Subgroups — #28592112

Order 15104.

882 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273663	C₂	1	1	yes	maximal, characteristic
2	#21432660	C₂	1	2	yes	maximal
3	C₂×#16837357	C₂	1	1	yes	maximal, characteristic
4	#21424850	C₂	1	4	yes	maximal, complemented
5	#21424845	C₂	1	4	yes	maximal, complemented
6	#21423805	C₂	1	2	yes	maximal
7	C₂×C₅₉×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11745664	C₂²	1	2	yes	—
9	C₂²×#11745664	C₂²	1	1	yes	characteristic
10	C₂×#14273686	C₂²	1	1	yes	characteristic
11	C₂×#14273663	—	2	4	no	—
12	C₂×#14273663	C₂²	1	2	yes	—
13	C₂×#14273663	C₂²	1	2	yes	—
14	#16837429	—	2	4	no	—
15	#16837429	C₂²	1	8	yes	—
16	#16837426	—	2	4	no	—
17	#16837357	—	2	4	no	—
18	C₂×#14273637	C₂²	1	2	yes	—
19	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
20	#16837161	C₂²	1	8	yes	—
21	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#117×C₅₉	C₂²	1	2	yes	—
23	C₅₉×#427	C₄	1	2	yes	complemented
24	C₅₉×#427	C₄	1	2	yes	complemented
25	C₅₉×#427	C₂²	1	2	yes	—
26	C₅₉×#427	C₂²	1	2	yes	—
27	C₂×#107×C₅₉	C₄	1	2	yes	complemented
28	C₂×#106×C₅₉	C₄	1	2	yes	complemented
29	C₂×#11745664	—	2	2	no	—
30	C₂×#11745664	—	2	2	no	—
31	C₂×#11745664	—	2	2	no	—
32	C₂×#11745664	—	2	4	no	—
33	C₂×#11745664	—	2	4	no	—
34	C₂×#11745664	D₄	1	2	yes	—
35	C₂×#11745664	D₄	1	2	yes	—
36	C₂×#11745664	C₂³	1	4	yes	—
37	C₂²×C₅₉⋊C₈	—	2	2	no	—
38	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
39	#14273686	—	2	2	no	—
40	#14273686	Q₈	1	4	yes	—
41	C₂×C₄⋊Dic₅₉	—	2	1	no	—
42	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
43	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
44	#14273663	—	2	4	no	—
45	#14273663	—	2	4	no	—
46	C₄⋊(C₅₉⋊C₈)	—	2	16	no	—
47	#14273637	—	2	4	no	—
48	C₈×Dic₅₉	—	2	4	no	—
49	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₉	D₄	1	4	yes	—
55	C₂²×C₈×C₅₉	Q₈	1	4	yes	—
56	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
58	#117×C₅₉	—	2	2	no	—
59	#117×C₅₉	—	2	2	no	—
60	#117×C₅₉	C₂×C₄	1	4	yes	—
61	#117×C₅₉	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
63	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
64	C₂×C₄²×C₅₉	—	2	2	no	—
65	#107×C₅₉	C₂×C₄	1	4	yes	—
66	#107×C₅₉	C₂×C₄	1	4	yes	—
67	#106×C₅₉	C₂×C₄	1	4	yes	—
68	#106×C₅₉	C₂×C₄	1	4	yes	—
69	C₂⁴×C₅₉	#41	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	—	2	4	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	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
79	C₂²×C₄×C₅₉	#41	1	2	yes	—
80	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
83	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
84	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
85	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
86	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
91	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
94	C₄⋊C₄×C₅₉	—	2	4	no	—
95	C₄⋊C₄×C₅₉	C₂²×C₄	1	8	yes	—
96	C₂²⋊C₄×C₅₉	—	2	4	no	—
97	C₂²⋊C₄×C₅₉	—	2	4	no	—
98	C₄²×C₅₉	—	2	4	no	—
99	C₄²×C₅₉	—	2	4	no	—
100	C₂²⋊Dic₅₉	—	2	4	no	—
101	C₄⋊Dic₅₉	—	2	4	no	—
102	C₄×Dic₅₉	—	2	2	no	—
103	C₄×Dic₅₉	C₂×Q₈	1	4	yes	—
104	#11745664	—	2	4	no	—
105	#11745664	—	2	4	no	—
106	#11745664	—	2	4	no	—
107	#11745664	—	2	4	no	—
108	#11745664	—	4	4	no	—
109	C₂×C₅₉⋊C₈	—	2	2	no	—
110	C₂×C₅₉⋊C₈	—	2	2	no	—
111	C₂×C₅₉⋊C₈	—	2	4	no	—
112	C₂×C₅₉⋊C₈	—	4	4	no	—
113	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
114	C₂³×C₅₉	—	4	2	no	—
115	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
116	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
117	C₂³×C₅₉	C₂²⋊Q₈	1	2	yes	—
118	C₂³×C₅₉	#125	1	2	yes	—
119	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₅₉	—	2	8	no	—
121	C₂×C₄×C₅₉	—	8	4	no	—
122	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₅₉	#121	1	2	yes	—
