Subgroups — #28743439

Order 15104.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434627	C₂	1	1	yes	maximal, characteristic, complemented
2	#21426272	C₂	1	4	yes	maximal, complemented
3	#21426267	C₂	1	4	yes	maximal, complemented
4	#21425870	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837226	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425862	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425832	C₂	1	1	yes	maximal, characteristic
8	C₂×#16837212	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#2957	C₂	1	1	yes	maximal, characteristic
10	#16838030	—	2	1	no	—
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	C₂×#14273667	C₂²	1	1	yes	characteristic
13	#16837811	—	2	2	no	—
14	#16837804	C₂²	1	2	yes	complemented
15	#16837800	C₂²	1	2	yes	complemented
16	#16837444	—	2	1	no	—
17	#16837440	C₂²	1	1	yes	characteristic
18	C₂×#14273659	C₂²	1	1	yes	characteristic
19	#16837226	C₂²	1	2	yes	complemented
20	#16837226	C₂²	1	4	yes	complemented
21	#16837226	C₂²	1	2	yes	complemented
22	#16837223	C₂²	1	4	yes	complemented
23	#16837212	C₂²	1	2	yes	—
24	#16837212	C₂²	1	4	yes	—
25	#16837212	C₂²	1	2	yes	—
26	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#568	—	2	1	no	—
28	C₅₉×#444	—	2	2	no	—
29	C₅₉×#443	C₂²	1	4	yes	—
30	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
31	C₅₉×#421	—	2	1	no	—
32	C₅₉×#405	C₂²	1	1	yes	characteristic
33	#14273727	—	2	2	no	—
34	#14273726	—	4	1	no	—
35	#14273719	—	2	2	no	—
36	#14273718	—	4	1	no	—
37	#14273704	—	4	2	no	—
38	#14273686	—	2	1	no	—
39	C₂×C₄⋊Dic₅₉	—	2	1	no	—
40	#14273667	—	2	1	no	—
41	#14273667	—	2	1	no	—
42	#14273667	C₂³	1	2	yes	—
43	#14273667	C₂³	1	2	yes	—
44	C₂×C₅₉⋊C₁₆	—	2	1	no	—
45	C₂×C₅₉⋊C₁₆	—	2	1	no	—
46	#14273661	—	2	1	no	—
47	#14273661	—	2	2	no	—
48	#14273659	—	2	1	no	—
49	#14273659	—	2	2	no	—
50	#14273659	C₂³	1	2	yes	—
51	#14273659	C₂³	1	1	yes	characteristic
52	#14273659	C₂³	1	1	yes	characteristic
53	C₂×#41×C₅₉	—	2	1	no	—
54	#135×C₅₉	—	2	1	no	—
55	#135×C₅₉	—	2	1	no	—
56	#135×C₅₉	Q₈	1	2	yes	complemented
57	#135×C₅₉	C₂³	1	2	yes	—
58	C₂×Q₁₆×C₅₉	Q₈	1	1	yes	characteristic, complemented
59	C₂×SD₁₆×C₅₉	—	2	1	no	—
60	C₂×D₈×C₅₉	Q₈	1	1	yes	characteristic, complemented
61	#131×C₅₉	—	2	4	no	—
62	#131×C₅₉	—	4	2	no	—
63	C₂×M₁₆×C₅₉	—	2	1	no	—
64	C₂×M₁₆×C₅₉	—	4	2	no	—
65	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	#117×C₅₉	D₄	1	1	yes	characteristic
68	#108×C₅₉	—	2	1	no	—
69	#108×C₅₉	C₂³	1	2	yes	—
70	#104×C₅₉	—	2	4	no	—
71	#97×C₅₉	D₄	1	2	yes	—
72	C₄×C₈×C₅₉	C₂³	1	2	yes	—
73	#41×C₅₉	—	2	2	no	—
74	#41×C₅₉	—	2	2	no	—
75	#41×C₅₉	—	4	1	no	—
76	#41×C₅₉	—	4	1	no	—
77	C₂×Q₈×C₅₉	—	2	1	no	—
78	C₂×D₄×C₅₉	—	2	1	no	—
79	C₂×D₄×C₅₉	—	4	1	no	—
80	C₂²×C₄×C₅₉	—	4	1	no	—
81	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₅₉	—	4	1	no	—
83	C₂×Dic₁₁₈	—	4	2	no	—
84	Q₁₆×C₅₉	—	2	1	no	—
85	Q₁₆×C₅₉	C₂×Q₈	1	2	yes	—
86	SD₁₆×C₅₉	—	2	2	no	—
87	SD₁₆×C₅₉	—	2	2	no	—
88	D₈×C₅₉	—	2	1	no	—
89	D₈×C₅₉	C₂×Q₈	1	2	yes	—
90	M₁₆×C₅₉	—	2	2	no	—
91	M₁₆×C₅₉	—	4	1	no	—
92	M₁₆×C₅₉	—	4	4	no	—
93	M₁₆×C₅₉	SD₁₆	1	2	yes	—
94	M₁₆×C₅₉	SD₁₆	1	2	yes	—
95	C₂×C₈×C₅₉	—	4	1	no	—
96	C₂×C₈×C₅₉	—	4	4	no	—
97	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
102	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
103	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₅₉	—	2	1	no	—
105	C₂²⋊C₄×C₅₉	—	2	1	no	—
106	C₄²×C₅₉	C₂×D₄	1	2	yes	—
107	C₂²⋊Dic₅₉	—	4	1	no	—
