Subgroups — #28483340

Order 15104.

534 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433177	C₂	1	1	yes	maximal, characteristic
2	#21422491	C₂	1	2	yes	maximal
3	#21422420	C₂	1	1	yes	maximal, characteristic
4	#21421929	C₂	1	2	yes	maximal
5	C₄×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₅₉⋊C₈	—	2	4	no	—
7	#16837455	—	2	2	no	—
8	#16837455	C₂²	1	2	yes	—
9	#16837455	C₂²	1	2	yes	—
10	Q₈⋊(C₅₉⋊C₈)	—	2	4	no	—
11	#16837128	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₅₉	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#391	C₄	1	1	yes	characteristic
16	C₅₉×#391	C₄	1	1	yes	characteristic
17	C₅₉×#389	—	2	2	no	—
18	C₅₉×#384	C₄	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₅₉	—	2	2	no	—
20	C₄³×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	D₄	1	4	yes	—
24	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₄×C₅₉⋊C₈	—	2	4	no	—
27	C₄×C₅₉⋊C₈	—	2	2	no	—
28	C₄×C₅₉⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₉	—	2	2	no	—
32	C₄⋊Q₈×C₅₉	C₈	1	4	yes	complemented
33	C₄×Q₈×C₅₉	—	2	2	no	—
34	C₄×Q₈×C₅₉	—	2	2	no	—
35	C₄×Q₈×C₅₉	—	2	2	no	—
36	C₄×Q₈×C₅₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
43	C₂×C₄²×C₅₉	—	2	1	no	—
44	C₂×C₄²×C₅₉	—	2	1	no	—
45	C₂×C₄²×C₅₉	—	2	1	no	—
46	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₉	D₄	1	2	yes	—
49	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
50	#95×C₅₉	—	2	2	no	—
51	#95×C₅₉	—	2	2	no	—
52	C₂×Q₈×C₅₉	—	2	2	no	—
53	C₂×Q₈×C₅₉	—	2	1	no	—
54	C₂×Q₈×C₅₉	—	2	2	no	—
55	C₂×Q₈×C₅₉	—	2	2	no	—
56	C₂×Q₈×C₅₉	—	2	2	no	—
57	C₂×Q₈×C₅₉	C₂×C₈	1	2	yes	—
58	C₂×Q₈×C₅₉	C₂×C₈	1	2	yes	—
59	C₂×Q₈×C₅₉	M₁₆	1	2	yes	—
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂²×C₄×C₅₉	—	2	1	no	—
63	C₂²×C₄×C₅₉	—	2	1	no	—
64	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₅₉	—	2	8	no	—
70	C₄⋊C₄×C₅₉	—	2	2	no	—
71	C₄⋊C₄×C₅₉	—	2	4	no	—
72	C₄⋊C₄×C₅₉	—	2	2	no	—
73	C₄⋊C₄×C₅₉	—	4	4	no	—
74	C₄⋊C₄×C₅₉	—	4	2	no	—
75	C₄²×C₅₉	—	2	4	no	—
76	C₄²×C₅₉	—	2	2	no	—
77	C₄²×C₅₉	—	2	2	no	—
78	C₄²×C₅₉	—	2	1	no	—
79	C₄²×C₅₉	—	2	2	no	—
80	C₄²×C₅₉	—	4	2	no	—
81	C₄²×C₅₉	—	4	1	no	—
82	C₄²×C₅₉	—	4	1	no	—
83	C₄²×C₅₉	C₂×C₈	1	2	yes	—
84	C₄²×C₅₉	C₂×D₄	1	2	yes	—
85	C₄²×C₅₉	#41	1	2	yes	—
86	C₂×C₅₉⋊C₈	—	2	4	no	—
87	C₂×C₅₉⋊C₈	—	4	2	no	—
88	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₅₉	—	2	4	no	—
90	Q₈×C₅₉	—	2	4	no	—
91	Q₈×C₅₉	—	2	4	no	—
92	Q₈×C₅₉	—	4	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	2	2	no	—
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	—	2	2	no	—
101	C₂×C₄×C₅₉	—	2	2	no	—
102	C₂×C₄×C₅₉	—	4	1	no	—
103	C₂×C₄×C₅₉	—	4	1	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	1	no	—
110	C₂×C₄×C₅₉	C₂²⋊C₈	1	4	yes	—
111	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
113	C₂×C₄×C₅₉	#121	1	2	yes	—
114	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅₉	#131	1	2	yes	—
118	C₅₉⋊C₈	—	4	4	no	—
119	C₅₉⋊C₈	—	8	2	no	—
120	#8958	—	59	1	no	maximal, Sylow
121	C₂²×C₅₉	—	2	1	no	—
122	C₂²×C₅₉	—	2	1	no	—
123	C₂²×C₅₉	#386	1	1	yes	characteristic
124	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₂²×C₅₉	#408	1	1	yes	characteristic
126	C₄×C₅₉	—	2	4	no	—
127	C₄×C₅₉	—	2	4	no	—
128	C₄×C₅₉	—	2	2	no	—
129	C₄×C₅₉	—	2	1	no	—
130	C₄×C₅₉	—	4	2	no	—
131	C₄×C₅₉	—	4	2	no	—
132	C₄×C₅₉	—	4	2	no	—
133	C₄×C₅₉	—	8	1	no	—
134	C₄×C₅₉	C₈×D₄	1	2	yes	—
135	C₄×C₅₉	#435	1	2	yes	—
136	C₄×C₅₉	#456	1	2	yes	—
137	#2982	—	59	1	no	—
138	C₄×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
