Subgroups — C₂×#21422152 #28499968

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×#16837807	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#21422152	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₉×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273727	—	2	1	no	—
6	C₂²×C₅₉⋊C₁₆	—	2	1	no	—
7	#16837807	—	2	1	no	—
8	#16837807	—	2	1	no	—
9	#16837807	C₄	1	2	yes	—
10	#16837807	C₂²	1	2	yes	—
11	C₂×#14273661	—	2	1	no	—
12	C₂×#14273661	C₄	1	2	yes	—
13	C₄⋊(C₅₉⋊C₁₆)	—	2	1	no	—
14	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#335	—	2	1	no	—
18	C₅₉×#335	C₂²	1	2	yes	—
19	C₂²×Dic₁₁₈	—	4	1	no	—
20	#14273727	—	2	2	no	—
21	#14273727	—	2	2	no	—
22	#14273727	—	4	1	no	—
23	#14273727	—	4	1	no	—
24	C₂×C₄⋊Dic₅₉	—	2	2	no	—
25	C₂×C₄⋊Dic₅₉	—	4	1	no	—
26	C₂×C₅₉⋊C₁₆	—	2	2	no	—
27	C₂×C₅₉⋊C₁₆	—	4	2	no	—
28	#14273661	—	2	2	no	—
29	#14273661	—	2	2	no	—
30	#14273661	—	2	2	no	—
31	#14273661	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₉	—	4	1	no	—
33	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₉	—	2	2	no	—
37	C₄⋊C₈×C₅₉	—	4	1	no	—
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	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
45	C₂²×Dic₅₉	—	4	2	no	—
46	C₂×Dic₁₁₈	—	4	2	no	—
47	C₂×Dic₁₁₈	—	4	2	no	—
48	C₂×Dic₁₁₈	—	8	2	no	—
49	C₂×Dic₁₁₈	—	8	2	no	—
50	C₂×C₈×C₅₉	—	2	2	no	—
51	C₂×C₈×C₅₉	—	4	2	no	—
52	C₂×C₈×C₅₉	—	8	2	no	—
53	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
57	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₉	D₈	1	1	yes	characteristic
61	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
62	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₄⋊Dic₅₉	—	2	4	no	—
64	C₄⋊Dic₅₉	—	4	2	no	—
65	C₄⋊Dic₅₉	—	4	2	no	—
66	C₄⋊Dic₅₉	—	4	2	no	—
67	C₅₉⋊C₁₆	—	4	2	no	—
68	C₂³×C₅₉	#102	1	1	yes	characteristic
69	C₂×C₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	#101	1	2	yes	—
71	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
73	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#103	1	2	yes	—
75	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
76	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
77	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×Dic₅₉	—	4	4	no	—
81	C₂×Dic₅₉	—	8	4	no	—
82	Dic₁₁₈	—	8	4	no	—
83	Dic₁₁₈	—	8	2	no	—
84	Dic₁₁₈	—	8	2	no	—
85	C₈×C₅₉	—	2	1	no	—
86	C₈×C₅₉	—	2	1	no	—
87	C₈×C₅₉	—	8	2	no	—
88	C₈×C₅₉	#103	1	2	yes	—
89	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
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	#332	1	2	yes	—
99	C₄×C₅₉	#357	1	2	yes	—
100	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
101	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
102	C₄×C₅₉	C₂×D₁₆	1	1	yes	characteristic
103	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
104	Dic₅₉	—	8	4	no	—
105	C₂×#500	—	59	1	no	—
106	C₂×C₄⋊C₁₆	—	59	1	no	—
107	C₂×#335	D₅₉	1	1	yes	characteristic
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	—	118	1	no	—
114	C₂²×C₁₆	—	118	1	no	—
115	#500	—	59	2	no	—
116	#500	—	59	2	no	—
117	#500	—	118	1	no	—
118	#500	—	118	1	no	—
119	C₂×#107	—	59	2	no	—
120	C₂×#107	—	118	1	no	—
121	C₂×C₄⋊C₈	—	2	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	—	2	1	no	—
126	#335	C₂×D₅₉	1	2	yes	—
127	C₅₉	C₂×#1364	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	236	1	no	—
129	C₂²×C₈	—	4	1	no	—
130	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
131	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
