Subgroups — #28619568

Order 15104.

616 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433000	C₂	1	1	yes	maximal, characteristic
2	#21425676	C₂	1	1	yes	maximal, characteristic, complemented
3	#21425671	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425547	C₂	1	2	yes	maximal, complemented
5	#21425402	C₂	1	1	yes	maximal, characteristic
6	#21425383	C₂	1	1	yes	maximal, characteristic
7	#21425265	C₂	1	2	yes	maximal, complemented
8	#21423933	C₂	1	2	yes	maximal, complemented
9	#21423897	C₂	1	2	yes	maximal, complemented
10	#21423851	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838202	C₂²	1	1	yes	characteristic
13	C₂×#14273733	C₂²	1	2	yes	—
14	#16837525	—	2	2	no	—
15	#16837520	—	2	2	no	—
16	#16837451	C₂²	1	1	yes	characteristic
17	#16837420	C₂²	1	2	yes	—
18	C₂×#14273655	C₂²	1	2	yes	—
19	#16837399	C₂²	1	2	yes	—
20	#16837384	C₂²	1	2	yes	—
21	#16837375	C₂²	1	2	yes	—
22	#16837357	C₂²	1	2	yes	—
23	#16837174	C₂²	1	2	yes	—
24	#16837172	C₂²	1	2	yes	—
25	#16837167	C₂²	1	2	yes	—
26	#16837162	C₂²	1	2	yes	—
27	C₅₉⋊#338	C₂²	1	1	yes	characteristic
28	#16837159	C₂²	1	1	yes	characteristic
29	#16837158	C₂²	1	2	yes	—
30	C₅₉×#533	C₂²	1	1	yes	characteristic
31	C₅₉×#446	—	2	2	no	—
32	C₈×Q₈×C₅₉	—	2	2	no	—
33	C₅₉×#424	C₂²	1	1	yes	characteristic
34	C₅₉×#424	C₂²	1	1	yes	characteristic
35	C₅₉×#423	C₂²	1	1	yes	characteristic
36	C₅₉×#405	C₂²	1	1	yes	characteristic
37	C₅₉×#405	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₂×#11745685	C₂³	1	2	yes	—
40	#14273760	—	2	2	no	—
41	#14273755	—	2	2	no	—
42	#14273749	—	2	2	no	—
43	#14273733	—	2	2	no	—
44	#14273733	—	2	2	no	—
45	#14273733	—	2	2	no	—
46	#14273733	—	2	2	no	—
47	C₂×#11745664	C₂³	1	2	yes	—
48	C₂×#11745664	C₂³	1	2	yes	—
49	C₂×#11745659	—	2	2	no	—
50	C₂×C₈×D₅₉	—	2	2	no	—
51	#14273686	C₂³	1	2	yes	—
52	#14273663	—	2	2	no	—
53	#14273655	—	2	4	no	—
54	#14273655	—	2	2	no	—
55	#14273655	—	2	2	no	—
56	#14273652	—	2	2	no	—
57	#14273649	—	2	2	no	—
58	#14273645	—	2	2	no	—
59	#14273643	—	2	2	no	—
60	#14273641	—	2	4	no	—
61	#14273637	—	2	2	no	—
62	C₈×Dic₅₉	—	2	2	no	—
63	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₅₉	—	2	1	no	—
68	#125×C₅₉	—	2	1	no	—
69	C₂²⋊Q₈×C₅₉	—	2	2	no	—
70	C₄×Q₈×C₅₉	—	2	2	no	—
71	#117×C₅₉	C₂³	1	1	yes	characteristic
72	#117×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₅₉	—	2	2	no	—
75	C₄⋊C₈×C₅₉	—	2	2	no	—
76	C₄⋊C₈×C₅₉	—	2	2	no	—
77	#97×C₅₉	—	2	1	no	—
78	#97×C₅₉	D₄	1	2	yes	—
79	C₄×C₈×C₅₉	—	2	1	no	—
80	C₄×C₈×C₅₉	—	2	1	no	—
81	C₄×C₈×C₅₉	—	2	1	no	—
82	C₄×C₈×C₅₉	D₄	1	2	yes	—
83	C₂×Q₈×C₅₉	—	2	2	no	—
84	C₂²×C₄×C₅₉	—	2	1	no	—
