Subgroups — C₂×#21422914 #28558081

Order 15104.

718 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837526	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21422914	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837132	C₂	1	2	yes	maximal
5	C₂×#16837123	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₉×#334	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₃₆	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₁₈	C₂²	1	2	yes	—
9	#16837526	—	2	4	no	—
10	#16837526	C₂²	1	8	yes	—
11	C₄×Dic₂₃₆	C₄	1	4	yes	—
12	C₄×Dic₂₃₆	C₂²	1	4	yes	—
13	C₂×#14273661	C₄	1	1	yes	characteristic
14	C₂×Dic₅₉⋊C₈	—	2	2	no	—
15	C₂×#14273652	C₄	1	2	yes	—
16	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₅₉	—	2	2	no	—
19	#16837132	C₂²	1	8	yes	—
20	#16837123	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#334	C₂²	1	4	yes	—
24	C₂²×Dic₁₁₈	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₃₆	C₈	1	8	yes	complemented
26	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
27	C₂²×C₅₉⋊C₈	—	2	1	no	—
28	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
29	C₄×Dic₁₁₈	—	2	2	no	—
30	C₄×Dic₁₁₈	—	2	2	no	—
31	C₄×Dic₁₁₈	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₁₈	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
34	C₂×Dic₅₉⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₅₉	—	2	2	no	—
36	#14273661	C₂×C₄	1	4	yes	—
37	Dic₅₉⋊C₈	—	2	8	no	—
38	#14273652	C₂×C₄	1	8	yes	—
39	C₄×C₅₉⋊C₈	—	2	1	no	—
40	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
41	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
42	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
43	C₈×Dic₅₉	—	2	8	no	—
44	C₂²×C₈×C₅₉	—	2	2	no	—
45	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₉	—	2	2	no	—
49	C₄⋊C₈×C₅₉	C₂³	1	4	yes	—
50	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₅₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
55	C₂²×Dic₅₉	—	2	2	no	—
56	C₂²×Dic₅₉	—	2	2	no	—
57	C₂×Dic₁₁₈	—	2	2	no	—
58	C₂×Dic₁₁₈	—	2	4	no	—
59	C₂×Dic₁₁₈	C₂×C₈	1	8	yes	—
60	C₂×Dic₁₁₈	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	—	2	4	no	—
63	C₂×C₈×C₅₉	—	4	4	no	—
64	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
65	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
67	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₉	#41	1	1	yes	characteristic
70	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
71	C₄⋊Dic₅₉	—	2	2	no	—
72	C₄⋊Dic₅₉	C₂²×C₄	1	4	yes	—
73	Dic₅₉⋊C₄	—	2	4	no	—
74	Dic₅₉⋊C₄	—	2	4	no	—
75	C₄×Dic₅₉	—	2	4	no	—
76	C₄×Dic₅₉	—	2	4	no	—
77	C₂×C₅₉⋊C₈	—	2	4	no	—
78	C₂×C₅₉⋊C₈	—	2	2	no	—
79	C₂×C₅₉⋊C₈	—	4	2	no	—
80	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
81	Dic₂₃₆	C₂×C₈	1	16	yes	complemented
82	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₅₉	#131	1	2	yes	—
89	C₂×C₄×C₅₉	#137	1	4	yes	—
90	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
93	C₂×Dic₅₉	—	2	4	no	—
94	C₂×Dic₅₉	—	2	8	no	—
95	C₂×Dic₅₉	—	2	4	no	—
96	C₂×Dic₅₉	—	4	4	no	—
97	Dic₁₁₈	—	2	4	no	—
98	Dic₁₁₈	—	2	8	no	—
99	Dic₁₁₈	C₂²×C₈	1	8	yes	—
100	C₈×C₅₉	—	2	2	no	—
101	C₈×C₅₉	—	4	4	no	—
102	C₈×C₅₉	C₂²×C₈	1	4	yes	—
103	C₅₉⋊C₈	—	2	4	no	—
104	C₅₉⋊C₈	—	4	2	no	—
105	C₂×#1579	—	59	1	no	maximal, Sylow
106	C₂²×C₅₉	C₈×D₄	1	2	yes	—
107	C₂²×C₅₉	#441	1	2	yes	—
108	C₂²×C₅₉	#444	1	2	yes	—
109	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₅₉	—	2	1	no	—
