Subgroups — #28745500

Order 15104.

636 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437247	C₂	1	2	yes	maximal, complemented
2	#21435901	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426348	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426219	C₂	1	2	yes	maximal, complemented
5	#21426177	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426174	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426074	C₂	1	2	yes	maximal, complemented
8	#21426072	C₂	1	2	yes	maximal, complemented
9	#21425975	C₂	1	1	yes	maximal, characteristic, complemented
10	#21425933	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#3293	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838219	C₂²	1	2	yes	—
13	#16838216	—	2	1	no	—
14	#16837843	—	2	2	no	—
15	#16837843	C₂²	1	2	yes	—
16	#16837842	—	2	2	no	—
17	#16837842	C₂²	1	2	yes	—
18	#16837832	—	2	2	no	—
19	#16837832	C₂²	1	1	yes	characteristic
20	C₂×#14273673	—	2	2	no	—
21	C₂×#14273673	C₂²	1	2	yes	—
22	C₂×#14273673	C₂²	1	2	yes	—
23	#16837819	C₂²	1	2	yes	—
24	#16837782	C₂²	1	1	yes	characteristic
25	#16837678	—	2	1	no	—
26	#16837677	C₂²	1	2	yes	—
27	#16837586	—	2	2	no	—
28	#16837520	C₂²	1	1	yes	characteristic
29	#16837231	C₂²	1	1	yes	characteristic
30	#16837223	C₂²	1	1	yes	characteristic
31	#16837218	C₂²	1	2	yes	—
32	#16837217	C₂²	1	2	yes	—
33	#16837209	C₂²	1	1	yes	characteristic
34	#16837208	C₂²	1	1	yes	characteristic
35	#16837186	C₂²	1	2	yes	—
36	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₅₉×#578	—	2	1	no	—
38	C₅₉×#578	C₂²	1	2	yes	—
39	C₅₉×#494	C₂²	1	1	yes	characteristic
40	C₅₉×#473	—	2	1	no	—
41	C₅₉×#473	C₂²	1	2	yes	—
42	C₅₉×#456	—	2	2	no	—
43	C₅₉×#443	C₂²	1	1	yes	characteristic
44	C₅₉×#443	C₂²	1	1	yes	characteristic
45	#14273774	—	2	2	no	—
46	#14273760	—	2	2	no	—
47	#14273760	—	4	2	no	—
48	#14273757	—	2	2	no	—
49	#14273756	—	4	2	no	—
50	#14273749	—	2	2	no	—
51	#14273749	C₂³	1	1	yes	characteristic
52	#14273746	—	4	1	no	—
53	C₂×Dic₂₃₆	—	2	2	no	—
54	C₂×Dic₂₃₆	—	2	1	no	—
55	C₂×Dic₂₃₆	C₂³	1	2	yes	—
56	Q₁₆×D₅₉	—	4	1	no	—
57	#14273733	—	2	1	no	—
58	#14273727	—	2	1	no	—
59	#14273673	—	2	2	no	—
60	#14273673	—	2	2	no	—
61	#14273673	—	2	2	no	—
62	#14273673	—	2	2	no	—
63	#14273673	—	4	2	no	—
64	#14273673	D₄	1	1	yes	characteristic
65	#14273671	—	2	2	no	—
66	#14273671	D₄	1	1	yes	characteristic
67	#14273670	—	2	2	no	—
68	#14273670	D₄	1	1	yes	characteristic
69	#14273668	D₄	1	1	yes	characteristic
70	#14273667	—	2	2	no	—
71	#14273667	C₂³	1	2	yes	—