108	C₄⋊Dic₅₉	—	2	2	no	—
109	C₄⋊Dic₅₉	—	2	1	no	—
110	C₄⋊Dic₅₉	—	2	1	no	—
111	C₄⋊Dic₅₉	—	4	2	no	—
112	C₄⋊Dic₅₉	—	4	1	no	—
113	Dic₅₉⋊C₄	—	4	2	no	—
114	Dic₅₉⋊C₄	—	4	2	no	—
115	C₄×Dic₅₉	—	4	1	no	—
116	C₅₉⋊C₁₆	—	2	2	no	—
117	C₅₉⋊C₁₆	—	2	2	no	—
118	C₂³×C₅₉	—	4	1	no	—
119	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₅₉	—	2	2	no	—
121	Q₈×C₅₉	—	4	1	no	—
122	D₄×C₅₉	—	2	2	no	—
123	D₄×C₅₉	—	4	2	no	—
124	D₄×C₅₉	—	4	2	no	—
125	D₄×C₅₉	—	4	1	no	—
126	C₂×C₄×C₅₉	—	2	2	no	—
127	C₂×C₄×C₅₉	—	4	1	no	—
128	C₂×C₄×C₅₉	—	4	2	no	—
129	C₂×C₄×C₅₉	—	4	2	no	—
130	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₅₉	—	4	1	no	—
137	C₂×Dic₅₉	—	4	1	no	—
138	C₂×Dic₅₉	—	4	2	no	—
139	C₂×Dic₅₉	—	16	1	no	—
140	Dic₁₁₈	—	16	2	no	—
141	C₈×C₅₉	—	4	2	no	—
142	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₅₉	C₂×SD₁₆	1	4	yes	—
145	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
146	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
147	#33638	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	—	4	1	no	—
149	C₂²×C₅₉	—	4	2	no	—
150	C₂²×C₅₉	—	16	1	no	—
151	C₂²×C₅₉	#476	1	1	yes	characteristic
152	C₂²×C₅₉	#476	1	1	yes	characteristic
153	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₄×C₅₉	—	4	2	no	—
155	C₄×C₅₉	—	4	2	no	—
156	C₄×C₅₉	#476	1	1	yes	characteristic
157	C₄×C₅₉	#476	1	1	yes	characteristic
158	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
160	Dic₅₉	—	8	1	no	—
161	Dic₅₉	—	8	2	no	—
162	Dic₅₉	—	8	1	no	—
163	#3163	—	59	1	no	—
164	#2957	D₅₉	1	1	yes	characteristic
165	#2231	—	59	4	no	—
166	#2230	—	59	4	no	—
167	#2155	—	59	1	no	—
168	C₂×#365	—	59	1	no	—
169	#2153	—	59	1	no	—
170	#2145	—	59	1	no	—
171	C₂×#360	—	59	1	no	—
172	C₂×C₅₉	—	2	1	no	—
173	C₂×C₅₉	—	2	1	no	—
174	C₂×C₅₉	—	2	1	no	—
175	C₂×C₅₉	—	8	2	no	—
176	C₂×C₅₉	C₂×#476	1	1	yes	characteristic
177	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
178	#568	—	2	1	no	—
179	#544	—	118	1	no	—
180	C₂×M₃₂	—	59	1	no	—
181	C₂×M₃₂	—	59	1	no	—
182	#501	—	118	2	no	—
183	#499	—	59	2	no	—
184	#498	—	59	2	no	—
185	#444	—	2	2	no	—
186	#443	C₂×D₅₉	1	4	yes	—
187	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
188	#428	—	118	1	no	—
189	#427	—	59	1	no	—
190	C₂×#106	—	59	1	no	—
191	#421	—	2	1	no	—
192	#405	C₂×D₅₉	1	1	yes	characteristic
193	#365	—	59	4	no	—
194	#365	—	59	2	no	—
195	#365	—	59	2	no	—
196	#364	—	59	4	no	—
197	#360	—	59	4	no	—
198	#360	—	59	2	no	—
199	#360	—	59	2	no	—
200	C₅₉	#33638	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	Dic₁₁₈	1	2	yes	complemented
205	#135	C₂²×D₅₉	1	2	yes	—
206	C₂×Q₁₆	Dic₁₁₈	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	Dic₁₁₈	1	1	yes	characteristic, complemented
209	#131	—	2	4	no	—
210	#131	—	4	2	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	4	2	no	—
213	C₂×M₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
215	C₄⋊Q₈	—	118	2	no	—
216	C₄⋊Q₈	—	236	1	no	—
217	#125	—	118	2	no	—
218	#125	—	236	1	no	—
219	C₂²⋊Q₈	—	236	2	no	—
220	#117	—	118	1	no	—
221	#117	C₂²⋊D₅₉	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	118	1	no	—
223	M₃₂	—	59	2	no	—
224	M₃₂	—	59	2	no	—
225	M₃₂	—	118	1	no	—
226	M₃₂	—	118	1	no	—
227	C₂×C₁₆	—	118	1	no	—
228	C₂×C₁₆	—	118	1	no	—
229	#108	—	2	1	no	—
230	#108	C₂²×D₅₉	1	2	yes	—
231	#107	—	118	1	no	—
232	#107	—	118	2	no	—
233	#106	—	59	2	no	—
234	#106	—	59	1	no	—
235	#106	—	59	1	no	—
236	#106	—	118	1	no	—
237	#106	—	118	2	no	—
