Subgroups — #28743193

Order 15104.

650 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#21433751	C₂	1	2	yes	maximal
3	#21426204	C₂	1	4	yes	maximal, complemented
4	#21426187	C₂	1	4	yes	maximal, complemented
5	#21425880	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425817	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₉×#504	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
9	#16837973	—	2	2	no	—
10	C₂×Dic₄₇₂	—	2	4	no	—
11	C₂×Dic₄₇₂	C₂²	1	2	yes	—
12	C₂×Dic₄₇₂	C₂²	1	2	yes	—
13	#16837660	—	2	4	no	—
14	#16837643	C₂²	1	8	yes	—
15	#16837440	C₂²	1	2	yes	—
16	#16837412	—	2	2	no	—
17	#16837230	C₂²	1	2	yes	—
18	#16837228	C₂²	1	2	yes	—
19	#16837207	C₂²	1	8	yes	—
20	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#504	—	2	4	no	—
22	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
23	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₁₈	—	2	2	no	—
27	C₂×Dic₂₃₆	—	2	4	no	—
28	C₂×Dic₂₃₆	—	2	4	no	—
29	C₂×Dic₂₃₆	—	2	2	no	—
30	C₂×Dic₂₃₆	C₂³	1	4	yes	—
31	#14273725	—	4	2	no	—
32	#14273714	—	4	2	no	—
33	#14273708	—	4	4	no	—
34	#14273706	—	2	8	no	—
35	#14273686	—	2	2	no	—
36	Dic₄₇₂	—	2	4	no	—
37	Dic₄₇₂	—	2	4	no	—
38	#14273661	C₂³	1	4	yes	—
39	#14273659	—	2	2	no	—
40	#14273652	—	2	8	no	—
41	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
42	#131×C₅₉	—	2	2	no	—
43	#131×C₅₉	—	2	2	no	—
44	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
45	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	M₃₂×C₅₉	—	2	2	no	—
49	M₃₂×C₅₉	—	2	4	no	—
50	C₂×C₁₆×C₅₉	—	2	4	no	—
51	C₂×C₁₆×C₅₉	D₄	1	4	yes	—
52	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
53	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
55	#41×C₅₉	—	2	2	no	—
56	#41×C₅₉	—	2	2	no	—
57	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
58	C₂×D₄×C₅₉	D₈	1	2	yes	—
59	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₅₉	—	4	2	no	—
63	C₂×Dic₁₁₈	—	2	4	no	—
64	C₂×Dic₁₁₈	—	4	4	no	—
65	C₂×Dic₁₁₈	—	4	4	no	—
66	C₂×Dic₁₁₈	—	4	2	no	—
67	M₁₆×C₅₉	—	2	2	no	—
68	M₁₆×C₅₉	—	2	4	no	—
69	C₂×C₈×C₅₉	—	2	2	no	—
70	C₂×C₈×C₅₉	—	2	2	no	—
71	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₅₉	#41	1	2	yes	—
75	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
76	C₂²⋊Dic₅₉	—	4	2	no	—
77	C₂²⋊Dic₅₉	—	4	8	no	—
78	C₄⋊Dic₅₉	—	2	4	no	—
79	Dic₅₉⋊C₄	—	4	8	no	—
80	C₄×Dic₅₉	—	4	2	no	—
81	Dic₂₃₆	—	2	4	no	—
82	Dic₂₃₆	—	2	4	no	—
83	Dic₂₃₆	—	4	4	no	—
84	C₁₆×C₅₉	—	2	2	no	—
85	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
86	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
87	C₂³×C₅₉	C₂×D₈	1	2	yes	—
88	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₅₉	—	2	2	no	—
90	D₄×C₅₉	—	2	4	no	—
91	D₄×C₅₉	—	2	2	no	—
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	#121	1	2	yes	—
95	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#135	1	2	yes	—
98	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₅₉	—	4	4	no	—
100	C₂×Dic₅₉	—	4	4	no	—
101	C₂×Dic₅₉	—	8	4	no	—
102	Dic₁₁₈	—	4	4	no	—
103	Dic₁₁₈	—	4	4	no	—
104	Dic₁₁₈	—	8	4	no	—
105	C₈×C₅₉	—	2	2	no	—
106	C₈×C₅₉	#121	1	2	yes	—
107	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
109	#33608	—	59	1	no	maximal, Sylow
110	C₂²×C₅₉	—	2	2	no	—
111	C₂²×C₅₉	—	2	4	no	—
112	C₂²×C₅₉	—	2	2	no	—
113	C₂²×C₅₉	#466	1	2	yes	—
114	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₅₉	—	2	2	no	—
116	C₄×C₅₉	#466	1	2	yes	—
117	C₄×C₅₉	C₂×#121	1	1	yes	characteristic
118	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
119	Dic₅₉	—	8	4	no	—
120	Dic₅₉	—	8	4	no	—
121	C₂²×Q₃₂	—	59	1	no	—
122	C₂×#504	D₅₉	1	1	yes	characteristic
123	#3059	—	59	2	no	—
124	#2219	—	59	4	no	—
125	#2216	—	59	4	no	—
126	#2158	—	59	1	no	—
127	#2141	—	59	2	no	—
128	C₂×C₅₉	—	2	2	no	—
129	C₂×C₅₉	—	4	2	no	—
130	C₂×C₅₉	C₂×#466	1	1	yes	characteristic
131	C₂×C₅₉	#3417	1	2	yes	—
132	C₂²×Q₁₆	—	59	2	no	—
133	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
134	#536	—	118	2	no	—
