Subgroups — #28745993

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21436519	C₂	1	1	yes	maximal, characteristic
2	#21426323	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426301	C₂	1	2	yes	maximal, complemented
4	#21426248	C₂	1	1	yes	maximal, characteristic, complemented
5	#21426211	C₂	1	2	yes	maximal, complemented
6	#21426138	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426013	C₂	1	2	yes	maximal, complemented
8	C₄×#14273670	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425928	C₂	1	1	yes	maximal, characteristic, complemented
10	#21425816	C₂	1	2	yes	maximal, complemented
11	D₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838117	—	2	1	no	—
13	C₂×#14273670	C₂²	1	1	yes	characteristic
14	C₂×#14273670	C₂²	1	2	yes	—
15	C₂×#14273667	C₂²	1	2	yes	—
16	#16837807	C₂²	1	1	yes	characteristic
17	#16837727	—	2	2	no	—
18	#16837693	—	2	1	no	—
19	#16837638	C₂²	1	2	yes	—
20	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
21	#16837440	C₂²	1	2	yes	—
22	#16837229	C₂²	1	1	yes	characteristic
23	#16837227	C₂²	1	2	yes	—
24	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#16837206	C₂²	1	1	yes	characteristic
26	#16837206	C₂²	1	2	yes	—
27	#16837206	C₂²	1	2	yes	—
28	#16837206	C₂²	1	2	yes	—
29	#16837206	C₂²	1	2	yes	—
30	#16837202	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
34	D₄×Q₈×C₅₉	—	2	1	no	—
35	C₅₉×#494	C₂²	1	1	yes	characteristic
36	C₅₉×#467	C₂²	1	2	yes	—
37	C₅₉×#462	—	2	1	no	—
38	C₅₉×#451	—	2	2	no	—
39	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×D₄₇₂	C₂³	1	1	yes	characteristic
42	#14273727	—	2	1	no	—
43	#14273713	—	4	2	no	—
44	#14273701	—	2	2	no	—
45	#14273698	—	4	1	no	—
46	C₄×D₂₃₆	—	2	1	no	—
47	#14273686	—	2	2	no	—
48	#14273685	—	4	1	no	—
49	#14273670	—	2	2	no	—
50	#14273670	—	4	2	no	—
51	#14273667	—	4	2	no	—
52	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
55	#14273661	—	2	1	no	—
56	#14273661	C₂³	1	2	yes	—
57	#14273661	C₂³	1	1	yes	characteristic
58	#14273659	—	2	2	no	—
59	#14273649	—	2	2	no	—
60	#14273649	—	2	1	no	—
61	C₂²×Q₈×C₅₉	—	2	2	no	—
62	C₂×Q₁₆×C₅₉	—	2	4	no	—
63	C₂×Q₁₆×C₅₉	—	2	2	no	—
64	C₂×Q₁₆×C₅₉	—	2	1	no	—
65	C₂×Q₁₆×C₅₉	—	4	2	no	—
66	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₅₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₅₉	—	2	1	no	—
70	C₄⋊Q₈×C₅₉	—	4	1	no	—
71	C₂²⋊Q₈×C₅₉	—	2	2	no	—
72	C₂²⋊Q₈×C₅₉	—	4	2	no	—
73	C₄×Q₈×C₅₉	—	2	1	no	—
74	C₄×D₄×C₅₉	—	4	1	no	—
75	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
76	#107×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
78	#103×C₅₉	—	2	1	no	—
79	#103×C₅₉	—	2	2	no	—
80	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
81	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₅₉	—	2	1	no	—
83	C₂×Q₈×C₅₉	—	2	2	no	—
84	C₂×Q₈×C₅₉	—	4	4	no	—
85	C₂×Q₈×C₅₉	—	4	2	no	—
86	C₂×Q₈×C₅₉	—	8	2	no	—
87	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₅₉	—	4	2	no	—
89	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
90	C₂×C₂²⋊D₅₉	—	4	2	no	—
91	C₂×D₂₃₆	—	2	1	no	—
92	C₂×C₄×D₅₉	—	4	1	no	—
93	C₂×Dic₁₁₈	—	4	1	no	—
94	Q₁₆×C₅₉	—	2	2	no	—
95	Q₁₆×C₅₉	—	2	1	no	—
96	Q₁₆×C₅₉	—	2	2	no	—
97	Q₁₆×C₅₉	—	4	4	no	—
98	Q₁₆×C₅₉	—	8	1	no	—
99	Q₁₆×C₅₉	#41	1	2	yes	—
100	C₂×C₈×C₅₉	—	2	2	no	—
101	C₂×C₈×C₅₉	—	2	2	no	—
102	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₅₉	—	2	1	no	—
106	C₄⋊C₄×C₅₉	—	4	1	no	—
107	C₄⋊C₄×C₅₉	—	4	1	no	—
108	C₄⋊C₄×C₅₉	—	4	2	no	—
109	C₄⋊C₄×C₅₉	—	4	1	no	—
110	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
