Subgroups — #28476908

Order 15104.

462 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424239	C₂	1	1	yes	maximal, characteristic
2	#21423949	C₂	1	1	yes	maximal, characteristic
3	#21421919	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
6	C₂×#14273638	—	2	1	no	—
7	#16837157	—	2	1	no	—
8	#16837157	—	2	1	no	—
9	#16837157	C₂²	1	2	yes	—
10	#16837157	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₅₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₉	—	2	1	no	—
13	C₅₉×#391	—	2	1	no	—
14	C₅₉×#389	—	2	1	no	—
15	C₅₉×#389	C₂²	1	2	yes	—
16	C₅₉×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
18	C₄×C₄⋊C₄×C₅₉	C₄	1	2	yes	—
19	C₄³×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	C₂²×C₅₉⋊C₈	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
23	#14273638	—	4	2	no	—
24	C₂²×Q₈×C₅₉	—	2	1	no	—
25	C₄⋊Q₈×C₅₉	—	2	4	no	—
26	C₄×Q₈×C₅₉	—	4	2	no	—
27	C₄×Q₈×C₅₉	—	4	2	no	—
28	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₉	—	2	2	no	—
35	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
42	#95×C₅₉	—	2	2	no	—
43	#95×C₅₉	—	2	2	no	—
44	C₂×Q₈×C₅₉	—	2	4	no	—
45	C₂×Q₈×C₅₉	—	4	2	no	—
46	C₂×Q₈×C₅₉	—	4	2	no	—
47	C₂²×C₄×C₅₉	—	2	2	no	—
48	C₂²×C₄×C₅₉	—	2	2	no	—
49	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₅₉	—	2	4	no	—
57	C₄⋊C₄×C₅₉	—	2	2	no	—
58	C₄⋊C₄×C₅₉	—	4	2	no	—
59	C₄⋊C₄×C₅₉	—	4	2	no	—
60	C₄⋊C₄×C₅₉	—	4	2	no	—
61	C₄⋊C₄×C₅₉	—	4	2	no	—
62	C₄⋊C₄×C₅₉	—	4	2	no	—
63	C₄⋊C₄×C₅₉	M₁₆	1	4	yes	—
64	C₄²×C₅₉	—	2	2	no	—
65	C₄²×C₅₉	—	2	2	no	—
66	C₄²×C₅₉	—	2	2	no	—
67	C₄²×C₅₉	—	2	2	no	—
68	C₄²×C₅₉	—	4	1	no	—
69	C₄²×C₅₉	—	4	1	no	—
70	C₄²×C₅₉	—	4	4	no	—
71	C₄²×C₅₉	D₈	1	2	yes	—
72	C₄²×C₅₉	SD₁₆	1	2	yes	—
73	C₂×C₅₉⋊C₈	—	2	4	no	—
74	C₂×C₅₉⋊C₈	—	4	2	no	—
75	C₂×C₅₉⋊C₈	—	4	2	no	—
76	C₂×C₅₉⋊C₈	—	4	2	no	—
77	C₂×C₅₉⋊C₈	—	4	2	no	—
78	C₂×C₅₉⋊C₈	—	4	2	no	—
79	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	Q₈×C₅₉	—	4	8	no	—
81	C₂×C₄×C₅₉	—	2	1	no	—
82	C₂×C₄×C₅₉	—	2	1	no	—
83	C₂×C₄×C₅₉	—	2	1	no	—
84	C₂×C₄×C₅₉	—	2	1	no	—
85	C₂×C₄×C₅₉	—	2	1	no	—
86	C₂×C₄×C₅₉	—	2	1	no	—
87	C₂×C₄×C₅₉	—	2	1	no	—
88	C₂×C₄×C₅₉	—	2	1	no	—
89	C₂×C₄×C₅₉	—	2	1	no	—
90	C₂×C₄×C₅₉	—	2	1	no	—
91	C₂×C₄×C₅₉	—	2	1	no	—
92	C₂×C₄×C₅₉	—	2	1	no	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	2	4	no	—
96	C₂×C₄×C₅₉	—	4	2	no	—
97	C₂×C₄×C₅₉	—	4	2	no	—
98	C₂×C₄×C₅₉	—	4	2	no	—
99	C₂×C₄×C₅₉	—	4	2	no	—
100	C₂×C₄×C₅₉	—	4	2	no	—
101	C₂×C₄×C₅₉	#101	1	2	yes	—
102	C₂×C₄×C₅₉	#102	1	2	yes	—
103	C₂×C₄×C₅₉	#102	1	2	yes	—
104	C₂×C₄×C₅₉	#104	1	2	yes	—
105	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#137	1	2	yes	—
109	C₅₉⋊C₈	—	4	4	no	—
110	C₅₉⋊C₈	—	8	2	no	—
111	#7625	—	59	1	no	maximal, Sylow
112	C₂²×C₅₉	#407	1	1	yes	characteristic
113	C₂²×C₅₉	#410	1	1	yes	characteristic
114	C₂²×C₅₉	C₂×#101	1	1	yes	characteristic
115	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
116	C₂²×C₅₉	#419	1	1	yes	characteristic
117	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
118	C₂²×C₅₉	#421	1	1	yes	characteristic
119	C₄×C₅₉	—	2	2	no	—
120	C₄×C₅₉	—	2	2	no	—
121	C₄×C₅₉	—	2	2	no	—
122	C₄×C₅₉	—	2	2	no	—
123	C₄×C₅₉	—	2	1	no	—
124	C₄×C₅₉	—	2	1	no	—
125	C₄×C₅₉	—	2	2	no	—
126	C₄×C₅₉	—	2	2	no	—
127	C₄×C₅₉	—	4	4	no	—
