Subgroups — #28746102

Order 15104.

494 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21436499	C₂	1	2	yes	maximal, complemented
2	#21426375	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426288	C₂	1	2	yes	maximal, complemented
4	#21426282	C₂	1	2	yes	maximal, complemented
5	#21426254	C₂	1	2	yes	maximal, complemented
6	#21426182	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426182	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₉₄₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425880	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16837228	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838098	—	2	2	no	—
13	C₂×D₉₄₄	C₂²	1	1	yes	characteristic
14	#16837804	C₂²	1	2	yes	—
15	#16837727	—	2	2	no	—
16	#16837721	—	2	2	no	—
17	#16837703	—	2	2	no	—
18	#16837638	C₂²	1	2	yes	—
19	#16837638	C₂²	1	2	yes	—
20	C₄×D₄₇₂	C₂²	1	2	yes	—
21	#16837440	C₂²	1	2	yes	—
22	C₂×#14273661	C₂²	1	2	yes	—
23	#16837230	C₂²	1	2	yes	—
24	#16837230	C₂²	1	2	yes	—
25	#16837228	C₂²	1	1	yes	characteristic
26	#16837228	C₂²	1	1	yes	characteristic
27	#16837228	C₂²	1	1	yes	characteristic
28	#16837228	C₂²	1	2	yes	—
29	#16837204	C₂²	1	2	yes	—
30	#16837204	C₂²	1	2	yes	—
31	#16837204	C₂²	1	2	yes	—
32	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
35	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₅₉×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₅₉×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
39	C₂×D₄₇₂	C₂³	1	2	yes	—
40	#14273722	—	4	2	no	—
41	#14273719	—	2	2	no	—
42	#14273709	—	4	2	no	—
43	#14273701	—	2	2	no	—
44	#14273701	—	2	2	no	—
45	#14273699	—	4	2	no	—
46	C₄×D₂₃₆	—	2	2	no	—
47	(C₂²⋊D₅₉)⋊C₄	—	4	2	no	—
48	#14273686	—	2	2	no	—
49	C₂×C₄⋊Dic₅₉	—	2	2	no	—
50	D₉₄₄	—	2	2	no	—
51	#14273661	—	2	2	no	—
52	#14273661	C₂³	1	2	yes	—
53	#14273661	C₂³	1	2	yes	—
54	#14273661	C₂³	1	2	yes	—
55	#14273659	—	2	2	no	—
56	#14273659	—	2	2	no	—
57	#14273649	—	2	2	no	—
58	#14273649	—	2	2	no	—
59	#14273649	—	2	2	no	—
60	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
63	C₂×C₁₆×C₅₉	—	2	2	no	—
64	C₂×C₁₆×C₅₉	—	2	2	no	—
65	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
66	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
70	C₂²⋊C₈×C₅₉	D₄	1	1	yes	characteristic
71	C₂²⋊C₈×C₅₉	D₄	1	1	yes	characteristic
72	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₅₉	—	4	2	no	—
77	C₂×C₂²⋊D₅₉	—	4	2	no	—
78	C₂²×Dic₅₉	—	4	2	no	—
79	C₂×D₂₃₆	—	2	2	no	—
80	C₂×C₄×D₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	—	2	1	no	—
82	C₂×C₈×C₅₉	—	2	1	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	—	2	1	no	—
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
90	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
91	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂²⋊Dic₅₉	—	4	2	no	—
94	(C₂×D₅₉)⋊C₄	—	4	2	no	—
95	(C₂×D₅₉)⋊C₄	—	4	2	no	—
96	(C₂×D₅₉)⋊C₄	—	4	2	no	—
97	C₄⋊Dic₅₉	—	2	2	no	—
98	C₄⋊Dic₅₉	—	2	2	no	—
99	C₄⋊Dic₅₉	—	2	2	no	—
100	C₄⋊Dic₅₉	—	4	2	no	—
101	Dic₅₉⋊C₄	—	4	2	no	—
102	Dic₅₉⋊C₄	—	4	2	no	—
103	C₄×Dic₅₉	—	4	2	no	—
104	D₄₇₂	—	2	2	no	—
105	D₄₇₂	—	2	2	no	—
106	C₁₆×C₅₉	—	2	1	no	—
107	C₁₆×C₅₉	—	2	1	no	—
