Subgroups — #28479476

Order 15104.

502 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273650	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837132	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837124	C₂	1	1	yes	maximal, characteristic
5	#21421937	C₂	1	1	yes	maximal, characteristic
6	#21421930	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₁₈	—	2	2	no	—
10	C₈×Dic₁₁₈	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₉⋊C₈	—	2	1	no	—
13	C₂×#14273650	C₄	1	2	yes	—
14	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₉	—	2	1	no	—
17	#16837164	C₄	1	2	yes	—
18	#16837132	C₄	1	4	yes	complemented
19	#16837124	—	2	2	no	—
20	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₉	—	2	1	no	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#336	—	2	1	no	—
25	C₂²×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅₉⋊C₈	—	2	1	no	—
27	C₂²×C₅₉⋊C₈	—	2	1	no	—
28	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₁₈	—	2	1	no	—
30	C₄×Dic₁₁₈	—	2	1	no	—
31	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×Dic₅₉⋊C₄	—	2	1	no	—
35	C₂×C₄×Dic₅₉	—	2	1	no	—
36	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
37	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
38	Dic₅₉⋊C₈	—	2	4	no	—
39	#14273650	C₈	1	8	yes	complemented
40	C₄×C₅₉⋊C₈	—	2	1	no	—
41	C₄×C₅₉⋊C₈	—	2	1	no	—
42	C₄×C₅₉⋊C₈	—	2	1	no	—
43	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
44	C₈×Dic₅₉	—	2	4	no	—
45	C₂²×C₈×C₅₉	—	2	1	no	—
46	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅₉	—	2	1	no	—
50	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₅₉	—	2	1	no	—
52	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₅₉	—	4	2	no	—
54	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₅₉	D₄	1	2	yes	—
56	#95×C₅₉	—	2	1	no	—
57	C₂²×C₄×C₅₉	—	2	2	no	—
58	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
61	C₂²×Dic₅₉	—	2	1	no	—
62	C₂²×Dic₅₉	—	2	1	no	—
63	C₂×Dic₁₁₈	—	2	1	no	—
64	C₂×Dic₁₁₈	—	2	2	no	—
65	C₂×Dic₁₁₈	C₄²	1	2	yes	—
66	C₂×Dic₁₁₈	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₅₉	—	2	1	no	—
68	C₂×C₈×C₅₉	—	2	1	no	—
69	C₂×C₈×C₅₉	—	2	1	no	—
70	C₂×C₈×C₅₉	—	2	1	no	—
71	C₂×C₈×C₅₉	—	2	2	no	—
72	C₂×C₈×C₅₉	—	4	1	no	—
73	C₂×C₈×C₅₉	—	4	1	no	—
74	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
76	C₄⋊C₄×C₅₉	—	2	2	no	—
77	C₄⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
78	C₄²×C₅₉	—	4	1	no	—
79	C₄²×C₅₉	—	4	1	no	—
80	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₉	#41	1	1	yes	characteristic
84	C₄⋊Dic₅₉	—	2	1	no	—
85	C₄⋊Dic₅₉	C₄²	1	2	yes	—
86	Dic₅₉⋊C₄	—	2	2	no	—
87	Dic₅₉⋊C₄	—	2	2	no	—
88	C₄×Dic₅₉	—	2	2	no	—
89	C₄×Dic₅₉	—	2	2	no	—
90	C₂×C₅₉⋊C₈	—	2	2	no	—
91	C₂×C₅₉⋊C₈	—	2	2	no	—
92	C₂×C₅₉⋊C₈	—	2	2	no	—
93	C₂×C₅₉⋊C₈	—	2	2	no	—
94	C₂×C₅₉⋊C₈	—	4	2	no	—
95	C₂×C₅₉⋊C₈	—	4	1	no	—
96	C₂×C₅₉⋊C₈	—	4	1	no	—
97	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
98	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	—	2	4	no	—
101	C₂×C₄×C₅₉	—	4	2	no	—
102	C₂×C₄×C₅₉	—	4	2	no	—
