Subgroups — #28742105

Order 15104.

642 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437230	C₂	1	1	yes	maximal, characteristic
2	#21433013	C₂	1	1	yes	maximal, characteristic, complemented
3	#21433012	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425979	C₂	1	4	yes	maximal, complemented
5	#21425974	C₂	1	4	yes	maximal, complemented
6	#21425890	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16837210	C₂	1	1	yes	maximal, characteristic, complemented
8	#21425692	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₉×#359	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14273749	C₂²	1	1	yes	characteristic
11	#16838169	—	2	1	no	—
12	#16838168	—	2	1	no	—
13	#16837843	C₄	1	2	yes	complemented
14	#16837842	C₄	1	2	yes	complemented
15	#16837838	C₄	1	2	yes	complemented
16	#16837837	C₄	1	2	yes	complemented
17	#16837832	C₂²	1	4	yes	—
18	C₂×#14273667	C₂²	1	1	yes	characteristic
19	#16837524	—	2	2	no	—
20	#16837523	—	2	2	no	—
21	#16837519	C₂²	1	4	yes	complemented
22	#16837518	C₂²	1	4	yes	complemented
23	C₂×#14273663	C₂²	1	1	yes	characteristic
24	#16837419	—	2	1	no	—
25	#16837418	—	2	1	no	—
26	#16837357	C₂²	1	1	yes	characteristic
27	#16837231	C₂²	1	4	yes	complemented
28	#16837210	C₂²	1	2	yes	complemented
29	#16837210	C₂²	1	2	yes	complemented
30	#16837177	C₂²	1	2	yes	complemented
31	C₁₆×Dic₅₉	C₂²	1	2	yes	complemented
32	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
34	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
35	C₅₉×#359	C₂²	1	2	yes	complemented
36	C₅₉×#359	C₂²	1	2	yes	complemented
37	C₂×#11745685	—	2	1	no	—
38	#14273749	—	2	1	no	—
39	#14273749	—	2	1	no	—
40	#14273749	C₂×C₄	1	2	yes	—
41	#14273749	C₂³	1	2	yes	—
42	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
43	C₂×#11745660	—	2	1	no	—
44	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
45	#14273735	—	2	4	no	—
46	#14273734	—	2	4	no	—
47	#14273734	—	4	2	no	—
48	#14273733	—	4	2	no	—
49	C₂×#11745664	—	2	1	no	—
50	C₂×#11745664	—	4	2	no	—
51	C₂×#11745664	D₄	1	1	yes	characteristic
52	M₁₆×D₅₉	—	4	2	no	—
53	#14273686	D₄	1	1	yes	characteristic
54	#14273673	C₂×C₄	1	4	yes	—
55	#14273671	C₂×C₄	1	4	yes	—
56	#14273670	C₂×C₄	1	4	yes	—
57	#14273668	C₂×C₄	1	4	yes	—
58	#14273667	C₂×C₄	1	2	yes	—
59	#14273667	C₂×C₄	1	2	yes	—
60	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
61	#14273663	—	2	1	no	—
62	#14273663	C₂³	1	2	yes	—
63	#14273654	—	2	4	no	—
64	#14273653	—	2	4	no	—
65	#14273637	D₄	1	2	yes	—
66	C₈×Dic₅₉	C₂³	1	2	yes	—
67	C₂×#41×C₅₉	—	2	1	no	—
68	#135×C₅₉	—	2	1	no	—
69	#135×C₅₉	—	2	1	no	—
70	#135×C₅₉	C₂×C₄	1	2	yes	—
71	#135×C₅₉	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
73	C₂×SD₁₆×C₅₉	—	2	1	no	—
74	C₂×D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
75	C₂×M₁₆×C₅₉	—	2	1	no	—
76	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₅₉	—	2	1	no	—
78	C₂×C₁₆×C₅₉	—	2	1	no	—
79	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
81	#108×C₅₉	—	2	1	no	—
82	#108×C₅₉	C₂³	1	2	yes	—
83	#41×C₅₉	—	2	2	no	—
84	#41×C₅₉	—	2	2	no	—
85	#41×C₅₉	—	4	1	no	—
86	#41×C₅₉	—	4	1	no	—
87	C₂×Q₈×C₅₉	—	2	1	no	—
88	C₂×D₄×C₅₉	—	2	1	no	—
89	C₂×D₄×C₅₉	—	4	1	no	—
90	C₂²×C₄×C₅₉	—	4	1	no	—
91	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₅₉	—	4	1	no	—
93	#11745685	—	2	2	no	—
94	#11745685	—	2	2	no	—
95	#11745685	—	4	1	no	—
96	#11745685	—	4	1	no	—
97	C₂×D₂₃₆	—	2	1	no	—
98	C₂×C₄×D₅₉	—	4	1	no	—
99	C₂×Dic₁₁₈	—	2	1	no	—
100	Q₁₆×C₅₉	—	2	1	no	—
101	Q₁₆×C₅₉	C₂²×C₄	1	2	yes	—
102	SD₁₆×C₅₉	—	2	2	no	—
103	SD₁₆×C₅₉	—	2	2	no	—
104	D₈×C₅₉	—	2	1	no	—
105	D₈×C₅₉	C₂²×C₄	1	2	yes	—
106	M₁₆×C₅₉	—	2	2	no	—
107	M₁₆×C₅₉	—	4	1	no	—
108	C₂×C₈×C₅₉	—	4	1	no	—
109	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
115	C₂²⋊Dic₅₉	—	2	1	no	—
116	C₄⋊Dic₅₉	—	2	1	no	—
117	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
118	#11745664	—	2	2	no	—
119	#11745664	—	4	1	no	—
