Subgroups — #28507758

Order 15104.

414 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837807	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837206	C₂	1	2	yes	maximal
3	#21422187	C₂	1	2	yes	maximal
4	#21422174	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1969	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273727	—	2	1	no	—
7	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
8	#16837807	—	2	4	no	—
9	C₂×#14273661	—	2	1	no	—
10	C₂×#14273661	C₂²	1	2	yes	—
11	#16837206	—	2	4	no	—
12	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#103×C₅₉	—	2	1	no	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#391	—	2	1	no	—
17	C₂²×Dic₁₁₈	—	4	1	no	—
18	#14273727	—	2	4	no	—
19	#14273727	—	4	2	no	—
20	C₂×C₄⋊Dic₅₉	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	4	1	no	—
22	C₂×C₅₉⋊C₁₆	—	2	2	no	—
23	C₂×C₅₉⋊C₁₆	—	2	2	no	—
24	C₂×C₅₉⋊C₁₆	—	2	2	no	—
25	#14273661	—	2	4	no	—
26	#14273661	—	2	2	no	—
27	#14273661	D₄	1	4	yes	—
28	C₂²×Q₈×C₅₉	—	2	1	no	—
29	C₂×Q₁₆×C₅₉	—	2	2	no	—
30	C₂×Q₁₆×C₅₉	—	2	2	no	—
31	C₂×Q₁₆×C₅₉	—	2	1	no	—
32	C₂×Q₁₆×C₅₉	Q₈	1	2	yes	complemented
33	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
36	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
37	#107×C₅₉	—	2	2	no	—
38	#103×C₅₉	—	4	2	no	—
39	C₄×C₈×C₅₉	—	2	2	no	—
40	#95×C₅₉	—	4	1	no	—
41	#95×C₅₉	—	4	1	no	—
42	C₂×Q₈×C₅₉	—	2	2	no	—
43	C₂×Q₈×C₅₉	—	4	1	no	—
44	C₂×Q₈×C₅₉	—	4	1	no	—
45	C₂×Q₈×C₅₉	—	4	1	no	—
46	C₂×Q₈×C₅₉	—	4	1	no	—
47	C₂×Q₈×C₅₉	—	4	1	no	—
48	C₂²×C₄×C₅₉	—	4	1	no	—
49	C₂²×C₄×C₅₉	—	4	1	no	—
50	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₅₉	—	4	2	no	—
54	C₂×Dic₁₁₈	—	4	4	no	—
55	C₂×Dic₁₁₈	—	8	2	no	—
56	C₂×Dic₁₁₈	—	8	2	no	—
57	Q₁₆×C₅₉	—	2	4	no	—
58	Q₁₆×C₅₉	—	4	2	no	—
59	C₂×C₈×C₅₉	—	2	1	no	—
60	C₂×C₈×C₅₉	—	2	1	no	—
61	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
62	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₅₉	#41	1	2	yes	—
65	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₅₉	—	4	1	no	—
67	C₄⋊C₄×C₅₉	—	4	1	no	—
68	C₄²×C₅₉	—	2	1	no	—
69	C₄²×C₅₉	—	2	1	no	—
70	C₄⋊Dic₅₉	—	2	4	no	—
71	C₄⋊Dic₅₉	—	4	4	no	—
72	C₄⋊Dic₅₉	—	4	2	no	—
73	C₅₉⋊C₁₆	—	4	2	no	—
74	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₅₉	—	4	2	no	—
76	Q₈×C₅₉	—	4	2	no	—
77	Q₈×C₅₉	—	8	1	no	—
78	Q₈×C₅₉	—	8	1	no	—
79	C₂×C₄×C₅₉	—	2	1	no	—
80	C₂×C₄×C₅₉	—	2	1	no	—
81	C₂×C₄×C₅₉	—	2	1	no	—
82	C₂×C₄×C₅₉	—	2	1	no	—
83	C₂×C₄×C₅₉	—	4	2	no	—
84	C₂×C₄×C₅₉	—	8	1	no	—
85	C₂×C₄×C₅₉	—	8	1	no	—
86	C₂×C₄×C₅₉	—	8	1	no	—
87	C₂×C₄×C₅₉	—	8	1	no	—
88	C₂×C₄×C₅₉	—	8	1	no	—
89	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
90	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
91	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
97	C₂×Dic₅₉	—	4	4	no	—
98	C₂×Dic₅₉	—	8	2	no	—
99	C₂×Dic₅₉	—	8	2	no	—
100	Dic₁₁₈	—	8	8	no	—
101	C₈×C₅₉	—	2	2	no	—
102	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₅₉	#124	1	2	yes	—
104	#13364	—	59	1	no	maximal, Sylow
105	C₂²×C₅₉	#447	1	1	yes	characteristic
106	C₂²×C₅₉	#474	1	1	yes	characteristic
107	C₂²×C₅₉	#486	1	1	yes	characteristic
108	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
109	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
110	C₂²×C₅₉	#509	1	1	yes	characteristic
111	C₂²×C₅₉	#510	1	1	yes	characteristic
112	C₄×C₅₉	—	2	2	no	—
113	C₄×C₅₉	—	2	2	no	—
114	C₄×C₅₉	—	8	2	no	—
115	C₄×C₅₉	—	16	1	no	—
116	C₄×C₅₉	#393	1	1	yes	characteristic
117	C₄×C₅₉	#448	1	1	yes	characteristic
118	C₄×C₅₉	#476	1	1	yes	characteristic
119	C₄×C₅₉	#489	1	1	yes	characteristic
120	Dic₅₉	—	8	4	no	—
121	C₂×#500	—	59	1	no	—
122	C₂×#358	—	59	2	no	—
123	#1969	D₅₉	1	1	yes	characteristic
124	#1380	—	59	2	no	—
125	#1374	—	59	1	no	—
126	C₂×C₅₉	#2023	1	1	yes	characteristic
127	C₂×C₅₉	#2187	1	1	yes	characteristic
128	C₂×C₅₉	#2188	1	1	yes	characteristic
129	C₂×C₅₉	#2225	1	1	yes	characteristic
