Subgroups — #28608392

Order 15104.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273749	C₂	1	1	yes	maximal, characteristic
2	#21432761	C₂	1	2	yes	maximal, complemented
3	#21424989	C₂	1	2	yes	maximal
4	#21424745	C₂	1	4	yes	maximal, complemented
5	#21424268	C₂	1	1	yes	maximal, characteristic, complemented
6	#21424117	C₂	1	2	yes	maximal, complemented
7	#21423980	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₂²	1	1	yes	characteristic
10	C₄×#11745685	C₂²	1	2	yes	—
11	C₄×Dic₂₃₆	C₄	1	1	yes	characteristic, complemented
12	C₄×Dic₂₃₆	C₄	1	1	yes	characteristic, complemented
13	C₄×#11745660	C₄	1	2	yes	complemented
14	C₄×#11745660	C₄	1	2	yes	complemented
15	C₄×D₄₇₂	C₄	1	1	yes	characteristic, complemented
16	C₄×D₄₇₂	C₄	1	1	yes	characteristic, complemented
17	#16837462	—	2	4	no	—
18	#16837458	—	2	2	no	—
19	#16837454	—	2	2	no	—
20	C₂×#14273663	C₂²	1	2	yes	—
21	C₂×#14273663	C₂²	1	2	yes	—
22	#16837440	C₂²	1	1	yes	characteristic
23	#16837426	C₂²	1	2	yes	—
24	C₂×#14273655	C₂²	1	4	yes	—
25	#16837399	C₂²	1	2	yes	—
26	#16837368	C₂²	1	4	yes	—
27	#16837357	C₂²	1	4	yes	—
28	#16837162	C₂²	1	4	yes	—
29	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×#97×C₅₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
33	C₅₉×#336	C₂²	1	2	yes	—
34	C₂×#11745685	C₂³	1	2	yes	—
35	#14273749	—	2	2	no	—
36	#14273749	—	2	2	no	—
37	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
38	C₂×#11745660	C₂×C₄	1	2	yes	—
39	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
40	C₂×#11745664	C₂³	1	4	yes	—
41	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
43	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
44	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
45	C₄×C₂²⋊D₅₉	—	2	2	no	—
46	C₄×C₂²⋊D₅₉	—	2	2	no	—
47	#14273686	C₂³	1	2	yes	—
48	#14273684	—	2	2	no	—
49	C₄²×D₅₉	—	2	2	no	—
50	#14273663	—	2	4	no	—
51	#14273663	—	4	2	no	—
52	#14273661	C₂×C₄	1	1	yes	characteristic
53	#14273661	C₂×C₄	1	1	yes	characteristic
54	#14273659	C₂×C₄	1	2	yes	—
55	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
56	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
57	#14273655	—	4	4	no	—
58	#14273652	C₂×C₄	1	2	yes	—
59	#14273652	C₂×C₄	1	2	yes	—
60	#14273649	C₂×C₄	1	2	yes	—
61	#14273649	C₂×C₄	1	2	yes	—
62	(C₂×D₅₉)⋊C₈	—	2	4	no	—
63	(C₂×D₅₉)⋊C₈	—	2	4	no	—
64	#14273637	—	2	4	no	—
65	C₈×Dic₅₉	—	2	4	no	—
66	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
68	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
70	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
71	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
73	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
74	#97×C₅₉	—	2	4	no	—
75	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
79	#95×C₅₉	—	2	1	no	—
80	C₂²×C₄×C₅₉	—	2	2	no	—
81	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
82	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
