Subgroups — #29453034

Order 15104.

2604 conjugacy classes of subgroups.

The subgroup lattice has 340 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×#16837523	C₂	1	6	yes	maximal, complemented
7	#21432641	C₂	1	2	yes	maximal, complemented
8	#21432522	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	4	yes	complemented
23	#16837523	C₂²	1	12	yes	complemented
24	#16837523	C₂²	1	12	yes	complemented
25	#16837523	C₂²	1	12	yes	complemented
26	#16837518	C₂²	1	24	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×#14273637	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	8	yes	—
57	#14273733	—	2	24	no	—
58	C₂×#11745664	C₂³	1	3	yes	—
59	C₂×#11745659	—	2	12	no	—
60	M₁₆×D₅₉	—	2	24	no	—
61	C₂²×C₅₉⋊C₈	C₂³	1	3	yes	—
62	C₂×C₈×D₅₉	—	2	12	no	—
63	Q₈×Dic₅₉	—	2	1	no	—
64	D₄×Dic₅₉	—	2	3	no	—
65	#14273686	C₂³	1	3	yes	—
66	C₂×C₄×Dic₅₉	C₂³	1	3	yes	—
67	#14273653	C₂³	1	8	yes	—
68	#14273653	C₂³	1	24	yes	—
69	#14273637	C₂³	1	1	yes	characteristic
70	#14273637	C₂³	1	3	yes	—
71	#14273637	C₂³	1	3	yes	—
72	#14273637	C₂³	1	3	yes	—
73	C₈×Dic₅₉	C₂³	1	6	yes	—
74	C₂×#41×C₅₉	C₂³	1	1	yes	characteristic
75	#131×C₅₉	—	2	8	no	—
76	#131×C₅₉	C₂×C₄	1	4	yes	—
77	#131×C₅₉	C₂×C₄	1	4	yes	—
78	C₂×M₁₆×C₅₉	C₂³	1	12	yes	—
79	C₂×M₁₆×C₅₉	C₂³	1	3	yes	—
80	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
81	#108×C₅₉	C₂³	1	4	yes	—
82	#108×C₅₉	C₂³	1	12	yes	—
83	#41×C₅₉	C₂²×C₄	1	4	yes	—
84	#41×C₅₉	C₂×D₄	1	4	yes	—
85	C₂×Q₈×C₅₉	#41	1	1	yes	characteristic
86	C₂×D₄×C₅₉	#41	1	3	yes	—
87	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
88	C₂×C₂²⋊D₅₉	—	2	12	no	—
89	C₂²×Dic₅₉	—	2	3	no	—
90	#11745689	—	2	8	no	—
91	Q₈×D₅₉	—	2	8	no	—
92	#11745687	—	2	24	no	—
93	D₄×D₅₉	—	2	24	no	—
94	#11745685	—	2	12	no	—
95	#11745685	—	2	12	no	—
96	#11745685	C₂²×C₄	1	24	yes	—
97	#11745685	C₂⁴	1	8	yes	—
98	C₂×D₂₃₆	C₂²×C₄	1	12	yes	—
99	C₂×C₄×D₅₉	—	2	12	no	—
100	C₂×Dic₁₁₈	C₂²×C₄	1	12	yes	—
101	M₁₆×C₅₉	—	2	12	no	—
102	M₁₆×C₅₉	C₂²×C₄	1	12	yes	—
103	M₁₆×C₅₉	C₂⁴	1	8	yes	—
104	C₂×C₈×C₅₉	—	2	12	no	—
105	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
106	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
107	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₅₉	C₂⁴	1	3	yes	—
109	C₂²⋊Dic₅₉	—	2	3	no	—
110	C₄⋊Dic₅₉	—	2	3	no	—