120	#11745664	—	4	4	no	—
121	#11745664	D₈	1	2	yes	—
122	#11745664	D₈	1	2	yes	—
123	C₂×C₅₉⋊C₈	—	4	1	no	—
124	C₂×C₅₉⋊C₈	—	4	4	no	—
125	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
126	Dic₂₃₆	—	2	1	no	—
127	Dic₂₃₆	C₂²×C₄	1	2	yes	—
128	D₄₇₂	—	2	1	no	—
129	D₄₇₂	C₂²×C₄	1	2	yes	—
130	#11745660	—	2	2	no	—
131	#11745660	—	2	2	no	—
132	#11745659	—	4	4	no	—
133	C₈×D₅₉	—	4	4	no	—
134	C₁₆×C₅₉	—	2	1	no	—
135	C₁₆×C₅₉	—	2	1	no	—
136	C₅₉⋊C₁₆	C₂²×C₄	1	4	yes	—
137	C₂³×C₅₉	—	4	1	no	—
138	C₂³×C₅₉	#135	1	1	yes	characteristic
139	C₂²×D₅₉	—	4	1	no	—
140	Q₈×C₅₉	—	2	2	no	—
141	Q₈×C₅₉	—	4	1	no	—
142	D₄×C₅₉	—	2	2	no	—
143	D₄×C₅₉	—	4	2	no	—
144	D₄×C₅₉	—	4	2	no	—
145	D₄×C₅₉	—	4	1	no	—
146	C₂×C₄×C₅₉	—	4	1	no	—
147	C₂×C₄×C₅₉	—	4	2	no	—
148	C₂×C₄×C₅₉	—	4	2	no	—
149	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
154	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
155	C₂²⋊D₅₉	—	4	2	no	—
156	C₂²⋊D₅₉	—	4	2	no	—
157	C₂×Dic₅₉	—	2	2	no	—
158	C₂×Dic₅₉	—	4	1	no	—
159	D₂₃₆	—	2	2	no	—
160	D₂₃₆	—	4	1	no	—
161	C₄×D₅₉	—	4	2	no	—
162	C₄×D₅₉	—	4	2	no	—
163	Dic₁₁₈	—	2	2	no	—
164	Dic₁₁₈	—	4	1	no	—
165	C₈×C₅₉	—	4	2	no	—
166	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
167	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
168	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
169	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
170	C₅₉⋊C₈	—	4	2	no	—
171	C₅₉⋊C₈	C₂×D₈	1	4	yes	—
172	#33464	—	59	1	no	maximal, Sylow
173	C₂²×C₅₉	—	4	1	no	—
174	C₂²×C₅₉	—	4	2	no	—
175	C₂²×C₅₉	—	16	1	no	—
176	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
177	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
178	C₂²×C₅₉	C₂×#135	1	1	yes	characteristic
179	C₂×D₅₉	—	4	2	no	—
180	C₂×D₅₉	—	16	1	no	—
181	C₄×C₅₉	—	4	2	no	—
182	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
183	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
184	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
185	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
186	Dic₅₉	—	4	2	no	—
187	Dic₅₉	—	4	2	no	—
188	#3414	—	59	1	no	—
189	#2957	—	59	1	no	—
190	#2957	—	59	1	no	—
191	#2179	—	59	4	no	—
192	#2178	—	59	4	no	—
193	#2161	—	59	1	no	—
194	C₂×#359	—	59	1	no	—
195	C₂×#359	D₅₉	1	1	yes	characteristic, complemented
196	#2108	—	59	1	no	—
197	C₂×C₅₉	—	2	1	no	—
198	C₂×C₅₉	—	2	1	no	—
199	C₂×C₅₉	—	2	1	no	—
200	C₂×C₅₉	—	8	2	no	—
201	C₂×C₅₉	C₂×C₄×D₈	1	1	yes	characteristic
202	D₅₉	—	8	2	no	—
203	C₂×#135	—	59	1	no	—
204	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
205	#568	—	118	1	no	—
206	#568	—	118	1	no	—
207	#510	—	59	2	no	—
208	#510	—	59	2	no	—
209	#509	—	59	2	no	—
210	#509	—	59	2	no	—
211	#508	—	59	4	no	—
212	C₂×M₃₂	—	59	1	no	—
213	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
214	#444	—	118	2	no	—
215	#444	—	118	2	no	—
216	#443	—	59	4	no	—
217	#443	—	59	4	no	—
218	C₂×#108	—	59	1	no	—
219	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
220	#421	—	118	1	no	—
221	#421	—	118	1	no	—
222	#405	—	59	1	no	—
223	#368	—	59	4	no	—
224	#359	—	59	2	no	—
225	#359	—	59	2	no	—
226	#359	C₂×D₅₉	1	2	yes	complemented
227	#359	C₂×D₅₉	1	2	yes	complemented
228	#346	—	59	2	no	—
229	C₄×C₁₆	—	59	2	no	—
230	C₅₉	#33464	1	1	yes	characteristic, complemented, Sylow
231	C₂×#41	—	2	1	no	—
232	C₂×#41	—	118	1	no	—
233	#135	—	2	1	no	—
234	#135	—	2	1	no	—
235	#135	—	59	2	no	—
236	#135	—	59	2	no	—
237	#135	—	118	1	no	—
238	#135	—	118	1	no	—
239	#135	C₄×D₅₉	1	2	yes	—
240	#135	C₂²×D₅₉	1	2	yes	—
241	C₂×Q₁₆	—	59	1	no	—
242	C₂×Q₁₆	C₄×D₅₉	1	1	yes	characteristic
243	C₂×SD₁₆	—	2	1	no	—
244	C₂×SD₁₆	—	118	1	no	—
245	C₂×D₈	—	59	1	no	—
246	C₂×D₈	C₄×D₅₉	1	1	yes	characteristic
247	#131	—	118	4	no	—
248	#131	—	118	4	no	—
249	#131	—	236	2	no	—
250	#131	—	236	2	no	—
251	C₂×M₁₆	—	2	1	no	—
252	C₂×M₁₆	—	59	1	no	—
253	C₂×M₁₆	—	118	1	no	—
254	C₂×M₁₆	—	236	2	no	—
255	C₂×M₁₆	—	236	2	no	—
