Subgroups — #28481813

Order 15104.

482 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424268	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422844	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422521	C₂	1	2	yes	maximal
4	#21421993	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273663	—	2	1	no	—
8	C₂×#14273663	—	2	1	no	—
9	#16837426	C₂²	1	2	yes	—
10	#16837155	—	2	1	no	—
11	#16837153	—	2	1	no	—
12	D₄⋊(C₅₉⋊C₈)	—	2	4	no	—
13	C₂³×C₈×C₅₉	—	2	1	no	—
14	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₅₉	—	2	1	no	—
16	C₈×D₄×C₅₉	—	2	1	no	—
17	C₈×D₄×C₅₉	C₄	1	2	yes	—
18	C₈×D₄×C₅₉	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₅₉	—	2	1	no	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#11745664	—	2	2	no	—
23	#14273663	—	4	2	no	—
24	#14273663	—	4	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
29	C₄×C₅₉⋊C₈	—	2	2	no	—
30	C₂²×D₄×C₅₉	Q₈	1	1	yes	characteristic
31	C₂³×C₄×C₅₉	—	2	1	no	—
32	C₂²×C₈×C₅₉	—	2	1	no	—
33	C₂²×C₈×C₅₉	—	2	2	no	—
34	C₂²×C₈×C₅₉	—	2	2	no	—
35	C₂²×C₈×C₅₉	—	4	2	no	—
36	C₂²×C₈×C₅₉	—	4	2	no	—
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₄×D₄×C₅₉	—	2	1	no	—
40	C₄×D₄×C₅₉	—	2	1	no	—
41	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
42	C₄×D₄×C₅₉	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₉	Q₈	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
45	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₅₉	—	2	1	no	—
47	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
48	C₂²⋊C₈×C₅₉	—	2	2	no	—
49	C₂²⋊C₈×C₅₉	—	2	2	no	—
50	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
54	C₂⁴×C₅₉	—	2	1	no	—
55	C₂×D₄×C₅₉	—	2	4	no	—
56	C₂×D₄×C₅₉	—	2	1	no	—
57	C₂×D₄×C₅₉	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₉	—	2	1	no	—
59	C₂²×C₄×C₅₉	—	2	2	no	—
60	C₂²×C₄×C₅₉	—	2	2	no	—
61	C₂²×C₄×C₅₉	—	4	1	no	—
62	C₂²×C₄×C₅₉	—	4	1	no	—
63	C₂²×C₄×C₅₉	—	4	1	no	—
64	C₂²×C₄×C₅₉	—	4	1	no	—
65	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
68	C₂×C₈×C₅₉	—	2	1	no	—
69	C₂×C₈×C₅₉	—	2	1	no	—
70	C₂×C₈×C₅₉	—	2	2	no	—
71	C₂×C₈×C₅₉	—	4	1	no	—
72	C₂×C₈×C₅₉	—	4	1	no	—
73	C₂×C₈×C₅₉	—	4	4	no	—
74	C₂×C₈×C₅₉	—	4	4	no	—
75	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	D₈	1	2	yes	—
80	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
81	C₄⋊C₄×C₅₉	—	2	1	no	—
82	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
83	C₂²⋊C₄×C₅₉	—	2	2	no	—
84	C₂²⋊C₄×C₅₉	—	2	2	no	—
85	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅₉	#41	1	1	yes	characteristic
89	#11745664	—	4	2	no	—
90	#11745664	—	4	2	no	—
91	C₂×C₅₉⋊C₈	—	4	2	no	—
92	C₂×C₅₉⋊C₈	—	4	2	no	—
93	C₂³×C₅₉	—	2	4	no	—
94	C₂³×C₅₉	—	2	2	no	—
95	C₂³×C₅₉	—	4	2	no	—
96	C₂³×C₅₉	—	4	1	no	—
97	C₂³×C₅₉	—	4	1	no	—
98	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
99	D₄×C₅₉	—	2	4	no	—
100	D₄×C₅₉	—	2	2	no	—
101	D₄×C₅₉	#108	1	4	yes	complemented
102	C₂×C₄×C₅₉	—	2	2	no	—
103	C₂×C₄×C₅₉	—	2	2	no	—
104	C₂×C₄×C₅₉	—	4	1	no	—
105	C₂×C₄×C₅₉	—	4	1	no	—
106	C₂×C₄×C₅₉	—	4	1	no	—
107	C₂×C₄×C₅₉	—	4	2	no	—
108	C₂×C₄×C₅₉	—	4	2	no	—
109	C₂×C₄×C₅₉	—	4	2	no	—
110	C₂×C₄×C₅₉	—	4	2	no	—
111	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
121	C₈×C₅₉	—	2	1	no	—
122	C₈×C₅₉	—	2	1	no	—
123	C₈×C₅₉	—	4	2	no	—
124	C₈×C₅₉	#102	1	2	yes	—
125	C₈×C₅₉	#102	1	2	yes	—
126	C₅₉⋊C₈	—	8	2	no	—
127	C₅₉⋊C₈	—	8	2	no	—
128	#8658	—	59	1	no	maximal, Sylow
129	C₂²×C₅₉	—	2	1	no	—
130	C₂²×C₅₉	—	2	1	no	—
131	C₂²×C₅₉	—	2	4	no	—
132	C₂²×C₅₉	—	4	4	no	—
133	C₂²×C₅₉	—	4	4	no	—
134	C₂²×C₅₉	—	4	2	no	—
135	C₂²×C₅₉	—	4	2	no	—
136	C₂²×C₅₉	#380	1	1	yes	characteristic
137	C₂²×C₅₉	#474	1	1	yes	characteristic
138	C₂²×C₅₉	#476	1	1	yes	characteristic
139	C₄×C₅₉	—	2	1	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	—	4	2	no	—
143	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
144	C₄×C₅₉	#417	1	1	yes	characteristic
145	C₄×C₅₉	C₂×#108	1	2	yes	—
146	C₄×C₅₉	#459	1	1	yes	characteristic
147	C₄×C₅₉	#461	1	1	yes	characteristic
148	C₂×C₈×D₄	D₅₉	1	1	yes	characteristic
149	#1848	—	59	1	no	—
150	#1564	—	59	1	no	—
151	#1486	—	59	2	no	—