139	#1479	—	59	2	no	—
140	#1461	—	59	1	no	—
141	#1276	—	59	2	no	—
142	C₂×C₅₉	—	2	1	no	—
143	C₂×C₅₉	—	2	1	no	—
144	C₂×C₅₉	#1869	1	1	yes	characteristic
145	C₂×C₅₉	#1884	1	1	yes	characteristic
146	C₂×C₅₉	#1886	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	Dic₅₉	1	1	yes	characteristic
148	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
149	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
150	C₈×Q₈	—	118	4	no	—
151	#431	—	59	2	no	—
152	#431	—	59	2	no	—
153	#431	—	118	2	no	—
154	#391	Dic₅₉	1	1	yes	characteristic
155	#391	Dic₅₉	1	1	yes	characteristic
156	#389	—	2	2	no	—
157	#384	Dic₅₉	1	1	yes	characteristic
158	C₄×C₄⋊C₄	—	2	2	no	—
159	C₄³	C₂×D₅₉	1	1	yes	characteristic
160	Q₈⋊C₈	—	118	4	no	—
161	#324	—	118	2	no	—
162	C₅₉	#8958	1	1	yes	characteristic, complemented, Sylow
163	C₂²×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
164	C₂²×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	C₅₉⋊C₈	1	4	yes	complemented
167	C₄×Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	C₂×Dic₅₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×Dic₅₉	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	—	2	1	no	—
180	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
183	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
184	C₄⋊C₈	—	59	4	no	—
185	C₄⋊C₈	—	118	4	no	—
186	C₄⋊C₈	—	118	2	no	—
187	C₂²⋊C₈	—	118	2	no	—
188	C₂²⋊C₈	—	118	2	no	—
189	C₄×C₈	—	118	4	no	—
190	C₄×C₈	—	118	2	no	—
191	C₄×C₈	—	118	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	1	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	C₂×C₅₉⋊C₈	1	2	yes	—
200	C₂×Q₈	C₂×C₅₉⋊C₈	1	2	yes	—
201	C₂×Q₈	#11745664	1	2	yes	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
207	C₂²×C₄	C₂²⋊Dic₅₉	1	2	yes	—
208	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
211	C₂×C₈	—	118	4	no	—
212	C₂×C₈	—	236	2	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
228	C₄²	D₄×D₅₉	1	2	yes	—
229	C₄²	#11745687	1	2	yes	—
230	C₂³	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	Q₈	—	2	4	no	—
233	Q₈	—	2	4	no	—
234	Q₈	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	4	yes	—
253	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
254	C₂×C₄	#14273695	1	1	yes	characteristic
255	C₂×C₄	#14273699	1	2	yes	—
256	C₂×C₄	#14273708	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
258	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
259	C₂×C₄	#14273734	1	2	yes	—
260	C₈	—	472	2	no	—
261	C₈	—	236	4	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	#16837282	1	1	yes	characteristic
265	C₂²	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
266	C₂²	#16837369	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	8	1	no	—
275	C₄	D₄×C₅₉⋊C₈	1	2	yes	—
276	C₄	#16837474	1	2	yes	—
277	C₄	#16837584	1	2	yes	—
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	#21424389	1	1	yes	characteristic
281	C₂	#21424480	1	1	yes	characteristic
282	C₂	#21424490	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273695
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×C₄⋊Q₈×C₅₉	C₂
socle	C₂²×C₅₉	#386
prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#8958
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
32. C₄⋊Q₈×C₅₉	C₈	semidirect
162. C₅₉	#8958	semidirect
166. C₄⋊Q₈	C₅₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21433177	2
#21422491	2
#21422420	2
#21421929	2
C₄×C₄⋊Q₈×C₅₉	2
#8958	59