111	C₂²⋊C₄×C₅₉	—	4	2	no	—
112	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
113	C₄²×C₅₉	—	4	1	no	—
114	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂²⋊Dic₅₉	—	4	2	no	—
116	(C₂×D₅₉)⋊C₄	—	4	2	no	—
117	(C₂×D₅₉)⋊C₄	—	4	1	no	—
118	C₄⋊Dic₅₉	—	2	1	no	—
119	C₄⋊Dic₅₉	—	2	2	no	—
120	C₄⋊Dic₅₉	—	4	1	no	—
121	C₄×Dic₅₉	—	4	2	no	—
122	D₄₇₂	—	2	1	no	—
123	D₄₇₂	—	2	1	no	—
124	C₅₉⋊C₁₆	—	2	1	no	—
125	C₅₉⋊C₁₆	—	2	1	no	—
126	C₅₉⋊C₁₆	—	2	2	no	—
127	C₂³×C₅₉	C₂×D₈	1	2	yes	—
128	C₂²×D₅₉	—	4	1	no	—
129	Q₈×C₅₉	—	2	2	no	—
130	Q₈×C₅₉	—	4	1	no	—
131	Q₈×C₅₉	—	4	2	no	—
132	Q₈×C₅₉	—	4	2	no	—
133	Q₈×C₅₉	—	8	4	no	—
134	D₄×C₅₉	—	2	2	no	—
135	C₂×C₄×C₅₉	—	2	2	no	—
136	C₂×C₄×C₅₉	—	2	2	no	—
137	C₂×C₄×C₅₉	—	4	1	no	—
138	C₂×C₄×C₅₉	—	4	1	no	—
139	C₂×C₄×C₅₉	—	4	2	no	—
140	C₂×C₄×C₅₉	—	8	4	no	—
141	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₅₉	—	16	2	no	—
146	C₂×Dic₅₉	—	4	1	no	—
147	C₂×Dic₅₉	—	4	2	no	—
148	D₂₃₆	—	4	1	no	—
149	D₂₃₆	—	4	1	no	—
150	C₄×D₅₉	—	8	2	no	—
151	Dic₁₁₈	—	16	1	no	—
152	C₈×C₅₉	—	2	2	no	—
153	C₈×C₅₉	—	2	1	no	—
154	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
155	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
156	#33896	—	59	1	no	maximal, Sylow
157	C₂²×C₅₉	—	2	2	no	—
158	C₂²×C₅₉	—	2	2	no	—
159	C₂²×C₅₉	—	2	2	no	—
160	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
161	C₂×D₅₉	—	8	1	no	—
162	C₂×D₅₉	—	8	1	no	—
163	C₂×D₅₉	—	8	1	no	—
164	C₄×C₅₉	—	2	1	no	—
165	C₄×C₅₉	—	2	2	no	—
166	C₄×C₅₉	—	4	2	no	—
167	C₄×C₅₉	—	8	1	no	—
168	C₄×C₅₉	—	8	2	no	—
169	C₄×C₅₉	#510	1	2	yes	—
170	C₄×C₅₉	#546	1	1	yes	characteristic
171	C₄×C₅₉	#577	1	1	yes	characteristic
172	Dic₅₉	—	8	1	no	—
173	Dic₅₉	—	8	2	no	—
174	#3341	—	59	1	no	—
175	D₄×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
176	#2241	—	59	1	no	—
177	#2237	—	59	2	no	—
178	#2227	—	59	1	no	—
179	#2220	—	59	2	no	—
180	#2208	—	59	1	no	—
181	#2185	—	59	2	no	—
182	C₄×SD₃₂	—	59	1	no	—
183	#2169	—	59	1	no	—
184	#2141	—	59	2	no	—
185	C₂×C₅₉	—	4	2	no	—
186	C₂×C₅₉	#3294	1	1	yes	characteristic
187	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
188	C₂×C₅₉	#3416	1	1	yes	characteristic
189	D₅₉	—	16	1	no	—
190	C₂²×Q₁₆	C₂×D₅₉	1	2	yes	—
191	#555	—	118	1	no	—
192	D₄×Q₈	—	2	1	no	—
193	C₂×SD₃₂	—	59	1	no	—
194	C₂×SD₃₂	—	59	2	no	—
195	C₂×M₃₂	—	59	2	no	—
196	#500	—	59	1	no	—
197	#494	C₂×D₅₉	1	1	yes	characteristic
198	#482	—	118	2	no	—
199	#476	—	118	1	no	—
200	#467	C₂×D₅₉	1	2	yes	—
201	#466	—	59	2	no	—
202	#462	—	2	1	no	—
203	#451	—	2	2	no	—
204	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
205	C₄×D₈	—	59	1	no	—
206	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
207	#427	—	59	2	no	—
208	#367	—	59	1	no	—
209	#366	—	59	2	no	—
210	C₄⋊C₁₆	—	59	1	no	—
211	#358	—	59	1	no	—
212	#358	—	59	2	no	—
213	#358	—	59	2	no	—
214	#358	—	59	2	no	—
215	#358	—	59	2	no	—
216	#357	—	59	1	no	—
217	C₂²⋊C₁₆	—	59	2	no	—
218	C₄×C₁₆	—	59	1	no	—
219	C₅₉	#33896	1	1	yes	characteristic, complemented, Sylow
220	C₂²×Q₈	—	2	2	no	—
221	C₂×Q₁₆	—	2	4	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	4	2	no	—
225	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
226	C₂×Q₁₆	C₂²×D₅₉	1	1	yes	characteristic
227	C₂×D₈	—	59	1	no	—
228	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	4	1	no	—
231	C₄⋊Q₈	—	118	1	no	—
232	#124	—	236	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	4	2	no	—
235	#121	—	118	2	no	—
236	#121	—	236	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×D₄	—	4	1	no	—
239	C₄×D₄	—	118	1	no	—
