Subgroups — #29453025

Order 15104.

2612 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	#21438870	C₂	1	1	yes	maximal, characteristic
2	#21438809	C₂	1	2	yes	maximal, complemented
3	#21433012	C₂	1	6	yes	maximal, complemented
4	#21433005	C₂	1	6	yes	maximal, complemented
5	#21432998	C₂	1	6	yes	maximal, complemented
6	C₂×#16837518	C₂	1	6	yes	maximal, complemented
7	#21432641	C₂	1	2	yes	maximal, complemented
8	#131×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#2916	C₂	1	1	yes	maximal, characteristic, complemented
10	#16838238	C₂²	1	2	yes	—
11	#16838219	C₄	1	4	yes	complemented
12	#16838214	C₄	1	8	yes	complemented
13	#16838207	C₄	1	4	yes	complemented
14	#16838202	C₂²	1	6	yes	—
15	C₂×#14273749	C₂²	1	6	yes	—
16	#16838168	C₂²	1	12	yes	complemented
17	#16838167	—	2	8	no	—
18	C₂×#14273735	C₂²	1	12	yes	complemented
19	C₂×#14273734	C₂²	1	1	yes	characteristic
20	#131×D₅₉	—	2	8	no	—
21	#41×Dic₅₉	C₂²	1	1	yes	characteristic
22	#16837523	C₂²	1	24	yes	complemented
23	#16837518	C₂²	1	4	yes	complemented
24	#16837518	C₂²	1	12	yes	complemented
25	#16837518	C₂²	1	12	yes	complemented
26	#16837518	C₂²	1	12	yes	complemented
27	#16837418	C₂²	1	12	yes	complemented
28	C₂×#14273653	C₂²	1	12	yes	complemented
29	#16837360	C₂²	1	3	yes	complemented
30	#16837357	C₂²	1	3	yes	complemented
31	M₁₆×Dic₅₉	C₂²	1	3	yes	complemented
32	C₂×C₈×Dic₅₉	C₂²	1	3	yes	complemented
33	C₅₉×#568	C₂²	1	2	yes	—
34	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
35	C₅₉×#430	C₂²	1	6	yes	complemented
36	C₂×#108×C₅₉	C₂²	1	6	yes	complemented
37	#14273775	—	2	2	no	—
38	#14273774	C₂×C₄	1	8	yes	—
39	#14273773	C₂×C₄	1	8	yes	—
40	#14273772	—	2	6	no	—
41	C₂×#11745685	C₂³	1	12	yes	—
42	#41×D₅₉	—	2	8	no	—
43	#14273760	C₂×C₄	1	24	yes	complemented
44	#14273755	C₂×C₄	1	24	yes	complemented
45	#14273749	C₂×C₄	1	12	yes	complemented
46	#14273749	C₂×C₄	1	12	yes	complemented
47	#14273749	C₂³	1	4	yes	—
48	#14273749	C₂³	1	12	yes	—
49	C₂×Dic₂₃₆	C₂×C₄	1	6	yes	complemented
50	C₂×#11745660	C₂×C₄	1	12	yes	complemented
51	C₂×D₄₇₂	C₂×C₄	1	6	yes	complemented
52	#14273735	—	2	12	no	—
53	#14273735	—	2	12	no	—
54	#14273735	C₂³	1	8	yes	—
55	#14273735	C₂³	1	24	yes	—
56	#14273734	D₄	1	4	yes	—
57	#14273734	D₄	1	4	yes	—
58	#14273733	—	2	24	no	—
59	C₂×#11745664	C₂³	1	3	yes	—
60	C₂×#11745659	—	2	12	no	—
61	M₁₆×D₅₉	—	2	24	no	—
62	C₂²×C₅₉⋊C₈	C₂³	1	3	yes	—
63	C₂×C₈×D₅₉	—	2	12	no	—
