Subgroups — #28500081

Order 15104.

394 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425895	C₂	1	1	yes	maximal, characteristic
2	#21422855	C₂	1	1	yes	maximal, characteristic
3	#21422145	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic
5	#16837426	—	2	1	no	—
6	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
7	C₂²⋊(C₅₉⋊C₁₆)	—	2	1	no	—
8	C₄×C₅₉⋊C₁₆	—	2	1	no	—
9	#16837155	—	2	1	no	—
10	#16837155	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₅₉	—	2	1	no	—
12	C₂²×Q₁₆×C₅₉	C₄	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
14	C₅₉×#494	—	2	1	no	—
15	C₅₉×#494	—	2	1	no	—
16	C₅₉×#494	C₄	1	2	yes	—
17	C₅₉×#494	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×#11745664	—	4	1	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
23	C₂²×Q₈×C₅₉	—	2	2	no	—
24	C₂×Q₁₆×C₅₉	—	2	2	no	—
25	C₂×Q₁₆×C₅₉	—	2	2	no	—
26	C₂×Q₁₆×C₅₉	—	2	2	no	—
27	C₂×Q₁₆×C₅₉	—	4	4	no	—
28	C₂×Q₁₆×C₅₉	—	4	4	no	—
29	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₅₉	—	2	2	no	—
33	C₄⋊Q₈×C₅₉	—	2	2	no	—
34	C₄⋊Q₈×C₅₉	—	4	1	no	—
35	C₄⋊Q₈×C₅₉	—	4	1	no	—
36	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
37	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₅₉	—	2	4	no	—
44	C₂×Q₈×C₅₉	—	4	4	no	—
45	C₂×Q₈×C₅₉	—	4	4	no	—
46	C₂×Q₈×C₅₉	—	4	2	no	—
47	C₂²×C₄×C₅₉	—	4	2	no	—
48	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₅₉	—	4	4	no	—
52	Q₁₆×C₅₉	—	4	4	no	—
53	Q₁₆×C₅₉	—	4	4	no	—
54	Q₁₆×C₅₉	—	4	4	no	—
55	C₂×C₈×C₅₉	—	2	2	no	—
56	C₂×C₈×C₅₉	—	2	2	no	—
57	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₅₉	—	4	2	no	—
62	C₄⋊C₄×C₅₉	—	4	2	no	—
63	C₄⋊C₄×C₅₉	—	4	2	no	—
64	C₄⋊C₄×C₅₉	—	4	2	no	—
65	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₉	D₈	1	1	yes	characteristic
67	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
68	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
69	#11745664	—	8	2	no	—
70	C₂×C₅₉⋊C₈	—	4	2	no	—
71	C₅₉⋊C₁₆	—	4	2	no	—
72	C₂³×C₅₉	#102	1	1	yes	characteristic
73	Q₈×C₅₉	—	4	4	no	—
74	Q₈×C₅₉	—	4	4	no	—
75	Q₈×C₅₉	—	8	4	no	—
76	C₂×C₄×C₅₉	—	2	2	no	—
77	C₂×C₄×C₅₉	—	2	1	no	—
78	C₂×C₄×C₅₉	—	2	1	no	—
79	C₂×C₄×C₅₉	—	4	4	no	—
80	C₂×C₄×C₅₉	—	8	4	no	—
81	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
85	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
86	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₅₉	—	2	1	no	—
90	C₈×C₅₉	—	2	1	no	—
91	C₈×C₅₉	#100	1	2	yes	—
92	C₈×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
93	C₅₉⋊C₈	—	8	2	no	—
94	#11962	—	59	1	no	maximal, Sylow
95	C₂²×C₅₉	—	2	1	no	—
96	C₂²×C₅₉	—	2	1	no	—
97	C₂²×C₅₉	#358	1	2	yes	—
98	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
99	C₄×C₅₉	—	2	1	no	—
100	C₄×C₅₉	—	2	1	no	—
101	C₄×C₅₉	—	8	4	no	—
102	C₄×C₅₉	#358	1	2	yes	—
103	C₄×C₅₉	#362	1	2	yes	—
104	C₄×C₅₉	C₂×#100	1	1	yes	characteristic
105	C₄×C₅₉	#418	1	1	yes	characteristic
106	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
107	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
108	C₂×#494	D₅₉	1	1	yes	characteristic
109	#2162	—	59	1	no	—
110	#1567	—	59	1	no	—
111	#1362	—	59	4	no	—
112	C₂×C₅₉	—	2	2	no	—
113	C₂×C₅₉	#1545	1	1	yes	characteristic
114	C₂×C₅₉	C₂×#358	1	1	yes	characteristic
115	C₂×C₅₉	C₂×#362	1	1	yes	characteristic
