Subgroups — #28505309

Order 15104.

398 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837231	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837209	C₂	1	1	yes	maximal, characteristic
3	#21425803	C₂	1	1	yes	maximal, characteristic
4	#21424844	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422193	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422174	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₉×#443	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#14273663	C₂²	1	1	yes	characteristic
11	#16837440	C₂²	1	1	yes	characteristic
12	#16837428	—	2	1	no	—
13	#16837231	—	2	2	no	—
14	#16837209	—	2	2	no	—
15	#16837206	C₄	1	1	yes	characteristic, complemented
16	#16837206	C₄	1	1	yes	characteristic, complemented
17	#16837203	C₄	1	1	yes	characteristic, complemented
18	#16837203	C₄	1	1	yes	characteristic, complemented
19	#16837162	—	2	1	no	—
20	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#131×C₅₉	—	2	1	no	—
22	C₅₉×#443	—	2	2	no	—
23	C₅₉×#443	—	2	2	no	—
24	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#104×C₅₉	—	2	1	no	—
26	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#11745664	—	2	1	no	—
28	#14273686	—	2	1	no	—
29	C₂×C₅₉⋊C₁₆	—	2	1	no	—
30	C₂×C₅₉⋊C₁₆	—	2	1	no	—
31	C₂×C₅₉⋊C₁₆	—	2	1	no	—
32	C₂×C₅₉⋊C₁₆	—	2	1	no	—
33	C₂×C₅₉⋊C₁₆	—	2	1	no	—
34	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
36	#14273663	—	2	1	no	—
37	#14273663	—	2	1	no	—
38	#14273661	C₂×C₄	1	1	yes	characteristic
39	#14273661	C₂×C₄	1	1	yes	characteristic
40	#14273659	—	2	1	no	—
41	Dic₅₉⋊C₈	—	4	1	no	—
42	Dic₅₉⋊C₈	—	4	1	no	—
43	C₂×#41×C₅₉	—	2	1	no	—
44	#135×C₅₉	—	2	1	no	—
45	#135×C₅₉	—	2	1	no	—
46	#135×C₅₉	D₄	1	2	yes	—
47	#135×C₅₉	Q₈	1	2	yes	—
48	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂×SD₁₆×C₅₉	—	2	1	no	—
50	C₂×D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
51	#131×C₅₉	—	2	4	no	—
52	#131×C₅₉	—	4	2	no	—
53	C₂×M₁₆×C₅₉	—	2	1	no	—
54	C₂×M₁₆×C₅₉	—	4	1	no	—
55	C₂²×C₈×C₅₉	—	4	1	no	—
56	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
59	#108×C₅₉	—	2	1	no	—
60	#108×C₅₉	D₄	1	2	yes	—
61	#104×C₅₉	—	2	4	no	—
62	C₄×C₈×C₅₉	—	2	1	no	—
63	C₄×C₈×C₅₉	—	2	1	no	—
64	#41×C₅₉	—	2	2	no	—
65	#41×C₅₉	—	2	2	no	—
66	#41×C₅₉	—	4	1	no	—
67	#41×C₅₉	—	4	1	no	—
68	C₂×Q₈×C₅₉	—	2	1	no	—
69	C₂×D₄×C₅₉	—	2	1	no	—
70	C₂×D₄×C₅₉	—	4	1	no	—
71	C₂²×C₄×C₅₉	—	2	1	no	—
72	C₂²×C₄×C₅₉	—	4	1	no	—
73	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₅₉	—	2	1	no	—
75	Q₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
76	SD₁₆×C₅₉	—	2	2	no	—
77	SD₁₆×C₅₉	—	2	2	no	—
78	D₈×C₅₉	—	2	1	no	—
79	D₈×C₅₉	C₄⋊C₄	1	2	yes	—
80	M₁₆×C₅₉	—	2	2	no	—
81	M₁₆×C₅₉	—	4	4	no	—
82	M₁₆×C₅₉	—	4	1	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	—	2	1	no	—
85	C₂×C₈×C₅₉	—	4	4	no	—
86	C₂×C₈×C₅₉	—	4	1	no	—
87	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
88	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
89	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
94	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
95	C₄²×C₅₉	—	2	1	no	—
96	C₄²×C₅₉	—	2	1	no	—
97	C₂²⋊Dic₅₉	—	4	1	no	—
98	C₄⋊Dic₅₉	—	2	1	no	—
