Subgroups — #28504886

Order 15104.

358 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837232	C₂	1	1	yes	maximal, characteristic, complemented
2	#21425880	C₂	1	1	yes	maximal, characteristic, complemented
3	#21424150	C₂	1	2	yes	maximal
4	#21422211	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273663	C₂²	1	2	yes	—
7	#16837440	C₂²	1	2	yes	—
8	#16837419	—	2	2	no	—
9	#16837232	—	2	1	no	—
10	#16837232	—	2	1	no	—
11	#16837230	—	2	1	no	—
12	#16837228	—	2	1	no	—
13	#16837174	—	2	2	no	—
14	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉×#504	—	2	1	no	—
16	C₅₉×#504	—	2	1	no	—
17	C₅₉×#504	C₄	1	2	yes	complemented
18	C₅₉×#504	C₄	1	2	yes	complemented
19	C₂×M₃₂×C₅₉	—	2	1	no	—
20	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₉	—	2	1	no	—
22	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×#11745664	—	2	2	no	—
24	#14273686	—	2	2	no	—
25	#14273663	—	2	2	no	—
26	#14273663	—	2	2	no	—
27	#14273661	—	2	2	no	—
28	#14273659	—	2	2	no	—
29	#14273654	—	4	4	no	—
30	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
31	#131×C₅₉	—	2	1	no	—
32	#131×C₅₉	—	2	1	no	—
33	#131×C₅₉	C₂×C₄	1	2	yes	—
34	#131×C₅₉	Q₈	1	2	yes	—
35	C₂×M₁₆×C₅₉	—	2	1	no	—
36	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	—	2	1	no	—
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	M₃₂×C₅₉	—	2	1	no	—
40	M₃₂×C₅₉	—	2	2	no	—
41	M₃₂×C₅₉	—	2	2	no	—
42	M₃₂×C₅₉	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₅₉	—	2	1	no	—
44	C₂×C₁₆×C₅₉	—	2	2	no	—
45	C₂×C₁₆×C₅₉	—	2	2	no	—
46	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
49	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
52	#41×C₅₉	—	2	1	no	—
53	#41×C₅₉	—	2	1	no	—
54	#41×C₅₉	C₄⋊C₄	1	2	yes	—
55	#41×C₅₉	D₈	1	2	yes	—
56	C₂×Q₈×C₅₉	Q₁₆	1	1	yes	characteristic
57	C₂×D₄×C₅₉	—	2	1	no	—
58	C₂×D₄×C₅₉	Q₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	—	2	1	no	—
60	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	M₁₆×C₅₉	—	2	1	no	—
62	M₁₆×C₅₉	—	2	2	no	—
63	M₁₆×C₅₉	—	2	2	no	—
64	M₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₉	—	2	1	no	—
66	C₂×C₈×C₅₉	—	2	2	no	—
67	C₂×C₈×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
74	C₂²⋊Dic₅₉	—	4	2	no	—
75	C₄⋊Dic₅₉	—	4	2	no	—
76	C₄×Dic₅₉	—	4	2	no	—
77	#11745664	—	4	2	no	—
78	#11745664	—	4	2	no	—
79	C₂×C₅₉⋊C₈	—	4	2	no	—
80	C₁₆×C₅₉	—	2	2	no	—
81	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₁₆×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂³×C₅₉	—	2	1	no	—
86	C₂³×C₅₉	#135	1	1	yes	characteristic
87	Q₈×C₅₉	—	2	1	no	—
88	Q₈×C₅₉	#107	1	2	yes	—
89	D₄×C₅₉	—	2	2	no	—
90	D₄×C₅₉	—	2	2	no	—
91	D₄×C₅₉	—	2	1	no	—
92	D₄×C₅₉	#107	1	2	yes	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
102	C₂×Dic₅₉	—	8	2	no	—
103	C₈×C₅₉	—	2	2	no	—
104	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₅₉	#117	1	1	yes	characteristic
106	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₅₉⋊C₈	—	8	2	no	—
109	#12841	—	59	1	no	maximal, Sylow
110	C₂²×C₅₉	—	2	2	no	—
111	C₂²×C₅₉	—	4	1	no	—
112	C₂²×C₅₉	—	4	2	no	—
113	C₂²×C₅₉	#427	1	1	yes	characteristic
114	C₂²×C₅₉	#466	1	1	yes	characteristic
115	C₂²×C₅₉	#467	1	1	yes	characteristic
116	C₄×C₅₉	—	2	2	no	—
117	C₄×C₅₉	#380	1	1	yes	characteristic
118	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
119	C₄×C₅₉	#466	1	1	yes	characteristic
120	C₄×C₅₉	#467	1	1	yes	characteristic
121	Dic₅₉	—	16	2	no	—
122	C₂×#504	D₅₉	1	1	yes	characteristic
123	C₂×#368	—	59	1	no	—
124	#2158	—	59	1	no	—
125	#1814	—	59	2	no	—
126	#1389	—	59	2	no	—
127	C₂×C₅₉	—	2	1	no	—
128	C₂×C₅₉	—	2	1	no	—
129	C₂×C₅₉	—	2	1	no	—
130	C₂×C₅₉	—	4	2	no	—
131	C₂×C₅₉	#1808	1	1	yes	characteristic
132	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
133	#504	—	2	1	no	—
134	#504	—	2	1	no	—
135	#504	Dic₅₉	1	2	yes	complemented
136	#504	Dic₅₉	1	2	yes	complemented
137	C₂×M₃₂	—	2	1	no	—
138	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