256	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
257	#117	—	59	1	no	—
258	Q₃₂	—	59	4	no	—
259	SD₃₂	—	59	4	no	—
260	SD₃₂	—	59	4	no	—
261	D₁₆	—	59	4	no	—
262	M₃₂	—	59	2	no	—
263	M₃₂	—	59	2	no	—
264	C₂×C₁₆	—	2	1	no	—
265	C₂×C₁₆	—	2	1	no	—
266	C₂×C₁₆	—	59	2	no	—
267	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
269	#108	—	2	1	no	—
270	#108	—	59	2	no	—
271	#108	—	118	1	no	—
272	#108	C₂²×D₅₉	1	2	yes	—
273	#104	—	118	4	no	—
274	#104	—	118	4	no	—
275	#97	—	59	2	no	—
276	C₄×C₈	—	59	2	no	—
277	#41	—	2	2	no	—
278	#41	—	2	2	no	—
279	#41	—	4	1	no	—
280	#41	—	4	1	no	—
281	#41	—	118	2	no	—
282	#41	—	118	2	no	—
283	#41	—	236	1	no	—
284	#41	—	236	1	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	118	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	118	1	no	—
290	C₂×D₄	—	236	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	236	1	no	—
293	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
294	Q₁₆	—	2	1	no	—
295	Q₁₆	—	59	2	no	—
296	Q₁₆	—	118	1	no	—
297	Q₁₆	C₂×C₄×D₅₉	1	2	yes	—
298	SD₁₆	—	2	2	no	—
299	SD₁₆	—	2	2	no	—
300	SD₁₆	—	118	2	no	—
301	SD₁₆	—	118	2	no	—
302	D₈	—	2	1	no	—
303	D₈	—	59	2	no	—
304	D₈	—	118	1	no	—
305	D₈	C₂×C₄×D₅₉	1	2	yes	—
306	M₁₆	—	2	2	no	—
307	M₁₆	—	4	1	no	—
308	M₁₆	—	59	2	no	—
309	M₁₆	—	59	2	no	—
310	M₁₆	—	118	2	no	—
311	M₁₆	—	236	4	no	—
312	M₁₆	—	236	4	no	—
313	M₁₆	—	236	1	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	59	2	no	—
316	C₂×C₈	—	236	4	no	—
317	C₂×C₈	—	236	4	no	—
318	C₂×C₈	—	236	1	no	—
319	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
320	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
321	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
322	C₂×C₈	#11745687	1	1	yes	characteristic
323	C₂×C₈	#11745687	1	1	yes	characteristic
324	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
325	C₄⋊C₄	—	118	1	no	—
326	C₂²⋊C₄	—	118	1	no	—
327	C₄²	—	59	2	no	—
328	C₁₆	—	2	1	no	—
329	C₁₆	—	2	1	no	—
330	C₁₆	—	59	4	no	—
331	C₂³	—	4	1	no	—
332	C₂³	—	236	1	no	—
333	C₂³	#14273748	1	1	yes	characteristic
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	118	2	no	—
337	Q₈	—	236	1	no	—
338	D₄	—	2	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	1	no	—
342	D₄	—	118	2	no	—
343	D₄	—	236	2	no	—
344	D₄	—	236	2	no	—
345	D₄	—	236	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	118	2	no	—
350	C₂×C₄	—	236	1	no	—
351	C₂×C₄	—	236	2	no	—
352	C₂×C₄	—	236	2	no	—
353	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
354	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
355	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
356	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
357	C₂×C₄	#14273748	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	59	4	no	—
361	C₈	—	236	2	no	—
362	C₈	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
363	C₈	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
364	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
365	C₈	C₂×#11745687	1	1	yes	characteristic
366	C₂²	—	4	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	16	1	no	—
369	C₂²	—	944	1	no	—
370	C₂²	—	236	2	no	—
371	C₂²	#16837482	1	1	yes	characteristic
372	C₂²	#16837482	1	1	yes	characteristic
373	C₂²	C₂×#14273748	1	1	yes	characteristic
374	C₄	—	4	2	no	—
375	C₄	—	236	2	no	—
376	C₄	—	236	2	no	—
377	C₄	#16837482	1	1	yes	characteristic
378	C₄	#16837482	1	1	yes	characteristic
379	C₄	C₂×(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
380	C₄	C₂×D₈×D₅₉	1	1	yes	characteristic
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	8	2	no	—
385	C₂	—	472	2	no	—
386	C₂	C₂×#16837482	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₅₉
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#359	C₂
socle	C₂×C₅₉	C₂×C₄×D₈
prime radical	#21437230	C₂
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#359	D₅₉
hypocenter	C₅₉	#33464
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#359