111	C₄×Dic₅₉	—	2	3	no	—
112	C₄×Dic₅₉	C₂⁴	1	3	yes	—
113	C₄×Dic₅₉	C₂⁴	1	1	yes	characteristic
114	#11745664	C₂×D₄	1	12	yes	—
115	C₂×C₅₉⋊C₈	C₂×D₄	1	12	yes	—
116	C₂×C₅₉⋊C₈	C₂⁴	1	3	yes	—
117	C₂×C₅₉⋊C₈	C₂⁴	1	1	yes	characteristic
118	Dic₂₃₆	C₂²×C₄	1	4	yes	—
119	Dic₂₃₆	C₂²×C₄	1	12	yes	—
120	D₄₇₂	C₂²×C₄	1	4	yes	—
121	D₄₇₂	C₂²×C₄	1	12	yes	—
122	#11745660	C₂²×C₄	1	8	yes	—
123	#11745660	C₂²×C₄	1	24	yes	—
124	#11745659	—	2	24	no	—
125	#11745659	—	2	8	no	—
126	C₈×D₅₉	—	2	8	no	—
127	C₈×D₅₉	—	2	24	no	—
128	C₂³×C₅₉	C₂×#41	1	3	yes	—
129	C₂²×D₅₉	—	2	12	no	—
130	Q₈×C₅₉	C₄×D₄	1	4	yes	—
131	D₄×C₅₉	C₄×D₄	1	12	yes	—
132	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
133	C₂×C₄×C₅₉	C₂²×D₄	1	6	yes	—
134	C₂×C₄×C₅₉	C₂×#41	1	3	yes	—
135	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
136	C₂²⋊D₅₉	—	2	24	no	—
137	C₂²⋊D₅₉	—	2	8	no	—
138	C₂×Dic₅₉	—	2	12	no	—
139	C₂×Dic₅₉	—	2	1	no	—
140	C₂×Dic₅₉	—	2	3	no	—
141	C₂×Dic₅₉	—	8	3	no	—
142	D₂₃₆	—	2	12	no	—
143	D₂₃₆	C₂³×C₄	1	8	yes	—
144	C₄×D₅₉	—	2	24	no	—
145	C₄×D₅₉	—	2	8	no	—
146	Dic₁₁₈	—	2	12	no	—
147	Dic₁₁₈	C₂³×C₄	1	8	yes	—
148	C₈×C₅₉	—	2	8	no	—
149	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
150	C₈×C₅₉	C₂³×C₄	1	6	yes	—
151	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
152	C₅₉⋊C₈	C₂²×D₄	1	2	yes	—
153	C₅₉⋊C₈	C₂²×D₄	1	6	yes	—
154	#60184	—	59	1	no	maximal, Sylow
155	C₂²×C₅₉	—	4	3	no	—
156	C₂²×C₅₉	C₂×C₄×D₄	1	6	yes	—
157	C₂²×C₅₉	C₂²×#41	1	1	yes	characteristic
158	C₂×D₅₉	—	2	8	no	—
159	C₂×D₅₉	—	8	12	no	—
160	C₄×C₅₉	C₂×C₄×D₄	1	6	yes	—
161	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
162	C₄×C₅₉	C₂³×D₄	1	1	yes	characteristic
163	Dic₅₉	—	2	8	no	—
164	Dic₅₉	—	4	3	no	—
165	Dic₅₉	—	4	1	no	—
166	#3584	—	59	1	no	—
167	#3573	—	59	2	no	—
168	#2957	—	59	6	no	—
169	#2956	—	59	6	no	—
170	#2955	—	59	6	no	—
171	C₂×#444	—	59	6	no	—
172	#2916	D₅₉	1	1	yes	characteristic, complemented
173	#2901	—	59	2	no	—
174	#2875	—	59	1	no	—
175	C₂×C₅₉	—	2	1	no	—
176	C₂×C₅₉	—	2	6	no	—
177	C₂×C₅₉	C₂²×C₄×D₄	1	1	yes	characteristic
178	D₅₉	—	4	8	no	—
179	#585	—	59	2	no	—
180	#578	—	59	4	no	—
181	#577	—	59	8	no	—
182	#576	—	59	4	no	—
183	#575	—	59	6	no	—
184	C₂×#135	—	59	6	no	—
185	#568	—	59	12	no	—
186	#568	—	118	8	no	—
187	#568	C₂×D₅₉	1	2	yes	—
188	C₂×#131	—	59	1	no	—