64	Q₈×Dic₅₉	—	2	1	no	—
65	D₄×Dic₅₉	—	2	3	no	—
66	#14273686	C₂³	1	3	yes	—
67	C₂×C₄×Dic₅₉	C₂³	1	3	yes	—
68	#14273653	C₂³	1	8	yes	—
69	#14273653	C₂³	1	24	yes	—
70	#14273637	C₂³	1	6	yes	—
71	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
72	C₈×Dic₅₉	C₂³	1	3	yes	—
73	C₈×Dic₅₉	C₂³	1	3	yes	—
74	C₈×Dic₅₉	C₂³	1	3	yes	—
75	C₂×#41×C₅₉	C₂³	1	1	yes	characteristic
76	#131×C₅₉	—	2	8	no	—
77	#131×C₅₉	C₂×C₄	1	4	yes	—
78	#131×C₅₉	C₂×C₄	1	4	yes	—
79	C₂×M₁₆×C₅₉	C₂³	1	12	yes	—
80	C₂×M₁₆×C₅₉	C₂³	1	3	yes	—
81	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
82	#108×C₅₉	C₂³	1	4	yes	—
83	#108×C₅₉	C₂³	1	12	yes	—
84	#41×C₅₉	C₂²×C₄	1	4	yes	—
85	#41×C₅₉	C₂×D₄	1	4	yes	—
86	C₂×Q₈×C₅₉	#41	1	1	yes	characteristic
87	C₂×D₄×C₅₉	#41	1	3	yes	—
88	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
89	C₂×C₂²⋊D₅₉	—	2	12	no	—
90	C₂²×Dic₅₉	—	2	3	no	—
91	#11745689	—	2	8	no	—
92	Q₈×D₅₉	—	2	8	no	—
93	#11745687	—	2	24	no	—
94	D₄×D₅₉	—	2	24	no	—
95	#11745685	—	2	12	no	—
96	#11745685	—	2	12	no	—
97	#11745685	C₂²×C₄	1	24	yes	—
98	#11745685	C₂⁴	1	8	yes	—
99	C₂×D₂₃₆	C₂²×C₄	1	12	yes	—
100	C₂×C₄×D₅₉	—	2	12	no	—
101	C₂×Dic₁₁₈	C₂²×C₄	1	12	yes	—
102	M₁₆×C₅₉	—	2	12	no	—
103	M₁₆×C₅₉	C₂²×C₄	1	12	yes	—
104	M₁₆×C₅₉	C₂⁴	1	8	yes	—
105	C₂×C₈×C₅₉	—	2	12	no	—
106	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
107	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
108	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₅₉	C₂⁴	1	3	yes	—
110	C₂²⋊Dic₅₉	—	2	3	no	—
111	C₄⋊Dic₅₉	—	2	3	no	—
112	C₄×Dic₅₉	—	2	3	no	—
113	C₄×Dic₅₉	C₂⁴	1	1	yes	characteristic
114	C₄×Dic₅₉	C₂⁴	1	3	yes	—
115	#11745664	C₂×D₄	1	12	yes	—
116	C₂×C₅₉⋊C₈	C₂×D₄	1	6	yes	—
117	C₂×C₅₉⋊C₈	C₂×D₄	1	6	yes	—
118	C₂×C₅₉⋊C₈	C₂⁴	1	1	yes	characteristic
119	C₂×C₅₉⋊C₈	C₂⁴	1	3	yes	—
120	Dic₂₃₆	C₂²×C₄	1	4	yes	—
121	Dic₂₃₆	C₂²×C₄	1	12	yes	—
122	D₄₇₂	C₂²×C₄	1	4	yes	—
123	D₄₇₂	C₂²×C₄	1	12	yes	—
124	#11745660	C₂²×C₄	1	8	yes	—
125	#11745660	C₂²×C₄	1	24	yes	—
126	#11745659	—	2	24	no	—
127	#11745659	—	2	8	no	—
128	C₈×D₅₉	—	2	8	no	—
129	C₈×D₅₉	—	2	24	no	—
130	C₂³×C₅₉	C₂×#41	1	3	yes	—
131	C₂²×D₅₉	—	2	12	no	—
132	Q₈×C₅₉	C₄×D₄	1	4	yes	—
133	D₄×C₅₉	C₄×D₄	1	12	yes	—
134	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
135	C₂×C₄×C₅₉	C₂²×D₄	1	6	yes	—
136	C₂×C₄×C₅₉	C₂×#41	1	3	yes	—
137	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
138	C₂²⋊D₅₉	—	2	24	no	—
139	C₂²⋊D₅₉	—	2	8	no	—
140	C₂×Dic₅₉	—	2	12	no	—
