Subgroups — #28504082

Order 15104.

310 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837231	C₂	1	1	yes	maximal, characteristic
2	#21425878	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837177	C₂	1	1	yes	maximal, characteristic
4	#21423978	C₂	1	1	yes	maximal, characteristic
5	#21423977	C₂	1	1	yes	maximal, characteristic
6	#21422192	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1382	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#14273663	C₂²	1	1	yes	characteristic
10	#16837440	C₂²	1	1	yes	characteristic
11	#16837360	—	2	1	no	—
12	#16837358	—	2	1	no	—
13	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
14	#16837231	C₄	1	4	yes	complemented
15	#16837229	C₄	1	2	yes	complemented
16	#16837227	C₄	1	2	yes	complemented
17	#16837177	C₄	1	4	yes	complemented
18	#16837163	—	2	1	no	—
19	#16837161	—	2	1	no	—
20	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#427	C₂²	1	1	yes	characteristic
23	C₂×#11745664	—	2	1	no	—
24	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
25	#14273686	—	2	1	no	—
26	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
27	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
31	#14273663	—	2	1	no	—
32	#14273663	C₂×C₄	1	2	yes	—
33	#14273661	C₂×C₄	1	2	yes	—
34	#14273659	—	2	1	no	—
35	#14273638	—	4	1	no	—
36	#14273637	—	4	1	no	—
37	C₄×C₅₉⋊C₈	—	4	1	no	—
38	C₈×Dic₅₉	—	4	1	no	—
39	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
40	C₈×Dic₅₉	Q₈	1	2	yes	—
41	C₂×M₁₆×C₅₉	—	2	1	no	—
42	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	#117×C₅₉	—	2	1	no	—
44	C₂×C₁₆×C₅₉	—	2	1	no	—
45	C₂×C₁₆×C₅₉	—	2	1	no	—
46	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
47	#108×C₅₉	—	2	1	no	—
48	#108×C₅₉	C₂×C₄	1	2	yes	—
49	#107×C₅₉	C₂×C₄	1	2	yes	—
50	#106×C₅₉	—	2	1	no	—
51	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅₉	—	2	1	no	—
53	M₁₆×C₅₉	—	2	2	no	—
54	M₁₆×C₅₉	—	4	1	no	—
55	C₂×C₈×C₅₉	—	4	1	no	—
56	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
61	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₅₉	—	2	2	no	—
63	C₂²⋊C₄×C₅₉	—	4	1	no	—
64	C₄²×C₅₉	—	4	1	no	—
65	C₂²⋊Dic₅₉	—	4	1	no	—
66	C₄⋊Dic₅₉	—	2	2	no	—
67	C₄×Dic₅₉	—	2	1	no	—
68	C₄×Dic₅₉	—	4	1	no	—
69	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
70	#11745664	—	2	2	no	—
71	#11745664	—	4	1	no	—
72	C₂×C₅₉⋊C₈	—	4	1	no	—
73	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
74	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
75	C₂×C₅₉⋊C₈	D₈	1	1	yes	characteristic
76	C₂×C₅₉⋊C₈	Q₁₆	1	2	yes	—
77	C₁₆×C₅₉	—	2	2	no	—
78	C₅₉⋊C₁₆	C₄²	1	2	yes	—
79	C₅₉⋊C₁₆	C₄²	1	2	yes	—
80	C₂³×C₅₉	#135	1	1	yes	characteristic
81	C₂×C₄×C₅₉	—	4	2	no	—
82	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
88	C₂×Dic₅₉	—	2	2	no	—
89	C₂×Dic₅₉	—	4	2	no	—
90	C₂×Dic₅₉	—	4	2	no	—
91	C₈×C₅₉	—	4	2	no	—
92	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
93	C₈×C₅₉	#117	1	1	yes	characteristic
94	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
95	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
96	C₅₉⋊C₈	—	4	2	no	—
97	C₅₉⋊C₈	#107	1	2	yes	—
98	C₅₉⋊C₈	#107	1	2	yes	—
99	#12671	—	59	1	no	maximal, Sylow
100	C₂²×C₅₉	—	2	2	no	—
101	C₂²×C₅₉	#427	1	1	yes	characteristic
102	C₂²×C₅₉	C₄×D₈	1	1	yes	characteristic
103	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
104	C₄×C₅₉	—	8	2	no	—
105	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
107	C₄×C₅₉	C₄×D₈	1	1	yes	characteristic
108	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
109	Dic₅₉	—	4	2	no	—
110	Dic₅₉	—	8	2	no	—
111	C₂×#368	—	59	1	no	—
112	#2158	—	59	1	no	—
113	C₂×#346	—	59	1	no	—
114	#1772	—	59	1	no	—
115	#1772	—	59	1	no	—
116	#1382	—	59	1	no	—
117	#1382	D₅₉	1	1	yes	characteristic
118	C₂×C₅₉	—	2	1	no	—
119	C₂×C₅₉	—	2	1	no	—
120	C₂×C₅₉	C₄×#107	1	1	yes	characteristic