83	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
84	C₂×C₂²⋊D₅₉	—	2	2	no	—
85	#11745685	—	2	2	no	—
86	#11745685	—	2	2	no	—
87	#11745685	—	2	2	no	—
88	#11745685	—	2	2	no	—
89	C₂×D₂₃₆	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₅₉	—	2	2	no	—
91	C₂×C₄×D₅₉	—	2	2	no	—
92	C₂×C₄×D₅₉	—	2	2	no	—
93	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₅₉	—	2	2	no	—
95	C₂×C₈×C₅₉	—	2	1	no	—
96	C₂×C₈×C₅₉	—	2	1	no	—
97	C₂×C₈×C₅₉	—	2	2	no	—
98	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₅₉	#41	1	2	yes	—
104	C₂×C₈×C₅₉	#41	1	2	yes	—
105	C₄⋊C₄×C₅₉	—	4	2	no	—
106	C₄²×C₅₉	—	2	2	no	—
107	C₄²×C₅₉	—	2	2	no	—
108	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂²⋊Dic₅₉	—	2	2	no	—
113	(C₂×D₅₉)⋊C₄	—	2	2	no	—
114	(C₂×D₅₉)⋊C₄	—	2	2	no	—
115	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
116	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
117	Dic₅₉⋊C₄	—	2	2	no	—
118	Dic₅₉⋊C₄	—	2	2	no	—
119	C₄×Dic₅₉	—	2	2	no	—
120	C₄×Dic₅₉	—	2	2	no	—
121	C₄×Dic₅₉	—	2	2	no	—
122	#11745664	—	2	4	no	—
123	#11745664	—	2	4	no	—
124	C₂×C₅₉⋊C₈	—	2	4	no	—
125	Dic₂₃₆	—	2	2	no	—
126	D₄₇₂	—	2	2	no	—
127	#11745660	—	2	4	no	—
128	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
129	C₂²×D₅₉	—	2	2	no	—
130	C₂×C₄×C₅₉	—	2	2	no	—
131	C₂×C₄×C₅₉	—	4	1	no	—
132	C₂×C₄×C₅₉	—	4	2	no	—
133	C₂×C₄×C₅₉	—	4	2	no	—
134	C₂×C₄×C₅₉	—	4	2	no	—
135	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²⋊D₅₉	—	4	2	no	—
146	C₂²⋊D₅₉	—	4	2	no	—
147	C₂×Dic₅₉	—	2	2	no	—
148	C₂×Dic₅₉	—	2	2	no	—
149	C₂×Dic₅₉	—	2	2	no	—
150	D₂₃₆	—	2	2	no	—
151	D₂₃₆	—	2	2	no	—
152	C₄×D₅₉	—	4	2	no	—
153	C₄×D₅₉	—	4	2	no	—
154	C₄×D₅₉	—	4	4	no	—
155	C₄×D₅₉	—	8	2	no	—
156	Dic₁₁₈	—	2	2	no	—
157	Dic₁₁₈	—	2	2	no	—
158	C₈×C₅₉	—	2	1	no	—
159	C₈×C₅₉	—	2	1	no	—
160	C₈×C₅₉	—	2	4	no	—
161	C₈×C₅₉	#117	1	2	yes	—
162	C₈×C₅₉	#117	1	2	yes	—
163	C₅₉⋊C₈	—	4	4	no	—
164	#22803	—	59	1	no	maximal, Sylow
165	C₂²×C₅₉	—	2	1	no	—
166	C₂²×C₅₉	—	2	1	no	—
167	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₅₉	#555	1	1	yes	characteristic
169	C₂²×C₅₉	#556	1	1	yes	characteristic
170	C₂×D₅₉	—	4	2	no	—
171	C₂×D₅₉	—	4	2	no	—
172	C₂×D₅₉	—	4	2	no	—
173	C₄×C₅₉	—	2	1	no	—
174	C₄×C₅₉	—	2	1	no	—
175	C₄×C₅₉	—	2	1	no	—
176	C₄×C₅₉	—	8	2	no	—
177	C₄×C₅₉	#444	1	2	yes	—
178	C₄×C₅₉	C₂×#117	1	1	yes	characteristic
179	C₄×C₅₉	#520	1	1	yes	characteristic
180	C₄×C₅₉	#547	1	1	yes	characteristic
181	C₄×C₅₉	#550	1	1	yes	characteristic
182	Dic₅₉	—	4	2	no	—
183	Dic₅₉	—	4	2	no	—
184	Dic₅₉	—	4	2	no	—
185	C₄×#135	—	59	1	no	—
186	#2925	—	59	2	no	—
187	#1975	—	59	2	no	—
188	#1929	—	59	4	no	—
189	#1848	—	59	1	no	—
190	#1806	—	59	2	no	—
191	#1773	—	59	2	no	—
192	#1770	D₅₉	1	1	yes	characteristic, complemented
193	C₂×C₅₉	—	2	1	no	—
194	C₂×C₅₉	—	2	1	no	—
195	C₂×C₅₉	#2354	1	1	yes	characteristic
196	C₂×C₅₉	C₂×#444	1	1	yes	characteristic
197	C₂×C₅₉	#2957	1	1	yes	characteristic
