Subgroups — C₂×#21422087 #28498843

Order 15104.

480 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₉⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422087	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₉×#469	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11745664	—	2	1	no	—
6	C₂×#14273667	—	2	1	no	—
7	#16837184	—	2	1	no	—
8	#16837184	C₂²	1	2	yes	complemented
9	C₅₉⋊#338	—	2	1	no	—
10	C₅₉⋊#338	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₅₉	C₄	1	1	yes	characteristic
12	C₂²×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#121×C₅₉	—	2	1	no	—
14	C₅₉×#469	—	2	1	no	—
15	C₅₉×#469	—	2	1	no	—
16	C₅₉×#469	C₄	1	2	yes	—
17	C₅₉×#469	C₂²	1	2	yes	—
18	C₂×#106×C₅₉	C₄	1	1	yes	characteristic
19	C₂×#102×C₅₉	—	2	1	no	—
20	C₂×#11745664	—	2	2	no	—
21	C₂×#11745664	—	2	2	no	—
22	C₂×#11745664	—	4	1	no	—
23	C₂×#11745664	—	4	1	no	—
24	C₂×#11745664	—	4	1	no	—
25	C₂×#11745664	—	4	1	no	—
26	C₂²×C₅₉⋊C₈	—	2	2	no	—
27	#14273667	—	2	2	no	—
28	#14273667	—	4	1	no	—
29	C₂×C₅₉⋊C₁₆	—	4	1	no	—
30	C₂²×D₄×C₅₉	—	2	1	no	—
31	C₂²×D₄×C₅₉	—	4	1	no	—
32	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
33	C₂×D₈×C₅₉	—	2	1	no	—
34	C₂×D₈×C₅₉	—	4	1	no	—
35	C₂×D₈×C₅₉	—	4	1	no	—
36	C₂×D₈×C₅₉	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₅₉	—	4	1	no	—
38	C₂×M₁₆×C₅₉	—	4	1	no	—
39	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
42	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
45	#121×C₅₉	—	2	2	no	—
46	#121×C₅₉	—	2	2	no	—
47	#121×C₅₉	—	4	1	no	—
48	#121×C₅₉	—	4	1	no	—
49	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₅₉	—	4	1	no	—
51	#106×C₅₉	—	2	1	no	—
52	#106×C₅₉	C₂×C₄	1	2	yes	—
53	#102×C₅₉	—	2	2	no	—
54	#102×C₅₉	—	2	2	no	—
55	C₂⁴×C₅₉	—	4	1	no	—
56	C₂⁴×C₅₉	SD₁₆	1	1	yes	characteristic
57	C₂×D₄×C₅₉	—	2	2	no	—
58	C₂×D₄×C₅₉	—	4	2	no	—
59	C₂×D₄×C₅₉	—	4	2	no	—
60	C₂×D₄×C₅₉	—	4	1	no	—
61	C₂×D₄×C₅₉	—	4	1	no	—
62	C₂×D₄×C₅₉	—	4	1	no	—
63	C₂×D₄×C₅₉	—	4	1	no	—
64	C₂×D₄×C₅₉	—	4	1	no	—
65	C₂×D₄×C₅₉	—	8	4	no	—
66	C₂²×C₄×C₅₉	—	2	1	no	—
67	C₂²×C₄×C₅₉	—	2	1	no	—
68	C₂²×C₄×C₅₉	—	2	1	no	—
69	C₂²×C₄×C₅₉	—	2	1	no	—
70	C₂²×C₄×C₅₉	—	4	1	no	—
71	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	D₈×C₅₉	—	4	2	no	—
78	D₈×C₅₉	—	4	2	no	—
79	M₁₆×C₅₉	—	4	1	no	—
80	M₁₆×C₅₉	—	4	1	no	—
81	M₁₆×C₅₉	—	4	1	no	—
82	M₁₆×C₅₉	—	4	1	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	—	2	1	no	—
85	C₂×C₈×C₅₉	—	2	1	no	—
86	C₂×C₈×C₅₉	—	2	1	no	—
87	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₄⋊C₄×C₅₉	—	2	2	no	—
92	C₄⋊C₄×C₅₉	—	4	1	no	—
93	C₂²⋊C₄×C₅₉	—	4	2	no	—
94	C₂²⋊C₄×C₅₉	—	4	2	no	—
95	#11745664	—	4	2	no	—
96	#11745664	—	4	2	no	—
97	#11745664	—	4	2	no	—
98	#11745664	—	4	2	no	—
99	C₂×C₅₉⋊C₈	—	2	4	no	—
100	C₂×C₅₉⋊C₈	—	4	2	no	—
101	C₂×C₅₉⋊C₈	—	4	2	no	—
102	C₅₉⋊C₁₆	—	4	2	no	—
103	C₂³×C₅₉	—	2	1	no	—
104	C₂³×C₅₉	—	2	1	no	—
105	C₂³×C₅₉	—	4	2	no	—
106	C₂³×C₅₉	—	4	2	no	—
107	C₂³×C₅₉	—	8	2	no	—
108	C₂³×C₅₉	—	8	2	no	—
109	C₂³×C₅₉	—	8	1	no	—
110	C₂³×C₅₉	—	8	1	no	—
111	C₂³×C₅₉	#102	1	1	yes	characteristic
112	C₂³×C₅₉	#102	1	1	yes	characteristic
113	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