132	C₄⋊Q₈	—	118	2	no	—
133	C₄⋊Q₈	—	118	2	no	—
134	C₄⋊Q₈	—	236	1	no	—
135	C₄⋊Q₈	—	236	1	no	—
136	C₂×C₄⋊C₄	—	118	2	no	—
137	C₂×C₄⋊C₄	—	236	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	—	59	4	no	—
142	#107	—	118	2	no	—
143	#107	—	118	2	no	—
144	#107	—	118	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	4	1	no	—
147	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
148	C₄×C₈	D₂₃₆	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	—	472	2	no	—
152	C₂×Q₈	—	472	2	no	—
153	C₂×Q₈	—	236	2	no	—
154	C₂×Q₈	—	236	2	no	—
155	C₂²×C₄	—	236	2	no	—
156	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
157	C₂²×C₄	#11745669	1	1	yes	characteristic
158	C₂²×C₄	#11745671	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	8	2	no	—
162	C₂×C₈	#11745660	1	2	yes	—
163	C₂×C₈	Dic₂₃₆	1	2	yes	—
164	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
165	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
166	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
167	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
168	C₄⋊C₄	—	118	4	no	—
169	C₄⋊C₄	—	236	2	no	—
170	C₄⋊C₄	—	236	2	no	—
171	C₄⋊C₄	—	236	2	no	—
172	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
173	C₄²	#11745669	1	1	yes	characteristic
174	C₄²	#11745671	1	1	yes	characteristic
175	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
176	C₁₆	—	236	2	no	—
177	C₂³	#14273647	1	1	yes	characteristic
178	Q₈	—	472	4	no	—
179	Q₈	—	472	2	no	—
180	Q₈	—	472	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	472	4	no	—
183	C₂×C₄	—	236	4	no	—
184	C₂×C₄	#14273645	1	2	yes	—
185	C₂×C₄	#14273647	1	1	yes	characteristic
186	C₂×C₄	#14273647	1	1	yes	characteristic
187	C₂×C₄	#14273647	1	1	yes	characteristic
188	C₂×C₄	#14273652	1	2	yes	—
189	C₂×C₄	#14273668	1	2	yes	—
190	C₂×C₄	#14273670	1	2	yes	—
191	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
192	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
193	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	8	2	no	—
197	C₈	#14273652	1	2	yes	—
198	C₈	C₂×#11745660	1	1	yes	characteristic
199	C₈	C₂×Dic₂₃₆	1	1	yes	characteristic
200	C₂²	#16837149	1	2	yes	—
201	C₂²	#16837202	1	2	yes	—
202	C₂²	#16837217	1	2	yes	—
203	C₂²	C₂×#14273647	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	472	4	no	—
207	C₄	#16837149	1	2	yes	—
208	C₄	#16837202	1	2	yes	—
209	C₄	C₂×#14273645	1	1	yes	characteristic
210	C₄	C₂×#14273652	1	1	yes	characteristic
211	C₄	C₂×#14273668	1	1	yes	characteristic
212	C₄	C₂×#14273670	1	1	yes	characteristic
213	C₂	#21422152	1	4	yes	direct product factor
214	C₂	C₂×#16837149	1	1	yes	characteristic
215	C₂	C₂×#16837202	1	1	yes	characteristic
216	C₂	C₂×#16837217	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273647
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#335	C₂
socle, prime radical	C₂³×C₅₉	#102
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×#335	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×#335

Complements

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

N	H ≅ G/N	type
3. #21422152	C₂	direct (N × H)
109. C₂×C₅₉	#1364	semidirect
127. C₅₉	C₂×#1364	semidirect
213. C₂	#21422152	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×#16837807	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
#21422152	2
C₂×C₅₉×#335	2
C₂×#1364	59

Subgroups — C2×#21422152 #28499968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422152 #28499968