Subgroups — #28742166

Order 15104.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837840	C₂	1	1	yes	maximal, characteristic
2	#21433014	C₂	1	1	yes	maximal, characteristic, complemented
3	#21432949	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425996	C₂	1	8	yes	maximal, complemented
5	#21425869	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837216	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425833	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₂²	1	1	yes	characteristic
10	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
11	#16838168	—	2	1	no	—
12	#16837876	—	2	1	no	—
13	#16837840	C₄	1	4	yes	complemented
14	#16837840	C₂²	1	4	yes	—
15	C₂×#14273672	C₄	1	2	yes	complemented
16	C₂×D₉₄₄	C₄	1	2	yes	complemented
17	#16837525	C₂²	1	4	yes	complemented
18	#16837520	—	2	2	no	—
19	#16837515	C₂²	1	4	yes	complemented
20	C₄×D₄₇₂	—	2	2	no	—
21	C₂×#14273663	C₂²	1	1	yes	characteristic
22	#16837440	C₂²	1	1	yes	characteristic
23	#16837420	—	2	1	no	—
24	#16837408	—	2	1	no	—
25	#16837225	C₂²	1	4	yes	complemented
26	#16837216	C₂²	1	4	yes	complemented
27	#16837213	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
30	C₅₉×#405	C₂²	1	1	yes	characteristic
31	C₅₉×#347	C₂²	1	4	yes	complemented
32	C₂×#11745685	—	2	1	no	—
33	C₂²×D₂₃₆	—	2	1	no	—
34	#14273749	—	2	1	no	—
35	#14273749	—	2	1	no	—
36	#14273749	C₂×C₄	1	2	yes	complemented
37	#14273749	C₂³	1	2	yes	—
38	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×#11745660	—	2	1	no	—
40	C₂×D₄₇₂	—	2	1	no	—
41	C₂×D₄₇₂	—	2	1	no	—
42	C₂×D₄₇₂	—	2	1	no	—
43	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
44	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
45	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×D₄₇₂	C₂³	1	2	yes	—
47	#14273735	—	4	2	no	—
48	#14273733	—	2	4	no	—
49	C₂×#11745664	—	2	1	no	—
50	M₁₆×D₅₉	—	4	2	no	—
51	C₄×D₂₃₆	—	2	4	no	—
52	#14273689	—	4	2	no	—
53	#14273686	—	2	1	no	—
54	C₂²⋊C₄×D₅₉	—	4	2	no	—
55	#14273672	C₂×C₄	1	8	yes	complemented
56	D₉₄₄	C₂×C₄	1	8	yes	complemented
57	#14273663	—	2	1	no	—
58	#14273663	C₂³	1	2	yes	—
59	#14273661	—	2	1	no	—
60	#14273659	C₂³	1	2	yes	—
61	#14273655	—	2	4	no	—
62	#14273649	—	2	4	no	—
63	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
65	#117×C₅₉	D₄	1	1	yes	characteristic
66	M₃₂×C₅₉	—	2	1	no	—
67	M₃₂×C₅₉	C₂×C₄	1	2	yes	complemented
68	M₃₂×C₅₉	C₂³	1	2	yes	—
69	M₃₂×C₅₉	C₂³	1	2	yes	—
70	C₂×C₁₆×C₅₉	—	2	1	no	—
71	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	complemented
72	#97×C₅₉	C₂³	1	2	yes	—
73	C₄×C₈×C₅₉	D₄	1	2	yes	—
74	C₂³×D₅₉	—	4	1	no	—
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₅₉	—	4	1	no	—
77	#11745685	—	2	2	no	—
78	#11745685	—	2	2	no	—
79	#11745685	—	4	1	no	—
80	#11745685	—	4	1	no	—
81	C₂×D₂₃₆	—	2	2	no	—
82	C₂×D₂₃₆	—	2	2	no	—
83	C₂×D₂₃₆	—	2	1	no	—
84	C₂×D₂₃₆	—	2	2	no	—
85	C₂×D₂₃₆	—	4	1	no	—
86	C₂×D₂₃₆	—	4	1	no	—