114	D₄×C₅₉	—	4	2	no	—
115	D₄×C₅₉	—	4	2	no	—
116	D₄×C₅₉	—	8	4	no	—
117	D₄×C₅₉	—	8	4	no	—
118	D₄×C₅₉	—	8	1	no	—
119	D₄×C₅₉	—	8	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	2	1	no	—
125	C₂×C₄×C₅₉	—	2	1	no	—
126	C₂×C₄×C₅₉	—	2	1	no	—
127	C₂×C₄×C₅₉	—	2	1	no	—
128	C₂×C₄×C₅₉	—	2	1	no	—
129	C₂×C₄×C₅₉	—	2	1	no	—
130	C₂×C₄×C₅₉	—	4	2	no	—
131	C₂×C₄×C₅₉	—	8	2	no	—
132	C₂×C₄×C₅₉	#99	1	1	yes	characteristic
133	C₂×C₄×C₅₉	#99	1	1	yes	characteristic
134	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
135	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
136	C₂×C₄×C₅₉	#104	1	1	yes	characteristic
137	C₂×C₄×C₅₉	#104	1	1	yes	characteristic
138	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₈×C₅₉	—	2	1	no	—
143	C₈×C₅₉	—	2	1	no	—
144	C₅₉⋊C₈	—	4	4	no	—
145	C₂×#1341	—	59	1	no	maximal, Sylow
146	C₂²×C₅₉	—	2	2	no	—
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	4	1	no	—
150	C₂²×C₅₉	—	4	1	no	—
151	C₂²×C₅₉	—	4	2	no	—
152	C₂²×C₅₉	—	8	1	no	—
153	C₂²×C₅₉	—	8	2	no	—
154	C₂²×C₅₉	—	8	2	no	—
155	C₂²×C₅₉	—	8	2	no	—
156	C₂²×C₅₉	—	16	2	no	—
157	C₂²×C₅₉	—	16	2	no	—
158	C₂²×C₅₉	#327	1	1	yes	characteristic
159	C₂²×C₅₉	#327	1	1	yes	characteristic
160	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
161	C₄×C₅₉	—	2	1	no	—
162	C₄×C₅₉	—	2	1	no	—
163	C₄×C₅₉	—	8	2	no	—
164	C₄×C₅₉	#327	1	1	yes	characteristic
165	C₄×C₅₉	#327	1	1	yes	characteristic
166	C₄×C₅₉	C₂×#99	1	1	yes	characteristic
167	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
168	C₂×#469	D₅₉	1	1	yes	characteristic
169	C₂×#349	—	59	1	no	—
170	C₂×#338	—	59	1	no	—
171	#1341	—	59	4	no	—
172	C₂×C₅₉	—	2	1	no	—
173	C₂×C₅₉	—	2	1	no	—
174	C₂×C₅₉	—	4	1	no	—
175	C₂×C₅₉	—	4	1	no	—
176	C₂×C₅₉	—	16	2	no	—
177	C₂×C₅₉	#1341	1	2	yes	complemented
178	C₂×C₅₉	C₂×#327	1	1	yes	characteristic
179	C₂²×D₈	Dic₅₉	1	1	yes	characteristic
180	C₂²×M₁₆	—	118	1	no	—
181	C₂²×M₁₆	C₂×D₅₉	1	1	yes	characteristic
182	C₂×#121	—	2	1	no	—
183	C₂×M₃₂	—	118	1	no	—
184	#469	—	2	1	no	—
185	#469	—	2	1	no	—
186	#469	Dic₅₉	1	2	yes	—
187	#469	C₂×D₅₉	1	2	yes	—
188	C₂×#106	Dic₅₉	1	1	yes	characteristic
189	C₂×#102	—	2	1	no	—
190	#349	—	59	2	no	—
191	#349	—	118	1	no	—
192	#338	—	59	2	no	—
193	#338	—	118	1	no	—
194	C₅₉	C₂×#1341	1	1	yes	characteristic, complemented, Sylow
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	4	1	no	—
197	C₂³×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	4	1	no	—
200	C₂×D₈	—	4	1	no	—
201	C₂×D₈	C₂×Dic₅₉	1	2	yes	—
202	C₂×M₁₆	—	4	1	no	—
203	C₂×M₁₆	—	4	1	no	—
204	C₂×M₁₆	—	118	2	no	—
205	C₂×M₁₆	—	118	2	no	—
206	C₂×M₁₆	—	236	1	no	—
207	C₂×M₁₆	—	236	1	no	—
208	C₂×M₁₆	—	236	1	no	—
209	C₂×M₁₆	—	236	1	no	—
210	C₂×M₁₆	C₂×Dic₅₉	1	1	yes	characteristic
211	C₂×M₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
212	C₂×M₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
213	C₂×M₁₆	C₂²×D₅₉	1	1	yes	characteristic
214	C₂²×C₈	—	118	2	no	—
215	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
216	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	#121	—	4	1	no	—
