Subgroups — #28745410

Order 15104.

674 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837834	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#21435839	C₂	1	2	yes	maximal, complemented
4	#21426357	C₂	1	1	yes	maximal, characteristic, complemented
5	#21426187	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426182	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426159	C₂	1	2	yes	maximal, complemented
8	#21426038	C₂	1	2	yes	maximal, complemented
9	#21426015	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14273749	C₂²	1	2	yes	—
13	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
14	#16838061	—	2	2	no	—
15	#16837834	—	2	4	no	—
16	C₂×Dic₄₇₂	—	2	4	no	—
17	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₇₂	C₂²	1	2	yes	—
19	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
20	C₂×#14273672	C₂²	1	2	yes	—
21	C₂×#14273672	C₂²	1	2	yes	—
22	C₂×D₉₄₄	C₂²	1	1	yes	characteristic
23	#16837786	C₂²	1	2	yes	—
24	#16837643	C₂²	1	2	yes	—
25	#16837638	C₂²	1	2	yes	—
26	#16837625	—	2	2	no	—
27	#16837566	—	2	2	no	—
28	#16837543	—	2	2	no	—
29	C₄×Dic₂₃₆	C₂²	1	2	yes	—
30	#16837228	C₂²	1	1	yes	characteristic
31	#16837207	C₂²	1	2	yes	—
32	#16837207	C₂²	1	2	yes	—
33	#16837204	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
37	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₅₉×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₅₉×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×#11745685	—	2	2	no	—
42	C₂²×Dic₁₁₈	—	2	2	no	—
43	#14273749	—	2	4	no	—
44	#14273749	—	4	2	no	—
45	C₂×Dic₂₃₆	—	2	4	no	—
46	C₂×Dic₂₃₆	—	2	2	no	—
47	C₂×Dic₂₃₆	—	4	2	no	—
48	C₂×Dic₂₃₆	C₂³	1	2	yes	—
49	C₂×Dic₂₃₆	C₂³	1	2	yes	—
50	C₂×#11745660	—	2	2	no	—
51	C₂×#11745660	—	2	2	no	—
52	C₂×D₄₇₂	C₂³	1	2	yes	—
53	#14273715	—	2	2	no	—
54	#14273713	—	4	2	no	—
55	#14273706	—	2	2	no	—
56	#14273705	—	4	2	no	—
57	#14273701	—	2	2	no	—
58	#14273699	—	4	2	no	—
59	C₄×Dic₁₁₈	—	2	2	no	—
60	(C₂²⋊D₅₉)⋊C₄	—	4	2	no	—
61	Dic₄₇₂	—	2	4	no	—
62	Dic₄₇₂	—	2	2	no	—
63	Dic₄₇₂	D₄	1	4	yes	complemented
64	#14273672	—	2	4	no	—
65	#14273672	—	4	2	no	—
66	D₉₄₄	—	2	2	no	—
67	#14273661	C₂³	1	2	yes	—
68	#14273652	—	2	2	no	—
69	#14273652	—	2	2	no	—
70	#14273649	—	2	2	no	—
71	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
74	C₂×C₁₆×C₅₉	—	2	2	no	—
75	C₂×C₁₆×C₅₉	—	2	2	no	—
76	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
81	C₂²⋊C₈×C₅₉	D₄	1	1	yes	characteristic
82	C₂²⋊C₈×C₅₉	D₄	1	1	yes	characteristic
83	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
84	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₅₉	—	4	2	no	—
88	C₂×C₂²⋊D₅₉	—	4	2	no	—
89	C₂²×Dic₅₉	—	4	2	no	—
90	#11745685	—	4	4	no	—
91	#11745685	—	8	2	no	—
92	C₂×D₂₃₆	—	2	2	no	—
