Subgroups — #28507118

Order 15104.

422 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	C₂²×#14273672	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837232	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837219	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837216	C₂	1	1	yes	maximal, characteristic
5	#21425011	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425006	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
10	C₂×#14273672	—	2	2	no	—
11	C₂×#14273672	—	2	1	no	—
12	C₂×#14273672	—	2	1	no	—
13	C₂×#14273672	C₄	1	4	yes	complemented
14	C₂×#14273663	C₂²	1	1	yes	characteristic
15	C₂×#14273663	C₂²	1	1	yes	characteristic
16	#16837410	—	2	1	no	—
17	#16837406	—	2	1	no	—
18	C₂×#14273645	—	2	1	no	—
19	C₂×#14273643	—	2	1	no	—
20	#16837232	—	2	2	no	—
21	#16837219	—	2	2	no	—
22	#16837216	—	2	2	no	—
23	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#405	C₂²	1	1	yes	characteristic
26	C₅₉×#346	—	2	1	no	—
27	C₄×C₁₆×C₅₉	—	2	1	no	—
28	C₂²×Dic₁₁₈	—	2	1	no	—
29	C₂²×D₂₃₆	—	2	1	no	—
30	C₂×Dic₂₃₆	—	2	1	no	—
31	C₂×Dic₂₃₆	—	2	1	no	—
32	C₂×Dic₂₃₆	—	2	1	no	—
33	C₂×Dic₂₃₆	—	2	1	no	—
34	C₂×Dic₂₃₆	—	2	1	no	—
35	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
36	C₂×D₄₇₂	—	2	1	no	—
37	C₂×D₄₇₂	—	2	1	no	—
38	C₂×D₄₇₂	—	2	1	no	—
39	C₂×D₄₇₂	—	2	1	no	—
40	C₂×D₄₇₂	—	2	1	no	—
41	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
42	C₂×#11745664	—	2	1	no	—
43	C₂×#11745664	—	2	1	no	—
44	#14273672	—	2	4	no	—
45	#14273672	—	2	4	no	—
46	#14273663	—	2	1	no	—
47	#14273663	—	2	1	no	—
48	#14273663	—	2	1	no	—
49	#14273663	—	2	1	no	—
50	#14273650	—	4	2	no	—
51	#14273647	—	4	2	no	—
52	#14273645	—	4	2	no	—
53	#14273643	—	4	2	no	—
54	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
56	#117×C₅₉	D₄	1	1	yes	characteristic
57	M₃₂×C₅₉	—	2	1	no	—
58	M₃₂×C₅₉	D₄	1	2	yes	—
59	C₂×C₁₆×C₅₉	—	2	1	no	—
60	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
61	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
62	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
63	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
64	#97×C₅₉	—	2	1	no	—
65	C₄×C₈×C₅₉	—	2	1	no	—
66	C₂³×D₅₉	—	4	1	no	—
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₅₉	—	4	1	no	—
69	C₂×D₂₃₆	—	2	2	no	—
70	C₂×D₂₃₆	—	4	1	no	—
71	C₂×D₂₃₆	—	4	1	no	—
72	C₂×D₂₃₆	—	4	1	no	—
73	C₂×D₂₃₆	—	4	1	no	—
74	C₂×D₂₃₆	—	4	1	no	—
75	C₂×Dic₁₁₈	—	2	2	no	—
76	C₂×Dic₁₁₈	—	4	1	no	—
77	C₂×Dic₁₁₈	—	4	1	no	—
78	C₂×Dic₁₁₈	—	4	1	no	—
79	C₂×Dic₁₁₈	—	4	1	no	—
80	C₂×Dic₁₁₈	—	4	1	no	—
81	M₁₆×C₅₉	—	2	1	no	—
82	M₁₆×C₅₉	D₈	1	2	yes	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
89	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
90	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
91	C₂²⋊C₄×C₅₉	—	2	1	no	—
92	C₄²×C₅₉	—	2	1	no	—
93	#11745664	—	4	1	no	—
94	#11745664	—	4	1	no	—
95	#11745664	—	4	1	no	—
96	#11745664	—	4	1	no	—
97	C₂×C₅₉⋊C₈	—	4	1	no	—
98	C₂×C₅₉⋊C₈	—	4	1	no	—
99	Dic₂₃₆	—	2	2	no	—
100	Dic₂₃₆	—	2	2	no	—
101	Dic₂₃₆	—	4	1	no	—
102	Dic₂₃₆	—	4	1	no	—
103	D₄₇₂	—	2	2	no	—
104	D₄₇₂	—	2	2	no	—
105	D₄₇₂	—	4	1	no	—
106	D₄₇₂	—	4	1	no	—
107	C₁₆×C₅₉	—	2	2	no	—
108	C₁₆×C₅₉	C₂²⋊C₄	1	2	yes	—
109	C₁₆×C₅₉	C₂²⋊C₄	1	2	yes	—
