	#131	—	236	2	no	—
229	C₂×M₁₆	—	118	1	no	—
230	C₂×M₁₆	—	236	1	no	—
231	C₂²×C₈	—	236	1	no	—
232	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₄×Q₈	—	2	4	no	—
235	C₄×Q₈	—	4	2	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	4	1	no	—
238	C₂×C₄²	—	4	1	no	—
239	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
240	Q₃₂	—	59	8	no	—
241	SD₃₂	—	59	8	no	—
242	M₃₂	—	59	2	no	—
243	M₃₂	—	59	2	no	—
244	M₃₂	—	59	2	no	—
245	M₃₂	—	118	1	no	—
246	C₂×C₁₆	—	59	2	no	—
247	C₂×C₁₆	—	118	1	no	—
248	#108	—	59	2	no	—
249	#108	—	118	1	no	—
250	#106	—	2	1	no	—
251	#106	C₂²×D₅₉	1	2	yes	—
252	#104	—	118	4	no	—
253	#103	—	2	4	no	—
254	#97	C₂²⋊D₅₉	1	2	yes	—
255	#97	C₂²×D₅₉	1	2	yes	—
256	#41	—	118	2	no	—
257	#41	—	118	2	no	—
258	#41	—	236	1	no	—
259	#41	—	236	1	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	118	1	no	—
267	C₂×D₄	—	118	1	no	—
268	C₂×D₄	—	236	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	236	1	no	—
273	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	59	2	no	—
278	Q₁₆	—	118	1	no	—
279	Q₁₆	C₂×C₄×D₅₉	1	4	yes	—
280	SD₁₆	—	118	2	no	—
281	SD₁₆	—	118	2	no	—
282	D₈	—	59	2	no	—
283	D₈	—	118	1	no	—
284	M₁₆	—	118	2	no	—
285	M₁₆	—	236	4	no	—
286	M₁₆	—	236	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	236	4	no	—
290	C₂×C₈	—	236	1	no	—
291	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
293	C₂×C₈	#11745685	1	1	yes	characteristic
294	C₂×C₈	#11745685	1	1	yes	characteristic
295	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
296	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	4	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	4	4	no	—
303	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
304	C₁₆	—	59	4	no	—
305	C₁₆	—	118	2	no	—
306	C₂³	—	236	1	no	—
307	C₂³	#14273753	1	1	yes	characteristic
308	Q₈	—	2	4	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	118	2	no	—
313	Q₈	—	236	1	no	—
314	D₄	—	118	2	no	—
315	D₄	—	236	2	no	—
316	D₄	—	236	2	no	—
317	D₄	—	236	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	236	1	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	—	236	2	no	—
328	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
329	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
330	C₂×C₄	#14273752	1	1	yes	characteristic
331	C₂×C₄	#14273752	1	1	yes	characteristic
332	C₂×C₄	#14273753	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	236	2	no	—
336	C₈	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
337	C₈	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
339	C₈	C₂×#11745685	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	472	2	no	—
342	C₂²	—	236	2	no	—
343	C₂²	#16837512	1	1	yes	characteristic
344	C₂²	#16837512	1	1	yes	characteristic
345	C₂²	C₂×#14273753	1	1	yes	characteristic
346	C₄	—	4	2	no	—
347	C₄	—	4	4	no	—
348	C₄	—	8	2	no	—
349	C₄	—	236	2	no	—
350	C₄	#16837512	1	1	yes	characteristic
351	C₄	#16837512	1	1	yes	characteristic
352	C₄	C₂×C₄×C₂²⋊D₅₉	1	1	yes	characteristic
353	C₄	C₂×#14273752	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	472	2	no	—
357	C₂	#21425985	1	2	yes	direct product factor
358	C₂	C₂×#16837512	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837512
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#441	C₂
socle	C₂²×C₅₉	#442
prime radical	C₂×#14273749	C₂²
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#441	D₅₉
hypocenter	C₅₉	C₂×#2180
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#441

Complements

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

N	H ≅ G/N	type
3. #21425985	C₂	direct (N × H)
4. C₂×#16837224	C₂	semidirect
5. C₂×#16837218	C₂	semidirect
7. C₂×#16837180	C₂	semidirect
8. C₂×C₅₉×#441	C₂	semidirect
11. #16837843	C₄	semidirect
13. C₂×#14273673	C₄	semidirect
14. C₂×#14273670	C₄	semidirect
21. #16837224	C₂²	semidirect
22. #16837218	C₂²	semidirect
24. #16837180	C₂²	semidirect
28. C₅₉×#441	C₂²	semidirect
45. #14273673	C₂×C₄	semidirect
46. #14273670	C₂×C₄	semidirect
177. C₂×#441	D₅₉	semidirect
185. C₂×C₅₉	#2180	semidirect
201. #441	C₂×D₅₉	semidirect
211. C₅₉	C₂×#2180	semidirect
357. C₂	#21425985	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×#16837843	2
C₂×#16837525	2
#21425985	2
C₂×#16837224	2
C₂×#16837218	2
C₂×#16837211	2
C₂×#16837180	2
C₂×C₅₉×#441	2
C₂×#2180	59

Subgroups — C2×#21425985 #28742120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425985 #28742120