Subgroups — #28508306

Order 15104.

382 conjugacy classes of subgroups.

The subgroup lattice has 232 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×#16837229	C₂	1	2	yes	maximal
3	#21422187	C₂	1	2	yes	maximal
4	#21422174	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1942	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	#16837229	—	2	4	no	—
12	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#389	—	2	1	no	—
16	C₅₉×#340	—	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	Q₈	1	4	yes	complemented
25	#14273661	—	2	4	no	—
26	#14273661	—	2	2	no	—
27	#14273661	D₄	1	4	yes	—
28	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
33	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
34	#107×C₅₉	—	2	1	no	—
35	#107×C₅₉	—	2	2	no	—
36	#107×C₅₉	Q₈	1	2	yes	complemented
37	C₄×C₈×C₅₉	—	2	2	no	—
38	#95×C₅₉	—	4	1	no	—
39	C₂²×C₄×C₅₉	—	4	1	no	—
40	C₂²×C₄×C₅₉	—	4	1	no	—
41	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₉	—	4	2	no	—
45	C₂×Dic₁₁₈	—	4	4	no	—
46	C₂×Dic₁₁₈	—	8	2	no	—
47	C₂×Dic₁₁₈	—	8	2	no	—
48	C₂×C₈×C₅₉	—	2	1	no	—
49	C₂×C₈×C₅₉	—	2	1	no	—
50	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
51	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
55	C₄⋊C₄×C₅₉	—	2	2	no	—
56	C₄⋊C₄×C₅₉	—	4	1	no	—
57	C₄⋊C₄×C₅₉	—	4	1	no	—
58	C₄⋊C₄×C₅₉	—	4	1	no	—
59	C₄⋊C₄×C₅₉	—	8	2	no	—
60	C₄²×C₅₉	—	2	1	no	—
61	C₄²×C₅₉	—	2	1	no	—
62	C₄⋊Dic₅₉	—	2	4	no	—
63	C₄⋊Dic₅₉	—	4	4	no	—
64	C₄⋊Dic₅₉	—	4	2	no	—
65	C₅₉⋊C₁₆	—	2	4	no	—
66	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	—	2	1	no	—
69	C₂×C₄×C₅₉	—	2	1	no	—
70	C₂×C₄×C₅₉	—	2	1	no	—
71	C₂×C₄×C₅₉	—	4	2	no	—
72	C₂×C₄×C₅₉	—	8	1	no	—
73	C₂×C₄×C₅₉	—	8	1	no	—
74	C₂×C₄×C₅₉	—	8	1	no	—
75	C₂×C₄×C₅₉	—	8	1	no	—
76	C₂×C₄×C₅₉	—	8	1	no	—
77	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
78	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
79	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
85	C₂×Dic₅₉	—	4	4	no	—
86	C₂×Dic₅₉	—	8	2	no	—
87	C₂×Dic₅₉	—	8	2	no	—
88	Dic₁₁₈	—	8	8	no	—
89	C₈×C₅₉	—	2	2	no	—
90	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
91	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
92	#13452	—	59	1	no	maximal, Sylow
93	C₂²×C₅₉	#447	1	1	yes	characteristic
94	C₂²×C₅₉	#474	1	1	yes	characteristic
95	C₂²×C₅₉	#500	1	1	yes	characteristic
96	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
97	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
98	C₂²×C₅₉	#509	1	1	yes	characteristic
99	C₂²×C₅₉	#510	1	1	yes	characteristic
100	C₄×C₅₉	—	2	2	no	—
101	C₄×C₅₉	—	2	2	no	—
102	C₄×C₅₉	—	8	2	no	—
103	C₄×C₅₉	—	16	1	no	—
104	C₄×C₅₉	#395	1	1	yes	characteristic
105	C₄×C₅₉	#451	1	1	yes	characteristic
106	C₄×C₅₉	#477	1	1	yes	characteristic
107	C₄×C₅₉	#501	1	1	yes	characteristic
108	Dic₅₉	—	8	4	no	—
109	C₂×#500	—	59	1	no	—
110	C₂×#367	—	59	2	no	—
111	#1942	D₅₉	1	1	yes	characteristic
112	#1380	—	59	2	no	—
113	#1374	—	59	1	no	—
114	C₂×C₅₉	#2057	1	1	yes	characteristic
115	C₂×C₅₉	#2185	1	1	yes	characteristic
116	C₂×C₅₉	#2187	1	1	yes	characteristic
117	C₂×C₅₉	#2225	1	1	yes	characteristic
118	C₂×C₅₉	#2226	1	1	yes	characteristic
119	C₂×C₅₉	#2254	1	1	yes	characteristic
120	C₂×C₅₉	#2255	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	—	118	1	no	—
122	C₂²×C₁₆	—	59	2	no	—
123	#500	—	118	4	no	—
124	C₂×#107	—	59	2	no	—
125	C₂×#107	—	118	1	no	—
126	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