116	C₂²×Q₁₆	—	2	1	no	—
117	C₂²×Q₁₆	Dic₅₉	1	2	yes	—
118	C₂×C₄⋊Q₈	—	2	1	no	—
119	#494	—	2	1	no	—
120	#494	—	2	1	no	—
121	#494	Dic₅₉	1	2	yes	—
122	#494	C₂×D₅₉	1	2	yes	—
123	#423	—	118	1	no	—
124	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
125	C₄⋊C₁₆	—	59	2	no	—
126	C₂²⋊C₁₆	—	118	1	no	—
127	C₄×C₁₆	—	118	1	no	—
128	#335	—	59	2	no	—
129	#335	—	118	1	no	—
130	C₅₉	#11962	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₈	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	4	4	no	—
136	C₂×Q₁₆	—	4	4	no	—
137	C₂×Q₁₆	C₂×Dic₅₉	1	4	yes	—
138	C₂×M₁₆	—	236	1	no	—
139	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	4	1	no	—
144	C₄⋊Q₈	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄⋊C₄	—	4	1	no	—
147	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
148	C₂×C₁₆	—	118	2	no	—
149	C₄⋊C₈	—	118	2	no	—
150	C₄⋊C₈	—	236	1	no	—
151	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
154	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	#11745669	1	1	yes	characteristic
161	C₂²×C₄	#11745670	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
163	Q₁₆	—	4	4	no	—
164	Q₁₆	—	4	4	no	—
165	Q₁₆	—	4	4	no	—
166	Q₁₆	—	4	4	no	—
167	M₁₆	—	472	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	236	2	no	—
171	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
173	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
174	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄²	#11745669	1	1	yes	characteristic
180	C₄²	#11745670	1	1	yes	characteristic
181	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
183	C₁₆	—	236	2	no	—
184	C₂³	#14273648	1	1	yes	characteristic
185	Q₈	—	4	4	no	—
186	Q₈	—	4	4	no	—
187	Q₈	—	8	4	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#14273648	1	1	yes	characteristic
194	C₂×C₄	#14273648	1	1	yes	characteristic
195	C₂×C₄	#14273648	1	1	yes	characteristic
196	C₂×C₄	#14273670	1	2	yes	—
197	C₂×C₄	#14273673	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
199	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
200	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	472	2	no	—
204	C₈	#14273644	1	2	yes	—
205	C₈	#14273753	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16837206	1	2	yes	—
209	C₂²	C₂×#14273648	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	4	no	—
213	C₄	#16837206	1	2	yes	—
214	C₄	#16837218	1	2	yes	—
215	C₄	C₂×#14273644	1	1	yes	characteristic
216	C₄	#16837407	1	1	yes	characteristic
217	C₄	C₂×#14273670	1	1	yes	characteristic
218	C₄	C₂×#14273673	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#21422764	1	1	yes	characteristic
221	C₂	C₂×#16837206	1	1	yes	characteristic
222	C₂	C₂×#16837218	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14273648
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#494	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	#102
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×#494	D₅₉
hypocenter	C₅₉	#11962
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #21422145	C₂	semidirect
130. C₅₉	#11962	semidirect

Maximal subgroups

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

subgroup	index
#21425895	2
#21422855	2
#21422145	2
C₂×C₅₉×#494	2
#11962	59