72	C₂×C₅₉⋊C₁₆	—	2	2	no	—
73	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
75	#14273663	C₂³	1	1	yes	characteristic
76	#14273655	—	2	1	no	—
77	#14273645	—	2	2	no	—
78	2₋¹⁺⁴×C₅₉	—	2	2	no	—
79	#137×C₅₉	—	2	2	no	—
80	#137×C₅₉	—	2	2	no	—
81	#137×C₅₉	—	4	2	no	—
82	#137×C₅₉	—	4	2	no	—
83	#135×C₅₉	—	2	2	no	—
84	#135×C₅₉	—	4	1	no	—
85	#135×C₅₉	D₄	1	1	yes	characteristic
86	#135×C₅₉	C₂³	1	1	yes	characteristic
87	C₂×Q₁₆×C₅₉	—	2	2	no	—
88	C₂×Q₁₆×C₅₉	—	2	1	no	—
89	C₂×Q₁₆×C₅₉	—	4	1	no	—
90	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
91	#131×C₅₉	—	2	1	no	—
92	#131×C₅₉	C₂³	1	2	yes	—
93	C₄⋊Q₈×C₅₉	—	2	1	no	—
94	#108×C₅₉	D₄	1	1	yes	characteristic
95	#108×C₅₉	C₂³	1	1	yes	characteristic
96	#104×C₅₉	—	2	1	no	—
97	#104×C₅₉	D₄	1	2	yes	—
98	#101×C₅₉	—	2	2	no	—
99	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
100	#41×C₅₉	—	2	1	no	—
101	#41×C₅₉	—	4	2	no	—
102	#41×C₅₉	—	4	2	no	—
103	#41×C₅₉	—	8	2	no	—
104	#41×C₅₉	C₂×D₄	1	2	yes	—
105	C₂×Q₈×C₅₉	—	2	2	no	—
106	C₂×Q₈×C₅₉	—	4	2	no	—
107	C₂×Q₈×C₅₉	—	4	2	no	—
108	Q₈×D₅₉	—	4	2	no	—
109	#11745687	—	4	2	no	—
110	#11745687	—	8	2	no	—
111	#11745685	—	2	1	no	—
112	#11745685	—	4	2	no	—
113	C₂×Dic₁₁₈	—	2	2	no	—
114	C₂×Dic₁₁₈	—	4	2	no	—
115	Q₁₆×C₅₉	—	2	2	no	—
116	Q₁₆×C₅₉	—	2	2	no	—
117	Q₁₆×C₅₉	—	4	2	no	—
118	Q₁₆×C₅₉	—	4	2	no	—
119	Q₁₆×C₅₉	—	4	1	no	—
120	Q₁₆×C₅₉	—	4	1	no	—
121	Q₁₆×C₅₉	D₈	1	1	yes	characteristic
122	Q₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
123	SD₁₆×C₅₉	—	2	1	no	—
124	SD₁₆×C₅₉	—	4	2	no	—
125	SD₁₆×C₅₉	—	4	2	no	—
126	SD₁₆×C₅₉	D₈	1	2	yes	—
127	D₈×C₅₉	D₈	1	1	yes	characteristic
128	D₈×C₅₉	C₂×D₄	1	1	yes	characteristic
129	M₁₆×C₅₉	—	2	2	no	—
130	M₁₆×C₅₉	—	2	1	no	—
131	M₁₆×C₅₉	—	4	1	no	—
132	M₁₆×C₅₉	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₅₉	—	2	2	no	—
134	C₂×C₈×C₅₉	—	4	1	no	—
135	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
137	C₄⋊C₄×C₅₉	—	8	1	no	—
138	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
139	#11745672	—	4	2	no	—
140	#11745670	—	4	2	no	—
141	C₄⋊Dic₅₉	—	8	1	no	—
142	#11745664	—	2	1	no	—
143	Dic₂₃₆	—	2	2	no	—
144	Dic₂₃₆	—	2	2	no	—
145	Dic₂₃₆	—	4	2	no	—
146	Dic₂₃₆	—	4	1	no	—
147	Dic₂₃₆	—	4	1	no	—
148	Dic₂₃₆	C₂×D₄	1	1	yes	characteristic
149	D₄₇₂	C₂×D₄	1	1	yes	characteristic
150	#11745660	—	2	1	no	—
151	#11745660	—	4	2	no	—
152	#11745659	—	4	1	no	—
153	C₈×D₅₉	—	4	1	no	—
154	C₅₉⋊C₁₆	—	2	2	no	—
155	C₅₉⋊C₁₆	—	2	1	no	—
156	C₅₉⋊C₁₆	C₂×D₄	1	1	yes	characteristic
157	C₅₉⋊C₁₆	C₂×D₄	1	1	yes	characteristic
