Subgroups — #28742184

Order 15104.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21426005	C₂	1	4	yes	maximal, complemented
2	Q₈×C₅₉⋊C₁₆	C₂	1	4	yes	maximal, complemented
3	#21425855	C₂	1	2	yes	maximal
4	#21425852	C₂	1	1	yes	maximal, characteristic
5	#21425693	C₂	1	2	yes	maximal
6	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#2960	C₂	1	1	yes	maximal, characteristic
8	C₂×#14273667	C₂²	1	2	yes	—
9	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
12	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	8	yes	—
13	#16837178	C₂²	1	4	yes	—
14	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
15	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
16	C₄×C₅₉⋊C₁₆	C₂²	1	4	yes	—
17	C₅₉×#533	C₄	1	1	yes	characteristic
18	C₅₉×#446	C₄	1	4	yes	—
19	C₈×Q₈×C₅₉	C₂²	1	4	yes	—
20	C₅₉×#424	C₄	1	2	yes	—
21	C₅₉×#423	C₄	1	1	yes	characteristic
22	C₅₉×#405	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	#14273667	—	2	4	no	—
25	C₂×C₅₉⋊C₁₆	—	2	2	no	—
26	C₂×C₅₉⋊C₁₆	Q₈	1	4	yes	complemented
27	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
28	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
29	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
30	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₉	C₈	1	2	yes	—
35	#125×C₅₉	C₈	1	2	yes	—
36	C₂²⋊Q₈×C₅₉	C₈	1	4	yes	—
37	C₄×Q₈×C₅₉	C₂×C₄	1	4	yes	—
38	#117×C₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
41	C₄⋊C₈×C₅₉	C₂×C₄	1	8	yes	—
42	#97×C₅₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	C₂³	1	2	yes	—
45	C₄×C₈×C₅₉	C₂³	1	4	yes	—
46	C₂×Q₈×C₅₉	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₅₉	—	2	4	no	—
50	C₂×C₈×C₅₉	—	2	2	no	—
51	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₅₉	#41	1	2	yes	—
56	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
58	C₄⋊C₄×C₅₉	C₂×C₈	1	8	yes	—
59	C₄⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
60	C₂²⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
61	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
64	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
65	C₅₉⋊C₁₆	—	2	2	no	—
66	C₅₉⋊C₁₆	—	2	4	no	—
67	C₅₉⋊C₁₆	C₂×Q₈	1	2	yes	—
68	C₅₉⋊C₁₆	C₂×Q₈	1	2	yes	—
69	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
70	Q₈×C₅₉	—	2	8	no	—
71	C₂×C₄×C₅₉	—	2	4	no	—
72	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₅₉	C₂²×C₈	1	8	yes	—
75	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₅₉	#131	1	2	yes	—
78	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₅₉	—	2	2	no	—
80	C₈×C₅₉	—	2	4	no	—
81	C₈×C₅₉	C₄×Q₈	1	2	yes	—
82	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
84	#33474	—	59	1	no	maximal, Sylow
85	C₂²×C₅₉	—	2	2	no	—
86	C₂²×C₅₉	C₈×Q₈	1	2	yes	—
87	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₅₉	—	2	2	no	—
89	C₄×C₅₉	—	2	8	no	—
90	C₄×C₅₉	C₈×Q₈	1	2	yes	—
91	C₄×C₅₉	#504	1	4	yes	—
92	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
93	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
94	#2960	D₅₉	1	1	yes	characteristic
95	#2183	—	59	4	no	—
96	Q₈×C₁₆	—	59	4	no	—
97	#2151	—	59	2	no	—
98	#2150	—	59	1	no	—
99	#2108	—	59	2	no	—
100	C₂×C₄×C₁₆	—	59	1	no	—
101	C₂×C₅₉	—	2	2	no	—
102	C₂×C₅₉	C₂×C₈×Q₈	1	1	yes	characteristic
