Subgroups — #28475896

Order 15104.

226 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424268	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273663	—	2	1	no	—
5	C₂×#14273663	C₄	1	2	yes	—
6	#16837426	—	2	1	no	—
7	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
8	#16837155	—	2	1	no	—
9	#16837155	C₄	1	2	yes	—
10	C₂×M₃₂×C₅₉	—	2	1	no	—
11	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×C₄⋊C₁₆	—	2	1	no	—
13	C₅₉×C₄⋊C₁₆	C₄	1	2	yes	—
14	C₂×#11745664	—	2	2	no	—
15	C₂²×C₅₉⋊C₈	—	4	1	no	—
16	#14273663	—	4	1	no	—
17	#14273663	—	4	1	no	—
18	#14273663	—	4	2	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
20	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
21	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
22	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
26	M₃₂×C₅₉	—	4	1	no	—
27	M₃₂×C₅₉	—	4	1	no	—
28	C₂×C₁₆×C₅₉	—	2	2	no	—
29	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	—	2	1	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	D₈	1	1	yes	characteristic
43	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
44	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
46	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₉	D₈	1	1	yes	characteristic
49	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
50	#11745664	—	4	2	no	—
51	#11745664	—	4	2	no	—
52	C₂×C₅₉⋊C₈	—	4	2	no	—
53	C₂×C₅₉⋊C₈	—	4	2	no	—
54	C₂×C₅₉⋊C₈	—	8	2	no	—
55	C₁₆×C₅₉	—	4	2	no	—
56	C₂³×C₅₉	#103	1	1	yes	characteristic
57	C₂×C₄×C₅₉	—	2	1	no	—
58	C₂×C₄×C₅₉	—	2	1	no	—
59	C₂×C₄×C₅₉	—	4	1	no	—
60	C₂×C₄×C₅₉	#95	1	1	yes	characteristic
61	C₂×C₄×C₅₉	#100	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#101	1	1	yes	characteristic
63	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
64	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
65	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
67	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
70	C₈×C₅₉	—	2	1	no	—
71	C₈×C₅₉	—	2	1	no	—
72	C₈×C₅₉	#102	1	1	yes	characteristic
73	C₈×C₅₉	#103	1	1	yes	characteristic
74	C₈×C₅₉	#106	1	1	yes	characteristic
75	C₈×C₅₉	#107	1	1	yes	characteristic
76	C₅₉⋊C₈	—	8	2	no	—
77	C₅₉⋊C₈	—	8	2	no	—
78	#7281	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	—	2	1	no	—
80	C₂²×C₅₉	—	2	1	no	—
81	C₂²×C₅₉	#332	1	1	yes	characteristic
82	C₂²×C₅₉	#333	1	1	yes	characteristic
83	C₂²×C₅₉	#340	1	1	yes	characteristic
84	C₄×C₅₉	—	2	1	no	—
85	C₄×C₅₉	—	2	1	no	—
86	C₄×C₅₉	—	2	1	no	—
87	C₄×C₅₉	#331	1	1	yes	characteristic
88	C₄×C₅₉	#332	1	1	yes	characteristic
89	C₄×C₅₉	#340	1	1	yes	characteristic
90	C₄×C₅₉	#343	1	1	yes	characteristic
91	C₄×C₅₉	#361	1	1	yes	characteristic
92	C₄×C₅₉	#362	1	1	yes	characteristic
93	#2150	D₅₉	1	1	yes	characteristic
94	#1848	—	59	1	no	—
95	C₂×#335	—	59	1	no	—
96	C₂×C₅₉	—	2	1	no	—
97	C₂×C₅₉	—	2	1	no	—
98	C₂×C₅₉	#1296	1	1	yes	characteristic
99	C₂×C₅₉	#1385	1	1	yes	characteristic
100	C₂×C₅₉	#1387	1	1	yes	characteristic
101	C₂×M₃₂	—	2	1	no	—
102	C₂×#108	—	59	2	no	—
103	C₂×#108	—	118	1	no	—
104	#423	—	118	1	no	—
105	C₂×C₄⋊C₈	—	118	1	no	—
106	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
107	C₄⋊C₁₆	—	2	1	no	—
108	C₄⋊C₁₆	Dic₅₉	1	2	yes	—
109	#335	—	59	2	no	—
110	#335	—	118	1	no	—
111	C₅₉	#7281	1	1	yes	characteristic, complemented, Sylow
