Subgroups — #28741349

Order 15104.

716 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437199	C₂	1	1	yes	maximal, characteristic
2	#21432522	C₂	1	1	yes	maximal, characteristic
3	#21425700	C₂	1	6	yes	maximal, complemented
4	C₂×#16837179	C₂	1	6	yes	maximal, complemented
5	C₅₉×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273734	C₄	1	1	yes	characteristic
7	#41×Dic₅₉	C₄	1	1	yes	characteristic
8	#16837819	C₄	1	8	yes	complemented
9	C₂×#14273667	C₂²	1	6	yes	—
10	#16837360	C₄	1	3	yes	—
11	#16837357	C₂²	1	3	yes	—
12	M₁₆×Dic₅₉	C₄	1	3	yes	—
13	C₂×#14273637	C₂²	1	3	yes	—
14	#16837179	C₂²	1	4	yes	—
15	#16837179	C₂²	1	12	yes	—
16	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#504	C₄	1	8	yes	complemented
18	C₂×M₃₂×C₅₉	C₂²	1	6	yes	—
19	#14273734	—	2	4	no	—
20	C₂×#11745664	C₂×C₄	1	3	yes	—
21	C₂²×C₅₉⋊C₈	C₂×C₄	1	3	yes	—
22	Q₈×Dic₅₉	—	2	1	no	—
23	D₄×Dic₅₉	—	2	3	no	—
24	#14273686	C₂×C₄	1	3	yes	—
25	C₂×C₄×Dic₅₉	C₂×C₄	1	3	yes	—
26	#14273667	C₂×C₄	1	12	yes	complemented
27	#14273667	C₂³	1	4	yes	—
28	C₂×C₅₉⋊C₁₆	C₂×C₄	1	12	yes	complemented
29	#14273637	C₂×C₄	1	3	yes	—
30	#14273637	C₂×C₄	1	3	yes	—
31	#14273637	C₂³	1	1	yes	characteristic
32	#14273637	C₂³	1	3	yes	—
33	C₈×Dic₅₉	C₂×C₄	1	6	yes	—
34	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
35	#131×C₅₉	C₂×C₄	1	4	yes	—
36	#131×C₅₉	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₅₉	C₂×C₄	1	3	yes	—
38	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
39	M₃₂×C₅₉	C₂×C₄	1	12	yes	complemented
40	M₃₂×C₅₉	C₂³	1	4	yes	—
41	C₂×C₁₆×C₅₉	C₂×C₄	1	12	yes	complemented
42	#41×C₅₉	C₄²	1	4	yes	—
43	#41×C₅₉	M₁₆	1	4	yes	—
44	C₂×Q₈×C₅₉	M₁₆	1	1	yes	characteristic
45	C₂×D₄×C₅₉	M₁₆	1	3	yes	—
46	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₅₉	—	2	3	no	—
48	M₁₆×C₅₉	C₄²	1	12	yes	—
49	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
51	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
52	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
53	C₂²⋊Dic₅₉	—	2	3	no	—
54	C₄⋊Dic₅₉	—	2	3	no	—
55	C₄×Dic₅₉	—	2	3	no	—
56	C₄×Dic₅₉	C₂²×C₄	1	3	yes	—
57	C₄×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
58	#11745664	—	2	6	no	—
59	C₂×C₅₉⋊C₈	—	2	6	no	—
60	C₂×C₅₉⋊C₈	C₂²×C₄	1	3	yes	—
61	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
62	C₁₆×C₅₉	C₂²×C₄	1	8	yes	—
63	C₅₉⋊C₁₆	C₂²×C₄	1	8	yes	—
64	C₂³×C₅₉	C₂×M₁₆	1	3	yes	—
65	Q₈×C₅₉	#97	1	4	yes	—
66	D₄×C₅₉	#97	1	12	yes	—
67	C₂×C₄×C₅₉	C₂×C₄²	1	6	yes	—
68	C₂×C₄×C₅₉	C₂×M₁₆	1	6	yes	—
69	C₂×C₄×C₅₉	C₂×M₁₆	1	3	yes	—
70	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
71	C₂×Dic₅₉	—	2	1	no	—
72	C₂×Dic₅₉	—	2	3	no	—
73	C₂×Dic₅₉	—	8	3	no	—
74	C₈×C₅₉	C₂×C₄²	1	6	yes	—
75	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
76	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
77	C₅₉⋊C₈	—	2	1	no	—
78	C₅₉⋊C₈	—	2	3	no	—
79	#33329	—	59	1	no	maximal, Sylow
80	C₂²×C₅₉	—	4	3	no	—
81	C₂²×C₅₉	C₂×#97	1	6	yes	—
82	C₂²×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
83	C₄×C₅₉	C₂×#97	1	6	yes	—
84	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
85	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
86	Dic₅₉	—	4	3	no	—
87	Dic₅₉	—	4	1	no	—
88	#3407	—	59	1	no	—
89	#3407	D₅₉	1	1	yes	characteristic
90	#2875	—	59	1	no	—
91	#2110	—	59	6	no	—
92	C₂×#347	—	59	6	no	—
93	C₂×C₅₉	—	2	1	no	—
94	C₂×C₅₉	—	2	6	no	—
95	C₂×C₅₉	C₂²×#97	1	1	yes	characteristic
96	C₂×#131	—	59	1	no	—
