Subgroups — #108×C₅₉⋊C₈ #28479983

Order 15104.

374 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422844	C₂	1	1	yes	maximal, characteristic
2	M₁₆×C₅₉⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#21421934	C₂	1	2	yes	maximal
5	C₄×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	#16837155	C₄	1	2	yes	—
10	#16837153	C₄	1	2	yes	—
11	#16837123	—	2	2	no	—
12	C₈×C₅₉⋊C₈	—	2	2	no	—
13	C₈×C₅₉⋊C₈	C₄	1	2	yes	—
14	C₈×C₅₉⋊C₈	C₄	1	2	yes	—
15	C₂×#108×C₅₉	C₄	1	2	yes	—
16	C₄×M₁₆×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#341	C₄	1	2	yes	—
19	C₂²×C₅₉⋊C₈	—	2	1	no	—
20	C₂²×C₅₉⋊C₈	Q₈	1	2	yes	—
21	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
22	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄×C₅₉⋊C₈	—	2	2	no	—
24	C₄×C₅₉⋊C₈	—	2	1	no	—
25	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
28	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
29	C₂×M₁₆×C₅₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
33	#108×C₅₉	C₈	1	8	yes	complemented
34	#97×C₅₉	—	2	2	no	—
35	C₄×C₈×C₅₉	—	2	2	no	—
36	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
43	M₁₆×C₅₉	—	2	4	no	—
44	M₁₆×C₅₉	C₂×C₈	1	8	yes	—
45	C₂×C₈×C₅₉	—	2	4	no	—
46	C₂×C₈×C₅₉	—	2	2	no	—
47	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
52	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₅₉	#41	1	1	yes	characteristic
56	C₂×C₅₉⋊C₈	—	2	4	no	—
57	C₂×C₅₉⋊C₈	—	2	2	no	—
58	C₂×C₅₉⋊C₈	—	2	4	no	—
59	C₂×C₅₉⋊C₈	—	2	4	no	—
60	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
61	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
62	C₂³×C₅₉	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	2	2	no	—
65	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₅₉	#131	1	2	yes	—
73	C₈×C₅₉	—	2	1	no	—
74	C₈×C₅₉	—	2	1	no	—
75	C₈×C₅₉	—	2	8	no	—
76	C₈×C₅₉	C₄×C₈	1	2	yes	—
77	C₈×C₅₉	C₄×C₈	1	2	yes	—
78	C₅₉⋊C₈	—	2	4	no	—
79	C₅₉⋊C₈	—	2	2	no	—
80	C₅₉⋊C₈	—	4	4	no	—
81	C₅₉⋊C₈	#108	1	4	yes	direct product factor
82	C₈×#108	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	2	1	no	—
84	C₂²×C₅₉	—	2	1	no	—
85	C₂²×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₅₉	C₈×Q₈	1	2	yes	—
87	C₄×C₅₉	—	2	1	no	—
88	C₄×C₅₉	—	2	1	no	—
89	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₅₉	#405	1	1	yes	characteristic
91	C₄×C₅₉	C₂×#108	1	2	yes	—
92	C₄×C₅₉	C₈×D₄	1	2	yes	—
93	C₄×C₅₉	#443	1	2	yes	—
94	C₄×#108	D₅₉	1	1	yes	characteristic
95	#1564	—	59	1	no	—
96	C₈×M₁₆	—	59	2	no	—
97	C₂×C₈²	—	59	1	no	—
98	#1278	—	59	2	no	—
99	C₂×C₅₉	—	2	2	no	—
100	C₂×C₅₉	C₈×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₅₉	#1772	1	1	yes	characteristic
102	C₂×C₅₉	C₄×#108	1	1	yes	characteristic
103	#431	—	59	2	no	—
104	C₂×#108	Dic₅₉	1	2	yes	—
105	C₄×M₁₆	C₂×D₅₉	1	2	yes	—
106	C₂×C₄×C₈	—	59	1	no	—
107	C₂×C₄×C₈	—	59	1	no	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	#341	Dic₅₉	1	2	yes	—
110	#335	—	59	2	no	—
111	#334	—	59	2	no	—
112	#322	—	118	2	no	—
113	C₈²	—	59	2	no	—
114	C₈²	—	59	2	no	—
115	C₈²	—	118	2	no	—
116	C₅₉	C₈×#108	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	C₂×Dic₅₉	1	4	yes	—
118	C₂²×C₈	—	59	2	no	—
119	C₂²×C₈	—	118	1	no	—
120	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
123	#108	C₅₉⋊C₈	1	8	yes	direct product factor
124	C₄⋊C₈	—	59	4	no	—
125	C₂²⋊C₈	—	118	2	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	59	2	no	—
129	C₄×C₈	—	59	2	no	—
130	C₄×C₈	—	59	1	no	—
131	C₄×C₈	—	59	1	no	—
132	C₄×C₈	—	118	2	no	—
133	C₄×C₈	—	118	1	no	—
134	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
135	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
136	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
137	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
138	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
139	C₂²×C₄	#11745689	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
141	M₁₆	—	2	4	no	—
142	M₁₆	C₂×C₅₉⋊C₈	1	8	yes	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	59	2	no	—
146	C₂×C₈	—	59	2	no	—
147	C₂×C₈	—	118	4	no	—
148	C₂×C₈	—	118	4	no	—
149	C₂×C₈	—	118	2	no	—
150	C₂×C₈	—	118	4	no	—
151	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
152	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
155	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
157	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
158	C₄²	D₄×D₅₉	1	1	yes	characteristic
159	C₄²	#11745687	1	1	yes	characteristic
160	C₂³	Q₈×Dic₅₉	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
165	C₂×C₄	#14273685	1	1	yes	characteristic
166	C₂×C₄	#14273689	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
168	C₂×C₄	#14273692	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
170	C₂×C₄	#14273734	1	2	yes	—
171	C₈	—	2	8	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	59	4	no	—
175	C₈	—	118	4	no	—
176	C₈	—	118	2	no	—
177	C₈	—	236	4	no	—
178	C₈	C₄×C₅₉⋊C₈	1	2	yes	—
179	C₈	C₄×C₅₉⋊C₈	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
183	C₂²	Q₈×C₅₉⋊C₈	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
187	C₄	#16837358	1	1	yes	characteristic
188	C₄	#108×D₅₉	1	2	yes	—
189	C₄	D₄×C₅₉⋊C₈	1	2	yes	—
190	C₄	#16837518	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	C₄⋊C₄×C₅₉⋊C₈	1	1	yes	characteristic
193	C₂	#21423978	1	1	yes	characteristic
194	C₂	#108×Dic₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₉
commutator	C₄×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×#108×C₅₉	C₂
socle	C₂²×C₅₉	C₄×C₄⋊C₄
prime radical	C₂³×C₅₉	C₄×Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×#108	D₅₉
hypocenter	C₅₉	C₈×#108
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
33. #108×C₅₉	C₈	semidirect
81. C₅₉⋊C₈	#108	direct (N × H)
116. C₅₉	C₈×#108	semidirect
123. #108	C₅₉⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
#21422844	2
M₁₆×C₅₉⋊C₈	2
C₂×C₈×C₅₉⋊C₈	2
#21421934	2
C₄×#108×C₅₉	2
C₈×#108	59

Subgroups — #108×C59⋊C8 #28479983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #108×C₅₉⋊C₈ #28479983