139	C₂²×C₁₆	—	2	1	no	—
140	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
141	C₂×#108	—	59	2	no	—
142	#427	—	59	2	no	—
143	#421	—	118	2	no	—
144	#368	—	118	1	no	—
145	#368	—	118	1	no	—
146	#367	—	118	1	no	—
147	#366	—	118	1	no	—
148	#344	—	118	2	no	—
149	C₅₉	#12841	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	D₂₃₆	1	1	yes	characteristic
151	#131	—	2	1	no	—
152	#131	—	2	1	no	—
153	#131	Dic₁₁₈	1	2	yes	—
154	#131	C₂×Dic₅₉	1	2	yes	—
155	C₂×M₁₆	—	2	1	no	—
156	C₂×M₁₆	—	118	2	no	—
157	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
160	#117	—	118	2	no	—
161	M₃₂	—	2	1	no	—
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	C₂×Dic₅₉	1	2	yes	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
169	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
170	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
171	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
172	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
173	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
174	#108	—	118	2	no	—
175	#108	—	118	2	no	—
176	#107	—	118	2	no	—
177	#106	—	118	2	no	—
178	#104	—	236	4	no	—
179	#41	—	2	1	no	—
180	#41	—	2	1	no	—
181	#41	D₄₇₂	1	2	yes	—
182	#41	C₄⋊Dic₅₉	1	2	yes	—
183	C₂×Q₈	Dic₂₃₆	1	1	yes	characteristic
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	Dic₂₃₆	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
188	M₁₆	—	2	1	no	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	236	2	no	—
192	M₁₆	—	236	2	no	—
193	M₁₆	C₄⋊Dic₅₉	1	2	yes	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	236	2	no	—
198	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
199	C₂×C₈	#11745685	1	1	yes	characteristic
200	C₂×C₈	#11745685	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
204	C₄⋊C₄	—	236	2	no	—
205	C₂²⋊C₄	—	236	2	no	—
206	C₄²	—	236	2	no	—
207	C₁₆	—	2	2	no	—
208	C₁₆	C₂²⋊Dic₅₉	1	1	yes	characteristic
209	C₁₆	C₂²⋊Dic₅₉	1	1	yes	characteristic
210	C₁₆	C₂²⋊Dic₅₉	1	1	yes	characteristic
211	C₁₆	C₂²⋊Dic₅₉	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	#14273749	1	1	yes	characteristic
214	Q₈	—	2	1	no	—
215	Q₈	#14273661	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	#14273661	1	2	yes	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	472	2	no	—
224	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
225	C₂×C₄	#14273701	1	1	yes	characteristic
226	C₂×C₄	#14273701	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
228	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
229	C₂×C₄	#14273749	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	472	2	no	—
232	C₈	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
233	C₈	#14273686	1	1	yes	characteristic
234	C₈	#14273706	1	1	yes	characteristic
235	C₈	#14273706	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	#16837440	1	1	yes	characteristic
240	C₂²	#16837638	1	1	yes	characteristic
241	C₂²	#16837643	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	944	2	no	—
244	C₄	#16837262	1	1	yes	characteristic
245	C₄	C₂×#14273661	1	1	yes	characteristic
246	C₄	#16837638	1	1	yes	characteristic
247	C₄	#16837643	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	4	2	no	—
252	C₂	#21424125	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837638
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#504	C₂
socle	C₂×C₅₉	#1808
prime radical	C₂×#41×C₅₉	D₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#504	D₅₉
hypocenter	C₅₉	#12841
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16837232	C₂	semidirect
2. #21425880	C₂	semidirect
4. #21422211	C₂	semidirect
17. C₅₉×#504	C₄	semidirect
18. C₅₉×#504	C₄	semidirect
135. #504	Dic₅₉	semidirect
136. #504	Dic₅₉	semidirect
149. C₅₉	#12841	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837232	2
#21425880	2
#21424150	2
#21422211	2
C₂×C₅₉×#504	2
#12841	59