Subgroups — #28506796

Order 15104.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837232	C₂	1	2	yes	maximal
2	#21424268	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273663	C₂	1	1	yes	maximal, characteristic
4	#21422190	C₂	1	2	yes	maximal
5	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273663	—	2	1	no	—
7	C₂×#14273663	C₂²	1	2	yes	—
8	C₂×#14273663	C₂²	1	1	yes	characteristic
9	C₂×#14273663	C₂²	1	1	yes	characteristic
10	#16837426	—	2	1	no	—
11	C₄×#11745664	—	2	1	no	—
12	#16837232	—	2	4	no	—
13	#16837167	—	2	1	no	—
14	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×C₄⋊C₁₆	C₄	1	2	yes	—
17	C₅₉×C₄⋊C₁₆	C₄	1	2	yes	—
18	C₂×#11745664	—	2	1	no	—
19	C₂×#11745664	—	2	1	no	—
20	C₂×#11745664	—	2	2	no	—
21	#14273663	—	2	1	no	—
22	#14273663	—	2	1	no	—
23	#14273663	—	2	2	no	—
24	#14273663	—	2	2	no	—
25	#14273663	—	4	2	no	—
26	#14273663	D₄	1	2	yes	—
27	#14273663	Q₈	1	2	yes	—
28	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
29	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
30	#14273638	—	4	1	no	—
31	C₄×C₅₉⋊C₈	—	4	1	no	—
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₉	—	2	2	no	—
36	C₂×C₁₆×C₅₉	—	2	2	no	—
37	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	—	2	1	no	—
46	C₂×C₈×C₅₉	—	2	1	no	—
47	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
55	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
56	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₅₉	D₈	1	1	yes	characteristic
58	C₄²×C₅₉	D₈	1	1	yes	characteristic
59	#11745664	—	2	2	no	—
60	#11745664	—	2	2	no	—
61	#11745664	—	4	1	no	—
62	#11745664	—	4	1	no	—
63	#11745664	—	4	2	no	—
64	#11745664	—	4	2	no	—
65	C₂×C₅₉⋊C₈	—	4	1	no	—
66	C₂×C₅₉⋊C₈	—	4	1	no	—
67	C₂×C₅₉⋊C₈	—	4	2	no	—
68	C₁₆×C₅₉	—	2	4	no	—
69	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	4	1	no	—
72	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
73	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#107	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
82	C₈×C₅₉	—	2	1	no	—
83	C₈×C₅₉	—	2	1	no	—
84	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
85	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₅₉	#125	1	1	yes	characteristic
87	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₅₉⋊C₈	—	4	2	no	—
89	C₅₉⋊C₈	—	4	2	no	—
90	C₅₉⋊C₈	—	8	2	no	—
91	#13179	—	59	1	no	maximal, Sylow
92	C₂²×C₅₉	—	2	1	no	—
93	C₂²×C₅₉	—	2	1	no	—
94	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
95	C₂²×C₅₉	#462	1	1	yes	characteristic
96	C₂²×C₅₉	#477	1	1	yes	characteristic
97	C₄×C₅₉	—	2	1	no	—
98	C₄×C₅₉	—	2	1	no	—
99	C₄×C₅₉	—	2	1	no	—
100	C₄×C₅₉	#368	1	2	yes	complemented
101	C₄×C₅₉	#389	1	1	yes	characteristic
102	C₄×C₅₉	#428	1	1	yes	characteristic
103	C₄×C₅₉	#459	1	1	yes	characteristic
104	C₄×C₅₉	#474	1	1	yes	characteristic
105	C₂×#368	—	59	2	no	—
106	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
107	#1848	—	59	1	no	—
108	C₄×#108	—	59	1	no	—
109	#1381	—	59	2	no	—
110	C₂×C₅₉	—	2	1	no	—
111	C₂×C₅₉	—	2	1	no	—
112	C₂×C₅₉	#1947	1	1	yes	characteristic
113	C₂×C₅₉	C₂×#368	1	1	yes	characteristic
114	C₂×C₅₉	#2161	1	1	yes	characteristic
115	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
116	C₂×#108	—	59	1	no	—
117	C₂×#108	—	59	1	no	—
118	C₂×#108	—	59	2	no	—
119	C₂×#108	—	118	1	no	—
120	#423	—	118	1	no	—
121	C₄×M₁₆	—	118	1	no	—
122	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