97	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
98	C₄×#41	—	59	1	no	—
99	#504	—	59	8	no	—
100	#504	Dic₅₉	1	8	yes	complemented
101	C₂×M₃₂	—	59	6	no	—
102	C₂×M₃₂	C₂×D₅₉	1	6	yes	—
103	#405	—	59	3	no	—
104	#405	—	59	3	no	—
105	C₄×M₁₆	—	59	3	no	—
106	C₂×#97	—	59	3	no	—
107	#347	—	59	4	no	—
108	#347	—	59	12	no	—
109	C₅₉	#33329	1	1	yes	characteristic, complemented, Sylow
110	C₂×#41	C₄×D₅₉	1	1	yes	characteristic
111	#131	—	118	4	no	—
112	#131	C₄×D₅₉	1	4	yes	—
113	#131	C₂×Dic₅₉	1	4	yes	—
114	C₂×M₁₆	—	59	3	no	—
115	C₂×M₁₆	C₄×D₅₉	1	3	yes	—
116	C₂²×C₈	—	59	3	no	—
117	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
118	C₄×Q₈	—	118	1	no	—
119	C₄×D₄	—	118	3	no	—
120	#117	—	59	3	no	—
121	C₂×C₄²	—	59	3	no	—
122	M₃₂	—	59	12	no	—
123	M₃₂	—	59	4	no	—
124	M₃₂	C₂×Dic₅₉	1	12	yes	complemented
125	M₃₂	C₂²×D₅₉	1	4	yes	—
126	C₂×C₁₆	—	59	12	no	—
127	C₂×C₁₆	C₂×Dic₅₉	1	12	yes	complemented
128	#97	—	59	3	no	—
129	#97	—	59	3	no	—
130	#97	—	59	1	no	—
131	#97	—	59	3	no	—
132	C₄×C₈	—	59	6	no	—
133	#41	#11745659	1	4	yes	—
134	#41	C₄×Dic₅₉	1	4	yes	—
135	C₂×Q₈	#11745659	1	1	yes	characteristic
136	C₂×D₄	#11745659	1	3	yes	—
137	C₂²×C₄	—	118	3	no	—
138	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
139	M₁₆	—	118	6	no	—
140	M₁₆	C₄×Dic₅₉	1	12	yes	—
141	C₂×C₈	—	59	3	no	—
142	C₂×C₈	—	59	1	no	—
143	C₂×C₈	—	118	6	no	—
144	C₂×C₈	C₂×C₄×D₅₉	1	3	yes	—
145	C₂×C₈	C₂×C₄×D₅₉	1	6	yes	—
146	C₂×C₈	C₂²×Dic₅₉	1	6	yes	—
147	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
148	C₄⋊C₄	—	118	3	no	—
149	C₂²⋊C₄	—	118	3	no	—
150	C₄²	—	59	3	no	—
151	C₄²	—	59	1	no	—
152	C₄²	—	118	3	no	—
153	C₁₆	—	59	8	no	—
154	C₁₆	C₂²×Dic₅₉	1	8	yes	—
155	C₂³	C₂×#11745659	1	3	yes	—
156	Q₈	#14273637	1	4	yes	—
157	D₄	#14273637	1	12	yes	—
158	C₂×C₄	—	118	1	no	—
159	C₂×C₄	—	118	3	no	—
160	C₂×C₄	—	472	3	no	—
161	C₂×C₄	C₂×C₄×Dic₅₉	1	6	yes	—
162	C₂×C₄	C₂×#11745659	1	6	yes	—
163	C₂×C₄	C₂×#11745659	1	3	yes	—
164	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
165	C₈	—	118	1	no	—
166	C₈	—	118	3	no	—
167	C₈	C₂×C₄×Dic₅₉	1	6	yes	—
168	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
169	C₈	C₂³×Dic₅₉	1	1	yes	characteristic
170	C₂²	—	4	3	no	—
171	C₂²	C₂×#14273637	1	6	yes	—
172	C₂²	C₂²×#11745659	1	1	yes	characteristic
173	C₄	—	236	1	no	—
174	C₄	—	236	3	no	—
175	C₄	C₂×#14273637	1	6	yes	—
176	C₄	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
177	C₄	C₂²×#11745659	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	6	no	—
180	C₂	C₂²×#14273637	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₅₉
commutator	C₄×C₅₉	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3407	C₂
socle	C₂×C₅₉	C₂²×#97
prime radical	C₂×#41×C₅₉	C₂×C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#3407	D₅₉
hypocenter	C₅₉	#33329
LCS	G ▷ C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ #3407

Complements

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

N	H ≅ G/N	type
3. #21425700	C₂	semidirect
4. C₂×#16837179	C₂	semidirect
8. #16837819	C₄	semidirect
17. C₅₉×#504	C₄	semidirect
26. #14273667	C₂×C₄	semidirect
28. C₂×C₅₉⋊C₁₆	C₂×C₄	semidirect
39. M₃₂×C₅₉	C₂×C₄	semidirect
41. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
100. #504	Dic₅₉	semidirect
109. C₅₉	#33329	semidirect
124. M₃₂	C₂×Dic₅₉	semidirect
127. C₂×C₁₆	C₂×Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
#21437199	2
#21432522	2
#21425700	2
C₂×#16837179	2
C₅₉×#3407	2
#33329	59