110	C₂³×C₅₉	#135	1	1	yes	characteristic
111	C₂²×D₅₉	—	4	2	no	—
112	C₂²×D₅₉	—	8	1	no	—
113	C₂²×D₅₉	—	8	1	no	—
114	C₂²×D₅₉	—	16	2	no	—
115	C₂×C₄×C₅₉	—	2	2	no	—
116	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
122	C₂×Dic₅₉	—	4	2	no	—
123	C₂×Dic₅₉	—	8	1	no	—
124	C₂×Dic₅₉	—	8	1	no	—
125	D₂₃₆	—	4	2	no	—
126	D₂₃₆	—	4	2	no	—
127	D₂₃₆	—	8	1	no	—
128	D₂₃₆	—	8	1	no	—
129	Dic₁₁₈	—	4	2	no	—
130	Dic₁₁₈	—	4	2	no	—
131	Dic₁₁₈	—	8	1	no	—
132	Dic₁₁₈	—	8	1	no	—
133	C₈×C₅₉	—	2	2	no	—
134	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
136	C₈×C₅₉	#121	1	1	yes	characteristic
137	C₈×C₅₉	#124	1	1	yes	characteristic
138	C₅₉⋊C₈	—	8	1	no	—
139	C₅₉⋊C₈	—	8	1	no	—
140	#13246	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	—	2	2	no	—
142	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
143	C₂²×C₅₉	#466	1	1	yes	characteristic
144	C₂²×C₅₉	#495	1	1	yes	characteristic
145	C₂×D₅₉	—	8	2	no	—
146	C₂×D₅₉	—	16	2	no	—
147	C₂×D₅₉	—	16	2	no	—
148	C₂×D₅₉	—	16	2	no	—
149	C₄×C₅₉	—	4	2	no	—
150	C₄×C₅₉	#390	1	1	yes	characteristic
151	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
152	C₄×C₅₉	#466	1	1	yes	characteristic
153	C₄×C₅₉	#493	1	1	yes	characteristic
154	Dic₅₉	—	8	2	no	—
155	C₂²×SD₃₂	—	59	1	no	—
156	C₂×#368	—	59	1	no	—
157	C₂×#362	—	59	1	no	—
158	C₂×#361	—	59	1	no	—
159	#2108	D₅₉	1	1	yes	characteristic, complemented
160	#1979	—	59	1	no	—
161	#1978	—	59	1	no	—
162	C₂×C₅₉	—	2	1	no	—
163	C₂×C₅₉	—	2	1	no	—
164	C₂×C₅₉	#1970	1	1	yes	characteristic
165	C₂×C₅₉	#2223	1	2	yes	—
166	D₅₉	—	16	2	no	—
167	C₂²×Q₁₆	—	59	1	no	—
168	C₂²×D₈	—	59	1	no	—
169	C₂×SD₃₂	—	59	4	no	—
170	C₂×SD₃₂	—	118	2	no	—
171	C₂×SD₃₂	—	118	1	no	—
172	C₂×SD₃₂	—	118	1	no	—
173	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
174	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
175	C₂×#108	—	59	1	no	—
176	C₂×#108	—	59	1	no	—
177	#419	—	118	1	no	—
178	#418	—	118	1	no	—
179	C₂×#101	—	118	1	no	—
180	C₂×#100	—	118	1	no	—
181	#405	C₂×D₅₉	1	1	yes	characteristic
182	#368	—	118	2	no	—
183	#362	—	118	2	no	—
184	#361	—	118	2	no	—
185	#346	—	2	1	no	—
186	C₄×C₁₆	—	2	1	no	—
187	C₅₉	#13246	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	118	1	no	—
189	C₂²×D₄	—	118	1	no	—
190	C₂×Q₁₆	—	59	2	no	—
191	C₂×Q₁₆	—	118	1	no	—
192	C₂×Q₁₆	—	118	1	no	—
193	C₂×Q₁₆	—	118	1	no	—
194	C₂×Q₁₆	—	118	1	no	—
195	C₂×Q₁₆	—	118	1	no	—
196	C₂×D₈	—	59	2	no	—
197	C₂×D₈	—	118	1	no	—
198	C₂×D₈	—	118	1	no	—
199	C₂×D₈	—	118	1	no	—
200	C₂×D₈	—	118	1	no	—
201	C₂×D₈	—	118	1	no	—
202	C₂×M₁₆	—	118	1	no	—
203	C₂×M₁₆	—	118	1	no	—
204	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
206	#117	D₂₃₆	1	1	yes	characteristic
207	SD₃₂	—	118	4	no	—
208	SD₃₂	—	118	4	no	—
209	M₃₂	—	2	1	no	—
210	M₃₂	D₂₃₆	1	2	yes	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
213	C₂×C₁₆	D₂₃₆	1	2	yes	—
214	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
215	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
216	#108	—	118	1	no	—
217	#108	—	118	1	no	—
218	#108	—	118	1	no	—
219	#108	—	118	1	no	—
220	#103	—	236	2	no	—
221	#102	—	236	2	no	—
222	#101	—	236	2	no	—
223	#100	—	236	2	no	—
224	#97	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₂⁴	—	236	1	no	—
227	C₂×Q₈	—	118	2	no	—
