Subgroups — #28741308

Order 15104.

806 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837811	C₂	1	2	yes	maximal
2	C₂×#16837807	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837804	C₂	1	1	yes	maximal, characteristic
4	#21430215	C₂	1	2	yes	maximal
5	#21424264	C₂	1	4	yes	maximal
6	#21424260	C₂	1	2	yes	maximal
7	#21424256	C₂	1	2	yes	maximal
8	C₅₉×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×#14273727	C₂²	1	2	yes	—
10	C₂×#14273719	C₂²	1	2	yes	—
11	#16837811	—	2	8	no	—
12	#16837807	—	2	4	no	—
13	#16837804	—	2	4	no	—
14	C₂×#14273661	C₂²	1	1	yes	characteristic
15	C₂×#14273661	C₂²	1	2	yes	—
16	C₂×#14273661	C₂²	1	4	yes	—
17	C₂×#14273661	C₂²	1	1	yes	characteristic
18	C₂×#14273661	C₂²	1	2	yes	—
19	C₂×#14273659	C₂²	1	4	yes	—
20	C₂×#14273659	C₂²	1	2	yes	—
21	#16837334	—	2	4	no	—
22	#16837323	—	2	4	no	—
23	#16837272	C₂²	1	4	yes	—
24	#16837268	C₂²	1	4	yes	—
25	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#97×C₅₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
29	C₅₉×#376	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₁₈	—	2	2	no	—
31	#14273727	—	2	4	no	—
32	#14273727	—	2	4	no	—
33	#14273719	—	2	4	no	—
34	#14273719	—	2	4	no	—
35	C₂×C₄⋊Dic₅₉	—	2	2	no	—
36	C₂×C₄⋊Dic₅₉	—	2	8	no	—
37	C₂×C₄⋊Dic₅₉	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₅₉	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₅₉	C₂³	1	4	yes	—
40	C₂×Dic₅₉⋊C₄	—	2	2	no	—
41	C₂×Dic₅₉⋊C₄	—	2	2	no	—
42	#14273661	—	2	2	no	—
43	#14273661	—	2	2	no	—
44	#14273661	—	2	4	no	—
45	#14273661	—	2	4	no	—
46	#14273661	—	2	8	no	—
47	#14273659	—	2	4	no	—
48	#14273659	—	2	8	no	—
49	C₅₉⋊#95	—	2	8	no	—
50	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
55	#97×C₅₉	D₄	1	4	yes	—
56	#97×C₅₉	Q₈	1	8	yes	complemented
57	C₄×C₈×C₅₉	D₄	1	4	yes	—
58	C₄×C₈×C₅₉	Q₈	1	8	yes	complemented
59	#95×C₅₉	D₄	1	2	yes	—
60	#95×C₅₉	D₄	1	2	yes	—
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₅₉	—	2	2	no	—
66	C₂²×Dic₅₉	—	2	4	no	—
67	C₂²×Dic₅₉	—	2	2	no	—
68	C₂×Dic₁₁₈	—	4	4	no	—
69	C₂×Dic₁₁₈	—	8	4	no	—
70	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
76	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
77	C₄²×C₅₉	C₂×D₄	1	2	yes	—
78	C₄²×C₅₉	C₂×D₄	1	2	yes	—
79	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
80	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
81	C₄⋊Dic₅₉	—	2	2	no	—
82	C₄⋊Dic₅₉	—	2	2	no	—
83	C₄⋊Dic₅₉	—	2	2	no	—
84	C₄⋊Dic₅₉	—	2	4	no	—
85	C₄⋊Dic₅₉	—	2	2	no	—
86	C₄⋊Dic₅₉	—	2	4	no	—
87	C₄⋊Dic₅₉	—	4	4	no	—
88	C₄⋊Dic₅₉	—	8	8	no	—
89	Dic₅₉⋊C₄	—	4	4	no	—
90	Dic₅₉⋊C₄	—	4	4	no	—
91	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	—	2	4	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	4	no	—