121	C₂×C₅₉	#2179	1	2	yes	—
122	C₂²×C₁₆	—	59	1	no	—
123	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
124	C₂×#108	—	59	1	no	—
125	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
126	#427	—	59	1	no	—
127	#427	C₂×D₅₉	1	1	yes	characteristic
128	#405	—	118	1	no	—
129	#405	—	118	1	no	—
130	C₂×C₄×C₈	—	59	1	no	—
131	#368	—	59	4	no	—
132	#367	—	59	2	no	—
133	#366	—	59	2	no	—
134	#346	—	59	4	no	—
135	#339	—	118	1	no	—
136	#339	—	118	1	no	—
137	C₅₉	#12671	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	—	2	1	no	—
139	C₂×M₁₆	—	118	1	no	—
140	C₂²×C₈	—	59	1	no	—
141	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
142	#117	—	2	1	no	—
143	#117	—	118	1	no	—
144	C₂×C₄²	—	59	1	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	—	2	1	no	—
147	C₂×C₁₆	—	59	2	no	—
148	C₂×C₁₆	—	59	1	no	—
149	C₂×C₁₆	—	59	1	no	—
150	C₂×C₁₆	—	59	2	no	—
151	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
152	#108	—	2	1	no	—
153	#108	—	59	2	no	—
154	#108	—	118	1	no	—
155	#108	C₄×D₅₉	1	2	yes	—
156	#107	—	59	2	no	—
157	#107	C₄×D₅₉	1	2	yes	—
158	#106	—	2	1	no	—
159	#106	—	118	1	no	—
160	#97	—	236	1	no	—
161	#97	—	236	1	no	—
162	C₄×C₈	—	59	2	no	—
163	C₄×C₈	—	59	2	no	—
164	C₄×C₈	—	236	1	no	—
165	C₄×C₈	—	236	1	no	—
166	C₂²×C₄	—	118	1	no	—
167	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	4	1	no	—
170	M₁₆	—	118	2	no	—
171	M₁₆	—	236	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	59	2	no	—
174	C₂×C₈	—	59	1	no	—
175	C₂×C₈	—	59	1	no	—
176	C₂×C₈	—	59	2	no	—
177	C₂×C₈	—	236	1	no	—
178	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
180	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
181	C₂×C₈	#11745687	1	1	yes	characteristic
182	C₂×C₈	#11745687	1	1	yes	characteristic
183	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	118	2	no	—
186	C₂²⋊C₄	—	4	1	no	—
187	C₂²⋊C₄	—	236	1	no	—
188	C₄²	—	4	1	no	—
189	C₄²	—	59	2	no	—
190	C₄²	—	118	1	no	—
191	C₄²	—	236	1	no	—
192	C₁₆	—	2	2	no	—
193	C₁₆	—	59	2	no	—
194	C₁₆	—	59	2	no	—
195	C₂³	#14273748	1	1	yes	characteristic
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	118	2	no	—
198	C₂×C₄	—	236	2	no	—
199	C₂×C₄	—	236	2	no	—
200	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
201	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
202	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
204	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
205	C₂×C₄	#14273748	1	1	yes	characteristic
206	C₈	—	4	2	no	—
207	C₈	—	59	2	no	—
208	C₈	—	59	2	no	—
209	C₈	—	236	2	no	—
210	C₈	C₄²×D₅₉	1	1	yes	characteristic
211	C₈	#14273683	1	1	yes	characteristic
212	C₈	#14273692	1	1	yes	characteristic
213	C₈	#14273692	1	1	yes	characteristic
214	C₂²	—	2	2	no	—
215	C₂²	#16837437	1	1	yes	characteristic
216	C₂²	#16837482	1	1	yes	characteristic
217	C₂²	#16837494	1	1	yes	characteristic
218	C₄	—	8	2	no	—
219	C₄	—	472	2	no	—
220	C₄	—	236	2	no	—
221	C₄	#16837253	1	1	yes	characteristic
222	C₄	#107×D₅₉	1	1	yes	characteristic
223	C₄	#16837482	1	1	yes	characteristic
224	C₄	#16837494	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#21423988	1	1	yes	characteristic
228	C₂	#21425979	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₉
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1382	C₂
socle	C₂²×C₅₉	C₄×D₈
prime radical	C₂³×C₅₉	#135
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1382	D₅₉
hypocenter	C₅₉	#12671
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1382

Complements

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

N	H ≅ G/N	type
14. #16837231	C₄	semidirect
15. #16837229	C₄	semidirect
16. #16837227	C₄	semidirect
17. #16837177	C₄	semidirect
137. C₅₉	#12671	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837231	2
#21425878	2
C₂×#16837177	2
#21423978	2
#21423977	2
#21422192	2
C₅₉×#1382	2
#12671	59