93	C₂×C₄×D₅₉	—	4	2	no	—
94	C₂×Dic₁₁₈	—	2	2	no	—
95	C₂×Dic₁₁₈	—	2	2	no	—
96	C₂×Dic₁₁₈	—	4	4	no	—
97	C₂×Dic₁₁₈	—	4	2	no	—
98	C₂×Dic₁₁₈	—	8	2	no	—
99	C₂×C₈×C₅₉	—	2	1	no	—
100	C₂×C₈×C₅₉	—	2	1	no	—
101	C₂×C₈×C₅₉	—	2	1	no	—
102	C₂×C₈×C₅₉	—	2	1	no	—
103	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
109	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
110	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₂²⋊Dic₅₉	—	4	2	no	—
112	(C₂×D₅₉)⋊C₄	—	4	2	no	—
113	(C₂×D₅₉)⋊C₄	—	4	2	no	—
114	(C₂×D₅₉)⋊C₄	—	4	2	no	—
115	C₄⋊Dic₅₉	—	2	2	no	—
116	Dic₅₉⋊C₄	—	4	2	no	—
117	Dic₅₉⋊C₄	—	4	2	no	—
118	C₄×Dic₅₉	—	4	2	no	—
119	Dic₂₃₆	—	2	2	no	—
120	Dic₂₃₆	—	2	2	no	—
121	Dic₂₃₆	—	2	2	no	—
122	Dic₂₃₆	—	4	4	no	—
123	Dic₂₃₆	C₂×D₄	1	4	yes	—
124	D₄₇₂	—	2	2	no	—
125	D₄₇₂	—	2	2	no	—
126	#11745660	—	4	4	no	—
127	#11745660	—	8	2	no	—
128	C₁₆×C₅₉	—	2	1	no	—
129	C₁₆×C₅₉	—	2	1	no	—
130	C₁₆×C₅₉	—	2	1	no	—
131	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
132	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
133	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₅₉	—	4	2	no	—
135	D₄×C₅₉	—	2	2	no	—
136	C₂×C₄×C₅₉	—	2	1	no	—
137	C₂×C₄×C₅₉	—	2	1	no	—
138	C₂×C₄×C₅₉	—	2	1	no	—
139	C₂×C₄×C₅₉	—	2	1	no	—
140	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
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	—	8	4	no	—
146	C₂²⋊D₅₉	—	8	4	no	—
147	C₂×Dic₅₉	—	4	2	no	—
148	C₂×Dic₅₉	—	4	2	no	—
149	C₂×Dic₅₉	—	4	2	no	—
150	C₂×Dic₅₉	—	8	4	no	—
151	D₂₃₆	—	4	2	no	—
152	D₂₃₆	—	4	2	no	—
153	C₄×D₅₉	—	8	4	no	—
154	Dic₁₁₈	—	2	4	no	—
155	Dic₁₁₈	—	4	2	no	—
156	Dic₁₁₈	—	4	2	no	—
157	Dic₁₁₈	—	4	2	no	—
158	Dic₁₁₈	—	8	4	no	—
159	C₈×C₅₉	—	2	1	no	—
160	C₈×C₅₉	—	2	1	no	—
161	C₈×C₅₉	—	2	1	no	—
162	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
164	#33851	—	59	1	no	maximal, Sylow
165	C₂²×C₅₉	—	2	1	no	—
166	C₂²×C₅₉	—	2	1	no	—
167	C₂²×C₅₉	—	2	1	no	—
168	C₂²×C₅₉	—	2	1	no	—
169	C₂²×C₅₉	—	2	1	no	—
170	C₂²×C₅₉	#508	1	2	yes	complemented
171	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
172	C₂×D₅₉	—	8	2	no	—
173	C₂×D₅₉	—	8	2	no	—
174	C₂×D₅₉	—	8	2	no	—
175	C₄×C₅₉	—	2	1	no	—
176	C₄×C₅₉	—	2	1	no	—
177	C₄×C₅₉	—	2	1	no	—
178	C₄×C₅₉	—	2	1	no	—
179	C₄×C₅₉	D₄²	1	1	yes	characteristic
180	C₄×C₅₉	#576	1	1	yes	characteristic
181	Dic₅₉	—	4	4	no	—
182	Dic₅₉	—	8	2	no	—
183	Dic₅₉	—	8	2	no	—
184	C₂×#508	—	59	1	no	—
185	C₂²×Q₃₂	—	59	1	no	—
186	#3285	—	59	2	no	—
187	#2249	—	59	1	no	—
188	#2216	—	59	1	no	—
189	#2215	—	59	1	no	—
190	#2211	—	59	2	no	—
191	#2189	—	59	2	no	—
192	#2185	—	59	2	no	—
193	C₄×Q₃₂	—	59	1	no	—
194	D₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
195	C₂×C₅₉	—	2	2	no	—
196	C₂×C₅₉	—	4	1	no	—
197	C₂×C₅₉	D₄×D₈	1	1	yes	characteristic