99	C₄⋊Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	4	1	no	—
101	#11745664	—	4	1	no	—
102	#11745664	—	4	1	no	—
103	C₂×C₅₉⋊C₈	—	4	1	no	—
104	C₅₉⋊C₁₆	—	2	1	no	—
105	C₅₉⋊C₁₆	—	2	1	no	—
106	C₅₉⋊C₁₆	—	4	1	no	—
107	C₂³×C₅₉	—	4	1	no	—
108	C₂³×C₅₉	#135	1	1	yes	characteristic
109	Q₈×C₅₉	—	2	2	no	—
110	Q₈×C₅₉	—	4	1	no	—
111	D₄×C₅₉	—	2	2	no	—
112	D₄×C₅₉	—	4	2	no	—
113	D₄×C₅₉	—	4	2	no	—
114	D₄×C₅₉	—	4	1	no	—
115	C₂×C₄×C₅₉	—	2	2	no	—
116	C₂×C₄×C₅₉	—	4	1	no	—
117	C₂×C₄×C₅₉	—	4	1	no	—
118	C₂×C₄×C₅₉	—	4	1	no	—
119	C₂×C₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
127	C₂×Dic₅₉	—	4	1	no	—
128	C₂×Dic₅₉	—	4	1	no	—
129	C₈×C₅₉	—	2	1	no	—
130	C₈×C₅₉	—	2	1	no	—
131	C₈×C₅₉	—	4	2	no	—
132	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
134	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₅₉	#123	1	1	yes	characteristic
136	C₅₉⋊C₈	—	8	1	no	—
137	#12939	—	59	1	no	maximal, Sylow
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	—	2	1	no	—
140	C₂²×C₅₉	—	4	2	no	—
141	C₂²×C₅₉	—	8	1	no	—
142	C₂²×C₅₉	—	8	1	no	—
143	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
144	C₂²×C₅₉	#452	1	1	yes	characteristic
145	C₂²×C₅₉	#478	1	1	yes	characteristic
146	C₄×C₅₉	—	4	2	no	—
147	C₄×C₅₉	—	4	2	no	—
148	C₄×C₅₉	#385	1	1	yes	characteristic
149	C₄×C₅₉	#440	1	1	yes	characteristic
150	C₄×C₅₉	#447	1	1	yes	characteristic
151	C₄×C₅₉	#474	1	1	yes	characteristic
152	Dic₅₉	—	8	1	no	—
153	Dic₅₉	—	8	1	no	—
154	C₂×#443	D₅₉	1	1	yes	characteristic
155	C₂×#368	—	59	1	no	—
156	C₂×#359	—	59	1	no	—
157	#2138	—	59	1	no	—
158	#1945	—	59	1	no	—
159	#1382	—	59	1	no	—
160	#1374	—	59	1	no	—
161	C₂×C₅₉	—	2	1	no	—
162	C₂×C₅₉	—	2	1	no	—
163	C₂×C₅₉	—	8	2	no	—
164	C₂×C₅₉	#1862	1	1	yes	characteristic
165	C₂×C₅₉	#2178	1	1	yes	characteristic
166	C₂×C₅₉	#2179	1	1	yes	characteristic
167	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
168	C₂×#131	—	2	1	no	—
169	C₂²×C₁₆	—	59	1	no	—
170	C₂²×C₁₆	—	59	1	no	—
171	#443	—	2	2	no	—
172	#443	—	2	2	no	—
173	C₂×#108	—	59	1	no	—
174	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
175	#427	—	59	1	no	—
176	#424	—	118	1	no	—
177	C₂×#104	—	2	1	no	—
178	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
179	#368	—	118	2	no	—
180	#359	—	118	2	no	—
181	#358	—	59	1	no	—
182	#358	—	59	1	no	—
183	#357	—	59	1	no	—
184	#357	—	59	1	no	—
185	#339	—	118	1	no	—
186	C₅₉	#12939	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	2	1	no	—
188	#135	—	2	1	no	—
189	#135	—	2	1	no	—
190	#135	Dic₁₁₈	1	2	yes	—
191	#135	C₂²⋊D₅₉	1	2	yes	—
192	C₂×Q₁₆	C₄×D₅₉	1	1	yes	characteristic
193	C₂×SD₁₆	—	2	1	no	—
194	C₂×D₈	C₄×D₅₉	1	1	yes	characteristic
195	#131	—	2	4	no	—
196	#131	—	4	2	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	4	1	no	—
199	C₂×M₁₆	—	118	1	no	—
200	C₂²×C₈	—	4	1	no	—
201	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
203	#117	—	118	1	no	—
204	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
205	C₂×C₁₆	—	59	1	no	—
206	C₂×C₁₆	—	59	1	no	—
207	C₂×C₁₆	—	118	1	no	—
208	C₂×C₁₆	—	118	1	no	—
209	C₂×C₁₆	—	118	1	no	—
210	C₂×C₁₆	—	118	1	no	—
211	C₂×C₁₆	—	118	1	no	—
212	#108	—	2	1	no	—
213	#108	—	118	1	no	—
214	#108	—	118	1	no	—
215	#108	C₂²⋊D₅₉	1	2	yes	—