129	C₂×C₄×C₅₉	#121	1	2	yes	—
130	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅₉	#125	1	2	yes	—
132	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
134	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₅₉	—	2	4	no	—
141	C₂×Dic₅₉	—	4	4	no	—
142	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
143	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
144	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
145	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
146	C₅₉⋊C₈	—	2	4	no	—
147	C₅₉⋊C₈	—	4	4	no	—
148	#21163	—	59	1	no	maximal, Sylow
149	C₂²×C₅₉	—	2	2	no	—
150	C₂²×C₅₉	—	2	2	no	—
151	C₂²×C₅₉	—	2	2	no	—
152	C₂²×C₅₉	—	4	2	no	—
153	C₂²×C₅₉	—	4	2	no	—
154	C₂²×C₅₉	#389	1	2	yes	—
155	C₂²×C₅₉	#389	1	2	yes	—
156	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₅₉	C₂×#125	1	1	yes	characteristic
159	C₄×C₅₉	—	4	8	no	—
160	C₄×C₅₉	#389	1	2	yes	—
161	C₄×C₅₉	#389	1	2	yes	—
162	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₅₉	C₂×#121	1	1	yes	characteristic
165	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	Dic₅₉	—	4	4	no	—
167	C₂²×#108	—	59	1	no	—
168	C₂×#427	D₅₉	1	1	yes	characteristic
169	#2905	—	59	2	no	—
170	C₂×#405	—	59	1	no	—
171	#1946	—	59	4	no	—
172	#1945	—	59	4	no	—
173	#1733	—	59	2	no	—
174	C₂×C₅₉	—	2	1	no	—
175	C₂×C₅₉	—	2	1	no	—
176	C₂×C₅₉	—	2	2	no	—
177	C₂×C₅₉	—	2	2	no	—
178	C₂×C₅₉	C₂×#389	1	1	yes	characteristic
179	C₂×C₅₉	#2957	1	2	yes	—
180	C₂²×M₁₆	—	59	1	no	—
181	C₂²×M₁₆	—	59	2	no	—
182	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
183	C₂×#117	—	59	1	no	—
184	C₂×#117	C₂×D₅₉	1	2	yes	—
185	C₂×#108	—	59	2	no	—
186	C₂×#108	—	59	2	no	—
187	C₂×#108	—	118	4	no	—
188	#427	Dic₅₉	1	2	yes	complemented
189	#427	Dic₅₉	1	2	yes	complemented
190	#427	C₂×D₅₉	1	2	yes	—
191	#427	C₂×D₅₉	1	2	yes	—
192	C₂×#107	Dic₅₉	1	2	yes	complemented
193	C₂×#106	Dic₅₉	1	2	yes	complemented
194	#424	—	59	8	no	—
195	#424	—	118	4	no	—
196	#423	—	118	4	no	—
197	#405	—	118	4	no	—
198	C₂×#97	—	59	2	no	—
199	C₂×C₄×C₈	—	59	2	no	—
200	#339	—	59	8	no	—
201	C₅₉	#21163	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
203	C₂×M₁₆	—	59	2	no	—
204	C₂×M₁₆	—	59	2	no	—
205	C₂×M₁₆	—	59	4	no	—
206	C₂×M₁₆	—	118	2	no	—
207	C₂×M₁₆	—	118	2	no	—
208	C₂×M₁₆	—	118	2	no	—
209	C₂×M₁₆	—	118	4	no	—
210	C₂×M₁₆	—	118	4	no	—
211	C₂²×C₈	—	59	2	no	—
212	C₂²×C₈	—	118	2	no	—
213	C₂²×C₈	Dic₁₁₈	1	4	yes	—
214	C₂²×C₈	D₂₃₆	1	4	yes	—
215	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
216	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
217	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
218	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
221	#117	—	2	2	no	—
222	#117	—	2	2	no	—
223	#117	—	59	4	no	—
224	#117	—	118	2	no	—
225	#117	C₂×Dic₅₉	1	4	yes	—
226	#117	C₂²×D₅₉	1	4	yes	—
227	C₂×C₄⋊C₄	—	118	1	no	—
228	C₂×C₄⋊C₄	C₂×Dic₅₉	1	4	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	118	1	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	59	2	no	—
233	#108	—	118	4	no	—
234	#108	—	118	4	no	—
235	#107	C₂×Dic₅₉	1	4	yes	—
236	#107	C₂×Dic₅₉	1	4	yes	—
237	#106	C₂×Dic₅₉	1	4	yes	—
238	#106	C₂×Dic₅₉	1	4	yes	—
239	C₄⋊C₈	—	118	16	no	—
240	#97	—	118	4	no	—
241	C₄×C₈	—	118	4	no	—
242	C₂⁴	#11745687	1	1	yes	characteristic
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	118	2	no	—
245	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
246	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
247	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
249	C₂²×C₄	#11745687	1	1	yes	characteristic
250	C₂²×C₄	Q₈×D₅₉	1	2	yes	—
251	C₂²×C₄	#11745689	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
254	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