103	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅₉	#131	1	2	yes	—
111	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
113	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
114	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
115	C₂×Dic₅₉	—	2	2	no	—
116	C₂×Dic₅₉	—	2	4	no	—
117	C₂×Dic₅₉	—	2	2	no	—
118	C₂×Dic₅₉	—	4	2	no	—
119	Dic₁₁₈	—	2	2	no	—
120	Dic₁₁₈	—	2	4	no	—
121	Dic₁₁₈	C₄×C₈	1	4	yes	complemented
122	C₈×C₅₉	—	2	2	no	—
123	C₈×C₅₉	—	2	2	no	—
124	C₈×C₅₉	—	4	2	no	—
125	C₅₉⋊C₈	—	2	4	no	—
126	C₅₉⋊C₈	—	4	2	no	—
127	C₅₉⋊C₈	—	4	2	no	—
128	#8201	—	59	1	no	maximal, Sylow
129	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₅₉	C₈×D₄	1	2	yes	—
131	C₂²×C₅₉	#440	1	1	yes	characteristic
132	C₂²×C₅₉	#441	1	1	yes	characteristic
133	C₂²×C₅₉	#444	1	2	yes	—
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	4	4	no	—
137	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₅₉	#405	1	1	yes	characteristic
139	C₄×C₅₉	#417	1	1	yes	characteristic
140	C₄×C₅₉	#419	1	1	yes	characteristic
141	C₄×C₅₉	C₈×D₄	1	2	yes	—
142	C₄×C₅₉	#440	1	1	yes	characteristic
143	C₄×C₅₉	#441	1	1	yes	characteristic
144	Dic₅₉	—	2	4	no	—
145	Dic₅₉	—	4	2	no	—
146	C₂×C₈×Q₈	—	59	1	no	—
147	C₈×C₄⋊C₄	D₅₉	1	1	yes	characteristic
148	C₄×#103	—	59	1	no	—
149	C₂×Q₈⋊C₈	—	59	1	no	—
150	C₂×#322	—	59	1	no	—
151	#1279	—	59	1	no	—
152	#1277	—	59	1	no	—
153	C₂×C₅₉	SD₁₆⋊C₈	1	2	yes	—
154	C₂×C₅₉	#1579	1	2	yes	—
155	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₅₉	#1763	1	1	yes	characteristic
157	C₂×C₅₉	#1765	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	59	1	no	—
159	C₈×Q₈	—	59	4	no	—
160	C₈×Q₈	—	118	2	no	—
161	C₂×C₄⋊C₈	—	59	1	no	—
162	C₂×C₄⋊C₈	—	118	1	no	—
163	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
164	C₂×#103	—	59	2	no	—
165	C₂×C₄×C₈	—	2	1	no	—
166	C₂×C₄×C₈	—	59	1	no	—
167	C₂×C₄×C₈	—	59	1	no	—
168	C₂×C₄×C₈	—	118	1	no	—
169	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
170	C₄×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
171	#340	—	59	2	no	—
172	#336	—	2	1	no	—
173	Q₈⋊C₈	—	59	4	no	—
174	#322	—	118	2	no	—
175	C₅₉	#8201	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	59	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	59	2	no	—
179	C₂²×C₈	—	118	1	no	—
180	C₂²×C₈	—	118	1	no	—
181	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
183	C₄×Q₈	—	59	2	no	—
184	C₄×Q₈	—	59	2	no	—
185	C₄×Q₈	—	118	1	no	—
186	C₄×Q₈	—	118	1	no	—
187	C₂×C₄⋊C₄	—	59	1	no	—
188	C₂×C₄⋊C₄	—	118	1	no	—
189	C₂×C₄⋊C₄	C₄×D₅₉	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	118	1	no	—
192	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	59	2	no	—
195	C₄⋊C₈	—	118	4	no	—
196	C₄⋊C₈	—	118	1	no	—
197	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
198	#103	—	59	8	no	—
199	C₄×C₈	—	4	2	no	—
200	C₄×C₈	—	59	2	no	—
201	C₄×C₈	—	118	4	no	—
202	C₄×C₈	—	118	1	no	—
203	C₄×C₈	—	118	1	no	—
204	C₄×C₈	—	118	1	no	—
205	C₄×C₈	C₄×D₅₉	1	2	yes	—
206	C₄×C₈	D₂₃₆	1	2	yes	—
207	#95	—	2	1	no	—
208	C₂×Q₈	—	59	2	no	—