96	C₂×C₄×C₅₉	#127	1	2	yes	—
97	C₂×C₄×C₅₉	#127	1	2	yes	—
98	C₂×C₄×C₅₉	C₄⋊Q₈	1	8	yes	—
99	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
102	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
103	C₂×Dic₅₉	—	4	2	no	—
104	C₂×Dic₅₉	—	4	2	no	—
105	C₂×Dic₅₉	—	4	4	no	—
106	C₂×Dic₅₉	—	4	4	no	—
107	C₂×Dic₅₉	—	4	8	no	—
108	C₂×Dic₅₉	—	4	4	no	—
109	Dic₁₁₈	—	8	8	no	—
110	C₈×C₅₉	C₄⋊Q₈	1	4	yes	—
111	C₈×C₅₉	C₄⋊Q₈	1	4	yes	—
112	C₈×C₅₉	C₄⋊Q₈	1	8	yes	—
113	#33319	—	59	1	no	maximal, Sylow
114	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
115	C₂²×C₅₉	C₂×C₄⋊Q₈	1	2	yes	—
116	C₂²×C₅₉	#576	1	2	yes	—
117	C₂²×C₅₉	#578	1	2	yes	—
118	C₄×C₅₉	—	2	2	no	—
119	C₄×C₅₉	—	2	2	no	—
120	C₄×C₅₉	—	2	8	no	—
121	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₄×C₅₉	C₂×C₄⋊Q₈	1	2	yes	—
124	Dic₅₉	—	8	4	no	—
125	Dic₅₉	—	8	2	no	—
126	Dic₅₉	—	8	2	no	—
127	C₂×#501	—	59	2	no	—
128	C₂×#500	—	59	1	no	—
129	C₂×#499	—	59	1	no	—
130	#2660	—	59	2	no	—
131	#1847	—	59	4	no	—
132	#1846	—	59	2	no	—
133	#1845	—	59	2	no	—
134	#1726	D₅₉	1	1	yes	characteristic
135	C₂×C₅₉	#2868	1	1	yes	characteristic
136	C₂×C₅₉	#3155	1	1	yes	characteristic
137	C₂×C₅₉	#3156	1	1	yes	characteristic
138	C₂×C₅₉	#3164	1	4	yes	—
139	C₂×C₄⋊Q₈	—	59	2	no	—
140	C₂×#125	—	59	2	no	—
141	#501	—	118	8	no	—
142	#500	—	118	4	no	—
143	#499	—	118	4	no	—
144	C₂×#107	—	59	1	no	—
145	C₂×#107	—	59	2	no	—
146	C₂×#107	—	59	4	no	—
147	C₂×#107	—	59	1	no	—
148	C₂×#107	—	59	2	no	—
149	C₂×#106	—	59	4	no	—
150	C₂×#106	—	59	2	no	—
151	C₂×#97	C₂×D₅₉	1	2	yes	—
152	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
154	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
155	#398	—	118	4	no	—
156	#395	—	118	4	no	—
157	#384	—	59	4	no	—
158	#382	—	59	4	no	—
159	#376	C₂×D₅₉	1	1	yes	characteristic
160	C₅₉	#33319	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	118	2	no	—
162	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
164	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
165	C₄⋊Q₈	—	118	4	no	—
166	C₄⋊Q₈	—	118	4	no	—
167	#125	—	118	4	no	—
168	#125	—	118	4	no	—
169	C₂×C₄⋊C₄	—	59	2	no	—
170	C₂×C₄⋊C₄	—	59	2	no	—
171	C₂×C₄⋊C₄	—	59	4	no	—
172	C₂×C₄⋊C₄	—	118	8	no	—
173	C₂×C₄⋊C₄	—	118	2	no	—
174	C₂×C₄⋊C₄	—	118	2	no	—
175	C₂×C₄⋊C₄	—	118	2	no	—
176	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
177	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
178	#107	—	118	2	no	—
179	#107	—	118	2	no	—
180	#107	—	118	4	no	—
181	#107	—	118	4	no	—
182	#107	—	118	8	no	—
183	#106	—	118	4	no	—
184	#106	—	118	8	no	—
185	#97	Dic₁₁₈	1	8	yes	complemented
186	#97	D₂₃₆	1	4	yes	—
187	C₄×C₈	Dic₁₁₈	1	8	yes	complemented
188	C₄×C₈	D₂₃₆	1	4	yes	—
189	#95	—	118	8	no	—
190	#95	D₂₃₆	1	2	yes	—
191	#95	D₂₃₆	1	2	yes	—