141	C₂×Dic₅₉	—	2	1	no	—
142	C₂×Dic₅₉	—	2	3	no	—
143	C₂×Dic₅₉	—	4	6	no	—
144	D₂₃₆	—	2	12	no	—
145	D₂₃₆	C₂³×C₄	1	8	yes	—
146	C₄×D₅₉	—	2	24	no	—
147	C₄×D₅₉	—	2	8	no	—
148	Dic₁₁₈	—	2	12	no	—
149	Dic₁₁₈	C₂³×C₄	1	8	yes	—
150	C₈×C₅₉	—	2	8	no	—
151	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
152	C₈×C₅₉	C₂³×C₄	1	6	yes	—
153	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
154	C₅₉⋊C₈	C₂²×D₄	1	1	yes	characteristic
155	C₅₉⋊C₈	C₂²×D₄	1	6	yes	—
156	C₅₉⋊C₈	C₂²×D₄	1	1	yes	characteristic
157	#60183	—	59	1	no	maximal, Sylow
158	C₂²×C₅₉	—	4	3	no	—
159	C₂²×C₅₉	C₂×C₄×D₄	1	6	yes	—
160	C₂²×C₅₉	C₂²×#41	1	1	yes	characteristic
161	C₂×D₅₉	—	2	8	no	—
162	C₂×D₅₉	—	8	12	no	—
163	C₄×C₅₉	C₂×C₄×D₄	1	6	yes	—
164	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
165	C₄×C₅₉	C₂³×D₄	1	1	yes	characteristic
166	Dic₅₉	—	2	2	no	—
167	Dic₅₉	—	2	8	no	—
168	Dic₅₉	—	4	3	no	—
169	#3584	—	59	1	no	—
170	#3573	—	59	2	no	—
171	#2957	—	59	6	no	—
172	#2956	—	59	6	no	—
173	#2955	—	59	6	no	—
174	C₂×#443	—	59	6	no	—
175	#2916	D₅₉	1	1	yes	characteristic, complemented
176	#2901	—	59	2	no	—
177	C₄×#131	—	59	1	no	—
178	C₂×C₅₉	—	2	1	no	—
179	C₂×C₅₉	—	2	6	no	—
180	C₂×C₅₉	C₂²×C₄×D₄	1	1	yes	characteristic
181	D₅₉	—	4	8	no	—
182	#585	—	59	2	no	—
183	#578	—	59	4	no	—
184	#577	—	59	8	no	—
185	#576	—	59	4	no	—
186	#575	—	59	6	no	—
187	C₂×#135	—	59	6	no	—
188	#568	—	59	12	no	—
189	#568	—	118	8	no	—
190	#568	C₂×D₅₉	1	2	yes	—
191	C₂×#131	—	59	1	no	—
192	C₂×#131	—	59	12	no	—
193	C₂×#131	—	118	8	no	—
194	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
195	C₄×#41	—	59	1	no	—
196	#444	—	59	24	no	—
197	#443	—	59	4	no	—
198	#443	—	59	12	no	—
199	#443	—	59	12	no	—
200	#443	—	59	12	no	—
201	#430	C₂×D₅₉	1	6	yes	complemented
202	C₂×#108	C₂×D₅₉	1	6	yes	complemented
203	#421	—	59	12	no	—
204	C₂×#104	—	59	12	no	—
205	#405	—	59	3	no	—
206	#405	—	59	3	no	—
207	C₄×M₁₆	—	59	3	no	—
208	C₂×C₄×C₈	—	59	3	no	—
209	C₅₉	#60183	1	1	yes	characteristic, complemented, Sylow
210	2₋¹⁺⁴	—	59	8	no	—
211	2₋¹⁺⁴	—	118	2	no	—
212	H(5,𝔽₂)	—	59	8	no	—
213	H(5,𝔽₂)	—	118	6	no	—
214	C₂×#41	—	59	12	no	—
215	C₂×#41	—	118	8	no	—
216	C₂×#41	C₂²×D₅₉	1	1	yes	characteristic
217	#137	—	59	24	no	—
218	AGL(1,ℤ/8)	—	59	24	no	—
219	#135	—	59	12	no	—
220	#135	—	59	12	no	—
221	#135	—	59	4	no	—
222	#135	—	59	12	no	—
223	C₂×Q₁₆	—	59	6	no	—
224	C₂×SD₁₆	—	59	12	no	—
225	C₂×D₈	—	59	6	no	—