152	#1301	—	59	2	no	—
153	C₂×C₅₉	—	2	1	no	—
154	C₂×C₅₉	—	2	1	no	—
155	C₂×C₅₉	—	4	4	no	—
156	C₂×C₅₉	#1802	1	1	yes	characteristic
157	C₂×C₅₉	#1810	1	1	yes	characteristic
158	C₂×C₅₉	#1814	1	1	yes	characteristic
159	C₂³×C₈	—	2	1	no	—
160	C₂×C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
161	C₈×D₄	—	2	1	no	—
162	C₈×D₄	—	2	1	no	—
163	C₈×D₄	Dic₅₉	1	2	yes	—
164	C₈×D₄	Dic₅₉	1	2	yes	—
165	#431	—	59	2	no	—
166	C₂×#108	—	118	1	no	—
167	C₂×#108	—	118	1	no	—
168	#423	—	59	2	no	—
169	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
170	C₂×C₂²⋊C₈	—	2	1	no	—
171	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
172	#335	—	118	1	no	—
173	#334	—	118	1	no	—
174	D₄⋊C₈	—	118	4	no	—
175	C₅₉	#8658	1	1	yes	characteristic, complemented, Sylow
176	C₂²×D₄	Dic₁₁₈	1	1	yes	characteristic
177	C₂³×C₄	—	2	1	no	—
178	C₂×M₁₆	—	118	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	4	2	no	—
183	C₂²×C₈	—	4	2	no	—
184	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	D₂₃₆	1	2	yes	—
189	C₄×D₄	C₂×Dic₅₉	1	2	yes	—
190	C₂×C₄⋊C₄	Dic₁₁₈	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
193	#108	—	236	2	no	—
194	#108	—	236	2	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	118	2	no	—
197	C₄⋊C₈	—	118	2	no	—
198	C₄⋊C₈	—	118	2	no	—
199	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
200	C₂²⋊C₈	—	2	2	no	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₂²⋊C₈	—	118	2	no	—
203	C₄×C₈	—	118	2	no	—
204	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
205	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
208	C₂⁴	—	2	1	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	C₄⋊Dic₅₉	1	2	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
220	C₂²×C₄	#11745685	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
222	M₁₆	—	236	2	no	—
223	M₁₆	—	236	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	236	2	no	—
232	C₂×C₈	—	236	2	no	—
233	C₂×C₈	#11745669	1	2	yes	—
234	C₂×C₈	#11745670	1	2	yes	—
235	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
244	C₄²	#11745685	1	1	yes	characteristic
245	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
247	C₂³	—	2	4	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	#14273706	1	1	yes	characteristic
253	D₄	—	2	4	no	—
254	D₄	—	2	2	no	—
255	D₄	#14273663	1	4	yes	complemented
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
267	C₂×C₄	#14273686	1	1	yes	characteristic
268	C₂×C₄	#14273701	1	1	yes	characteristic
269	C₂×C₄	#14273701	1	1	yes	characteristic
270	C₂×C₄	#14273706	1	1	yes	characteristic
271	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
272	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
273	C₂×C₄	#14273752	1	1	yes	characteristic
274	C₂×C₄	#14273757	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	4	2	no	—
278	C₈	—	472	2	no	—
279	C₈	—	472	2	no	—
280	C₈	#14273648	1	2	yes	—
281	C₈	#14273648	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#16837262	1	1	yes	characteristic
290	C₂²	#16837682	1	1	yes	characteristic
291	C₂²	#16837692	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	4	2	no	—
296	C₄	C₂×#14273648	1	1	yes	characteristic
297	C₄	#16837403	1	1	yes	characteristic
298	C₄	C₂×#14273663	1	2	yes	—
299	C₄	#16837597	1	1	yes	characteristic
300	C₄	#16837607	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	4	no	—
304	C₂	#21424096	1	1	yes	characteristic
305	C₂	#21424134	1	1	yes	characteristic
306	C₂	#21424150	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈×D₄×C₅₉	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂²×D₄×C₅₉	Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈×D₄	D₅₉
hypocenter	C₅₉	#8658
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #21424268	C₂	semidirect
2. #21422844	C₂	semidirect
4. #21421993	C₂	semidirect
101. D₄×C₅₉	#108	semidirect
175. C₅₉	#8658	semidirect
255. D₄	#14273663	semidirect

Maximal subgroups

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

subgroup	index
#21424268	2
#21422844	2
#21422521	2
#21421993	2
C₂×C₈×D₄×C₅₉	2
#8658	59