158	Q₈×C₅₉	—	2	1	no	—
159	Q₈×C₅₉	—	4	2	no	—
160	Q₈×C₅₉	—	4	2	no	—
161	Q₈×C₅₉	—	8	2	no	—
162	Q₈×C₅₉	C₂×D₈	1	2	yes	—
163	D₄×C₅₉	—	2	2	no	—
164	D₄×C₅₉	—	2	1	no	—
165	D₄×C₅₉	—	4	1	no	—
166	D₄×C₅₉	—	8	2	no	—
167	D₄×C₅₉	C₂×D₈	1	2	yes	—
168	C₂×C₄×C₅₉	—	2	1	no	—
169	C₂×C₄×C₅₉	—	2	2	no	—
170	C₂×C₄×C₅₉	—	4	1	no	—
171	C₂×C₄×C₅₉	—	4	2	no	—
172	C₂×C₄×C₅₉	—	8	2	no	—
173	C₂×C₄×C₅₉	—	8	2	no	—
174	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
175	C₂²⋊D₅₉	—	4	1	no	—
176	C₂²⋊D₅₉	—	8	2	no	—
177	C₂×Dic₅₉	—	4	2	no	—
178	C₂×Dic₅₉	—	8	2	no	—
179	D₂₃₆	—	2	1	no	—
180	C₄×D₅₉	—	4	1	no	—
181	C₄×D₅₉	—	8	2	no	—
182	Dic₁₁₈	—	2	1	no	—
183	Dic₁₁₈	—	4	2	no	—
184	Dic₁₁₈	—	4	2	no	—
185	Dic₁₁₈	—	8	2	no	—
186	C₈×C₅₉	—	2	2	no	—
187	C₈×C₅₉	—	4	1	no	—
188	C₈×C₅₉	C₂×D₈	1	1	yes	characteristic
189	C₈×C₅₉	C₂×D₈	1	1	yes	characteristic
190	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
191	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
192	C₅₉⋊C₈	—	4	1	no	—
193	#33860	—	59	1	no	maximal, Sylow
194	C₂²×C₅₉	—	2	2	no	—
195	C₂²×C₅₉	—	4	1	no	—
196	C₂²×C₅₉	#576	1	1	yes	characteristic
197	C₂×D₅₉	—	4	1	no	—
198	C₄×C₅₉	—	2	2	no	—
199	C₄×C₅₉	—	2	2	no	—
200	C₄×C₅₉	—	4	1	no	—
201	C₄×C₅₉	—	8	2	no	—
202	C₄×C₅₉	D₄²	1	1	yes	characteristic
203	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
204	Dic₅₉	—	4	1	no	—
205	Dic₅₉	—	8	2	no	—
206	#3417	—	59	2	no	—
207	#3293	—	59	1	no	—
208	#3293	D₅₉	1	1	yes	characteristic, complemented
209	#2247	—	59	1	no	—
210	#2222	—	59	2	no	—
211	#2214	—	59	1	no	—
212	#2214	—	59	1	no	—
213	#2195	—	59	2	no	—
214	#2195	—	59	2	no	—
215	#2178	—	59	1	no	—
216	#2170	—	59	1	no	—
217	C₂×C₅₉	—	2	1	no	—
218	C₂×C₅₉	—	4	2	no	—
219	C₂×C₅₉	—	8	1	no	—
220	C₂×C₅₉	D₄×D₈	1	1	yes	characteristic
221	D₅₉	—	8	1	no	—
222	#578	—	2	1	no	—
223	#578	—	59	2	no	—
224	#578	—	118	1	no	—
225	#578	C₂×D₅₉	1	2	yes	—
226	#510	—	59	2	no	—
227	#510	—	59	2	no	—
228	#510	—	118	2	no	—
229	#510	—	118	2	no	—
230	#508	—	59	1	no	—
231	#508	—	118	2	no	—
232	C₂×Q₃₂	—	59	2	no	—
233	C₂×Q₃₂	—	59	2	no	—
234	C₂×Q₃₂	—	118	2	no	—
235	#504	—	59	2	no	—
236	#494	—	59	1	no	—
237	#494	C₂×D₅₉	1	1	yes	characteristic
238	#473	—	2	1	no	—
239	#473	—	59	2	no	—
240	#473	—	118	1	no	—
241	#473	C₂×D₅₉	1	2	yes	—
242	#456	—	2	2	no	—
243	#456	—	118	2	no	—
244	#443	—	59	1	no	—
245	#443	C₂×D₅₉	1	1	yes	characteristic
246	#443	C₂×D₅₉	1	1	yes	characteristic
247	#368	—	59	1	no	—
248	#364	—	59	1	no	—
249	#362	—	59	2	no	—
250	#362	—	59	2	no	—
251	#359	—	59	1	no	—
252	#359	—	59	1	no	—