226	#131	—	2	8	no	—
227	#131	—	59	4	no	—
228	#131	—	59	4	no	—
229	#131	—	59	8	no	—
230	#131	—	59	24	no	—
231	#131	—	118	24	no	—
232	#131	—	118	12	no	—
233	#131	—	118	12	no	—
234	#131	C₄×D₅₉	1	4	yes	—
235	#131	C₄×D₅₉	1	4	yes	—
236	C₂×M₁₆	—	59	3	no	—
237	C₂×M₁₆	—	118	12	no	—
238	C₂×M₁₆	—	118	24	no	—
239	C₂×M₁₆	C₂²×D₅₉	1	12	yes	—
240	C₂×M₁₆	C₂²×D₅₉	1	3	yes	—
241	C₂²×C₈	—	59	3	no	—
242	C₂²×C₈	—	118	12	no	—
243	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
244	C₄×Q₈	—	118	1	no	—
245	C₄×D₄	—	118	3	no	—
246	#117	—	59	3	no	—
247	C₂×C₄²	—	59	3	no	—
248	#108	C₂²×D₅₉	1	4	yes	—
249	#108	C₂²×D₅₉	1	12	yes	—
250	#104	—	59	8	no	—
251	#104	—	59	24	no	—
252	#97	—	59	6	no	—
253	C₄×C₈	—	59	1	no	—
254	C₄×C₈	—	59	3	no	—
255	C₄×C₈	—	59	3	no	—
256	C₄×C₈	—	59	3	no	—
257	#41	—	59	24	no	—
258	#41	—	59	8	no	—
259	#41	—	118	8	no	—
260	#41	—	118	24	no	—
261	#41	—	118	12	no	—
262	#41	—	118	12	no	—
263	#41	C₂×C₄×D₅₉	1	4	yes	—
264	#41	D₄×D₅₉	1	4	yes	—
265	C₂×Q₈	—	59	12	no	—
266	C₂×Q₈	—	118	8	no	—
267	C₂×Q₈	#11745689	1	1	yes	characteristic
268	C₂×D₄	—	59	12	no	—
269	C₂×D₄	—	118	12	no	—
270	C₂×D₄	—	118	24	no	—
271	C₂×D₄	#11745689	1	3	yes	—
272	C₂²×C₄	—	118	12	no	—
273	C₂²×C₄	—	118	3	no	—
274	C₂²×C₄	C₂³×D₅₉	1	3	yes	—
275	Q₁₆	—	59	4	no	—
276	Q₁₆	—	59	12	no	—
277	SD₁₆	—	59	8	no	—
278	SD₁₆	—	59	24	no	—
279	D₈	—	59	4	no	—
280	D₈	—	59	12	no	—
281	M₁₆	—	2	12	no	—
282	M₁₆	—	59	12	no	—
283	M₁₆	—	118	8	no	—
284	M₁₆	—	118	24	no	—
285	M₁₆	C₂×C₄×D₅₉	1	12	yes	—
286	M₁₆	C₂³×D₅₉	1	8	yes	—
287	C₂×C₈	—	2	12	no	—
288	C₂×C₈	—	59	6	no	—
289	C₂×C₈	—	59	6	no	—
290	C₂×C₈	—	59	1	no	—
291	C₂×C₈	—	59	3	no	—
292	C₂×C₈	—	118	8	no	—
293	C₂×C₈	—	118	24	no	—
294	C₂×C₈	C₂×C₄×D₅₉	1	6	yes	—
295	C₂×C₈	C₂×C₄×D₅₉	1	6	yes	—
296	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₅₉	1	3	yes	—
298	C₄⋊C₄	—	118	3	no	—
299	C₂²⋊C₄	—	118	3	no	—
300	C₄²	—	59	1	no	—
301	C₄²	—	59	3	no	—
302	C₄²	—	118	3	no	—
303	C₂³	—	118	12	no	—
304	C₂³	C₂×#11745689	1	3	yes	—
305	Q₈	—	59	8	no	—
306	Q₈	—	118	12	no	—
307	Q₈	#14273689	1	4	yes	—
308	D₄	—	59	8	no	—
309	D₄	—	118	8	no	—
310	D₄	—	118	24	no	—
311	D₄	—	118	12	no	—
312	D₄	#14273689	1	12	yes	—
313	C₂×C₄	—	118	1	no	—
314	C₂×C₄	—	118	12	no	—
315	C₂×C₄	—	118	8	no	—
316	C₂×C₄	—	118	24	no	—
317	C₂×C₄	—	118	3	no	—
318	C₂×C₄	—	236	6	no	—
319	C₂×C₄	C₂²×C₄×D₅₉	1	6	yes	—
320	C₂×C₄	C₂×D₄×D₅₉	1	6	yes	—
321	C₂×C₄	C₂×#11745689	1	3	yes	—