87	C₂×D₂₃₆	—	4	1	no	—
88	C₂×C₄×D₅₉	—	4	1	no	—
89	C₂×C₄×D₅₉	—	4	4	no	—
90	C₂×Dic₁₁₈	—	2	1	no	—
91	M₁₆×C₅₉	—	2	1	no	—
92	M₁₆×C₅₉	—	2	1	no	—
93	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
98	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
99	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₅₉	—	2	1	no	—
101	C₂²⋊C₄×C₅₉	—	2	1	no	—
102	C₄²×C₅₉	C₂×D₄	1	2	yes	—
103	C₂²⋊Dic₅₉	—	4	1	no	—
104	(C₂×D₅₉)⋊C₄	—	4	4	no	—
105	C₄⋊Dic₅₉	—	2	2	no	—
106	C₄×Dic₅₉	—	4	1	no	—
107	#11745664	—	2	2	no	—
108	#11745664	—	4	1	no	—
109	C₂×C₅₉⋊C₈	—	4	1	no	—
110	Dic₂₃₆	—	2	1	no	—
111	Dic₂₃₆	C₂²×C₄	1	2	yes	—
112	D₄₇₂	—	2	2	no	—
113	D₄₇₂	—	2	2	no	—
114	D₄₇₂	—	2	1	no	—
115	D₄₇₂	—	2	2	no	—
116	D₄₇₂	C₂²×C₄	1	2	yes	—
117	D₄₇₂	C₂²×C₄	1	4	yes	—
118	#11745660	—	2	2	no	—
119	#11745660	—	2	2	no	—
120	#11745659	—	4	4	no	—
121	C₈×D₅₉	—	4	4	no	—
122	C₁₆×C₅₉	—	2	2	no	—
123	C₁₆×C₅₉	C₂²×C₄	1	4	yes	—
124	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₅₉	—	4	2	no	—
126	C₂²×D₅₉	—	4	1	no	—
127	C₂²×D₅₉	—	4	2	no	—
128	C₂²×D₅₉	—	4	2	no	—
129	C₂²×D₅₉	—	16	2	no	—
130	C₂×C₄×C₅₉	—	2	2	no	—
131	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₅₉	—	4	2	no	—
138	C₂²⋊D₅₉	—	4	2	no	—
139	C₂×Dic₅₉	—	4	1	no	—
140	C₂×Dic₅₉	—	4	2	no	—
141	D₂₃₆	—	2	2	no	—
142	D₂₃₆	—	2	4	no	—
143	D₂₃₆	—	4	2	no	—
144	D₂₃₆	—	4	2	no	—
145	D₂₃₆	—	4	2	no	—
146	D₂₃₆	—	4	1	no	—
147	C₄×D₅₉	—	4	2	no	—
148	C₄×D₅₉	—	4	2	no	—
149	C₄×D₅₉	—	16	4	no	—
150	Dic₁₁₈	—	2	2	no	—
151	Dic₁₁₈	—	4	1	no	—
152	C₈×C₅₉	—	2	2	no	—
153	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
154	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
155	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
156	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
157	C₅₉⋊C₈	—	4	2	no	—
158	#33471	—	59	1	no	maximal, Sylow
159	C₂²×C₅₉	—	4	1	no	—
160	C₂²×C₅₉	#442	1	1	yes	characteristic
161	C₂²×C₅₉	#442	1	1	yes	characteristic
162	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×D₅₉	—	4	2	no	—
164	C₂×D₅₉	—	4	4	no	—
165	C₂×D₅₉	—	8	4	no	—
166	C₂×D₅₉	—	16	1	no	—
167	C₂×D₅₉	—	16	2	no	—
168	C₂×D₅₉	—	16	2	no	—
169	C₄×C₅₉	—	4	2	no	—
170	C₄×C₅₉	#442	1	1	yes	characteristic
171	C₄×C₅₉	#442	1	1	yes	characteristic
172	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	Dic₅₉	—	4	2	no	—
175	Dic₅₉	—	8	2	no	—
176	C₂×#509	—	59	1	no	—
177	#2957	—	59	1	no	—
178	#2948	—	59	1	no	—
179	AGL(1,ℤ/16)	—	59	8	no	—
180	#2155	—	59	1	no	—
181	C₂×#361	—	59	1	no	—
182	#2145	—	59	1	no	—
183	#2110	D₅₉	1	1	yes	characteristic, complemented
184	C₂×C₅₉	—	2	1	no	—
185	C₂×C₅₉	—	2	1	no	—
186	C₂×C₅₉	—	2	1	no	—
187	C₂×C₅₉	C₂×#442	1	1	yes	characteristic
188	D₅₉	—	8	4	no	—
189	D₅₉	—	8	2	no	—
190	C₂×#135	—	59	1	no	—
191	C₂²×D₈	—	59	1	no	—
192	#568	—	118	1	no	—
193	#518	—	118	1	no	—
194	#509	—	59	4	no	—
195	#509	—	59	4	no	—
196	C₂×SD₃₂	—	59	2	no	—
197	C₂×D₁₆	—	59	2	no	—