Complements

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

N	H ≅ G/N	type
2. #21433013	C₂	semidirect
3. #21433012	C₂	semidirect
4. #21425979	C₂	semidirect
5. #21425974	C₂	semidirect
6. #21425890	C₂	semidirect
7. C₂×#16837210	C₂	semidirect
8. #21425692	C₂	semidirect
9. C₂×C₅₉×#359	C₂	semidirect
13. #16837843	C₄	semidirect
14. #16837842	C₄	semidirect
15. #16837838	C₄	semidirect
16. #16837837	C₄	semidirect
21. #16837519	C₂²	semidirect
22. #16837518	C₂²	semidirect
27. #16837231	C₂²	semidirect
28. #16837210	C₂²	semidirect
29. #16837210	C₂²	semidirect
30. #16837177	C₂²	semidirect
31. C₁₆×Dic₅₉	C₂²	semidirect
35. C₅₉×#359	C₂²	semidirect
36. C₅₉×#359	C₂²	semidirect
195. C₂×#359	D₅₉	semidirect
226. #359	C₂×D₅₉	semidirect
227. #359	C₂×D₅₉	semidirect
230. C₅₉	#33464	semidirect

Maximal subgroups

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

subgroup	index
#21437230	2
#21433013	2
#21433012	2
#21425979	2
#21425974	2
#21425890	2
C₂×#16837210	2
#21425692	2
C₂×C₅₉×#359	2
#33464	59