85	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₂×C₂²⋊D₅₉	—	2	2	no	—
87	#11745685	—	2	2	no	—
88	#11745685	—	2	2	no	—
89	#11745685	—	2	2	no	—
90	#11745685	—	2	2	no	—
91	C₂×D₂₃₆	#41	1	2	yes	—
92	C₂×C₄×D₅₉	—	2	2	no	—
93	C₂×Dic₁₁₈	#41	1	2	yes	—
94	M₁₆×C₅₉	—	2	2	no	—
95	M₁₆×C₅₉	—	2	2	no	—
96	C₂×C₈×C₅₉	—	2	1	no	—
97	C₂×C₈×C₅₉	—	2	1	no	—
98	C₂×C₈×C₅₉	—	2	2	no	—
99	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₅₉	#41	1	2	yes	—
103	C₂×C₈×C₅₉	#41	1	2	yes	—
104	C₄⋊C₄×C₅₉	—	2	4	no	—
105	C₄⋊C₄×C₅₉	—	2	2	no	—
106	C₄⋊C₄×C₅₉	—	2	1	no	—
107	C₄⋊C₄×C₅₉	—	2	1	no	—
108	C₂²⋊C₄×C₅₉	—	2	1	no	—
109	C₂²⋊C₄×C₅₉	#41	1	2	yes	—
110	C₄²×C₅₉	—	2	1	no	—
111	C₄²×C₅₉	—	2	1	no	—
112	C₄²×C₅₉	C₂×D₄	1	2	yes	—
113	C₄²×C₅₉	#41	1	1	yes	characteristic
114	C₄²×C₅₉	#41	1	1	yes	characteristic
115	C₂²⋊Dic₅₉	—	2	2	no	—
116	C₄⋊Dic₅₉	—	2	2	no	—
117	C₄×Dic₅₉	—	2	2	no	—
118	#11745664	—	2	2	no	—
119	#11745664	—	2	2	no	—
120	#11745664	—	2	2	no	—
121	#11745664	—	2	2	no	—
122	C₂×C₅₉⋊C₈	—	2	2	no	—
123	C₂×C₅₉⋊C₈	—	2	2	no	—
124	Dic₂₃₆	—	2	2	no	—
125	D₄₇₂	—	2	2	no	—
126	#11745660	—	2	4	no	—
127	#11745659	—	4	2	no	—
128	#11745659	—	4	2	no	—
129	C₈×D₅₉	—	4	2	no	—
130	C₈×D₅₉	—	4	2	no	—
131	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
132	C₂²×D₅₉	—	2	2	no	—
133	Q₈×C₅₉	—	4	4	no	—
134	C₂×C₄×C₅₉	—	2	1	no	—
135	C₂×C₄×C₅₉	—	2	1	no	—
136	C₂×C₄×C₅₉	—	2	2	no	—
137	C₂×C₄×C₅₉	—	2	2	no	—
138	C₂×C₄×C₅₉	—	4	2	no	—
139	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
141	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
143	C₂²⋊D₅₉	—	4	2	no	—
144	C₂²⋊D₅₉	—	4	2	no	—
145	C₂×Dic₅₉	—	2	2	no	—
146	C₂×Dic₅₉	—	4	2	no	—
147	D₂₃₆	—	2	2	no	—
148	D₂₃₆	—	2	2	no	—
149	C₄×D₅₉	—	4	2	no	—
150	C₄×D₅₉	—	4	2	no	—
151	Dic₁₁₈	—	2	2	no	—
152	Dic₁₁₈	—	2	2	no	—
153	C₈×C₅₉	—	2	2	no	—
154	C₈×C₅₉	—	4	1	no	—
155	C₈×C₅₉	—	4	1	no	—
156	C₈×C₅₉	#124	1	2	yes	—
157	C₈×C₅₉	#124	1	1	yes	characteristic
158	C₈×C₅₉	#124	1	1	yes	characteristic
159	C₅₉⋊C₈	—	4	2	no	—
160	C₅₉⋊C₈	—	4	2	no	—
161	#23833	—	59	1	no	maximal, Sylow
162	C₂²×C₅₉	—	4	1	no	—
163	C₂²×C₅₉	#525	1	1	yes	characteristic
164	C₂²×C₅₉	#529	1	2	yes	—
165	C₂×D₅₉	—	4	2	no	—
166	C₂×D₅₉	—	8	2	no	—
167	C₄×C₅₉	—	2	2	no	—
168	C₄×C₅₉	—	4	1	no	—
169	C₄×C₅₉	—	4	2	no	—
170	C₄×C₅₉	—	4	2	no	—
171	C₄×C₅₉	#525	1	1	yes	characteristic
172	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
173	C₄×C₅₉	#529	1	2	yes	—
174	Dic₅₉	—	4	2	no	—
175	Dic₅₉	—	8	2	no	—
176	#2960	D₅₉	1	1	yes	characteristic, complemented
177	#2955	—	59	1	no	—
178	#2103	—	59	1	no	—
179	#2102	—	59	1	no	—
180	#2080	—	59	2	no	—
181	#2053	—	59	1	no	—
182	#2050	—	59	1	no	—