198	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
199	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
200	#444	—	59	4	no	—
201	#443	—	118	2	no	—
202	#442	—	59	4	no	—
203	C₄×D₈	—	118	2	no	—
204	C₂×#108	—	59	1	no	—
205	#427	—	59	1	no	—
206	#421	—	118	1	no	—
207	#418	—	118	1	no	—
208	#405	C₂×D₅₉	1	1	yes	characteristic
209	#365	—	59	4	no	—
210	#361	—	59	4	no	—
211	#360	—	59	4	no	—
212	#347	C₂×D₅₉	1	4	yes	complemented
213	C₅₉	#33471	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	118	1	no	—
215	C₂²×D₄	—	118	1	no	—
216	#135	—	59	2	no	—
217	#135	—	59	2	no	—
218	#135	—	118	1	no	—
219	#135	—	118	1	no	—
220	C₂×Q₁₆	—	59	1	no	—
221	C₂×SD₁₆	—	118	1	no	—
222	C₂×D₈	—	59	2	no	—
223	C₂×D₈	—	59	2	no	—
224	C₂×D₈	—	59	1	no	—
225	C₂×D₈	—	59	2	no	—
226	C₂×D₈	—	118	1	no	—
227	C₂×D₈	—	118	1	no	—
228	C₂×D₈	—	118	1	no	—
229	#131	—	118	4	no	—
230	#131	—	236	2	no	—
231	C₂×M₁₆	—	118	1	no	—
232	C₂×M₁₆	—	236	2	no	—
233	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₄×D₄	—	118	4	no	—
236	C₄×D₄	—	236	2	no	—
237	#117	—	118	1	no	—
238	#117	D₂₃₆	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	—	236	2	no	—
240	SD₃₂	—	59	8	no	—
241	D₁₆	—	59	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	C₄×D₅₉	1	2	yes	complemented
244	M₃₂	C₂²×D₅₉	1	2	yes	—
245	M₃₂	C₂²×D₅₉	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	C₄×D₅₉	1	2	yes	complemented
248	#108	—	59	2	no	—
249	#108	—	118	1	no	—
250	#107	—	118	1	no	—
251	#106	—	59	2	no	—
252	#104	—	118	4	no	—
253	#102	—	118	4	no	—
254	#97	C₂²×D₅₉	1	2	yes	—
255	C₄×C₈	D₂₃₆	1	2	yes	—
256	C₂⁴	—	236	1	no	—
257	#41	—	118	2	no	—
258	#41	—	118	2	no	—
259	#41	—	236	1	no	—
260	#41	—	236	1	no	—
261	C₂×Q₈	—	118	1	no	—
262	C₂×D₄	—	118	2	no	—
263	C₂×D₄	—	118	2	no	—
264	C₂×D₄	—	118	1	no	—
265	C₂×D₄	—	118	2	no	—
266	C₂×D₄	—	236	1	no	—
267	C₂×D₄	—	236	1	no	—
268	C₂×D₄	—	236	1	no	—
269	C₂×D₄	—	236	1	no	—
270	C₂²×C₄	—	236	4	no	—
271	C₂²×C₄	—	236	1	no	—
272	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
273	Q₁₆	—	59	2	no	—
274	Q₁₆	—	118	1	no	—
275	SD₁₆	—	118	2	no	—
276	SD₁₆	—	118	2	no	—
277	D₈	—	59	2	no	—
278	D₈	—	59	4	no	—
279	D₈	—	118	2	no	—
280	D₈	—	118	2	no	—
281	D₈	—	118	1	no	—
282	D₈	—	118	2	no	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	118	2	no	—
286	M₁₆	—	236	4	no	—
287	M₁₆	—	236	1	no	—
288	C₂×C₈	—	236	4	no	—
289	C₂×C₈	—	236	1	no	—
290	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
292	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
293	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
294	C₂×C₈	#11745685	1	1	yes	characteristic
295	C₂×C₈	#11745685	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	118	2	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	236	4	no	—
301	C₂²⋊C₄	—	236	1	no	—
302	C₄²	—	236	1	no	—
303	C₄²	C₂×D₂₃₆	1	2	yes	—
304	C₁₆	—	2	2	no	—
305	C₁₆	C₂×C₄×D₅₉	1	4	yes	—