240	C₄×D₄	C₂²⋊D₅₉	1	1	yes	characteristic
241	#117	—	118	2	no	—
242	#117	—	236	1	no	—
243	SD₃₂	—	118	2	no	—
244	SD₃₂	—	236	2	no	—
245	M₃₂	—	236	2	no	—
246	C₂×C₁₆	—	59	1	no	—
247	C₂×C₁₆	—	59	1	no	—
248	C₂×C₁₆	—	59	2	no	—
249	#107	—	59	2	no	—
250	#107	—	59	1	no	—
251	#107	—	118	1	no	—
252	#107	C₂²×D₅₉	1	1	yes	characteristic
253	#106	—	118	2	no	—
254	C₄⋊C₈	C₂²⋊D₅₉	1	1	yes	characteristic
255	#103	—	2	1	no	—
256	#103	—	2	2	no	—
257	#102	—	118	1	no	—
258	#102	—	118	2	no	—
259	C₂²⋊C₈	C₂²⋊D₅₉	1	2	yes	—
260	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	4	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	—	8	2	no	—
266	C₂×Q₈	—	236	1	no	—
267	C₂×D₄	—	118	1	no	—
268	C₂×D₄	—	236	2	no	—
269	C₂×D₄	#11745669	1	1	yes	characteristic
270	C₂²×C₄	—	4	2	no	—
271	C₂²×C₄	—	236	1	no	—
272	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
273	Q₁₆	—	2	2	no	—
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	4	4	no	—
277	Q₁₆	—	8	1	no	—
278	Q₁₆	#11745687	1	2	yes	—
279	D₈	—	118	1	no	—
280	D₈	—	118	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
284	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
285	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	118	1	no	—
292	C₄⋊C₄	—	118	2	no	—
293	C₄⋊C₄	—	236	1	no	—
294	C₄⋊C₄	#11745669	1	1	yes	characteristic
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	236	2	no	—
297	C₂²⋊C₄	—	236	2	no	—
298	C₂²⋊C₄	—	236	1	no	—
299	C₂²⋊C₄	#11745669	1	2	yes	—
300	C₄²	—	4	1	no	—
301	C₄²	—	236	2	no	—
302	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
303	C₁₆	—	118	1	no	—
304	C₁₆	—	118	1	no	—
305	C₁₆	—	118	2	no	—
306	C₂³	—	236	1	no	—
307	C₂³	C₂×#11745669	1	2	yes	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	4	no	—
313	Q₈	—	944	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	944	2	no	—
316	D₄	—	236	1	no	—
317	D₄	—	236	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	—	472	2	no	—
325	C₂×C₄	—	236	1	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	C₂×#11745669	1	2	yes	—
328	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
329	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	2	no	—
333	C₈	C₂×#11745687	1	1	yes	characteristic
334	C₈	#14273772	1	1	yes	characteristic
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	472	1	no	—
339	C₂²	—	472	1	no	—
340	C₂²	—	472	1	no	—
341	C₂²	C₂²×#11745669	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	—	472	1	no	—
348	C₄	—	472	2	no	—
349	C₄	#16837843	1	2	yes	—
350	C₄	#16838046	1	1	yes	characteristic
351	C₄	#16838213	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	944	1	no	—
354	C₂	#21435909	1	1	yes	characteristic
355	C₂	C₂×#16837843	1	1	yes	characteristic
356	C₂	#21437240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	D₄×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#21436519	C₂
hypercenter	D₄×Q₁₆	D₅₉
hypocenter	C₅₉	#33896
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #21426323	C₂	semidirect
3. #21426301	C₂	semidirect
4. #21426248	C₂	semidirect
5. #21426211	C₂	semidirect
6. #21426138	C₂	semidirect
7. #21426013	C₂	semidirect
8. C₄×#14273670	C₂	semidirect
9. #21425928	C₂	semidirect
10. #21425816	C₂	semidirect
11. D₄×Q₁₆×C₅₉	C₂	semidirect
175. D₄×Q₁₆	D₅₉	semidirect
219. C₅₉	#33896	semidirect

Maximal subgroups

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

subgroup	index
#21436519	2
#21426323	2
#21426301	2
#21426248	2
#21426211	2
#21426138	2
#21426013	2
C₄×#14273670	2
#21425928	2
#21425816	2
D₄×Q₁₆×C₅₉	2
#33896	59