Subgroups — C₂×#21422151 #28499967

Order 15104.

350 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
3	#21422151	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₉×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
9	#16837155	—	2	1	no	—
10	#16837155	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
12	C₅₉×#500	—	2	1	no	—
13	C₅₉×#500	—	2	1	no	—
14	C₅₉×#500	C₄	1	2	yes	—
15	C₅₉×#500	C₂²	1	2	yes	—
16	C₂×#107×C₅₉	—	2	1	no	—
17	C₂×#107×C₅₉	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₅₉⋊C₈	—	4	1	no	—
20	C₂×C₅₉⋊C₁₆	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	—	4	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
24	C₂²×Q₈×C₅₉	—	4	1	no	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₅₉	—	2	2	no	—
28	C₄⋊Q₈×C₅₉	—	2	2	no	—
29	C₄⋊Q₈×C₅₉	—	4	1	no	—
30	C₄⋊Q₈×C₅₉	—	4	1	no	—
31	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
33	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
34	#107×C₅₉	—	2	2	no	—
35	#107×C₅₉	—	2	2	no	—
36	#107×C₅₉	—	2	2	no	—
37	#107×C₅₉	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
42	C₂×Q₈×C₅₉	—	4	2	no	—
43	C₂×Q₈×C₅₉	—	4	2	no	—
44	C₂×Q₈×C₅₉	—	8	2	no	—
45	C₂×Q₈×C₅₉	—	8	2	no	—
46	C₂²×C₄×C₅₉	—	4	2	no	—
47	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₅₉	—	2	2	no	—
51	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
54	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
55	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₉	—	2	4	no	—
58	C₄⋊C₄×C₅₉	—	4	2	no	—
59	C₄⋊C₄×C₅₉	—	4	2	no	—
60	C₄⋊C₄×C₅₉	—	4	2	no	—
61	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₉	D₈	1	1	yes	characteristic
63	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
64	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₅₉⋊C₈	—	4	2	no	—
66	C₂×C₅₉⋊C₈	—	8	2	no	—
67	C₅₉⋊C₁₆	—	4	2	no	—
68	C₂³×C₅₉	#102	1	1	yes	characteristic
69	Q₈×C₅₉	—	8	4	no	—
70	Q₈×C₅₉	—	8	2	no	—
71	Q₈×C₅₉	—	8	2	no	—
72	C₂×C₄×C₅₉	—	2	2	no	—
73	C₂×C₄×C₅₉	—	4	4	no	—
74	C₂×C₄×C₅₉	—	8	4	no	—
75	C₂×C₄×C₅₉	#101	1	2	yes	—
76	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
78	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#103	1	2	yes	—
80	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
81	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
82	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
85	C₈×C₅₉	—	2	1	no	—
86	C₈×C₅₉	—	2	1	no	—
87	C₈×C₅₉	#103	1	2	yes	—
88	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
90	C₅₉⋊C₈	—	8	2	no	—
91	C₂×#1364	—	59	1	no	maximal, Sylow
92	C₂²×C₅₉	#332	1	2	yes	—
93	C₂²×C₅₉	#357	1	2	yes	—
94	C₂²×C₅₉	#362	1	2	yes	—
95	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
96	C₄×C₅₉	—	2	1	no	—
97	C₄×C₅₉	—	2	1	no	—
98	C₄×C₅₉	—	8	4	no	—
99	C₄×C₅₉	#332	1	2	yes	—
100	C₄×C₅₉	#357	1	2	yes	—
101	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
102	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
103	C₄×C₅₉	C₂×D₁₆	1	1	yes	characteristic
104	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
105	C₂×#500	D₅₉	1	1	yes	characteristic
106	C₂×C₄⋊C₁₆	—	59	1	no	—
107	C₂×#335	—	59	1	no	—
108	#1364	—	59	4	no	—
109	C₂×C₅₉	#1364	1	4	yes	complemented
110	C₂×C₅₉	C₂×#332	1	1	yes	characteristic
111	C₂×C₅₉	C₂×#357	1	1	yes	characteristic
112	C₂×C₅₉	C₂×#362	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂²×C₁₆	—	118	1	no	—
115	#500	—	2	1	no	—
116	#500	—	2	1	no	—
117	#500	Dic₅₉	1	2	yes	—