189	C₂×#131	—	59	12	no	—
190	C₂×#131	—	118	8	no	—
191	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
192	C₄×#41	—	59	1	no	—
193	#444	—	59	4	no	—
194	#444	—	59	12	no	—
195	#444	—	59	12	no	—
196	#444	—	59	12	no	—
197	#443	—	59	24	no	—
198	#430	C₂×D₅₉	1	6	yes	complemented
199	C₂×#108	C₂×D₅₉	1	6	yes	complemented
200	#421	—	59	12	no	—
201	C₂×#104	—	59	12	no	—
202	#405	—	59	3	no	—
203	#405	—	59	3	no	—
204	C₄×M₁₆	—	59	3	no	—
205	C₂×#97	—	59	3	no	—
206	C₅₉	#60184	1	1	yes	characteristic, complemented, Sylow
207	2₋¹⁺⁴	—	59	8	no	—
208	2₋¹⁺⁴	—	118	2	no	—
209	H(5,𝔽₂)	—	59	8	no	—
210	H(5,𝔽₂)	—	118	6	no	—
211	C₂×#41	—	59	12	no	—
212	C₂×#41	—	118	8	no	—
213	C₂×#41	C₂²×D₅₉	1	1	yes	characteristic
214	#137	—	59	24	no	—
215	AGL(1,ℤ/8)	—	59	24	no	—
216	#135	—	59	12	no	—
217	#135	—	59	12	no	—
218	#135	—	59	4	no	—
219	#135	—	59	12	no	—
220	C₂×Q₁₆	—	59	6	no	—
221	C₂×SD₁₆	—	59	12	no	—
222	C₂×D₈	—	59	6	no	—
223	#131	—	2	8	no	—
224	#131	—	59	8	no	—
225	#131	—	59	8	no	—
226	#131	—	59	24	no	—
227	#131	—	118	24	no	—
228	#131	—	118	12	no	—
229	#131	—	118	12	no	—
230	#131	C₄×D₅₉	1	4	yes	—
231	#131	C₄×D₅₉	1	4	yes	—
232	C₂×M₁₆	—	59	3	no	—
233	C₂×M₁₆	—	118	12	no	—
234	C₂×M₁₆	—	118	24	no	—
235	C₂×M₁₆	C₂²×D₅₉	1	12	yes	—
236	C₂×M₁₆	C₂²×D₅₉	1	3	yes	—
237	C₂²×C₈	—	59	3	no	—
238	C₂²×C₈	—	118	12	no	—
239	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
240	C₄×Q₈	—	118	1	no	—
241	C₄×D₄	—	118	3	no	—
242	#117	—	59	3	no	—
243	C₂×C₄²	—	59	3	no	—
244	#108	C₂²×D₅₉	1	4	yes	—
245	#108	C₂²×D₅₉	1	12	yes	—
246	#104	—	59	8	no	—
247	#104	—	59	24	no	—
248	#97	—	59	1	no	—
249	#97	—	59	3	no	—
250	#97	—	59	3	no	—
251	#97	—	59	3	no	—
252	C₄×C₈	—	59	6	no	—
253	#41	—	59	24	no	—
254	#41	—	59	8	no	—
255	#41	—	118	8	no	—
256	#41	—	118	24	no	—
257	#41	—	118	12	no	—
258	#41	—	118	12	no	—
259	#41	C₂×C₄×D₅₉	1	4	yes	—
260	#41	D₄×D₅₉	1	4	yes	—
261	C₂×Q₈	—	59	12	no	—
262	C₂×Q₈	—	118	8	no	—
263	C₂×Q₈	#11745689	1	1	yes	characteristic
264	C₂×D₄	—	59	12	no	—
265	C₂×D₄	—	118	12	no	—
266	C₂×D₄	—	118	24	no	—
267	C₂×D₄	#11745689	1	3	yes	—
268	C₂²×C₄	—	118	12	no	—
269	C₂²×C₄	—	118	3	no	—
270	C₂²×C₄	C₂³×D₅₉	1	3	yes	—
271	Q₁₆	—	59	4	no	—
272	Q₁₆	—	59	12	no	—
273	SD₁₆	—	59	8	no	—
274	SD₁₆	—	59	24	no	—
275	D₈	—	59	4	no	—
276	D₈	—	59	12	no	—
277	M₁₆	—	2	12	no	—