228	C₂×Q₈	—	236	1	no	—
229	C₂×Q₈	—	236	1	no	—
230	C₂×Q₈	—	236	1	no	—
231	C₂×Q₈	—	236	1	no	—
232	C₂×Q₈	—	236	1	no	—
233	C₂×D₄	—	118	2	no	—
234	C₂×D₄	—	236	1	no	—
235	C₂×D₄	—	236	1	no	—
236	C₂×D₄	—	236	1	no	—
237	C₂×D₄	—	236	1	no	—
238	C₂×D₄	—	236	1	no	—
239	C₂²×C₄	—	236	1	no	—
240	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
241	Q₁₆	—	118	2	no	—
242	Q₁₆	—	118	2	no	—
243	Q₁₆	—	236	1	no	—
244	Q₁₆	—	236	1	no	—
245	D₈	—	118	2	no	—
246	D₈	—	118	2	no	—
247	D₈	—	236	1	no	—
248	D₈	—	236	1	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	236	1	no	—
251	M₁₆	—	236	1	no	—
252	M₁₆	—	236	1	no	—
253	M₁₆	—	236	1	no	—
254	M₁₆	D₄₇₂	1	2	yes	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	236	1	no	—
257	C₂×C₈	—	236	1	no	—
258	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
259	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
260	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
261	C₂×C₈	#11745685	1	1	yes	characteristic
262	C₂×C₈	#11745685	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
264	C₄⋊C₄	D₄₇₂	1	2	yes	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	(C₂×D₅₉)⋊C₄	1	2	yes	—
269	C₁₆	(C₂×D₅₉)⋊C₄	1	2	yes	—
270	C₂³	—	944	2	no	—
271	C₂³	—	472	1	no	—
272	C₂³	—	472	1	no	—
273	C₂³	—	236	2	no	—
274	C₂³	#14273749	1	1	yes	characteristic
275	Q₈	—	472	1	no	—
276	Q₈	—	472	1	no	—
277	Q₈	—	236	2	no	—
278	Q₈	—	236	2	no	—
279	D₄	—	472	1	no	—
280	D₄	—	472	1	no	—
281	D₄	—	236	2	no	—
282	D₄	—	236	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	472	1	no	—
285	C₂×C₄	—	472	1	no	—
286	C₂×C₄	—	236	2	no	—
287	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
288	C₂×C₄	#14273701	1	1	yes	characteristic
289	C₂×C₄	#14273724	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
292	C₂×C₄	#14273749	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	472	1	no	—
295	C₈	—	472	1	no	—
296	C₈	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
297	C₈	C₄×D₂₃₆	1	1	yes	characteristic
298	C₈	#14273701	1	1	yes	characteristic
299	C₈	#14273715	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	944	2	no	—
302	C₂²	—	944	2	no	—
303	C₂²	—	944	2	no	—
304	C₂²	—	472	2	no	—
305	C₂²	C₄×D₄₇₂	1	1	yes	characteristic
306	C₂²	#16837638	1	1	yes	characteristic
307	C₂²	#16837786	1	1	yes	characteristic
308	C₄	—	4	2	no	—
309	C₄	—	472	2	no	—
310	C₄	#16837304	1	1	yes	characteristic
311	C₄	C₄×D₄₇₂	1	1	yes	characteristic
312	C₄	#16837638	1	1	yes	characteristic
313	C₄	#16837779	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	944	2	no	—
317	C₂	#21424962	1	1	yes	characteristic
318	C₂	#21426226	1	2	yes	—

	subgroup	G/H
center	C₂²	#16837638
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2108	C₂
socle	C₂²×C₅₉	#466
prime radical	C₂²×D₄₇₂	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#13246
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. C₂×#16837232	C₂	semidirect
3. C₂×#16837219	C₂	semidirect
5. #21425011	C₂	semidirect
7. C₅₉×#2108	C₂	semidirect
13. C₂×#14273672	C₄	semidirect
159. #2108	D₅₉	semidirect
187. C₅₉	#13246	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14273672	2
C₂×#16837232	2
C₂×#16837219	2
C₂×#16837216	2
#21425011	2
#21425006	2
C₅₉×#2108	2
#13246	59