Subgroups — #28500284

Order 15104.

220 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837223	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
3	#21422171	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273667	—	2	1	no	—
6	C₂×#14273667	C₄	1	1	yes	characteristic
7	C₂×#14273667	C₄	1	1	yes	characteristic
8	#16837223	—	2	1	no	—
9	#16837223	C₂²	1	2	yes	—
10	#16837178	C₄	1	2	yes	—
11	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	—
12	C₂×M₃₂×C₅₉	—	2	1	no	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#364	—	2	1	no	—
15	C₅₉×#364	C₂²	1	2	yes	—
16	#14273667	—	2	1	no	—
17	#14273667	—	2	1	no	—
18	#14273667	—	2	2	no	—
19	#14273667	—	4	1	no	—
20	#14273667	C₈	1	2	yes	—
21	#14273667	C₈	1	2	yes	—
22	C₂×C₅₉⋊C₁₆	—	4	1	no	—
23	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
28	M₃₂×C₅₉	—	2	2	no	—
29	M₃₂×C₅₉	—	4	1	no	—
30	C₂×C₁₆×C₅₉	—	4	1	no	—
31	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	—	2	1	no	—
36	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₅₉	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
44	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
45	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₉	—	2	1	no	—
50	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
52	C₁₆×C₅₉	—	4	2	no	—
53	C₅₉⋊C₁₆	—	2	2	no	—
54	C₅₉⋊C₁₆	—	4	2	no	—
55	C₅₉⋊C₁₆	C₂×C₈	1	4	yes	—
56	C₂³×C₅₉	#108	1	1	yes	characteristic
57	C₂×C₄×C₅₉	—	2	1	no	—
58	C₂×C₄×C₅₉	—	2	1	no	—
59	C₂×C₄×C₅₉	—	4	2	no	—
60	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#108	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
67	C₈×C₅₉	C₄⋊C₈	1	1	yes	characteristic
68	C₈×C₅₉	#107	1	1	yes	characteristic
69	C₈×C₅₉	#107	1	1	yes	characteristic
70	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
71	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₅₉	C₂×D₈	1	1	yes	characteristic
73	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
74	#12011	—	59	1	no	maximal, Sylow
75	C₂²×C₅₉	—	4	1	no	—
76	C₂²×C₅₉	#335	1	1	yes	characteristic
77	C₂²×C₅₉	#335	1	1	yes	characteristic
78	C₂²×C₅₉	C₂×#108	1	1	yes	characteristic
79	C₄×C₅₉	—	2	2	no	—
80	C₄×C₅₉	—	4	1	no	—
81	C₄×C₅₉	#335	1	1	yes	characteristic
82	C₄×C₅₉	#335	1	1	yes	characteristic
83	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
85	C₂×#364	—	59	1	no	—
86	C₂×#364	D₅₉	1	1	yes	characteristic
87	C₄×M₃₂	—	59	1	no	—
88	#1373	—	59	4	no	—
89	C₂×C₅₉	—	2	1	no	—
90	C₂×C₅₉	—	2	1	no	—
91	C₂×C₅₉	—	2	1	no	—
92	C₂×C₅₉	C₂×#335	1	1	yes	characteristic
93	C₂×M₃₂	—	2	1	no	—
94	C₂×M₃₂	—	59	1	no	—
95	C₂×M₃₂	—	59	1	no	—
96	C₂×M₃₂	—	118	1	no	—
97	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
98	#364	—	2	1	no	—
99	#364	—	59	2	no	—
100	#364	—	118	1	no	—
101	#364	C₂×D₅₉	1	2	yes	—
102	#346	—	59	2	no	—
103	C₄×C₁₆	—	59	2	no	—
104	C₅₉	#12011	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
107	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	4	1	no	—
110	M₃₂	—	59	2	no	—
111	M₃₂	—	59	2	no	—
112	M₃₂	—	118	1	no	—
113	M₃₂	—	118	1	no	—
114	M₃₂	—	118	2	no	—
115	M₃₂	—	236	1	no	—
116	C₂×C₁₆	—	4	1	no	—
117	C₂×C₁₆	—	59	2	no	—
118	C₂×C₁₆	—	59	2	no	—
119	C₂×C₁₆	—	236	1	no	—
120	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
121	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
126	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
127	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
128	C₂×C₈	#11745659	1	1	yes	characteristic
129	C₂×C₈	#11745659	1	1	yes	characteristic
130	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
131	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
132	C₂×C₈	#11745669	1	1	yes	characteristic
133	C₂×C₈	#11745669	1	1	yes	characteristic
134	C₂×C₈	#11745672	1	1	yes	characteristic
135	C₂×C₈	#11745672	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
138	C₄²	—	2	1	no	—
139	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
140	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
141	C₁₆	—	4	2	no	—
142	C₁₆	—	59	4	no	—
143	C₁₆	—	118	2	no	—
144	C₁₆	—	236	2	no	—
145	C₂³	#14273662	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
150	C₂×C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
151	C₂×C₄	#14273660	1	1	yes	characteristic
152	C₂×C₄	#14273660	1	1	yes	characteristic
153	C₂×C₄	#14273662	1	1	yes	characteristic
154	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
155	C₈	Dic₅₉⋊C₈	1	1	yes	characteristic
156	C₈	Dic₅₉⋊C₈	1	1	yes	characteristic
157	C₈	#14273660	1	1	yes	characteristic
158	C₈	#14273660	1	1	yes	characteristic
159	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
160	C₈	C₂×#11745659	1	1	yes	characteristic
161	C₈	C₂×#11745669	1	1	yes	characteristic
162	C₈	C₂×#11745672	1	1	yes	characteristic
163	C₂²	—	4	1	no	—
164	C₂²	#16837154	1	1	yes	characteristic
165	C₂²	#16837154	1	1	yes	characteristic
166	C₂²	C₂×#14273662	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	#16837154	1	1	yes	characteristic
170	C₄	#16837154	1	1	yes	characteristic
171	C₄	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
172	C₄	C₂×#14273660	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	C₂×#16837154	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×#11745669
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂×C₅₉	C₂×#335
prime radical	C₂³×C₅₉	#108
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	#12011
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #21422171	C₂	semidirect
104. C₅₉	#12011	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837223	2
C₄×#14273667	2
#21422171	2
C₂×C₅₉×#364	2
#12011	59