278	M₁₆	—	59	12	no	—
279	M₁₆	—	118	8	no	—
280	M₁₆	—	118	24	no	—
281	M₁₆	C₂×C₄×D₅₉	1	12	yes	—
282	M₁₆	C₂³×D₅₉	1	8	yes	—
283	C₂×C₈	—	2	12	no	—
284	C₂×C₈	—	59	12	no	—
285	C₂×C₈	—	59	3	no	—
286	C₂×C₈	—	59	1	no	—
287	C₂×C₈	—	118	8	no	—
288	C₂×C₈	—	118	24	no	—
289	C₂×C₈	C₂×C₄×D₅₉	1	6	yes	—
290	C₂×C₈	C₂×C₄×D₅₉	1	6	yes	—
291	C₂×C₈	C₂³×D₅₉	1	3	yes	—
292	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
293	C₄⋊C₄	—	118	3	no	—
294	C₂²⋊C₄	—	118	3	no	—
295	C₄²	—	59	3	no	—
296	C₄²	—	59	1	no	—
297	C₄²	—	118	3	no	—
298	C₂³	—	118	12	no	—
299	C₂³	C₂×#11745689	1	3	yes	—
300	Q₈	—	59	8	no	—
301	Q₈	—	118	12	no	—
302	Q₈	#14273689	1	4	yes	—
303	D₄	—	59	8	no	—
304	D₄	—	118	8	no	—
305	D₄	—	118	24	no	—
306	D₄	—	118	12	no	—
307	D₄	#14273689	1	12	yes	—
308	C₂×C₄	—	118	12	no	—
309	C₂×C₄	—	118	8	no	—
310	C₂×C₄	—	118	24	no	—
311	C₂×C₄	—	118	1	no	—
312	C₂×C₄	—	118	3	no	—
313	C₂×C₄	—	472	3	no	—
314	C₂×C₄	C₂²×C₄×D₅₉	1	6	yes	—
315	C₂×C₄	C₂×D₄×D₅₉	1	6	yes	—
316	C₂×C₄	C₂×#11745689	1	3	yes	—
317	C₂×C₄	C₂⁴×D₅₉	1	1	yes	characteristic
318	C₈	—	2	8	no	—
319	C₈	—	59	2	no	—
320	C₈	—	59	6	no	—
321	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
322	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
323	C₈	C₂²×C₄×D₅₉	1	6	yes	—
324	C₂²	—	4	3	no	—
325	C₂²	—	118	8	no	—
326	C₂²	—	472	12	no	—
327	C₂²	C₂×#14273689	1	6	yes	—
328	C₂²	C₂²×#11745689	1	1	yes	characteristic
329	C₄	—	118	8	no	—
330	C₄	—	236	1	no	—
331	C₄	—	236	3	no	—
332	C₄	C₂×#14273689	1	6	yes	—
333	C₄	C₂³×C₄×D₅₉	1	1	yes	characteristic
334	C₄	C₂²×D₄×D₅₉	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	6	no	—
337	C₂	—	236	8	no	—
338	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	#60184
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×#16837523	C₂	semidirect
7. #21432641	C₂	semidirect
8. #21432522	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. #16837523	C₂²	semidirect
24. #16837523	C₂²	semidirect
25. #16837523	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×#14273637	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
172. #2916	D₅₉	semidirect
198. #430	C₂×D₅₉	semidirect
199. C₂×#108	C₂×D₅₉	semidirect
206. C₅₉	#60184	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×#16837523	2
#21432641	2
#21432522	2
C₅₉×#2916	2
#60184	59