135	C₂×Q₃₂	—	59	2	no	—
136	C₂×Q₃₂	—	59	2	no	—
137	C₂×Q₃₂	—	118	4	no	—
138	#504	—	2	4	no	—
139	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
140	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
141	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
142	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
143	#470	—	118	4	no	—
144	#467	—	59	8	no	—
145	#427	—	59	2	no	—
146	#419	—	118	2	no	—
147	#367	—	59	2	no	—
148	#366	—	59	2	no	—
149	#358	—	59	8	no	—
150	C₅₉	#33608	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	D₂₃₆	1	1	yes	characteristic
152	C₂²×Q₈	—	118	2	no	—
153	C₂×Q₁₆	—	59	4	no	—
154	C₂×Q₁₆	—	118	2	no	—
155	C₂×Q₁₆	—	118	4	no	—
156	C₂×Q₁₆	—	118	4	no	—
157	#131	—	2	2	no	—
158	#131	—	2	2	no	—
159	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
160	C₂×M₁₆	D₂₃₆	1	2	yes	—
161	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
162	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
163	C₄⋊Q₈	—	236	2	no	—
164	#124	—	236	2	no	—
165	#123	—	236	4	no	—
166	C₂²⋊Q₈	—	118	8	no	—
167	#117	—	118	2	no	—
168	Q₃₂	—	118	4	no	—
169	Q₃₂	—	118	4	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	4	no	—
172	C₂×C₁₆	—	2	4	no	—
173	C₂×C₁₆	C₂²⋊D₅₉	1	4	yes	—
174	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
175	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
176	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
177	#107	—	59	4	no	—
178	#106	—	118	2	no	—
179	#103	—	118	8	no	—
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	—	118	4	no	—
183	C₂×Q₈	—	236	4	no	—
184	C₂×Q₈	—	236	4	no	—
185	C₂×Q₈	—	236	2	no	—
186	C₂×Q₈	D₄₇₂	1	1	yes	characteristic
187	C₂×D₄	D₄₇₂	1	2	yes	—
188	C₂×D₄	D₄₇₂	1	1	yes	characteristic
189	C₂²×C₄	—	236	2	no	—
190	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
191	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
192	Q₁₆	—	118	4	no	—
193	Q₁₆	—	118	4	no	—
194	Q₁₆	—	236	4	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
200	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
201	C₂×C₈	#11745685	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
203	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
204	C₄⋊C₄	—	118	4	no	—
205	C₄⋊C₄	—	236	8	no	—
206	C₂²⋊C₄	—	236	2	no	—
207	C₂²⋊C₄	—	236	8	no	—
208	C₄²	—	236	2	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	C₂×C₂²⋊D₅₉	1	2	yes	—
211	C₁₆	C₂×C₂²⋊D₅₉	1	2	yes	—
212	C₂³	C₂×D₄₇₂	1	2	yes	—
213	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	472	4	no	—
216	Q₈	—	236	4	no	—
217	Q₈	—	236	4	no	—
218	D₄	—	2	4	no	—
219	D₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	472	4	no	—
223	C₂×C₄	—	236	4	no	—
224	C₂×C₄	—	236	4	no	—
225	C₂×C₄	#14273701	1	2	yes	—
226	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
227	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
228	C₂×C₄	#14273749	1	2	yes	—
229	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	#14273701	1	2	yes	—
232	C₈	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
233	C₈	C₂×#11745685	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	#16837638	1	2	yes	—
238	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	472	4	no	—
241	C₄	—	472	4	no	—
242	C₄	#16837638	1	2	yes	—
243	C₄	C₂×#14273701	1	1	yes	characteristic
244	C₄	C₂×#14273749	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	4	2	no	—
247	C₂	C₂×#16837638	1	1	yes	characteristic
248	C₂	#21437249	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#504	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×#41×C₅₉	D₄
hypercenter	C₂×#504	D₅₉
hypocenter	C₅₉	#33608
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₄₇₂	C₂	semidirect
3. #21426204	C₂	semidirect
4. #21426187	C₂	semidirect
5. #21425880	C₂	semidirect
6. #21425817	C₂	semidirect
150. C₅₉	#33608	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₇₂	2
#21433751	2
#21426204	2
#21426187	2
#21425880	2
#21425817	2
C₂×C₅₉×#504	2
#33608	59