255	M₁₆	—	118	4	no	—
256	M₁₆	—	118	4	no	—
257	M₁₆	—	118	4	no	—
258	M₁₆	—	118	4	no	—
259	M₁₆	—	236	4	no	—
260	C₂×C₈	—	59	4	no	—
261	C₂×C₈	—	118	2	no	—
262	C₂×C₈	—	118	2	no	—
263	C₂×C₈	—	118	4	no	—
264	C₂×C₈	—	236	4	no	—
265	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
266	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
267	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
268	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
269	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
270	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
271	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
272	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
273	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
274	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
275	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
276	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	118	4	no	—
279	C₄⋊C₄	C₂²×Dic₅₉	1	8	yes	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	2	4	no	—
282	C₂²⋊C₄	—	118	4	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	59	4	no	—
286	C₄²	—	118	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
289	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
290	C₂³	#14273707	1	2	yes	—
291	C₂³	#14273718	1	2	yes	—
292	C₂³	C₂×#11745687	1	1	yes	characteristic
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	—	118	4	no	—
296	C₂×C₄	—	236	4	no	—
297	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
298	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
299	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
300	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
301	C₂×C₄	Q₈×Dic₅₉	1	2	yes	—
302	C₂×C₄	Q₈×Dic₅₉	1	2	yes	—
303	C₂×C₄	#14273702	1	2	yes	—
304	C₂×C₄	#14273702	1	2	yes	—
305	C₂×C₄	#14273707	1	2	yes	—
306	C₂×C₄	#14273718	1	2	yes	—
307	C₂×C₄	#14273726	1	2	yes	—
308	C₂×C₄	#14273726	1	2	yes	—
309	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
312	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
313	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
314	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
315	C₈	—	118	4	no	—
316	C₈	—	236	4	no	—
317	C₈	C₂×C₄⋊Dic₅₉	1	2	yes	—
318	C₈	C₂×C₄⋊Dic₅₉	1	2	yes	—
319	C₈	C₂×C₄⋊Dic₅₉	1	2	yes	—
320	C₈	C₂×C₄⋊Dic₅₉	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	#16837301	1	2	yes	—
327	C₂²	#16837301	1	2	yes	—
328	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
329	C₂²	C₂×#14273707	1	1	yes	characteristic
330	C₂²	C₂×#14273718	1	1	yes	characteristic
331	C₄	—	4	8	no	—
332	C₄	—	236	4	no	—
333	C₄	#16837301	1	2	yes	—
334	C₄	#16837301	1	2	yes	—
335	C₄	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
336	C₄	C₂×Q₈×Dic₅₉	1	1	yes	characteristic
337	C₄	C₂×#14273702	1	1	yes	characteristic
338	C₄	C₂×#14273726	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	2	2	no	—
342	C₂	—	2	2	no	—
343	C₂	C₂×#16837301	1	1	yes	characteristic
344	C₂	#21433013	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂×C₅₉×#427	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂⁴×C₅₉	#41
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂×#427	D₅₉
hypocenter	C₅₉	#21163
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #21424850	C₂	semidirect
5. #21424845	C₂	semidirect
23. C₅₉×#427	C₄	semidirect
24. C₅₉×#427	C₄	semidirect
27. C₂×#107×C₅₉	C₄	semidirect
28. C₂×#106×C₅₉	C₄	semidirect
188. #427	Dic₅₉	semidirect
189. #427	Dic₅₉	semidirect
192. C₂×#107	Dic₅₉	semidirect
193. C₂×#106	Dic₅₉	semidirect
201. C₅₉	#21163	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14273663	2
#21432660	2
C₂×#16837357	2
#21424850	2
#21424845	2
#21423805	2
C₂×C₅₉×#427	2
#21163	59