220	#121	—	4	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	4	1	no	—
223	M₃₂	—	118	2	no	—
224	M₃₂	—	236	1	no	—
225	C₂×C₁₆	—	236	1	no	—
226	#106	—	2	1	no	—
227	#106	C₂×Dic₅₉	1	2	yes	—
228	#102	—	2	2	no	—
229	#102	—	2	2	no	—
230	C₂⁴	—	4	1	no	—
231	C₂⁴	#11745670	1	1	yes	characteristic
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	8	4	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	#11745669	1	1	yes	characteristic
247	C₂²×C₄	#11745669	1	1	yes	characteristic
248	C₂²×C₄	#11745670	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
252	D₈	—	4	2	no	—
253	D₈	—	4	2	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	236	2	no	—
259	M₁₆	—	236	2	no	—
260	M₁₆	—	236	2	no	—
261	M₁₆	—	236	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	118	4	no	—
267	C₂×C₈	—	236	2	no	—
268	C₂×C₈	—	236	2	no	—
269	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₁₆	—	236	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	8	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	#14273648	1	1	yes	characteristic
287	C₂³	#14273648	1	1	yes	characteristic
288	C₂³	C₂×#11745670	1	1	yes	characteristic
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	4	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	1	no	—
294	D₄	—	8	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	#14273642	1	1	yes	characteristic
308	C₂×C₄	#14273642	1	1	yes	characteristic
309	C₂×C₄	#14273648	1	1	yes	characteristic
310	C₂×C₄	#14273648	1	1	yes	characteristic
311	C₂×C₄	#14273654	1	1	yes	characteristic
312	C₂×C₄	#14273654	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
314	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	236	4	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	16	2	no	—
331	C₂²	—	16	2	no	—
332	C₂²	#16837136	1	1	yes	characteristic
333	C₂²	#16837136	1	1	yes	characteristic
334	C₂²	C₂×#14273648	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	8	2	no	—
338	C₄	#16837136	1	1	yes	characteristic
339	C₄	#16837136	1	1	yes	characteristic
340	C₄	C₂×#14273642	1	1	yes	characteristic
341	C₄	C₂×#14273654	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	4	1	no	—
345	C₂	—	4	1	no	—
346	C₂	—	16	2	no	—
347	C₂	#21422087	1	2	yes	direct product factor
348	C₂	C₂×#16837136	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837136
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₅₉
Fitting, prime radical	C₂×C₅₉×#469	C₂
socle	C₂²×C₅₉	#327
cosocle	C₂×M₁₆×C₅₉	C₂³
hypercenter	C₂×#469	D₅₉
hypocenter	C₅₉	C₂×#1341
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#469

Complements

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

N	H ≅ G/N	type
1. C₂×#16837184	C₂	semidirect
2. C₂×C₅₉⋊#338	C₂	semidirect
3. #21422087	C₂	direct (N × H)
8. #16837184	C₂²	semidirect
10. C₅₉⋊#338	C₂²	semidirect
177. C₂×C₅₉	#1341	semidirect
194. C₅₉	C₂×#1341	semidirect
347. C₂	#21422087	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×#16837184	2
C₂×C₅₉⋊#338	2
#21422087	2
C₂×C₅₉×#469	2
C₂×#1341	59

Subgroups — C2×#21422087 #28498843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422087 #28498843