238	#104	—	2	4	no	—
239	#97	C₂²⋊D₅₉	1	2	yes	—
240	C₄×C₈	C₂²×D₅₉	1	2	yes	—
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	236	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	236	1	no	—
251	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
252	Q₁₆	—	2	1	no	—
253	Q₁₆	C₂×Dic₁₁₈	1	2	yes	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Dic₁₁₈	1	2	yes	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	4	no	—
261	M₁₆	#11745670	1	2	yes	—
262	M₁₆	#11745670	1	2	yes	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
266	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
267	C₂×C₈	#11745685	1	1	yes	characteristic
268	C₂×C₈	#11745685	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
271	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	118	2	no	—
274	C₄⋊C₄	—	118	1	no	—
275	C₄⋊C₄	—	118	1	no	—
276	C₄⋊C₄	—	236	2	no	—
277	C₄⋊C₄	—	236	2	no	—
278	C₄⋊C₄	—	236	2	no	—
279	C₄⋊C₄	—	236	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	236	1	no	—
282	C₄²	—	236	1	no	—
283	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
284	C₁₆	—	118	2	no	—
285	C₁₆	—	118	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	#14273752	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	944	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	944	1	no	—
300	C₂×C₄	—	236	1	no	—
301	C₂×C₄	—	236	1	no	—
302	C₂×C₄	—	236	2	no	—
303	C₂×C₄	#14273706	1	1	yes	characteristic
304	C₂×C₄	#14273706	1	1	yes	characteristic
305	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
306	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
307	C₂×C₄	#14273752	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	#14273706	1	1	yes	characteristic
311	C₈	#14273706	1	1	yes	characteristic
312	C₈	C₂×#11745670	1	4	yes	—
313	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
314	C₈	C₂×#11745685	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	#16837692	1	1	yes	characteristic
319	C₂²	#16837692	1	1	yes	characteristic
320	C₂²	C₂×#14273752	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	472	2	no	—
324	C₄	—	472	1	no	—
325	C₄	—	472	1	no	—
326	C₄	#16837692	1	1	yes	characteristic
327	C₄	#16837692	1	1	yes	characteristic
328	C₄	C₂×#14273706	1	1	yes	characteristic
329	C₄	C₂²×#11745670	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#16837692	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16837692
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2957	C₂
socle	C₂×C₅₉	C₂×#476
prime radical	C₂×#135×C₅₉	C₂²
hypercenter	#2957	D₅₉
hypocenter	C₅₉	#33638
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2957

Complements

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

N	H ≅ G/N	type
1. #21434627	C₂	semidirect
2. #21426272	C₂	semidirect
3. #21426267	C₂	semidirect
4. #21425870	C₂	semidirect
5. C₂×#16837226	C₂	semidirect
6. #21425862	C₂	semidirect
14. #16837804	C₂²	semidirect
15. #16837800	C₂²	semidirect
19. #16837226	C₂²	semidirect
20. #16837226	C₂²	semidirect
21. #16837226	C₂²	semidirect
22. #16837223	C₂²	semidirect
56. #135×C₅₉	Q₈	semidirect
58. C₂×Q₁₆×C₅₉	Q₈	semidirect
60. C₂×D₈×C₅₉	Q₈	semidirect
200. C₅₉	#33638	semidirect
204. #135	Dic₁₁₈	semidirect
206. C₂×Q₁₆	Dic₁₁₈	semidirect
208. C₂×D₈	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
#21434627	2
#21426272	2
#21426267	2
#21425870	2
C₂×#16837226	2
#21425862	2
#21425832	2
C₂×#16837212	2
C₅₉×#2957	2
#33638	59