198	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
199	C₂×C₅₉	#3417	1	1	yes	characteristic
200	D₅₉	—	16	2	no	—
201	C₂×#135	—	59	2	no	—
202	C₂²×Q₁₆	—	59	2	no	—
203	#548	—	118	2	no	—
204	#508	—	118	4	no	—
205	C₂×Q₃₂	—	59	1	no	—
206	C₂×Q₃₂	—	59	2	no	—
207	C₂×Q₃₂	—	59	1	no	—
208	C₂×Q₃₂	—	118	4	no	—
209	C₂×SD₃₂	—	59	2	no	—
210	C₂×SD₃₂	—	59	2	no	—
211	C₂×D₁₆	—	59	1	no	—
212	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
213	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
214	#495	—	59	2	no	—
215	#467	—	59	2	no	—
216	#466	—	59	2	no	—
217	#464	—	118	2	no	—
218	#452	—	118	2	no	—
219	#448	—	118	2	no	—
220	C₄×Q₁₆	—	59	2	no	—
221	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
222	#366	—	59	1	no	—
223	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
224	#358	—	59	2	no	—
225	#358	—	59	2	no	—
226	#357	—	59	2	no	—
227	C₂²⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
228	C₂²⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
229	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
230	C₅₉	#33851	1	1	yes	characteristic, complemented, Sylow
231	C₂×#41	—	118	2	no	—
232	C₂²×Q₈	—	118	2	no	—
233	#135	—	118	4	no	—
234	#135	—	236	2	no	—
235	C₂×Q₁₆	—	59	2	no	—
236	C₂×Q₁₆	—	59	2	no	—
237	C₂×Q₁₆	—	118	4	no	—
238	C₂×Q₁₆	—	118	2	no	—
239	C₂×Q₁₆	—	236	2	no	—
240	C₂×SD₁₆	—	118	2	no	—
241	C₂×SD₁₆	—	118	2	no	—
242	C₂×D₈	—	59	2	no	—
243	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
245	#124	—	118	2	no	—
246	#124	—	236	2	no	—
247	C₂²⋊Q₈	—	118	2	no	—
248	C₂²⋊Q₈	—	236	2	no	—
249	#121	—	118	2	no	—
250	#121	—	236	2	no	—
251	C₄×Q₈	—	118	2	no	—
252	C₄×D₄	—	236	2	no	—
253	C₄×D₄	D₂₃₆	1	1	yes	characteristic
254	Q₃₂	—	59	4	no	—
255	Q₃₂	—	118	4	no	—
256	Q₃₂	—	118	2	no	—
257	SD₃₂	—	118	4	no	—
258	SD₃₂	—	236	2	no	—
259	D₁₆	—	118	2	no	—
260	C₂×C₁₆	—	2	2	no	—
261	C₂×C₁₆	—	2	2	no	—
262	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
263	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
264	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
265	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
266	#107	—	59	2	no	—
267	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
268	#103	—	118	2	no	—
269	#103	—	118	2	no	—
270	#102	—	118	2	no	—
271	C₂²⋊C₈	D₂₃₆	1	1	yes	characteristic
272	C₂²⋊C₈	D₂₃₆	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
274	#41	—	472	2	no	—
275	#41	—	236	4	no	—
276	C₂×Q₈	—	118	2	no	—
277	C₂×Q₈	—	118	2	no	—
278	C₂×Q₈	—	472	2	no	—
279	C₂×Q₈	—	236	4	no	—
280	C₂×Q₈	—	236	2	no	—
281	C₂×D₄	—	118	2	no	—
282	C₂×D₄	—	236	2	no	—
283	C₂×D₄	—	236	2	no	—
284	C₂×D₄	D₄₇₂	1	1	yes	characteristic
285	C₂²×C₄	—	236	2	no	—
286	C₂²×C₄	—	236	2	no	—