183	#2031	—	59	2	no	—
184	#1764	—	59	2	no	—
185	#1757	—	59	2	no	—
186	#1747	—	59	1	no	—
187	C₂×C₅₉	—	2	2	no	—
188	C₂×C₅₉	—	2	1	no	—
189	C₂×C₅₉	#2398	1	1	yes	characteristic
190	D₅₉	—	8	2	no	—
191	#575	—	59	1	no	—
192	C₂×#131	—	59	2	no	—
193	#533	C₂×D₅₉	1	1	yes	characteristic
194	#446	—	2	2	no	—
195	C₈×Q₈	—	2	2	no	—
196	#444	—	118	2	no	—
197	#443	—	118	2	no	—
198	#430	—	59	1	no	—
199	#424	C₂×D₅₉	1	1	yes	characteristic
200	#424	C₂×D₅₉	1	1	yes	characteristic
201	#423	C₂×D₅₉	1	1	yes	characteristic
202	#421	—	59	2	no	—
203	C₂×#104	—	59	2	no	—
204	#416	—	59	2	no	—
205	#413	—	59	2	no	—
206	#410	—	59	2	no	—
207	#405	—	59	2	no	—
208	#405	C₂×D₅₉	1	1	yes	characteristic
209	#405	C₂×D₅₉	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
211	#344	—	59	2	no	—
212	#344	—	59	2	no	—
213	#341	—	59	2	no	—
214	#339	—	59	2	no	—
215	#338	—	59	1	no	—
216	H(3,ℤ/4)	—	59	1	no	—
217	H(3,ℤ/4)	—	59	2	no	—
218	C₅₉	#23833	1	1	yes	characteristic, complemented, Sylow
219	C₂×#41	—	59	2	no	—
220	#137	—	118	2	no	—
221	AGL(1,ℤ/8)	—	118	2	no	—
222	#135	—	118	2	no	—
223	#131	—	118	2	no	—
224	#131	—	118	2	no	—
225	#131	—	118	2	no	—
226	#131	—	118	2	no	—
227	C₂×M₁₆	—	59	2	no	—
228	C₂×M₁₆	—	59	2	no	—
229	C₂×M₁₆	—	118	2	no	—
230	C₂×M₁₆	C₂²×D₅₉	1	1	yes	characteristic
231	C₂×M₁₆	C₂²×D₅₉	1	1	yes	characteristic
232	C₂²×C₈	—	118	2	no	—
233	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₄⋊Q₈	—	2	1	no	—
236	#125	—	2	1	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	#117	—	59	2	no	—
240	#117	C₂²×D₅₉	1	1	yes	characteristic
241	#117	C₂²×D₅₉	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
243	#108	—	118	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	#104	—	118	2	no	—
248	#104	—	118	2	no	—
249	#104	—	118	4	no	—
250	#103	—	118	2	no	—
251	#102	—	118	2	no	—
252	#101	—	118	2	no	—
253	#100	—	118	2	no	—
254	#99	—	118	4	no	—
255	#97	—	2	1	no	—
256	#97	—	118	2	no	—
257	#97	C₂²⋊D₅₉	1	2	yes	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	118	2	no	—
262	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
263	#41	—	118	2	no	—
264	#41	—	118	2	no	—
265	#41	—	118	2	no	—
266	#41	—	118	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	59	2	no	—
269	C₂×D₄	—	59	2	no	—
270	C₂×D₄	—	118	2	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	118	2	no	—
273	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
274	Q₁₆	—	118	2	no	—
275	SD₁₆	—	118	4	no	—
276	D₈	—	118	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	118	2	no	—
280	M₁₆	—	118	2	no	—
281	M₁₆	—	118	2	no	—
282	M₁₆	—	118	2	no	—
283	M₁₆	—	236	2	no	—
