Subgroups — #28475624

Order 15104.

274 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837178	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273661	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×#334	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
5	C₂×#14273661	C₄	1	2	yes	—
6	C₄×C₄⋊Dic₅₉	—	2	1	no	—
7	#16837178	—	2	1	no	—
8	#16837178	C₄	1	2	yes	complemented
9	#16837165	—	2	1	no	—
10	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#334	—	2	1	no	—
13	C₅₉×#334	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₅₉	—	2	2	no	—
15	C₂×C₄×Dic₅₉	—	4	1	no	—
16	C₂×C₅₉⋊C₁₆	—	2	4	no	—
17	C₂×C₅₉⋊C₁₆	—	2	2	no	—
18	#14273661	—	2	2	no	—
19	#14273661	C₈	1	4	yes	complemented
20	C₅₉⋊#95	—	4	1	no	—
21	C₂²×C₈×C₅₉	—	4	1	no	—
22	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
25	C₄⋊C₈×C₅₉	—	2	2	no	—
26	C₄⋊C₈×C₅₉	—	4	1	no	—
27	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
34	C₂²×Dic₅₉	—	4	2	no	—
35	C₂×C₈×C₅₉	—	2	2	no	—
36	C₂×C₈×C₅₉	—	4	2	no	—
37	C₂×C₈×C₅₉	—	8	2	no	—
38	C₂×C₈×C₅₉	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
44	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
47	C₄²×C₅₉	Q₁₆	1	1	yes	characteristic
48	C₄⋊Dic₅₉	—	2	4	no	—
49	C₄⋊Dic₅₉	—	4	2	no	—
50	C₄×Dic₅₉	—	8	2	no	—
51	C₅₉⋊C₁₆	—	2	4	no	—
52	C₂³×C₅₉	#102	1	1	yes	characteristic
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	2	1	no	—
55	C₂×C₄×C₅₉	—	2	1	no	—
56	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
58	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#104	1	2	yes	—
62	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
64	C₂×Dic₅₉	—	4	4	no	—
65	C₂×Dic₅₉	—	8	2	no	—
66	C₂×Dic₅₉	—	8	2	no	—
67	C₈×C₅₉	—	2	1	no	—
68	C₈×C₅₉	—	2	1	no	—
69	C₈×C₅₉	—	8	2	no	—
70	C₈×C₅₉	C₄×C₈	1	1	yes	characteristic
71	C₈×C₅₉	#97	1	1	yes	characteristic
72	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
73	#7178	—	59	1	no	maximal, Sylow
74	C₂²×C₅₉	D₄⋊C₈	1	2	yes	—
75	C₂²×C₅₉	#340	1	1	yes	characteristic
76	C₂²×C₅₉	#360	1	2	yes	—
77	C₂²×C₅₉	#361	1	1	yes	characteristic
78	C₂²×C₅₉	#362	1	1	yes	characteristic
79	C₄×C₅₉	—	2	1	no	—
80	C₄×C₅₉	—	2	1	no	—
81	C₄×C₅₉	—	2	1	no	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	Q₈⋊C₈	1	2	yes	—
84	C₄×C₅₉	#336	1	1	yes	characteristic
85	C₄×C₅₉	#339	1	1	yes	characteristic
86	Dic₅₉	—	8	4	no	—
87	C₂×#346	—	59	1	no	—
88	C₄×#107	—	59	1	no	—
89	C₂×#334	D₅₉	1	1	yes	characteristic
90	C₂×C₅₉	#1279	1	1	yes	characteristic
91	C₂×C₅₉	#1333	1	2	yes	—
92	C₂×C₅₉	#1334	1	2	yes	—
93	C₂×C₅₉	#1386	1	1	yes	characteristic
94	C₂×C₅₉	#1387	1	1	yes	characteristic
95	C₂²×C₁₆	—	118	1	no	—
96	C₂×#107	—	59	2	no	—
97	C₂×C₄⋊C₈	—	2	1	no	—
98	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	118	1	no	—
100	#346	—	59	2	no	—
101	#346	—	118	1	no	—
102	#340	—	118	1	no	—
103	#334	—	2	1	no	—
104	#334	Dic₅₉	1	2	yes	complemented
105	C₅₉	#7178	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	4	1	no	—
107	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
108	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	118	2	no	—
110	C₂×C₄²	—	236	1	no	—
111	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
112	C₂×C₁₆	—	118	4	no	—
113	C₂×C₁₆	—	118	2	no	—
114	#107	—	59	4	no	—
115	#107	—	118	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
119	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
122	#95	—	236	1	no	—
123	C₂²×C₄	—	236	2	no	—
124	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
125	C₂²×C₄	#11745669	1	1	yes	characteristic
126	C₂²×C₄	#11745671	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	8	2	no	—
130	C₂×C₈	C₈×D₅₉	1	2	yes	—
131	C₂×C₈	#11745659	1	2	yes	—
132	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
133	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
135	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
136	C₄⋊C₄	—	118	4	no	—
137	C₄⋊C₄	—	236	2	no	—
138	C₄²	—	472	2	no	—
139	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
140	C₄²	#11745670	1	1	yes	characteristic
141	C₄²	#11745672	1	1	yes	characteristic
142	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
143	C₁₆	—	118	4	no	—
144	C₂³	#14273647	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	472	2	no	—
149	C₂×C₄	—	472	2	no	—
150	C₂×C₄	—	236	4	no	—
151	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
152	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
153	C₂×C₄	#14273648	1	1	yes	characteristic
154	C₂×C₄	#14273650	1	1	yes	characteristic
155	C₂×C₄	#14273651	1	1	yes	characteristic
156	C₂×C₄	#14273653	1	2	yes	—
157	C₂×C₄	#14273658	1	1	yes	characteristic
158	C₂×C₄	#14273660	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	8	2	no	—
162	C₈	C₈×Dic₅₉	1	1	yes	characteristic
163	C₈	#14273637	1	1	yes	characteristic
164	C₈	Dic₅₉⋊C₈	1	2	yes	—
165	C₂²	#16837129	1	2	yes	—
166	C₂²	#16837164	1	1	yes	characteristic
167	C₂²	#16837211	1	2	yes	—
168	C₂²	#16837214	1	1	yes	characteristic
169	C₂²	#16837217	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	472	4	no	—
175	C₄	#16837132	1	2	yes	—
176	C₄	#16837156	1	1	yes	characteristic
177	C₄	#16837163	1	1	yes	characteristic
178	C₂	#21421935	1	1	yes	characteristic
179	C₂	#21422062	1	2	yes	—
180	C₂	#21422065	1	2	yes	—
181	C₂	#21422203	1	1	yes	characteristic
182	C₂	#21422206	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273647
commutator	C₈×C₅₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₂×C₅₉×#334	C₂
socle, prime radical	C₂³×C₅₉	#102
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	C₂×#334	D₅₉
hypocenter	C₅₉	#7178
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #16837178	C₄	semidirect
13. C₅₉×#334	C₄	semidirect
19. #14273661	C₈	semidirect
104. #334	Dic₅₉	semidirect
105. C₅₉	#7178	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837178	2
C₄×#14273661	2
C₂×C₅₉×#334	2
#7178	59