128	C₄×C₅₉	—	8	2	no	—
129	C₄×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
130	#1839	—	59	1	no	—
131	#1767	—	59	1	no	—
132	#1271	—	59	4	no	—
133	C₂×C₅₉	#1467	1	1	yes	characteristic
134	C₂×C₅₉	#1486	1	1	yes	characteristic
135	C₂×C₅₉	#1491	1	1	yes	characteristic
136	C₂×C₅₉	#1502	1	1	yes	characteristic
137	C₂×C₅₉	#1515	1	1	yes	characteristic
138	C₂×C₅₉	#1531	1	1	yes	characteristic
139	C₂×C₅₉	#1546	1	1	yes	characteristic
140	C₂×C₄⋊Q₈	Dic₅₉	1	1	yes	characteristic
141	C₂×C₄×Q₈	—	2	1	no	—
142	C₂×C₄⋊C₈	—	118	1	no	—
143	C₂×#97	—	118	1	no	—
144	#391	—	2	1	no	—
145	#389	—	2	1	no	—
146	#389	C₂×D₅₉	1	2	yes	—
147	#384	Dic₅₉	1	1	yes	characteristic
148	C₄×C₄⋊C₄	—	2	1	no	—
149	C₄×C₄⋊C₄	Dic₅₉	1	2	yes	—
150	C₄³	C₂×D₅₉	1	1	yes	characteristic
151	#336	—	59	2	no	—
152	#336	—	59	2	no	—
153	#336	—	118	1	no	—
154	#336	—	118	1	no	—
155	C₅₉	#7625	1	1	yes	characteristic, complemented, Sylow
156	C₂²×Q₈	—	2	1	no	—
157	C₂²×C₈	—	118	2	no	—
158	C₂²×C₈	—	118	2	no	—
159	C₄⋊Q₈	—	2	4	no	—
160	C₄×Q₈	—	4	2	no	—
161	C₄×Q₈	—	4	2	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
176	C₄⋊C₈	—	118	4	no	—
177	#97	—	236	2	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
187	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
189	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
192	C₂×C₈	—	118	4	no	—
193	C₂×C₈	—	236	2	no	—
194	C₂×C₈	—	236	2	no	—
195	C₂×C₈	—	236	2	no	—
196	C₂×C₈	—	236	2	no	—
197	C₂×C₈	—	236	2	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	#11745664	1	4	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	4	1	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	4	no	—
213	C₄²	#11745669	1	2	yes	—
214	C₄²	#11745670	1	2	yes	—
215	C₂³	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
216	Q₈	—	4	8	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	#14273646	1	2	yes	—
238	C₂×C₄	#14273648	1	2	yes	—
239	C₂×C₄	#14273648	1	2	yes	—
240	C₂×C₄	#14273654	1	2	yes	—
241	C₂×C₄	C₂×#11745664	1	2	yes	—
242	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
243	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
244	C₂×C₄	#14273758	1	2	yes	—
245	C₈	—	472	2	no	—
246	C₈	—	236	4	no	—
247	C₂²	#16837366	1	1	yes	characteristic
248	C₂²	#16837376	1	1	yes	characteristic
249	C₂²	C₂×#14273646	1	1	yes	characteristic
250	C₂²	C₂×#14273648	1	1	yes	characteristic
251	C₂²	#16837411	1	1	yes	characteristic
252	C₂²	C₂×#14273654	1	1	yes	characteristic
253	C₂²	#16837419	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₂	#21422440	1	1	yes	characteristic
265	C₂	#21422521	1	1	yes	characteristic
266	C₂	#21422543	1	1	yes	characteristic
267	C₂	#21422591	1	1	yes	characteristic
268	C₂	#21422647	1	1	yes	characteristic
269	C₂	#21422711	1	1	yes	characteristic
270	C₂	#21422768	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₉
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×C₄⋊Q₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#7625
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
155. C₅₉	#7625	semidirect

Maximal subgroups

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

subgroup	index
#21424239	2
#21423949	2
#21421919	2
C₄×C₄⋊Q₈×C₅₉	2
#7625	59