198	D₅₉	—	8	2	no	—
199	C₂×#135	—	59	1	no	—
200	C₄×#41	—	59	2	no	—
201	C₄×Q₁₆	—	59	1	no	—
202	C₄×Q₁₆	—	59	1	no	—
203	C₄×SD₁₆	—	59	2	no	—
204	C₄×SD₁₆	—	59	2	no	—
205	C₄×D₈	—	59	1	no	—
206	C₄×D₈	—	59	1	no	—
207	#433	—	118	4	no	—
208	#432	—	118	2	no	—
209	#431	—	118	2	no	—
210	C₂×#108	—	59	2	no	—
211	C₂×#108	—	59	2	no	—
212	#427	—	59	1	no	—
213	#423	—	59	2	no	—
214	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
215	C₂×#104	—	59	4	no	—
216	#416	—	59	2	no	—
217	#408	—	59	4	no	—
218	#405	—	59	4	no	—
219	C₂×#97	C₂×D₅₉	1	2	yes	—
220	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
221	C₄×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
222	#339	—	59	4	no	—
223	#336	C₂×D₅₉	1	2	yes	—
224	C₅₉	#22803	1	1	yes	characteristic, complemented, Sylow
225	C₂×#41	—	59	2	no	—
226	#135	—	118	2	no	—
227	#135	—	118	2	no	—
228	C₂×Q₁₆	—	59	1	no	—
229	C₂×SD₁₆	—	59	2	no	—
230	C₂×D₈	—	59	1	no	—
231	C₂×M₁₆	—	59	4	no	—
232	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
234	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₄×Q₈	—	59	2	no	—
236	C₄×Q₈	—	59	2	no	—
237	C₄×D₄	—	59	2	no	—
238	C₄×D₄	—	59	2	no	—
239	C₄×D₄	—	118	2	no	—
240	C₄×D₄	—	118	2	no	—
241	#117	—	59	2	no	—
242	#117	—	118	2	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄²	—	118	2	no	—
245	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
246	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
247	#108	—	118	4	no	—
248	#108	—	236	2	no	—
249	#107	—	59	1	no	—
250	#107	—	59	1	no	—
251	#106	—	59	2	no	—
252	C₄⋊C₈	—	118	2	no	—
253	C₄⋊C₈	—	118	2	no	—
254	C₄⋊C₈	D₂₃₆	1	2	yes	—
255	C₄⋊C₈	D₂₃₆	1	2	yes	—
256	#104	—	236	4	no	—
257	#103	—	59	2	no	—
258	#103	—	59	2	no	—
259	#102	—	59	2	no	—
260	#102	—	59	2	no	—
261	C₂²⋊C₈	—	118	4	no	—
262	C₂²⋊C₈	—	118	4	no	—
263	#97	—	2	4	no	—
264	#97	—	118	4	no	—
265	C₄×C₈	—	118	4	no	—
266	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
267	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
268	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
269	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
270	#95	—	2	1	no	—
271	#41	—	118	2	no	—
272	#41	—	118	2	no	—
273	#41	—	118	2	no	—
274	#41	—	118	2	no	—
275	C₂×Q₈	—	59	2	no	—
276	C₂×D₄	—	59	2	no	—
277	C₂×D₄	—	118	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	118	2	no	—
280	C₂²×C₄	—	118	2	no	—
281	C₂²×C₄	—	118	2	no	—
282	C₂²×C₄	#11745685	1	1	yes	characteristic
283	C₂²×C₄	#11745685	1	1	yes	characteristic
284	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
285	Q₁₆	—	118	2	no	—
286	SD₁₆	—	118	4	no	—
287	D₈	—	118	2	no	—
288	M₁₆	—	118	4	no	—
289	M₁₆	—	118	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	118	4	no	—
295	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
299	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
300	C₂×C₈	#11745687	1	2	yes	—