253	#350	—	59	2	no	—
254	C₄×C₁₆	—	59	1	no	—
255	C₅₉	#33860	1	1	yes	characteristic, complemented, Sylow
256	2₋¹⁺⁴	—	2	2	no	—
257	2₋¹⁺⁴	—	118	2	no	—
258	#137	—	2	2	no	—
259	#137	—	2	2	no	—
260	#137	—	4	2	no	—
261	#137	—	4	2	no	—
262	#137	—	118	2	no	—
263	#137	—	118	2	no	—
264	#137	—	236	2	no	—
265	#137	—	236	2	no	—
266	#135	—	2	2	no	—
267	#135	—	4	1	no	—
268	#135	—	59	1	no	—
269	#135	—	118	2	no	—
270	#135	—	236	1	no	—
271	#135	C₂²⋊D₅₉	1	1	yes	characteristic
272	#135	C₂²×D₅₉	1	1	yes	characteristic
273	C₂×Q₁₆	—	2	2	no	—
274	C₂×Q₁₆	—	2	1	no	—
275	C₂×Q₁₆	—	4	1	no	—
276	C₂×Q₁₆	—	59	2	no	—
277	C₂×Q₁₆	—	118	2	no	—
278	C₂×Q₁₆	—	118	1	no	—
279	C₂×Q₁₆	—	236	1	no	—
280	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
281	#131	—	2	1	no	—
282	#131	—	118	1	no	—
283	#131	C₂²×D₅₉	1	2	yes	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	—	118	1	no	—
286	Q₃₂	—	59	1	no	—
287	Q₃₂	—	118	2	no	—
288	Q₃₂	—	118	2	no	—
289	Q₃₂	—	118	2	no	—
290	Q₃₂	—	118	2	no	—
291	Q₃₂	—	236	2	no	—
292	SD₃₂	—	59	1	no	—
293	SD₃₂	—	59	1	no	—
294	SD₃₂	—	118	2	no	—
295	SD₃₂	—	118	2	no	—
296	D₁₆	—	59	1	no	—
297	M₃₂	—	59	2	no	—
298	M₃₂	—	118	2	no	—
299	C₂×C₁₆	—	59	1	no	—
300	C₂×C₁₆	—	59	1	no	—
301	C₂×C₁₆	—	118	2	no	—
302	#108	—	59	1	no	—
303	#108	C₂²⋊D₅₉	1	1	yes	characteristic
304	#108	C₂²×D₅₉	1	1	yes	characteristic
305	#104	—	2	1	no	—
306	#104	—	118	1	no	—
307	#104	C₂²⋊D₅₉	1	2	yes	—
308	#101	—	2	2	no	—
309	#101	—	118	2	no	—
310	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
311	#41	—	2	1	no	—
312	#41	—	4	2	no	—
313	#41	—	4	2	no	—
314	#41	—	8	2	no	—
315	#41	—	118	1	no	—
316	#41	—	472	2	no	—
317	#41	—	236	2	no	—
318	#41	—	236	2	no	—
319	#41	C₂×C₂²⋊D₅₉	1	2	yes	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂×Q₈	—	118	2	no	—
324	C₂×Q₈	—	236	2	no	—
325	C₂×Q₈	—	236	2	no	—
326	Q₁₆	—	2	2	no	—
327	Q₁₆	—	2	2	no	—
328	Q₁₆	—	4	2	no	—
329	Q₁₆	—	4	2	no	—
330	Q₁₆	—	4	1	no	—
331	Q₁₆	—	4	1	no	—
332	Q₁₆	—	59	1	no	—
333	Q₁₆	—	118	2	no	—
334	Q₁₆	—	118	2	no	—
335	Q₁₆	—	236	2	no	—
336	Q₁₆	—	236	2	no	—
337	Q₁₆	—	236	1	no	—
338	Q₁₆	—	236	1	no	—
339	Q₁₆	#11745669	1	1	yes	characteristic
340	Q₁₆	D₄×D₅₉	1	1	yes	characteristic
341	SD₁₆	—	2	1	no	—
342	SD₁₆	—	4	2	no	—
343	SD₁₆	—	4	2	no	—
344	SD₁₆	—	118	1	no	—
345	SD₁₆	—	236	2	no	—
346	SD₁₆	—	236	2	no	—
347	SD₁₆	#11745669	1	2	yes	—
348	D₈	—	59	1	no	—
349	D₈	#11745669	1	1	yes	characteristic
350	D₈	D₄×D₅₉	1	1	yes	characteristic
351	M₁₆	—	2	2	no	—
352	M₁₆	—	2	1	no	—
353	M₁₆	—	4	1	no	—
