Subgroups — #28742190

Order 15104.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21426005	C₂	1	8	yes	maximal, complemented
2	#21425855	C₂	1	2	yes	maximal
3	#21425852	C₂	1	1	yes	maximal, characteristic
4	#21425693	C₂	1	2	yes	maximal
5	C₂×#16837178	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#2960	C₂	1	1	yes	maximal, characteristic
7	C₂×#14273667	C₂²	1	2	yes	—
8	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
11	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	8	yes	—
12	#16837178	C₂²	1	4	yes	—
13	#16837178	C₂²	1	4	yes	—
14	C₄×C₅₉⋊C₁₆	C₂²	1	4	yes	—
15	C₅₉×#533	C₄	1	1	yes	characteristic
16	C₅₉×#446	C₄	1	4	yes	—
17	C₈×Q₈×C₅₉	C₂²	1	4	yes	—
18	C₅₉×#424	C₄	1	2	yes	—
19	C₅₉×#423	C₄	1	1	yes	characteristic
20	C₅₉×#405	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	#14273667	—	2	4	no	—
23	C₂×C₅₉⋊C₁₆	—	2	2	no	—
24	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	complemented
25	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
26	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
27	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
28	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₅₉	C₈	1	2	yes	—
32	#125×C₅₉	C₈	1	2	yes	—
33	C₂²⋊Q₈×C₅₉	C₈	1	4	yes	—
34	C₄×Q₈×C₅₉	C₂×C₄	1	4	yes	—
35	#117×C₅₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
38	C₄⋊C₈×C₅₉	C₂×C₄	1	8	yes	—
39	#97×C₅₉	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₉	C₂³	1	4	yes	—
42	C₄×C₈×C₅₉	C₂³	1	2	yes	—
43	C₂×Q₈×C₅₉	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	M₁₆×C₅₉	—	2	4	no	—
47	C₂×C₈×C₅₉	—	2	2	no	—
48	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
50	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₉	#41	1	2	yes	—
53	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
55	C₄⋊C₄×C₅₉	C₂×C₈	1	8	yes	—
56	C₄⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
57	C₂²⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
58	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
61	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
62	C₅₉⋊C₁₆	—	2	2	no	—
63	C₅₉⋊C₁₆	—	2	4	no	—
64	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	—
65	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
66	Q₈×C₅₉	—	4	4	no	—
67	C₂×C₄×C₅₉	—	4	2	no	—
68	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₅₉	C₂²×C₈	1	8	yes	—
72	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₉	#131	1	2	yes	—
74	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
75	C₈×C₅₉	—	2	2	no	—
76	C₈×C₅₉	—	2	4	no	—
77	C₈×C₅₉	C₄×Q₈	1	2	yes	—
78	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
80	#33475	—	59	1	no	maximal, Sylow
81	C₂²×C₅₉	—	2	2	no	—
82	C₂²×C₅₉	C₈×Q₈	1	2	yes	—
83	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
84	C₄×C₅₉	—	2	2	no	—
85	C₄×C₅₉	—	2	2	no	—
86	C₄×C₅₉	—	2	8	no	—
87	C₄×C₅₉	C₈×Q₈	1	2	yes	—
88	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
90	#2960	D₅₉	1	1	yes	characteristic
91	#2183	—	59	8	no	—
92	#2151	—	59	2	no	—
93	#2150	—	59	1	no	—
94	#2108	—	59	2	no	—
95	C₂×#346	—	59	1	no	—
96	C₂×C₅₉	—	2	2	no	—
97	C₂×C₅₉	C₂×C₈×Q₈	1	1	yes	characteristic
98	C₂×C₅₉	#3407	1	2	yes	—
99	#533	Dic₅₉	1	1	yes	characteristic
100	C₂×M₃₂	—	59	2	no	—
101	C₂²×C₁₆	—	59	1	no	—
102	C₂²×C₁₆	—	59	1	no	—
103	#446	Dic₅₉	1	4	yes	—
104	C₈×Q₈	C₂×D₅₉	1	4	yes	—
105	#424	Dic₅₉	1	2	yes	—
106	#423	Dic₅₉	1	1	yes	characteristic
107	#405	C₂×D₅₉	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
109	C₄⋊C₁₆	—	59	4	no	—
110	C₄⋊C₁₆	—	59	8	no	—
111	#346	—	59	4	no	—
112	#346	—	59	4	no	—
113	C₄×C₁₆	—	59	4	no	—
114	C₅₉	#33475	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
116	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
118	C₄⋊Q₈	C₅₉⋊C₈	1	2	yes	—
119	#125	C₅₉⋊C₈	1	2	yes	—
120	C₂²⋊Q₈	C₅₉⋊C₈	1	4	yes	—
121	C₄×Q₈	C₂×Dic₅₉	1	4	yes	—
122	#117	C₂×Dic₅₉	1	2	yes	—
123	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
124	M₃₂	—	118	4	no	—
125	C₂×C₁₆	—	59	4	no	—
126	C₂×C₁₆	—	59	2	no	—
127	C₂×C₁₆	—	59	4	no	—
128	C₂×C₁₆	—	59	2	no	—
129	C₂×C₁₆	—	118	2	no	—
130	C₄⋊C₈	C₂×Dic₅₉	1	4	yes	—
131	C₄⋊C₈	C₂×Dic₅₉	1	8	yes	—
132	#97	C₂×Dic₅₉	1	4	yes	—
133	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
134	C₄×C₈	C₂²×D₅₉	1	4	yes	—
135	C₄×C₈	C₂²×D₅₉	1	2	yes	—
136	C₂×Q₈	C₂×C₅₉⋊C₈	1	4	yes	—
137	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	Q₈×D₅₉	1	2	yes	—
142	C₂×C₈	#11745689	1	2	yes	—
143	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
144	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
145	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
146	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
147	C₄⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
148	C₄⋊C₄	C₂×C₅₉⋊C₈	1	8	yes	—
149	C₄⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
150	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
151	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
152	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
153	C₄²	C₂²×Dic₅₉	1	2	yes	—
154	C₄²	C₂²×Dic₅₉	1	4	yes	—
155	C₁₆	—	59	4	no	—
156	C₁₆	—	118	2	no	—
157	C₁₆	—	118	4	no	—
158	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	Q₈×Dic₅₉	1	2	yes	—
162	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
163	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
164	C₂×C₄	C₂²×C₅₉⋊C₈	1	8	yes	—
165	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
166	C₂×C₄	#14273734	1	2	yes	—
167	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	4	no	—
170	C₈	Q₈×Dic₅₉	1	2	yes	—
171	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
172	C₈	C₂×#11745689	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×C₅₉⋊C₈	1	2	yes	—
175	C₂²	C₂³×C₅₉⋊C₈	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×C₅₉⋊C₈	1	2	yes	—
180	C₄	C₂×Q₈×Dic₅₉	1	1	yes	characteristic
181	C₄	C₂×#14273734	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×C₅₉⋊C₈	1	1	yes	characteristic
184	C₂	#21437199	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Fitting	C₅₉×#2960	C₂
prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2960	D₅₉
hypocenter	C₅₉	#33475
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #21426005	C₂	semidirect
24. C₂×C₅₉⋊C₁₆	Q₈	semidirect
114. C₅₉	#33475	semidirect

Maximal subgroups

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

subgroup	index
#21426005	2
#21425855	2
#21425852	2
#21425693	2
C₂×#16837178	2
C₅₉×#2960	2
#33475	59