Subgroups — #28503984

Order 15104.

438 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837525	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837211	C₂	1	1	yes	maximal, characteristic
3	#21425806	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837180	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422204	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422194	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1776	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273749	C₂²	1	1	yes	characteristic
9	C₂×#14273733	—	2	1	no	—
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	#16837525	—	2	2	no	—
13	#16837525	C₄	1	4	yes	complemented
14	C₂×#14273663	C₂²	1	1	yes	characteristic
15	C₂×#14273655	—	2	1	no	—
16	#16837211	C₄	1	4	yes	complemented
17	#16837205	C₄	1	2	yes	complemented
18	#16837202	C₄	1	2	yes	complemented
19	#16837180	—	2	2	no	—
20	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
24	C₅₉×#340	—	2	1	no	—
25	C₂×#11745685	—	2	1	no	—
26	#14273749	—	2	1	no	—
27	#14273749	—	2	1	no	—
28	#14273749	C₂×C₄	1	2	yes	—
29	#14273749	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
31	C₂×#11745660	—	2	1	no	—
32	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
33	#14273733	—	2	4	no	—
34	#14273733	—	4	2	no	—
35	C₂×#11745664	—	2	1	no	—
36	C₂×#11745659	—	4	1	no	—
37	C₂×C₈×D₅₉	—	4	1	no	—
38	#14273667	—	2	1	no	—
39	#14273667	—	2	1	no	—
40	#14273667	—	2	1	no	—
41	#14273667	C₂×C₄	1	2	yes	—
42	C₂×C₅₉⋊C₁₆	—	2	1	no	—
43	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
44	#14273663	—	2	1	no	—
45	#14273663	C₂×C₄	1	2	yes	—
46	#14273655	—	2	4	no	—
47	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
51	C₂×C₄²×C₅₉	—	4	1	no	—
52	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
53	#107×C₅₉	—	2	1	no	—
54	#107×C₅₉	C₂×C₄	1	2	yes	—
55	#106×C₅₉	—	2	1	no	—
56	#106×C₅₉	C₂×C₄	1	2	yes	—
57	#97×C₅₉	C₂×C₄	1	2	yes	—
58	#97×C₅₉	D₄	1	2	yes	—
59	#95×C₅₉	—	4	1	no	—
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	—	4	1	no	—
62	C₂²×C₄×C₅₉	—	4	1	no	—
63	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₅₉	—	4	1	no	—
65	#11745685	—	2	2	no	—
66	#11745685	—	2	2	no	—
67	#11745685	—	4	1	no	—
68	#11745685	—	4	1	no	—
69	C₂×D₂₃₆	—	2	1	no	—
70	C₂×C₄×D₅₉	—	4	1	no	—
71	C₂×Dic₁₁₈	—	2	1	no	—
72	C₂×C₈×C₅₉	—	2	1	no	—
73	C₂×C₈×C₅₉	—	2	1	no	—
74	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
80	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
81	C₄⋊C₄×C₅₉	—	2	2	no	—
82	C₄⋊C₄×C₅₉	—	2	2	no	—
83	C₄⋊C₄×C₅₉	—	4	1	no	—
84	C₄⋊C₄×C₅₉	—	4	1	no	—
85	C₄²×C₅₉	—	2	1	no	—
86	C₄²×C₅₉	—	8	2	no	—
87	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
88	#11745664	—	2	2	no	—
89	#11745664	—	4	1	no	—
90	C₂×C₅₉⋊C₈	—	4	1	no	—
91	Dic₂₃₆	—	2	1	no	—
92	Dic₂₃₆	C₄²	1	2	yes	complemented
93	D₄₇₂	—	2	1	no	—
94	D₄₇₂	C₄²	1	2	yes	complemented
95	#11745660	—	2	2	no	—
96	#11745660	—	2	2	no	—
97	#11745659	—	4	4	no	—
98	C₈×D₅₉	—	4	4	no	—
99	C₅₉⋊C₁₆	—	2	2	no	—
100	C₅₉⋊C₁₆	—	2	2	no	—