354	M₁₆	—	118	1	no	—
355	M₁₆	—	236	1	no	—
356	M₁₆	C₂×C₂²⋊D₅₉	1	2	yes	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	4	1	no	—
359	C₂×C₈	—	236	1	no	—
360	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
361	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
362	C₄⋊C₄	—	8	1	no	—
363	C₄⋊C₄	—	472	1	no	—
364	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
365	C₁₆	—	59	1	no	—
366	C₁₆	—	59	1	no	—
367	C₁₆	—	118	2	no	—
368	C₁₆	—	118	1	no	—
369	Q₈	—	2	1	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	8	2	no	—
373	Q₈	—	118	1	no	—
374	Q₈	—	472	2	no	—
375	Q₈	—	236	2	no	—
376	Q₈	—	236	2	no	—
377	Q₈	C₂×#11745669	1	2	yes	—
378	D₄	—	2	2	no	—
379	D₄	—	2	1	no	—
380	D₄	—	4	1	no	—
381	D₄	—	8	2	no	—
382	D₄	—	118	1	no	—
383	D₄	—	472	2	no	—
384	D₄	—	236	1	no	—
385	D₄	C₂×#11745669	1	2	yes	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	—	472	2	no	—
393	C₂×C₄	—	472	2	no	—
394	C₂×C₄	—	236	2	no	—
395	C₂×C₄	—	236	1	no	—
396	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	4	1	no	—
399	C₈	—	236	1	no	—
400	C₈	C₂×#11745669	1	1	yes	characteristic
401	C₈	C₂×#11745669	1	1	yes	characteristic
402	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
403	C₈	#14273772	1	1	yes	characteristic
404	C₂²	—	2	2	no	—
405	C₂²	—	4	1	no	—
406	C₂²	—	236	1	no	—
407	C₂²	#16838205	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	1	no	—
411	C₄	—	8	2	no	—
412	C₄	—	472	2	no	—
413	C₄	—	236	1	no	—
414	C₄	D₄×C₂²⋊D₅₉	1	1	yes	characteristic
415	C₄	C₂²×#11745669	1	1	yes	characteristic
416	C₂	—	2	1	no	—
417	C₂	—	4	2	no	—
418	C₂	—	8	1	no	—
419	C₂	—	472	1	no	—
420	C₂	D₄×#11745669	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3293	C₂
socle	C₂×C₅₉	D₄×D₈
hypercenter	#3293	D₅₉
hypocenter	C₅₉	#33860
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #3293

Complements

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

N	H ≅ G/N	type
1. #21437247	C₂	semidirect
2. #21435901	C₂	semidirect
3. #21426348	C₂	semidirect
4. #21426219	C₂	semidirect
5. #21426177	C₂	semidirect
6. #21426174	C₂	semidirect
7. #21426074	C₂	semidirect
8. #21426072	C₂	semidirect
9. #21425975	C₂	semidirect
10. #21425933	C₂	semidirect
11. C₅₉×#3293	C₂	semidirect
208. #3293	D₅₉	semidirect
255. C₅₉	#33860	semidirect

Maximal subgroups

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

subgroup	index
#21437247	2
#21435901	2
#21426348	2
#21426219	2
#21426177	2
#21426174	2
#21426074	2
#21426072	2
#21425975	2
#21425933	2
C₅₉×#3293	2
#33860	59