108	C₁₆×C₅₉	—	2	1	no	—
109	C₁₆×C₅₉	#41	1	2	yes	—
110	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
111	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
112	C₂²×D₅₉	—	4	2	no	—
113	D₄×C₅₉	—	2	2	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
123	C₂²⋊D₅₉	—	8	4	no	—
124	C₂²⋊D₅₉	—	16	2	no	—
125	C₂×Dic₅₉	—	4	2	no	—
126	C₂×Dic₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	4	2	no	—
128	C₂×Dic₅₉	—	16	2	no	—
129	D₂₃₆	—	4	2	no	—
130	D₂₃₆	—	4	2	no	—
131	C₄×D₅₉	—	16	2	no	—
132	C₈×C₅₉	—	2	1	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₈×C₅₉	—	2	1	no	—
135	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
136	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
137	#33906	—	59	1	no	maximal, Sylow
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	—	2	1	no	—
140	C₂²×C₅₉	—	2	1	no	—
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	#508	1	2	yes	—
144	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₅₉	—	8	2	no	—
146	C₂×D₅₉	—	8	2	no	—
147	C₂×D₅₉	—	8	2	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	#547	1	1	yes	characteristic
153	C₄×C₅₉	#578	1	1	yes	characteristic
154	Dic₅₉	—	8	2	no	—
155	Dic₅₉	—	8	2	no	—
156	Dic₅₉	—	8	2	no	—
157	#3339	—	59	2	no	—
158	#2253	—	59	1	no	—
159	#2234	—	59	2	no	—
160	#2233	—	59	2	no	—
161	#2228	—	59	2	no	—
162	#2215	—	59	1	no	—
163	#2215	—	59	1	no	—
164	C₄×D₁₆	—	59	1	no	—
165	D₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
166	#2158	—	59	1	no	—
167	C₂×#366	—	59	1	no	—
168	C₂×C₅₉	—	2	2	no	—
169	C₂×C₅₉	—	4	1	no	—
170	C₂×C₅₉	#3334	1	1	yes	characteristic
171	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
172	C₂×C₅₉	#3415	1	1	yes	characteristic
173	D₅₉	—	16	2	no	—
174	#553	—	118	2	no	—
175	C₂×D₁₆	—	59	1	no	—
176	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
177	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
178	#499	—	59	2	no	—
179	#482	—	118	2	no	—
180	#481	—	118	2	no	—
181	#478	—	118	2	no	—
182	#466	—	59	2	no	—
183	#466	—	59	2	no	—
184	C₄×D₈	—	59	2	no	—
185	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
186	#427	—	59	2	no	—
187	C₂×#107	—	59	2	no	—
188	#367	—	59	2	no	—
189	#367	—	59	2	no	—
190	#366	—	59	1	no	—
191	#366	—	59	1	no	—
192	#366	—	59	2	no	—
193	#366	—	59	1	no	—
194	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
195	#357	—	59	2	no	—
196	#357	—	59	2	no	—
197	#357	—	59	2	no	—
198	C₂²⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
199	C₂²⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
200	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
201	C₅₉	#33906	1	1	yes	characteristic, complemented, Sylow
202	C₂×D₈	—	59	2	no	—
203	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
205	#126	—	236	2	no	—
206	#125	—	118	2	no	—
207	#123	—	236	2	no	—
208	#121	—	118	2	no	—
209	#121	—	118	2	no	—
210	#121	—	236	2	no	—
211	C₄×D₄	—	118	2	no	—
212	C₄×D₄	—	236	2	no	—
213	C₄×D₄	D₂₃₆	1	1	yes	characteristic
214	#117	—	118	2	no	—
215	C₂×C₄⋊C₄	—	118	2	no	—
216	D₁₆	—	118	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	2	2	no	—
219	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
