Subgroups — #28559222

Order 15104.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434682	C₂	1	1	yes	maximal, characteristic
2	#21433030	C₂	1	2	yes	maximal
3	#21422994	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₂₃₆	C₂	1	4	yes	maximal, complemented
5	#21422855	C₂	1	1	yes	maximal, characteristic
6	#21422552	C₂	1	2	yes	maximal
7	C₂×C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
8	#16837959	C₂²	1	2	yes	—
9	#16837807	C₄	1	1	yes	characteristic
10	#16837782	C₄	1	1	yes	characteristic
11	#16837753	C₄	1	2	yes	—
12	#16837643	C₄	1	4	yes	—
13	#16837532	—	2	2	no	—
14	#16837532	C₂²	1	4	yes	—
15	C₈×Dic₁₁₈	—	2	2	no	—
16	C₈×Dic₁₁₈	C₂²	1	4	yes	—
17	C₄×Dic₂₃₆	C₂²	1	4	yes	—
18	#16837428	—	2	2	no	—
19	#16837426	C₂²	1	2	yes	—
20	#16837357	—	2	2	no	—
21	#16837155	C₂²	1	4	yes	—
22	#16837134	C₂²	1	8	yes	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₈²×C₅₉	C₂²	1	2	yes	—
27	C₈²×C₅₉	C₂²	1	2	yes	—
28	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
29	C₂×#11745664	—	2	2	no	—
30	#14273727	C₂×C₄	1	2	yes	—
31	#14273727	C₂×C₄	1	2	yes	—
32	#14273719	—	2	2	no	—
33	#14273706	—	2	2	no	—
34	#14273706	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₁₈	—	2	2	no	—
36	C₄×Dic₁₁₈	C₂³	1	4	yes	—
37	#14273686	—	2	2	no	—
38	#14273661	C₂×C₄	1	4	yes	—
39	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
40	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
41	Dic₅₉⋊C₈	—	2	4	no	—
42	Dic₅₉⋊C₈	—	2	4	no	—
43	#14273652	C₂×C₄	1	8	yes	—
44	#14273637	—	2	4	no	—
45	C₈×Dic₅₉	—	2	4	no	—
46	C₂²×C₈×C₅₉	—	2	1	no	—
47	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₅₉	—	2	1	no	—
52	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
56	C₄×C₈×C₅₉	C₂³	1	2	yes	—
57	C₄×C₈×C₅₉	C₂³	1	2	yes	—
58	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
63	C₂×Dic₁₁₈	—	2	2	no	—
64	C₂×Dic₁₁₈	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₅₉	—	2	4	no	—
66	C₂×C₈×C₅₉	—	2	2	no	—
67	C₂×C₈×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	—	2	4	no	—
69	C₂×C₈×C₅₉	Q₁₆	1	4	yes	complemented
70	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₉	#41	1	1	yes	characteristic
79	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
80	C₂²⋊Dic₅₉	—	2	4	no	—
81	C₄⋊Dic₅₉	—	2	2	no	—
82	C₄⋊Dic₅₉	—	2	2	no	—
83	C₄⋊Dic₅₉	—	2	2	no	—
84	C₄⋊Dic₅₉	C₂²×C₄	1	4	yes	—
85	Dic₅₉⋊C₄	—	2	4	no	—
86	Dic₅₉⋊C₄	—	2	4	no	—
87	C₄×Dic₅₉	—	2	4	no	—
88	#11745664	—	4	4	no	—
89	C₂×C₅₉⋊C₈	—	2	4	no	—
90	Dic₂₃₆	—	2	8	no	—
91	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₅₉	#135	1	2	yes	—
99	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₅₉	—	2	4	no	—
103	C₂×Dic₅₉	—	2	4	no	—
104	Dic₁₁₈	—	2	8	no	—
105	Dic₁₁₈	—	4	4	no	—
106	C₈×C₅₉	—	2	4	no	—
107	C₈×C₅₉	—	2	2	no	—
108	C₈×C₅₉	—	2	1	no	—
109	C₈×C₅₉	—	2	1	no	—
110	C₈×C₅₉	#131	1	4	yes	—
111	C₈×C₅₉	C₂×Q₁₆	1	2	yes	—
112	C₈×C₅₉	C₂×Q₁₆	1	2	yes	—
113	C₅₉⋊C₈	—	4	4	no	—
114	#17958	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	—	2	1	no	—
116	C₂²×C₅₉	—	2	1	no	—
117	C₂²×C₅₉	C₄×Q₁₆	1	2	yes	—
118	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
119	C₄×C₅₉	—	2	1	no	—
120	C₄×C₅₉	—	2	1	no	—
121	C₄×C₅₉	C₄×Q₁₆	1	2	yes	—
122	C₄×C₅₉	#443	1	2	yes	—
123	C₄×C₅₉	C₂×#131	1	2	yes	—
124	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
125	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
126	Dic₅₉	—	4	4	no	—
127	Dic₅₉	—	4	4	no	—
128	#3170	—	59	1	no	—
129	#2960	—	59	2	no	—
130	#1591	—	59	4	no	—
131	C₈×Q₁₆	—	59	4	no	—
132	#1567	—	59	1	no	—
133	#1493	—	59	2	no	—
134	C₂×C₈²	D₅₉	1	1	yes	characteristic
135	C₂×C₅₉	—	2	2	no	—
136	C₂×C₅₉	#2931	1	1	yes	characteristic
137	C₂×C₅₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
138	C₂×C₅₉	C₂×#443	1	1	yes	characteristic
139	#533	—	59	2	no	—
140	#500	—	59	1	no	—
141	#494	—	59	1	no	—
142	#487	—	59	2	no	—
143	#467	—	59	4	no	—
144	#446	—	59	4	no	—
145	#446	—	118	2	no	—
146	C₈×Q₈	—	59	4	no	—
147	C₈×Q₈	—	118	2	no	—
148	C₄×Q₁₆	—	59	4	no	—