101	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂²×D₅₉	—	4	1	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	2	no	—
107	C₂×C₄×C₅₉	—	4	2	no	—
108	C₂×C₄×C₅₉	—	8	1	no	—
109	C₂×C₄×C₅₉	—	8	1	no	—
110	C₂×C₄×C₅₉	—	8	1	no	—
111	C₂×C₄×C₅₉	—	8	1	no	—
112	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
118	C₂²⋊D₅₉	—	4	2	no	—
119	C₂²⋊D₅₉	—	4	2	no	—
120	C₂×Dic₅₉	—	4	1	no	—
121	D₂₃₆	—	2	2	no	—
122	D₂₃₆	—	4	1	no	—
123	C₄×D₅₉	—	4	2	no	—
124	C₄×D₅₉	—	4	2	no	—
125	Dic₁₁₈	—	2	2	no	—
126	Dic₁₁₈	—	4	1	no	—
127	C₈×C₅₉	—	2	2	no	—
128	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₅₉	#117	1	1	yes	characteristic
130	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
131	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
132	C₅₉⋊C₈	—	4	2	no	—
133	#12650	—	59	1	no	maximal, Sylow
134	C₂²×C₅₉	—	2	1	no	—
135	C₂²×C₅₉	—	2	1	no	—
136	C₂²×C₅₉	#417	1	1	yes	characteristic
137	C₂²×C₅₉	#442	1	1	yes	characteristic
138	C₂²×C₅₉	#442	1	1	yes	characteristic
139	C₂×D₅₉	—	4	2	no	—
140	C₂×D₅₉	—	8	1	no	—
141	C₂×D₅₉	—	8	1	no	—
142	C₄×C₅₉	—	4	2	no	—
143	C₄×C₅₉	—	8	2	no	—
144	C₄×C₅₉	—	8	2	no	—
145	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₄×C₅₉	#418	1	1	yes	characteristic
147	C₄×C₅₉	#440	1	1	yes	characteristic
148	C₄×C₅₉	#440	1	1	yes	characteristic
149	Dic₅₉	—	4	2	no	—
150	C₂×#444	—	59	1	no	—
151	C₂×#360	—	59	1	no	—
152	#2139	—	59	1	no	—
153	C₂×#347	—	59	1	no	—
154	#1776	D₅₉	1	1	yes	characteristic, complemented
155	#1387	—	59	1	no	—
156	#1383	—	59	1	no	—
157	C₂×C₅₉	—	2	1	no	—
158	C₂×C₅₉	—	2	1	no	—
159	C₂×C₅₉	#1762	1	1	yes	characteristic
160	C₂×C₅₉	#2180	1	1	yes	characteristic
161	C₂×C₅₉	AGL(1,ℤ/16)	1	1	yes	characteristic
162	D₅₉	—	8	2	no	—
163	C₂×#135	—	59	1	no	—
164	C₂×#131	—	118	1	no	—
165	C₂×M₃₂	—	59	1	no	—
166	C₂×M₃₂	—	59	1	no	—
167	#444	—	59	4	no	—
168	#444	—	118	2	no	—
169	C₂×#108	—	59	1	no	—
170	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
171	C₂×#106	C₂×D₅₉	1	1	yes	characteristic
172	C₂×#104	—	118	1	no	—
173	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	2	1	no	—
175	#360	—	59	4	no	—
176	#358	—	59	2	no	—
177	#357	—	59	2	no	—
178	#347	—	118	2	no	—
179	#340	—	2	1	no	—
180	C₅₉	#12650	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	—	118	1	no	—
182	#135	—	59	2	no	—
183	#135	—	59	2	no	—
184	#135	—	118	1	no	—
185	#135	—	118	1	no	—
186	C₂×Q₁₆	—	59	1	no	—
187	C₂×SD₁₆	—	118	1	no	—
188	C₂×D₈	—	59	1	no	—
189	#131	—	118	4	no	—
190	#131	—	236	2	no	—
191	C₂×M₁₆	—	118	1	no	—
192	C₂×M₁₆	—	236	1	no	—
193	C₂²×C₈	—	236	1	no	—
194	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄²	—	4	1	no	—
199	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
200	M₃₂	—	59	2	no	—
201	M₃₂	—	118	1	no	—
202	M₃₂	—	118	1	no	—
203	M₃₂	—	118	1	no	—
204	C₂×C₁₆	—	59	2	no	—
205	C₂×C₁₆	—	118	1	no	—
206	#108	—	59	2	no	—
207	#108	—	118	1	no	—
208	#107	—	2	1	no	—
209	#107	C₄×D₅₉	1	2	yes	—
210	#106	—	2	1	no	—
211	#106	C₄×D₅₉	1	2	yes	—
212	#104	—	118	4	no	—
213	#97	C₄×D₅₉	1	2	yes	—
214	#97	D₂₃₆	1	2	yes	—
215	#95	—	4	1	no	—
216	#41	—	118	2	no	—
217	#41	—	118	2	no	—
218	#41	—	236	1	no	—