216	#107	—	59	1	no	—
217	#107	—	59	1	no	—
218	#106	—	118	1	no	—
219	C₄⋊C₈	—	236	1	no	—
220	C₄⋊C₈	—	236	1	no	—
221	#104	—	2	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	#41	—	4	1	no	—
227	#41	—	4	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
234	Q₁₆	—	2	1	no	—
235	Q₁₆	Dic₅₉⋊C₄	1	2	yes	—
236	SD₁₆	—	2	2	no	—
237	SD₁₆	—	2	2	no	—
238	D₈	—	2	1	no	—
239	D₈	Dic₅₉⋊C₄	1	2	yes	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	4	4	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	236	1	no	—
244	M₁₆	—	236	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	236	1	no	—
250	C₂×C₈	#11745669	1	1	yes	characteristic
251	C₂×C₈	#11745669	1	1	yes	characteristic
252	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
254	C₂×C₈	#11745685	1	1	yes	characteristic
255	C₂×C₈	#11745685	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
258	C₄⋊C₄	—	118	1	no	—
259	C₄⋊C₄	—	118	1	no	—
260	C₂²⋊C₄	—	236	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	236	1	no	—
264	C₁₆	—	118	1	no	—
265	C₁₆	—	118	1	no	—
266	C₁₆	—	236	1	no	—
267	C₂³	—	4	1	no	—
268	C₂³	#14273747	1	1	yes	characteristic
269	Q₈	—	2	2	no	—
270	Q₈	—	4	1	no	—
271	D₄	—	2	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	236	1	no	—
282	C₂×C₄	—	236	1	no	—
283	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
284	C₂×C₄	#14273696	1	1	yes	characteristic
285	C₂×C₄	#14273706	1	1	yes	characteristic
286	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
287	C₂×C₄	#14273753	1	1	yes	characteristic
288	C₂×C₄	#14273757	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	4	2	no	—
292	C₈	—	472	1	no	—
293	C₈	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
294	C₈	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
295	C₈	#14273706	1	1	yes	characteristic
296	C₈	#14273710	1	1	yes	characteristic
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	1	no	—
302	C₂²	C₄×#11745669	1	1	yes	characteristic
303	C₂²	#16837564	1	1	yes	characteristic
304	C₂²	#16837702	1	1	yes	characteristic
305	C₄	—	4	2	no	—
306	C₄	—	4	2	no	—
307	C₄	—	472	1	no	—
308	C₄	—	472	1	no	—
309	C₄	#16837275	1	1	yes	characteristic
310	C₄	#16837502	1	1	yes	characteristic
311	C₄	#16837536	1	1	yes	characteristic
312	C₄	#16837682	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	8	2	no	—
316	C₂	#21424343	1	1	yes	characteristic
317	C₂	#21425975	1	1	yes	characteristic
318	C₂	#21425980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#443	C₂
socle	C₂²×C₅₉	C₄×D₈
prime radical	C₂×#135×C₅₉	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#443	D₅₉
hypocenter	C₅₉	#12939
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16837231	C₂	semidirect
4. #21424844	C₂	semidirect
5. #21422193	C₂	semidirect
6. #21422174	C₂	semidirect
15. #16837206	C₄	semidirect
16. #16837206	C₄	semidirect
17. #16837203	C₄	semidirect
18. #16837203	C₄	semidirect
186. C₅₉	#12939	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837231	2
C₂×#16837209	2
#21425803	2
#21424844	2
#21422193	2
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C₂×C₅₉×#443	2
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