301	C₂×C₈	#11745689	1	2	yes	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	59	2	no	—
304	C₄⋊C₄	—	59	2	no	—
305	C₄⋊C₄	—	118	2	no	—
306	C₄⋊C₄	—	118	2	no	—
307	C₂²⋊C₄	—	118	2	no	—
308	C₂²⋊C₄	—	118	2	no	—
309	C₂²⋊C₄	—	118	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	118	2	no	—
313	C₄²	—	118	2	no	—
314	C₄²	—	118	2	no	—
315	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
317	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
318	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
319	C₂³	—	118	2	no	—
320	C₂³	C₂×#11745685	1	1	yes	characteristic
321	Q₈	—	118	2	no	—
322	Q₈	—	118	2	no	—
323	D₄	—	118	2	no	—
324	D₄	—	118	2	no	—
325	D₄	—	236	2	no	—
326	D₄	—	236	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	118	2	no	—
333	C₂×C₄	—	118	2	no	—
334	C₂×C₄	—	118	2	no	—
335	C₂×C₄	—	472	2	no	—
336	C₂×C₄	—	236	4	no	—
337	C₂×C₄	—	236	2	no	—
338	C₂×C₄	—	236	2	no	—
339	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
340	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
341	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
342	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
345	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
346	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
347	C₂×C₄	#14273773	1	1	yes	characteristic
348	C₂×C₄	#14273774	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	236	4	no	—
353	C₈	#14273685	1	2	yes	—
354	C₈	#14273685	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	236	2	no	—
358	C₂²	—	236	2	no	—
359	C₂²	—	236	2	no	—
360	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
361	C₂²	#16838117	1	1	yes	characteristic
362	C₂²	#16838125	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	236	2	no	—
368	C₄	—	236	2	no	—
369	C₄	—	236	2	no	—
370	C₄	#16837525	1	2	yes	—
371	C₄	C₂×#14273685	1	1	yes	characteristic
372	C₄	#16837882	1	1	yes	characteristic
373	C₄	#16838056	1	1	yes	characteristic
374	C₄	#16838074	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	472	2	no	—
378	C₂	#21427043	1	1	yes	characteristic
379	C₂	C₂×#16837525	1	1	yes	characteristic
380	C₂	#21433014	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#1770	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂×#14273749	C₂²
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	#1770	D₅₉
hypocenter	C₅₉	#22803
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #21432761	C₂	semidirect
4. #21424745	C₂	semidirect
5. #21424268	C₂	semidirect
6. #21424117	C₂	semidirect
7. #21423980	C₂	semidirect
8. C₅₉×#1770	C₂	semidirect
11. C₄×Dic₂₃₆	C₄	semidirect
12. C₄×Dic₂₃₆	C₄	semidirect
13. C₄×#11745660	C₄	semidirect
14. C₄×#11745660	C₄	semidirect
15. C₄×D₄₇₂	C₄	semidirect
16. C₄×D₄₇₂	C₄	semidirect
192. #1770	D₅₉	semidirect
224. C₅₉	#22803	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273749	2
#21432761	2
#21424989	2
#21424745	2
#21424268	2
#21424117	2
#21423980	2
C₅₉×#1770	2
#22803	59