219	#41	—	236	1	no	—
220	C₂×Q₈	—	118	1	no	—
221	C₂×D₄	—	118	1	no	—
222	C₂×D₄	—	236	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	236	1	no	—
227	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
228	Q₁₆	—	59	2	no	—
229	Q₁₆	—	118	1	no	—
230	SD₁₆	—	118	2	no	—
231	SD₁₆	—	118	2	no	—
232	D₈	—	59	2	no	—
233	D₈	—	118	1	no	—
234	M₁₆	—	118	2	no	—
235	M₁₆	—	236	4	no	—
236	M₁₆	—	236	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	236	4	no	—
240	C₂×C₈	—	236	1	no	—
241	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
245	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
246	C₂×C₈	#11745685	1	1	yes	characteristic
247	C₂×C₈	#11745685	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	8	2	no	—
254	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
255	C₁₆	—	118	2	no	—
256	C₁₆	—	118	2	no	—
257	C₂³	—	236	1	no	—
258	C₂³	#14273753	1	1	yes	characteristic
259	Q₈	—	118	2	no	—
260	Q₈	—	236	1	no	—
261	D₄	—	118	2	no	—
262	D₄	—	236	2	no	—
263	D₄	—	236	2	no	—
264	D₄	—	236	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	236	1	no	—
275	C₂×C₄	—	236	2	no	—
276	C₂×C₄	—	236	2	no	—
277	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
279	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
280	C₂×C₄	#14273752	1	1	yes	characteristic
281	C₂×C₄	#14273752	1	1	yes	characteristic
282	C₂×C₄	#14273758	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	236	2	no	—
285	C₈	C₄²×D₅₉	1	1	yes	characteristic
286	C₈	#14273684	1	1	yes	characteristic
287	C₈	C₄×D₂₃₆	1	1	yes	characteristic
288	C₈	C₄×D₂₃₆	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	472	1	no	—
292	C₂²	—	472	1	no	—
293	C₂²	—	236	2	no	—
294	C₂²	#16837401	1	1	yes	characteristic
295	C₂²	#16837512	1	1	yes	characteristic
296	C₂²	#16837512	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	236	2	no	—
301	C₄	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
302	C₄	#16837406	1	1	yes	characteristic
303	C₄	#16837500	1	1	yes	characteristic
304	C₄	#16837500	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	472	2	no	—
308	C₂	#21423924	1	1	yes	characteristic
309	C₂	#21425985	1	1	yes	characteristic
310	C₂	#21425992	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837512
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1776	C₂
socle	C₂²×C₅₉	#442
prime radical	C₂×#14273749	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1776	D₅₉
hypocenter	C₅₉	#12650
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1776

Complements

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

N	H ≅ G/N	type
4. C₂×#16837180	C₂	semidirect
5. #21422204	C₂	semidirect
6. #21422194	C₂	semidirect
7. C₅₉×#1776	C₂	semidirect
13. #16837525	C₄	semidirect
16. #16837211	C₄	semidirect
17. #16837205	C₄	semidirect
18. #16837202	C₄	semidirect
92. Dic₂₃₆	C₄²	semidirect
94. D₄₇₂	C₄²	semidirect
154. #1776	D₅₉	semidirect
180. C₅₉	#12650	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837525	2
C₂×#16837211	2
#21425806	2
C₂×#16837180	2
#21422204	2
#21422194	2
C₅₉×#1776	2
#12650	59