112	C₂×M₁₆	—	118	2	no	—
113	C₂²×C₈	—	236	1	no	—
114	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
116	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
117	M₃₂	—	4	1	no	—
118	M₃₂	—	4	1	no	—
119	C₂×C₁₆	—	2	2	no	—
120	#108	—	236	2	no	—
121	#108	—	236	1	no	—
122	#108	—	236	1	no	—
123	C₄⋊C₈	—	118	2	no	—
124	C₄⋊C₈	—	236	1	no	—
125	C₄⋊C₈	—	236	1	no	—
126	C₄⋊C₈	—	236	1	no	—
127	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
128	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
129	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
131	C₂²×C₄	#11745660	1	1	yes	characteristic
132	C₂²×C₄	Dic₂₃₆	1	1	yes	characteristic
133	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
134	M₁₆	—	236	2	no	—
135	M₁₆	—	236	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	472	2	no	—
139	C₂×C₈	—	236	2	no	—
140	C₂×C₈	—	236	2	no	—
141	C₂×C₈	C₄×Dic₅₉	1	1	yes	characteristic
142	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
143	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
144	C₂×C₈	#11745669	1	1	yes	characteristic
145	C₂×C₈	#11745670	1	1	yes	characteristic
146	C₂×C₈	#11745671	1	1	yes	characteristic
147	C₂×C₈	#11745672	1	1	yes	characteristic
148	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
149	C₄²	#11745660	1	1	yes	characteristic
150	C₄²	D₄₇₂	1	1	yes	characteristic
151	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
152	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
153	C₁₆	—	4	2	no	—
154	C₂³	#14273652	1	1	yes	characteristic
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	C₅₉⋊#95	1	1	yes	characteristic
159	C₂×C₄	#14273643	1	1	yes	characteristic
160	C₂×C₄	#14273645	1	1	yes	characteristic
161	C₂×C₄	#14273647	1	1	yes	characteristic
162	C₂×C₄	#14273649	1	1	yes	characteristic
163	C₂×C₄	#14273649	1	1	yes	characteristic
164	C₂×C₄	#14273650	1	1	yes	characteristic
165	C₂×C₄	#14273652	1	1	yes	characteristic
166	C₂×C₄	#14273659	1	1	yes	characteristic
167	C₂×C₄	#14273661	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	472	2	no	—
171	C₈	—	472	2	no	—
172	C₈	#14273648	1	1	yes	characteristic
173	C₈	#14273651	1	1	yes	characteristic
174	C₈	#14273658	1	1	yes	characteristic
175	C₈	#14273660	1	1	yes	characteristic
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#16837149	1	1	yes	characteristic
179	C₂²	#16837150	1	1	yes	characteristic
180	C₂²	#16837165	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	#16837145	1	1	yes	characteristic
185	C₄	#16837147	1	1	yes	characteristic
186	C₄	#16837164	1	1	yes	characteristic
187	C₄	#16837170	1	1	yes	characteristic
188	C₄	#16837216	1	1	yes	characteristic
189	C₄	#16837219	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	#21421980	1	1	yes	characteristic
193	C₂	#21422200	1	1	yes	characteristic
194	C₂	#21422205	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837165
commutator	C₂×C₈×C₅₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	#340
prime radical	C₂³×C₅₉	#103
cosocle	C₂×C₄×C₈×C₅₉	C₂²
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#7281
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
111. C₅₉	#7281	semidirect

Maximal subgroups

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

subgroup	index
#21424268	2
C₂×#16837155	2
C₅₉×#2150	2
#7281	59