221	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
223	#107	—	59	2	no	—
224	#107	—	59	2	no	—
225	#107	—	59	2	no	—
226	#107	—	118	2	no	—
227	#106	—	118	2	no	—
228	#106	—	118	2	no	—
229	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
230	#102	—	118	2	no	—
231	#102	—	118	2	no	—
232	#102	—	118	2	no	—
233	C₂²⋊C₈	D₂₃₆	1	1	yes	characteristic
234	C₂²⋊C₈	D₂₃₆	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₂×D₄	—	118	2	no	—
237	C₂×D₄	—	236	2	no	—
238	C₂×D₄	—	236	2	no	—
239	C₂×D₄	D₄₇₂	1	1	yes	characteristic
240	C₂²×C₄	—	236	2	no	—
241	C₂²×C₄	—	236	2	no	—
242	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
244	D₈	—	118	2	no	—
245	D₈	—	118	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
251	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
252	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
254	C₄⋊C₄	—	118	2	no	—
255	C₄⋊C₄	—	118	2	no	—
256	C₄⋊C₄	—	118	2	no	—
257	C₄⋊C₄	—	236	2	no	—
258	C₄⋊C₄	—	236	2	no	—
259	C₄⋊C₄	—	236	2	no	—
260	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
261	C₂²⋊C₄	—	236	2	no	—
262	C₂²⋊C₄	—	236	2	no	—
263	C₂²⋊C₄	—	236	2	no	—
264	C₂²⋊C₄	—	236	2	no	—
265	C₂²⋊C₄	D₄₇₂	1	1	yes	characteristic
266	C₂²⋊C₄	D₄₇₂	1	1	yes	characteristic
267	C₄²	—	236	2	no	—
268	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	#11745687	1	2	yes	—
273	C₂³	—	236	2	no	—
274	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
275	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
276	D₄	—	2	2	no	—
277	D₄	—	944	2	no	—
278	D₄	—	472	4	no	—
279	D₄	—	236	2	no	—
280	D₄	—	236	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	944	2	no	—
286	C₂×C₄	—	944	2	no	—
287	C₂×C₄	—	236	2	no	—
288	C₂×C₄	—	236	2	no	—
289	C₂×C₄	—	236	2	no	—
290	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂×#11745687	1	1	yes	characteristic
299	C₈	#14273774	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	472	2	no	—
306	C₂²	—	472	2	no	—
307	C₂²	—	472	2	no	—
308	C₂²	#16837834	1	2	yes	—
309	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	472	2	no	—
315	C₄	—	472	2	no	—
316	C₄	—	472	2	no	—
317	C₄	#16838056	1	1	yes	characteristic
318	C₄	#16838219	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	—	4	1	no	—
321	C₂	—	944	2	no	—
322	C₂	#21436444	1	1	yes	characteristic
323	C₂	C₂×#16837834	1	1	yes	characteristic
324	C₂	#21437237	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	D₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	D₄×C₁₆	D₅₉
hypocenter	C₅₉	#33906
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #21436499	C₂	semidirect
2. #21426375	C₂	semidirect
3. #21426288	C₂	semidirect
4. #21426282	C₂	semidirect
5. #21426254	C₂	semidirect
6. #21426182	C₂	semidirect
7. #21426182	C₂	semidirect
8. C₄×D₉₄₄	C₂	semidirect
9. #21425880	C₂	semidirect
10. C₂×#16837228	C₂	semidirect
11. D₄×C₁₆×C₅₉	C₂	semidirect
165. D₄×C₁₆	D₅₉	semidirect
201. C₅₉	#33906	semidirect

Maximal subgroups

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

subgroup	index
#21436499	2
#21426375	2
#21426288	2
#21426282	2
#21426254	2
#21426182	2
C₄×D₉₄₄	2
#21425880	2
C₂×#16837228	2
D₄×C₁₆×C₅₉	2
#33906	59