287	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
289	Q₁₆	—	59	4	no	—
290	Q₁₆	—	118	2	no	—
291	Q₁₆	—	118	2	no	—
292	Q₁₆	—	118	2	no	—
293	Q₁₆	—	236	4	no	—
294	SD₁₆	—	472	2	no	—
295	SD₁₆	—	236	4	no	—
296	D₈	—	118	2	no	—
297	D₈	—	118	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
303	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
305	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
306	C₄⋊C₄	—	118	2	no	—
307	C₄⋊C₄	—	236	2	no	—
308	C₄⋊C₄	—	236	2	no	—
309	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
310	C₂²⋊C₄	—	236	2	no	—
311	C₂²⋊C₄	—	236	2	no	—
312	C₂²⋊C₄	—	236	2	no	—
313	C₂²⋊C₄	—	236	2	no	—
314	C₂²⋊C₄	D₄₇₂	1	1	yes	characteristic
315	C₂²⋊C₄	D₄₇₂	1	1	yes	characteristic
316	C₄²	—	236	2	no	—
317	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	2	1	no	—
321	C₁₆	D₄×D₅₉	1	2	yes	—
322	C₂³	—	236	2	no	—
323	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
324	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
325	Q₈	—	118	4	no	—
326	Q₈	—	472	4	no	—
327	Q₈	—	236	2	no	—
328	Q₈	—	236	2	no	—
329	Q₈	—	236	2	no	—
330	D₄	—	2	2	no	—
331	D₄	—	472	4	no	—
332	D₄	—	472	4	no	—
333	D₄	—	236	2	no	—
334	D₄	—	236	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	472	4	no	—
340	C₂×C₄	—	472	4	no	—
341	C₂×C₄	—	236	2	no	—
342	C₂×C₄	—	236	2	no	—
343	C₂×C₄	—	236	2	no	—
344	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
345	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
353	C₈	#14273773	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	472	2	no	—
360	C₂²	—	472	2	no	—
361	C₂²	—	472	2	no	—
362	C₂²	#16837834	1	2	yes	complemented
363	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	472	2	no	—
369	C₄	—	472	2	no	—
370	C₄	—	236	4	no	—
371	C₄	D₄×D₂₃₆	1	1	yes	characteristic
372	C₄	#16838207	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	4	1	no	—
375	C₂	—	944	2	no	—
376	C₂	D₄×D₄₇₂	1	1	yes	characteristic
377	C₂	C₂×#16837834	1	1	yes	characteristic
378	C₂	#21437249	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	#33851
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. C₂×#16837834	C₂	semidirect
2. C₂²×Dic₄₇₂	C₂	semidirect
3. #21435839	C₂	semidirect
4. #21426357	C₂	semidirect
5. #21426187	C₂	semidirect
6. #21426182	C₂	semidirect
7. #21426159	C₂	semidirect
8. #21426038	C₂	semidirect
9. #21426015	C₂	semidirect
10. C₄×Dic₄₇₂	C₂	semidirect
11. D₄×C₁₆×C₅₉	C₂	semidirect
63. Dic₄₇₂	D₄	semidirect
170. C₂²×C₅₉	#508	semidirect
194. D₄×C₁₆	D₅₉	semidirect
230. C₅₉	#33851	semidirect
362. C₂²	#16837834	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837834	2
C₂²×Dic₄₇₂	2
#21435839	2
#21426357	2
#21426187	2
#21426182	2
#21426159	2
#21426038	2
#21426015	2
C₄×Dic₄₇₂	2
D₄×C₁₆×C₅₉	2
#33851	59