123	#368	—	118	4	no	—
124	C₄⋊C₁₆	Dic₅₉	1	2	yes	—
125	C₄⋊C₁₆	Dic₅₉	1	2	yes	—
126	#341	—	118	1	no	—
127	C₅₉	#13179	1	1	yes	characteristic, complemented, Sylow
128	C₂×M₁₆	—	118	1	no	—
129	C₂×M₁₆	—	118	1	no	—
130	C₂×M₁₆	—	118	2	no	—
131	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
133	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
137	#108	—	59	2	no	—
138	#108	—	59	2	no	—
139	#108	—	118	2	no	—
140	#108	—	118	1	no	—
141	#108	—	118	1	no	—
142	#108	—	118	2	no	—
143	#108	—	236	2	no	—
144	C₄⋊C₈	—	236	1	no	—
145	C₄⋊C₈	—	236	1	no	—
146	#97	—	236	1	no	—
147	C₄×C₈	—	236	1	no	—
148	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
149	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
150	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
151	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
152	C₂²×C₄	Dic₂₃₆	1	1	yes	characteristic
153	C₂²×C₄	Dic₂₃₆	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
155	M₁₆	—	118	2	no	—
156	M₁₆	—	118	2	no	—
157	M₁₆	—	236	2	no	—
158	M₁₆	—	236	2	no	—
159	M₁₆	—	236	1	no	—
160	M₁₆	—	236	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	236	2	no	—
164	C₂×C₈	—	236	1	no	—
165	C₂×C₈	—	236	1	no	—
166	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
168	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
169	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
170	C₂×C₈	#11745687	1	1	yes	characteristic
171	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
172	C₂×C₈	#11745689	1	1	yes	characteristic
173	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
174	C₄²	D₄₇₂	1	1	yes	characteristic
175	C₄²	D₄₇₂	1	1	yes	characteristic
176	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
177	C₄²	C₄⋊Dic₅₉	1	1	yes	characteristic
178	C₁₆	—	2	4	no	—
179	C₂³	C₂×Dic₂₃₆	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	#14273661	1	1	yes	characteristic
183	C₂×C₄	#14273661	1	1	yes	characteristic
184	C₂×C₄	#14273661	1	1	yes	characteristic
185	C₂×C₄	#14273661	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
187	C₂×C₄	#14273702	1	1	yes	characteristic
188	C₂×C₄	#14273707	1	1	yes	characteristic
189	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
190	C₂×C₄	#14273755	1	1	yes	characteristic
191	C₂×C₄	#14273760	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	472	2	no	—
195	C₈	—	236	2	no	—
196	C₈	—	236	2	no	—
197	C₈	D₄×Dic₅₉	1	1	yes	characteristic
198	C₈	Q₈×Dic₅₉	1	1	yes	characteristic
199	C₈	#14273718	1	1	yes	characteristic
200	C₈	#14273726	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×#14273661	1	1	yes	characteristic
204	C₂²	#16837613	1	1	yes	characteristic
205	C₂²	#16837698	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	#16837232	1	2	yes	complemented
210	C₄	#16837301	1	1	yes	characteristic
211	C₄	#16837444	1	1	yes	characteristic
212	C₄	#16837598	1	1	yes	characteristic
213	C₄	#16837683	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	#21424855	1	1	yes	characteristic
217	C₂	C₂×#16837232	1	1	yes	characteristic
218	C₂	#21425891	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#13179
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
100. C₄×C₅₉	#368	semidirect
127. C₅₉	#13179	semidirect
209. C₄	#16837232	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837232	2
#21424268	2
C₄×#14273663	2
#21422190	2
C₂×C₅₉×C₄⋊C₁₆	2
#13179	59