192	C₂×Q₈	—	472	4	no	—
193	C₂×Q₈	—	236	4	no	—
194	C₂²×C₄	—	118	2	no	—
195	C₂²×C₄	—	118	4	no	—
196	C₂²×C₄	—	118	2	no	—
197	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₂₃₆	1	4	yes	—
200	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
201	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
202	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
203	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
204	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
205	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
206	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
207	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
208	C₄⋊C₄	—	118	2	no	—
209	C₄⋊C₄	—	118	2	no	—
210	C₄⋊C₄	—	118	2	no	—
211	C₄⋊C₄	—	118	4	no	—
212	C₄⋊C₄	—	118	2	no	—
213	C₄⋊C₄	—	118	4	no	—
214	C₄⋊C₄	—	472	8	no	—
215	C₄⋊C₄	—	236	4	no	—
216	C₄⋊C₄	—	236	4	no	—
217	C₄⋊C₄	—	236	4	no	—
218	C₄²	C₂×Dic₁₁₈	1	4	yes	—
219	C₄²	C₂×Dic₁₁₈	1	4	yes	—
220	C₄²	C₂×D₂₃₆	1	2	yes	—
221	C₄²	C₂×D₂₃₆	1	2	yes	—
222	C₂³	C₂²×D₂₃₆	1	1	yes	characteristic
223	Q₈	—	472	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	236	2	no	—
229	C₂×C₄	—	236	2	no	—
230	C₂×C₄	—	236	4	no	—
231	C₂×C₄	—	236	4	no	—
232	C₂×C₄	—	236	8	no	—
233	C₂×C₄	—	236	4	no	—
234	C₂×C₄	#14273724	1	2	yes	—
235	C₂×C₄	#14273724	1	2	yes	—
236	C₂×C₄	#14273727	1	8	yes	—
237	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
238	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
239	C₂×C₄	C₂²×Dic₁₁₈	1	2	yes	—
240	C₂×C₄	C₂²×Dic₁₁₈	1	2	yes	—
241	C₈	#14273727	1	4	yes	—
242	C₈	#14273727	1	8	yes	—
243	C₈	#14273727	1	4	yes	—
244	C₂²	C₂×#14273724	1	1	yes	characteristic
245	C₂²	C₂×#14273727	1	2	yes	—
246	C₂²	#16838207	1	2	yes	—
247	C₂²	#16838219	1	2	yes	—
248	C₄	—	2	8	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	472	4	no	—
252	C₄	—	472	2	no	—
253	C₄	—	472	2	no	—
254	C₄	C₂×#14273727	1	1	yes	characteristic
255	C₄	C₂×#14273727	1	2	yes	—
256	C₄	C₂×#14273727	1	1	yes	characteristic
257	C₂	#21432494	1	1	yes	characteristic
258	C₂	#21434582	1	1	yes	characteristic
259	C₂	#21434590	1	1	yes	characteristic
260	C₂	#21434635	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₃₆
commutator, cosocle	C₂²×C₄×C₅₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#1726	C₂
socle, prime radical	C₂³×C₅₉	C₂²×D₄
hypercenter	#1726	D₅₉
hypocenter	C₅₉	#33319
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
56. #97×C₅₉	Q₈	semidirect
58. C₄×C₈×C₅₉	Q₈	semidirect
160. C₅₉	#33319	semidirect
185. #97	Dic₁₁₈	semidirect
187. C₄×C₈	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837811	2
C₂×#16837807	2
C₂×#16837804	2
#21430215	2
#21424264	2
#21424260	2
#21424256	2
C₅₉×#1726	2
#33319	59