149	#424	—	118	2	no	—
150	#423	—	59	2	no	—
151	#405	—	118	2	no	—
152	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
155	#335	—	59	4	no	—
156	Q₈⋊C₈	—	59	8	no	—
157	C₈²	C₂×D₅₉	1	2	yes	—
158	C₈²	C₂×D₅₉	1	2	yes	—
159	C₅₉	#17958	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₁₆	—	59	4	no	—
161	C₂×M₁₆	—	118	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	C₄×D₅₉	1	2	yes	—
164	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
165	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
166	C₄⋊Q₈	—	59	2	no	—
167	C₄⋊Q₈	—	59	2	no	—
168	#125	—	118	2	no	—
169	C₂²⋊Q₈	—	59	4	no	—
170	C₂²⋊Q₈	—	118	2	no	—
171	C₄×Q₈	—	59	4	no	—
172	C₄×Q₈	—	118	2	no	—
173	#117	—	118	2	no	—
174	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
175	#107	—	59	4	no	—
176	C₄⋊C₈	—	59	4	no	—
177	C₄⋊C₈	—	118	4	no	—
178	C₄⋊C₈	—	118	4	no	—
179	C₄⋊C₈	—	118	2	no	—
180	#103	—	59	8	no	—
181	#97	—	118	4	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	118	4	no	—
184	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
185	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
186	C₄×C₈	D₂₃₆	1	1	yes	characteristic
187	C₄×C₈	D₂₃₆	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₅₉	1	2	yes	—
189	C₄×C₈	C₂²×D₅₉	1	2	yes	—
190	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
192	C₂×Q₈	—	59	4	no	—
193	C₂×Q₈	—	118	2	no	—
194	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
196	C₂²×C₄	#11745685	1	1	yes	characteristic
197	Q₁₆	—	118	8	no	—
198	M₁₆	—	236	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	118	4	no	—
204	C₂×C₈	Dic₂₃₆	1	4	yes	complemented
205	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
207	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
208	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
209	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
210	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
211	C₄⋊C₄	—	59	4	no	—
212	C₄⋊C₄	—	118	2	no	—
213	C₄⋊C₄	—	118	2	no	—
214	C₄⋊C₄	—	118	4	no	—
215	C₄⋊C₄	—	118	2	no	—
216	C₄⋊C₄	—	118	4	no	—
217	C₂²⋊C₄	—	118	4	no	—
218	C₄²	—	118	4	no	—
219	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
220	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
221	C₄²	#11745685	1	1	yes	characteristic
222	C₄²	C₂³×D₅₉	1	1	yes	characteristic
223	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
224	Q₈	—	118	8	no	—
225	Q₈	—	236	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	118	4	no	—
229	C₂×C₄	—	118	4	no	—
230	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
231	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
232	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
233	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
234	C₂×C₄	#14273749	1	2	yes	—
235	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
237	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	236	4	no	—
243	C₈	#14273733	1	4	yes	—
244	C₈	C₂×Dic₂₃₆	1	2	yes	—
245	C₈	C₂×Dic₂₃₆	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₄×Dic₂₃₆	1	2	yes	—
249	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	236	4	no	—
253	C₄	—	236	4	no	—
254	C₄	C₄×Dic₂₃₆	1	2	yes	—
255	C₄	#16837520	1	2	yes	—
256	C₄	C₂×#14273733	1	2	yes	—
257	C₄	C₂²×Dic₂₃₆	1	1	yes	characteristic
258	C₄	C₂×#14273749	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#21432809	1	1	yes	characteristic
261	C₂	C₂×C₄×Dic₂₃₆	1	1	yes	characteristic
262	C₂	C₂×#16837520	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₈²×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂×C₈²	D₅₉
hypocenter	C₅₉	#17958
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #21422994	C₂	semidirect
4. C₈×Dic₂₃₆	C₂	semidirect
69. C₂×C₈×C₅₉	Q₁₆	semidirect
159. C₅₉	#17958	semidirect
204. C₂×C₈	Dic₂₃₆	semidirect

Maximal subgroups

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

subgroup	index
#21434682	2
#21433030	2
#21422994	2
C₈×Dic₂₃₆	2
#21422855	2
#21422552	2
C₂×C₈²×C₅₉	2
#17958	59