306	C₂³	—	944	2	no	—
307	C₂³	—	236	1	no	—
308	C₂³	—	236	2	no	—
309	C₂³	—	236	2	no	—
310	C₂³	—	236	2	no	—
311	C₂³	#14273755	1	1	yes	characteristic
312	Q₈	—	118	2	no	—
313	Q₈	—	236	1	no	—
314	D₄	—	118	4	no	—
315	D₄	—	118	2	no	—
316	D₄	—	236	2	no	—
317	D₄	—	236	2	no	—
318	D₄	—	236	2	no	—
319	D₄	—	236	2	no	—
320	D₄	—	236	2	no	—
321	D₄	—	236	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	944	4	no	—
324	C₂×C₄	—	236	2	no	—
325	C₂×C₄	—	236	1	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	—	236	2	no	—
328	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
329	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
330	C₂×C₄	#14273755	1	1	yes	characteristic
331	C₂×C₄	#14273755	1	1	yes	characteristic
332	C₂×C₄	#14273755	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	236	2	no	—
336	C₈	C₄×D₂₃₆	1	1	yes	characteristic
337	C₈	C₄×D₂₃₆	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
339	C₈	C₂×#11745685	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	944	2	no	—
342	C₂²	—	944	2	no	—
343	C₂²	—	944	1	no	—
344	C₂²	—	472	4	no	—
345	C₂²	—	236	2	no	—
346	C₂²	—	236	4	no	—
347	C₂²	#16837515	1	1	yes	characteristic
348	C₂²	#16837515	1	1	yes	characteristic
349	C₂²	C₂×#14273755	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	472	2	no	—
352	C₄	—	236	2	no	—
353	C₄	#16837515	1	1	yes	characteristic
354	C₄	#16837515	1	1	yes	characteristic
355	C₄	C₂×C₄×D₂₃₆	1	1	yes	characteristic
356	C₄	C₂×#14273755	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	472	4	no	—
361	C₂	—	472	2	no	—
362	C₂	C₂×#16837515	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16837515
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2110	C₂
socle	C₂×C₅₉	C₂×#442
prime radical	C₂×#16837840	C₂
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2110	D₅₉
hypocenter	C₅₉	#33471
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
2. #21433014	C₂	semidirect
3. #21432949	C₂	semidirect
4. #21425996	C₂	semidirect
5. #21425869	C₂	semidirect
6. C₂×#16837216	C₂	semidirect
7. #21425833	C₂	semidirect
8. C₅₉×#2110	C₂	semidirect
13. #16837840	C₄	semidirect
15. C₂×#14273672	C₄	semidirect
16. C₂×D₉₄₄	C₄	semidirect
17. #16837525	C₂²	semidirect
19. #16837515	C₂²	semidirect
25. #16837225	C₂²	semidirect
26. #16837216	C₂²	semidirect
27. #16837213	C₂²	semidirect
31. C₅₉×#347	C₂²	semidirect
36. #14273749	C₂×C₄	semidirect
38. C₂×Dic₂₃₆	C₂×C₄	semidirect
45. C₂×D₄₇₂	C₂×C₄	semidirect
55. #14273672	C₂×C₄	semidirect
56. D₉₄₄	C₂×C₄	semidirect
67. M₃₂×C₅₉	C₂×C₄	semidirect
71. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
183. #2110	D₅₉	semidirect
212. #347	C₂×D₅₉	semidirect
213. C₅₉	#33471	semidirect
243. M₃₂	C₄×D₅₉	semidirect
247. C₂×C₁₆	C₄×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837840	2
#21433014	2
#21432949	2
#21425996	2
#21425869	2
C₂×#16837216	2
#21425833	2
C₅₉×#2110	2
#33471	59