103	C₂×C₅₉	C₂×#504	1	2	yes	—
104	#533	Dic₅₉	1	1	yes	characteristic
105	C₂×M₃₂	—	59	2	no	—
106	C₂²×C₁₆	—	59	1	no	—
107	C₂²×C₁₆	—	59	1	no	—
108	#446	Dic₅₉	1	4	yes	—
109	C₈×Q₈	C₂×D₅₉	1	4	yes	—
110	#424	Dic₅₉	1	2	yes	—
111	#423	Dic₅₉	1	1	yes	characteristic
112	#405	C₂×D₅₉	1	2	yes	—
113	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
114	C₄⋊C₁₆	—	59	4	no	—
115	C₄⋊C₁₆	—	59	8	no	—
116	#346	—	59	4	no	—
117	C₄×C₁₆	—	59	2	no	—
118	C₄×C₁₆	—	59	2	no	—
119	C₄×C₁₆	—	59	4	no	—
120	C₅₉	#33474	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
122	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
124	C₄⋊Q₈	C₅₉⋊C₈	1	2	yes	—
125	#125	C₅₉⋊C₈	1	2	yes	—
126	C₂²⋊Q₈	C₅₉⋊C₈	1	4	yes	—
127	C₄×Q₈	C₂×Dic₅₉	1	4	yes	—
128	#117	C₂×Dic₅₉	1	2	yes	—
129	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
130	M₃₂	—	118	4	no	—
131	C₂×C₁₆	—	59	4	no	—
132	C₂×C₁₆	—	59	1	no	—
133	C₂×C₁₆	—	59	1	no	—
134	C₂×C₁₆	—	59	2	no	—
135	C₂×C₁₆	—	59	4	no	—
136	C₂×C₁₆	—	118	2	no	—
137	C₄⋊C₈	C₂×Dic₅₉	1	4	yes	—
138	C₄⋊C₈	C₂×Dic₅₉	1	8	yes	—
139	#97	C₂×Dic₅₉	1	4	yes	—
140	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
141	C₄×C₈	C₂²×D₅₉	1	2	yes	—
142	C₄×C₈	C₂²×D₅₉	1	4	yes	—
143	C₂×Q₈	C₂×C₅₉⋊C₈	1	4	yes	—
144	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	Q₈×D₅₉	1	2	yes	—
149	C₂×C₈	#11745689	1	2	yes	—
150	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
151	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
152	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
153	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
155	C₄⋊C₄	C₂×C₅₉⋊C₈	1	8	yes	—
156	C₄⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
157	C₂²⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
158	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
159	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
160	C₄²	C₂²×Dic₅₉	1	2	yes	—
161	C₄²	C₂²×Dic₅₉	1	4	yes	—
162	C₁₆	—	59	2	no	—
163	C₁₆	—	59	2	no	—
164	C₁₆	—	118	2	no	—
165	C₁₆	—	118	4	no	—
166	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	Q₈×Dic₅₉	1	2	yes	—
170	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
171	C₂×C₄	C₂²×C₅₉⋊C₈	1	8	yes	—
172	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₅₉⋊C₈	1	2	yes	—
174	C₂×C₄	#14273734	1	2	yes	—
175	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	4	no	—
178	C₈	Q₈×Dic₅₉	1	2	yes	—
179	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
180	C₈	C₂×#11745689	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	Q₈×C₅₉⋊C₈	1	2	yes	—
183	C₂²	C₂³×C₅₉⋊C₈	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	Q₈×C₅₉⋊C₈	1	2	yes	—
187	C₄	#16837819	1	4	yes	—
188	C₄	C₂×Q₈×Dic₅₉	1	1	yes	characteristic
189	C₄	C₂×#14273734	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×Q₈×C₅₉⋊C₈	1	1	yes	characteristic
192	C₂	C₂×#16837819	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	#33474
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
2. Q₈×C₅₉⋊C₁₆	C₂	semidirect
26. C₂×C₅₉⋊C₁₆	Q₈	semidirect
120. C₅₉	#33474	semidirect

Maximal subgroups

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

subgroup	index
#21426005	2
Q₈×C₅₉⋊C₁₆	2
#21425855	2
#21425852	2
#21425693	2
C₂×C₄×C₅₉⋊C₁₆	2
C₅₉×#2960	2
#33474	59