284	M₁₆	—	236	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	118	2	no	—
289	C₂×C₈	—	118	2	no	—
290	C₂×C₈	—	236	2	no	—
291	C₂×C₈	—	236	2	no	—
292	C₂×C₈	#11745689	1	2	yes	—
293	C₂×C₈	#11745689	1	2	yes	—
294	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
295	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	118	2	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	118	2	no	—
304	C₂²⋊C₄	#11745685	1	2	yes	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	118	2	no	—
308	C₄²	#11745685	1	1	yes	characteristic
309	C₄²	#11745685	1	1	yes	characteristic
310	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
311	C₂³	—	118	2	no	—
312	C₂³	C₂×#11745685	1	1	yes	characteristic
313	Q₈	—	4	4	no	—
314	Q₈	—	118	2	no	—
315	Q₈	—	118	2	no	—
316	D₄	—	118	2	no	—
317	D₄	—	118	2	no	—
318	D₄	—	236	2	no	—
319	D₄	—	236	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	118	2	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	—	236	2	no	—
328	C₂×C₄	—	236	2	no	—
329	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
330	C₂×C₄	#41×D₅₉	1	2	yes	—
331	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
332	C₂×C₄	C₂×#11745689	1	2	yes	—
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	236	2	no	—
337	C₈	—	236	2	no	—
338	C₈	#14273716	1	2	yes	—
339	C₈	#14273716	1	1	yes	characteristic
340	C₈	#14273716	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	472	2	no	—
343	C₂²	—	236	2	no	—
344	C₂²	#16837913	1	1	yes	characteristic
345	C₂²	#16837939	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	472	2	no	—
351	C₄	—	236	2	no	—
352	C₄	#16837909	1	1	yes	characteristic
353	C₄	C₂×#14273716	1	1	yes	characteristic
354	C₄	#16837939	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	472	2	no	—
358	C₂	#21427379	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273716
commutator, cosocle	C₂²×C₄×C₅₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#2960	C₂
socle	C₂×C₅₉	#2398
prime radical	#16838202	C₂²
hypercenter	#2960	D₅₉
hypocenter	C₅₉	#23833
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
2. #21425676	C₂	semidirect
3. #21425671	C₂	semidirect
4. #21425547	C₂	semidirect
7. #21425265	C₂	semidirect
8. #21423933	C₂	semidirect
9. #21423897	C₂	semidirect
10. #21423851	C₂	semidirect
11. C₅₉×#2960	C₂	semidirect
176. #2960	D₅₉	semidirect
218. C₅₉	#23833	semidirect

Maximal subgroups

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

subgroup	index
#21433000	2
#21425676	2
#21425671	2
#21425547	2
#21425402	2
#21425383	2
#21425265	2
#21423933	2
#21423897	2
#21423851	2
C₅₉×#2960	2
#23833	59