130	C₂×C₅₉	#2227	1	1	yes	characteristic
131	C₂×C₅₉	#2241	1	1	yes	characteristic
132	C₂×C₅₉	#2243	1	1	yes	characteristic
133	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
134	C₂×C₄⋊Q₈	—	118	1	no	—
135	C₂²×C₁₆	—	59	2	no	—
136	#500	—	118	4	no	—
137	C₂×#107	—	59	2	no	—
138	C₂×#107	—	118	1	no	—
139	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
140	C₂×#103	—	2	1	no	—
141	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
142	#391	—	2	1	no	—
143	#358	—	118	4	no	—
144	C₅₉	#13364	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	—	236	1	no	—
147	C₂×Q₁₆	—	2	1	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	Dic₁₁₈	1	2	yes	complemented
151	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
153	C₄⋊Q₈	—	118	4	no	—
154	C₄⋊Q₈	—	236	2	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	118	2	no	—
157	C₂×C₄⋊C₄	—	236	1	no	—
158	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
159	C₂×C₁₆	—	118	2	no	—
160	C₂×C₁₆	—	118	2	no	—
161	C₂×C₁₆	—	118	2	no	—
162	#107	—	2	2	no	—
163	#107	—	59	4	no	—
164	#107	—	118	4	no	—
165	#107	—	118	2	no	—
166	#103	—	4	2	no	—
167	C₄×C₈	—	2	2	no	—
168	#95	—	4	1	no	—
169	#95	—	4	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	472	2	no	—
177	C₂×Q₈	—	472	2	no	—
178	C₂×Q₈	—	236	4	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	236	2	no	—
182	C₂²×C₄	#11745669	1	1	yes	characteristic
183	C₂²×C₄	#11745669	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	#11745670	1	2	yes	—
190	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
191	C₂×C₈	#11745685	1	1	yes	characteristic
192	C₂×C₈	D₄×D₅₉	1	2	yes	—
193	C₂×C₈	#11745687	1	2	yes	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	118	4	no	—
197	C₄⋊C₄	—	236	4	no	—
198	C₄⋊C₄	—	236	2	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₁₆	—	236	2	no	—
202	C₂³	C₂×#11745669	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	8	1	no	—
206	Q₈	—	8	1	no	—
207	Q₈	—	472	8	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	472	2	no	—
219	C₂×C₄	—	472	2	no	—
220	C₂×C₄	—	236	4	no	—
221	C₂×C₄	#14273670	1	2	yes	—
222	C₂×C₄	#14273673	1	2	yes	—
223	C₂×C₄	#14273695	1	1	yes	characteristic
224	C₂×C₄	#14273706	1	1	yes	characteristic
225	C₂×C₄	#14273714	1	1	yes	characteristic
226	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
227	C₂×C₄	#14273752	1	1	yes	characteristic
228	C₂×C₄	#14273757	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	#14273704	1	2	yes	—
231	C₈	#14273713	1	2	yes	—
232	C₂²	#16837535	1	1	yes	characteristic
233	C₂²	#16837682	1	1	yes	characteristic
234	C₂²	#16837748	1	1	yes	characteristic
235	C₂²	C₂×#14273670	1	1	yes	characteristic
236	C₂²	C₂×#14273673	1	1	yes	characteristic
237	C₂²	#16837837	1	1	yes	characteristic
238	C₂²	#16837842	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	8	2	no	—
242	C₄	—	16	1	no	—
243	C₄	—	472	4	no	—
244	C₄	#16837314	1	1	yes	characteristic
245	C₄	#16837540	1	1	yes	characteristic
246	C₄	#16837692	1	1	yes	characteristic
247	C₄	#16837761	1	1	yes	characteristic
248	C₂	#21425232	1	1	yes	characteristic
249	C₂	#21426024	1	1	yes	characteristic
250	C₂	#21426029	1	1	yes	characteristic
251	C₂	#21426238	1	1	yes	characteristic
252	C₂	#21426248	1	1	yes	characteristic
253	C₂	#21426323	1	1	yes	characteristic
254	C₂	#21426332	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator, cosocle	C₂²×C₈×C₅₉	C₂³
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1969	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	#1969	D₅₉
hypocenter	C₅₉	#13364
LCS	G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

Complements

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

N	H ≅ G/N	type
32. C₂×Q₁₆×C₅₉	Q₈	semidirect
144. C₅₉	#13364	semidirect
150. C₂×Q₁₆	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837807	2
C₂×#16837206	2
#21422187	2
#21422174	2
C₅₉×#1969	2
#13364	59