322	C₂×C₄	C₂⁴×D₅₉	1	1	yes	characteristic
323	C₈	—	2	8	no	—
324	C₈	—	59	1	no	—
325	C₈	—	59	1	no	—
326	C₈	—	59	6	no	—
327	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
328	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
329	C₈	C₂²×C₄×D₅₉	1	6	yes	—
330	C₂²	—	4	3	no	—
331	C₂²	—	118	8	no	—
332	C₂²	—	472	12	no	—
333	C₂²	C₂×#14273689	1	6	yes	—
334	C₂²	C₂²×#11745689	1	1	yes	characteristic
335	C₄	—	118	2	no	—
336	C₄	—	118	8	no	—
337	C₄	—	236	3	no	—
338	C₄	C₂×#14273689	1	6	yes	—
339	C₄	C₂³×C₄×D₅₉	1	1	yes	characteristic
340	C₄	C₂²×D₄×D₅₉	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	6	no	—
343	C₂	—	236	8	no	—
344	C₂	C₂²×#14273689	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₄×D₅₉
commutator	C₄×C₅₉	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₉
Fitting	C₅₉×#2916	C₂
socle	C₂×C₅₉	C₂²×C₄×D₄
prime radical	#21438870	C₂
cosocle	C₂×C₄×C₅₉	C₂⁵
hypercenter	#2916	D₅₉
hypocenter	C₅₉	#60183
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #2916

Complements

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

N	H ≅ G/N	type
2. #21438809	C₂	semidirect
3. #21433012	C₂	semidirect
4. #21433005	C₂	semidirect
5. #21432998	C₂	semidirect
6. C₂×#16837518	C₂	semidirect
7. #21432641	C₂	semidirect
8. #131×Dic₅₉	C₂	semidirect
9. C₅₉×#2916	C₂	semidirect
11. #16838219	C₄	semidirect
12. #16838214	C₄	semidirect
13. #16838207	C₄	semidirect
16. #16838168	C₂²	semidirect
18. C₂×#14273735	C₂²	semidirect
22. #16837523	C₂²	semidirect
23. #16837518	C₂²	semidirect
24. #16837518	C₂²	semidirect
25. #16837518	C₂²	semidirect
26. #16837518	C₂²	semidirect
27. #16837418	C₂²	semidirect
28. C₂×#14273653	C₂²	semidirect
29. #16837360	C₂²	semidirect
30. #16837357	C₂²	semidirect
31. M₁₆×Dic₅₉	C₂²	semidirect
32. C₂×C₈×Dic₅₉	C₂²	semidirect
35. C₅₉×#430	C₂²	semidirect
36. C₂×#108×C₅₉	C₂²	semidirect
43. #14273760	C₂×C₄	semidirect
44. #14273755	C₂×C₄	semidirect
45. #14273749	C₂×C₄	semidirect
46. #14273749	C₂×C₄	semidirect
49. C₂×Dic₂₃₆	C₂×C₄	semidirect
50. C₂×#11745660	C₂×C₄	semidirect
51. C₂×D₄₇₂	C₂×C₄	semidirect
175. #2916	D₅₉	semidirect
201. #430	C₂×D₅₉	semidirect
202. C₂×#108	C₂×D₅₉	semidirect
209. C₅₉	#60183	semidirect

Maximal subgroups

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

subgroup	index
#21438870	2
#21438809	2
#21433012	2
#21433005	2
#21432998	2
C₂×#16837518	2
#21432641	2
#131×Dic₅₉	2
C₅₉×#2916	2
#60183	59