127	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
129	#389	—	2	1	no	—
130	#367	—	118	4	no	—
131	#340	—	2	1	no	—
132	C₅₉	#13452	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	236	1	no	—
134	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
136	C₄⋊Q₈	—	118	4	no	—
137	C₄⋊Q₈	—	236	2	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	118	2	no	—
142	C₂×C₄⋊C₄	—	236	1	no	—
143	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
144	C₂×C₁₆	—	59	4	no	—
145	C₂×C₁₆	—	118	2	no	—
146	C₂×C₁₆	—	118	2	no	—
147	#107	—	2	1	no	—
148	#107	—	2	2	no	—
149	#107	—	59	4	no	—
150	#107	—	118	4	no	—
151	#107	—	118	2	no	—
152	#107	Dic₁₁₈	1	2	yes	complemented
153	C₄×C₈	—	2	2	no	—
154	#95	—	4	1	no	—
155	C₂×Q₈	—	472	2	no	—
156	C₂×Q₈	—	472	2	no	—
157	C₂×Q₈	—	236	4	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	236	2	no	—
161	C₂²×C₄	#11745669	1	1	yes	characteristic
162	C₂²×C₄	#11745669	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	#11745672	1	2	yes	—
167	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
168	C₂×C₈	#11745685	1	1	yes	characteristic
169	C₂×C₈	D₄×D₅₉	1	2	yes	—
170	C₂×C₈	Q₈×D₅₉	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	8	2	no	—
176	C₄⋊C₄	—	118	4	no	—
177	C₄⋊C₄	—	236	4	no	—
178	C₄⋊C₄	—	236	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₁₆	—	118	4	no	—
182	C₂³	C₂×#11745669	1	1	yes	characteristic
183	Q₈	—	472	8	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	8	1	no	—
190	C₂×C₄	—	8	1	no	—
191	C₂×C₄	—	8	1	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	472	2	no	—
195	C₂×C₄	—	472	2	no	—
196	C₂×C₄	—	236	4	no	—
197	C₂×C₄	#14273670	1	2	yes	—
198	C₂×C₄	#14273671	1	2	yes	—
199	C₂×C₄	#14273695	1	1	yes	characteristic
200	C₂×C₄	#14273706	1	1	yes	characteristic
201	C₂×C₄	#14273725	1	1	yes	characteristic
202	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
203	C₂×C₄	#14273752	1	1	yes	characteristic
204	C₂×C₄	#14273757	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	#14273705	1	2	yes	—
207	C₈	#14273726	1	2	yes	—
208	C₂²	#16837535	1	1	yes	characteristic
209	C₂²	#16837682	1	1	yes	characteristic
210	C₂²	#16837805	1	1	yes	characteristic
211	C₂²	C₂×#14273670	1	1	yes	characteristic
212	C₂²	C₂×#14273671	1	1	yes	characteristic
213	C₂²	#16837837	1	1	yes	characteristic
214	C₂²	#16837842	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	8	2	no	—
218	C₄	—	16	1	no	—
219	C₄	—	472	4	no	—
220	C₄	#16837322	1	1	yes	characteristic
221	C₄	#16837557	1	1	yes	characteristic
222	C₄	#16837697	1	1	yes	characteristic
223	C₄	#16837812	1	1	yes	characteristic
224	C₂	#21425424	1	1	yes	characteristic
225	C₂	#21426012	1	1	yes	characteristic
226	C₂	#21426024	1	1	yes	characteristic
227	C₂	#21426238	1	1	yes	characteristic
228	C₂	#21426243	1	1	yes	characteristic
229	C₂	#21426376	1	1	yes	characteristic
230	C₂	#21426383	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator, cosocle	C₂²×C₈×C₅₉	C₂³
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1942	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	#1942	D₅₉
hypocenter	C₅₉	#13452
LCS	G ▷ C₂²×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
24. C₂×C₅₉⋊C₁₆	Q₈	semidirect
36. #107×C₅₉	Q₈	semidirect
132. C₅₉	#13452	semidirect
152. #107	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×#16837229	2
#21422187	2
#21422174	2
C₅₉×#1942	2
#13452	59