118	#500	C₂×D₅₉	1	2	yes	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	Dic₅₉	1	2	yes	—
121	C₂×C₄⋊C₈	—	118	1	no	—
122	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
123	C₄⋊C₁₆	—	59	2	no	—
124	C₄⋊C₁₆	—	118	1	no	—
125	#335	—	59	2	no	—
126	#335	—	118	1	no	—
127	C₅₉	C₂×#1364	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	4	1	no	—
129	C₂²×C₈	—	236	1	no	—
130	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
131	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₄⋊Q₈	—	4	1	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
139	C₂×C₁₆	—	118	2	no	—
140	C₂×C₁₆	—	236	2	no	—
141	#107	—	2	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	C₂×Dic₅₉	1	4	yes	—
145	C₄⋊C₈	—	118	2	no	—
146	C₄⋊C₈	—	236	1	no	—
147	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
150	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	8	2	no	—
154	C₂×Q₈	—	8	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	#11745669	1	1	yes	characteristic
157	C₂²×C₄	#11745670	1	1	yes	characteristic
158	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	472	2	no	—
161	C₂×C₈	—	236	2	no	—
162	C₂×C₈	#11745671	1	2	yes	—
163	C₂×C₈	#11745672	1	2	yes	—
164	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
165	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
166	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄²	#11745669	1	1	yes	characteristic
173	C₄²	#11745670	1	1	yes	characteristic
174	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
176	C₁₆	—	236	2	no	—
177	C₂³	#14273648	1	1	yes	characteristic
178	Q₈	—	8	4	no	—
179	Q₈	—	8	2	no	—
180	Q₈	—	8	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	8	4	no	—
184	C₂×C₄	#14273646	1	2	yes	—
185	C₂×C₄	#14273648	1	1	yes	characteristic
186	C₂×C₄	#14273648	1	1	yes	characteristic
187	C₂×C₄	#14273648	1	1	yes	characteristic
188	C₂×C₄	#14273651	1	2	yes	—
189	C₂×C₄	#14273668	1	2	yes	—
190	C₂×C₄	#14273671	1	2	yes	—
191	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
192	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
193	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	472	2	no	—
197	C₈	#14273651	1	2	yes	—
198	C₈	C₂×#11745671	1	1	yes	characteristic
199	C₈	C₂×#11745672	1	1	yes	characteristic
200	C₂²	#16837148	1	2	yes	—
201	C₂²	#16837203	1	2	yes	—
202	C₂²	#16837218	1	2	yes	—
203	C₂²	C₂×#14273648	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	8	4	no	—
207	C₄	#16837148	1	2	yes	—
208	C₄	#16837203	1	2	yes	—
209	C₄	C₂×#14273646	1	1	yes	characteristic
210	C₄	C₂×#14273651	1	1	yes	characteristic
211	C₄	C₂×#14273668	1	1	yes	characteristic
212	C₄	C₂×#14273671	1	1	yes	characteristic
213	C₂	#21422151	1	4	yes	direct product factor
214	C₂	C₂×#16837148	1	1	yes	characteristic
215	C₂	C₂×#16837203	1	1	yes	characteristic
216	C₂	C₂×#16837218	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273648
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#500	C₂
socle, prime radical	C₂³×C₅₉	#102
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×#500	D₅₉
hypocenter	C₅₉	C₂×#1364
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #21422151	C₂	direct (N × H)
109. C₂×C₅₉	#1364	semidirect
127. C₅₉	C₂×#1364	semidirect
213. C₂	#21422151	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×#16837155	2
#21422151	2
C₂×C₅₉×#500	2
C₂×#1364	59

Subgroups — C2×#21422151 #28499967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422151 #28499967