209	C₂×Q₈	—	59	4	no	—
210	C₂×Q₈	—	118	1	no	—
211	C₂×Q₈	—	118	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	118	1	no	—
214	C₂²×C₄	—	118	1	no	—
215	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
216	C₂²×C₄	#11745685	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	59	4	no	—
226	C₂×C₈	—	118	2	no	—
227	C₂×C₈	—	118	2	no	—
228	C₂×C₈	—	118	2	no	—
229	C₂×C₈	—	118	2	no	—
230	C₂×C₈	—	236	2	no	—
231	C₂×C₈	—	236	1	no	—
232	C₂×C₈	—	236	1	no	—
233	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
234	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	59	2	no	—
237	C₄⋊C₄	—	118	2	no	—
238	C₄⋊C₄	—	118	1	no	—
239	C₄⋊C₄	—	118	2	no	—
240	C₄⋊C₄	C₈×D₅₉	1	4	yes	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	118	2	no	—
244	C₄²	—	118	2	no	—
245	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
247	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
248	C₄²	#11745685	1	1	yes	characteristic
249	C₂³	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
250	Q₈	—	59	4	no	—
251	Q₈	—	118	4	no	—
252	Q₈	—	118	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	118	4	no	—
258	C₂×C₄	—	118	2	no	—
259	C₂×C₄	—	118	2	no	—
260	C₂×C₄	—	236	2	no	—
261	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
262	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	#14273684	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
265	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
266	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
268	C₂×C₄	#14273733	1	2	yes	—
269	C₂×C₄	#14273753	1	1	yes	characteristic
270	C₂×C₄	#14273757	1	1	yes	characteristic
271	C₂×C₄	#14273757	1	1	yes	characteristic
272	C₂×C₄	#14273758	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	118	4	no	—
277	C₈	—	236	2	no	—
278	C₈	—	236	2	no	—
279	C₂²	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
280	C₂²	C₈×C₂²⋊D₅₉	1	2	yes	—
281	C₂²	#16837502	1	1	yes	characteristic
282	C₂²	#16837507	1	1	yes	characteristic
283	C₂²	#16837522	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	118	4	no	—
288	C₄	—	236	2	no	—
289	C₄	C₄×C₈×D₅₉	1	1	yes	characteristic
290	C₄	#16837356	1	1	yes	characteristic
291	C₄	#16837401	1	1	yes	characteristic
292	C₄	#16837410	1	1	yes	characteristic
293	C₄	C₈×D₂₃₆	1	2	yes	—
294	C₄	#16837502	1	1	yes	characteristic
295	C₄	#16837507	1	1	yes	characteristic
296	C₂	#21422908	1	2	yes	—
297	C₂	#21422913	1	2	yes	—
298	C₂	C₈×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
299	C₂	#21423929	1	1	yes	characteristic
300	C₂	#21423942	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₈×C₄⋊C₄×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₉
hypocenter	C₅₉	#8201
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₁₁₈	C₄	semidirect
18. #16837132	C₄	semidirect
39. #14273650	C₈	semidirect
121. Dic₁₁₈	C₄×C₈	semidirect
175. C₅₉	#8201	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₁₈	2
C₄×#14273650	2
C₂×#16837132	2
C₂×#16837124	2
#21421937	2
#21421930	2
C₈×C₄⋊C₄×C₅₉	2
#8201	59