111	C₄×C₅₉	—	2	1	no	—
112	C₄×C₅₉	C₈×D₄	1	2	yes	—
113	C₄×C₅₉	#441	1	2	yes	—
114	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
116	C₄×C₅₉	C₂×#137	1	2	yes	—
117	Dic₅₉	—	2	8	no	—
118	Dic₅₉	—	4	4	no	—
119	C₂×C₈×Q₈	—	59	2	no	—
120	C₂×C₄×Q₁₆	—	59	1	no	—
121	#1579	—	59	8	no	—
122	C₂×#334	D₅₉	1	1	yes	characteristic
123	C₂×Q₈⋊C₈	—	59	2	no	—
124	C₂×#322	—	59	1	no	—
125	C₂×C₅₉	#1579	1	4	yes	complemented
126	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
127	C₂×C₅₉	C₂×#441	1	1	yes	characteristic
128	C₂×C₅₉	C₂×#444	1	1	yes	characteristic
129	C₂²×Q₁₆	—	59	1	no	—
130	C₂×C₄×Q₈	—	59	2	no	—
131	C₈×Q₈	—	59	8	no	—
132	C₈×Q₈	—	118	4	no	—
133	C₄×Q₁₆	—	59	4	no	—
134	C₄×Q₁₆	—	59	4	no	—
135	C₂×#107	—	59	1	no	—
136	C₂×C₄⋊C₈	—	118	2	no	—
137	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
138	C₂×#103	—	59	2	no	—
139	C₂×C₄×C₈	—	59	1	no	—
140	C₂×C₄×C₈	—	59	1	no	—
141	C₂×C₄×C₈	—	118	2	no	—
142	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
143	#334	C₂×D₅₉	1	4	yes	—
144	Q₈⋊C₈	—	59	8	no	—
145	#322	—	59	4	no	—
146	C₅₉	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	59	2	no	—
148	C₂×Q₁₆	—	59	8	no	—
149	C₂×Q₁₆	—	59	4	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	59	2	no	—
152	C₂²×C₈	—	118	1	no	—
153	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
154	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
155	C₄×Q₈	—	59	4	no	—
156	C₄×Q₈	—	59	4	no	—
157	C₄×Q₈	—	118	2	no	—
158	C₄×Q₈	—	118	2	no	—
159	C₂×C₄⋊C₄	—	59	2	no	—
160	C₂×C₄⋊C₄	—	118	2	no	—
161	C₂×C₄²	—	118	2	no	—
162	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
163	#107	—	59	4	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	118	8	no	—
166	C₄⋊C₈	C₂²×D₅₉	1	4	yes	—
167	#103	—	59	8	no	—
168	C₄×C₈	—	59	2	no	—
169	C₄×C₈	—	59	2	no	—
170	C₄×C₈	—	59	2	no	—
171	C₄×C₈	—	118	8	no	—
172	C₄×C₈	—	118	1	no	—
173	C₄×C₈	C₄×D₅₉	1	2	yes	—
174	C₄×C₈	C₂²×D₅₉	1	2	yes	—
175	C₂×Q₈	—	59	8	no	—
176	C₂×Q₈	—	59	4	no	—
177	C₂×Q₈	—	118	2	no	—
178	C₂×Q₈	—	118	4	no	—
179	C₂²×C₄	—	118	2	no	—
180	C₂²×C₄	—	118	2	no	—
181	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
183	C₂²×C₄	#11745689	1	1	yes	characteristic
184	Q₁₆	—	59	16	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	59	4	no	—
189	C₂×C₈	—	118	4	no	—
190	C₂×C₈	—	118	2	no	—
191	C₂×C₈	—	236	2	no	—
192	C₂×C₈	C₈×D₅₉	1	4	yes	—
193	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
194	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
195	C₄⋊C₄	—	59	4	no	—
196	C₄⋊C₄	—	118	2	no	—
197	C₄⋊C₄	—	118	4	no	—
198	C₄⋊C₄	—	118	4	no	—
199	C₄²	—	118	4	no	—
200	C₄²	—	118	4	no	—
201	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
202	C₄²	D₄×D₅₉	1	1	yes	characteristic
203	C₄²	#11745689	1	1	yes	characteristic
204	C₄²	C₂³×D₅₉	1	1	yes	characteristic
205	C₂³	#14273689	1	1	yes	characteristic
206	Q₈	—	59	8	no	—
207	Q₈	—	118	4	no	—
208	Q₈	—	118	8	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	118	8	no	—
211	C₂×C₄	—	118	4	no	—
212	C₂×C₄	—	118	4	no	—
213	C₂×C₄	—	236	4	no	—
214	C₂×C₄	#14273689	1	1	yes	characteristic
215	C₂×C₄	#14273689	1	1	yes	characteristic
216	C₂×C₄	#14273689	1	1	yes	characteristic
217	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
218	C₂×C₄	#14273735	1	2	yes	—
219	C₂×C₄	#14273756	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
222	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	4	no	—
225	C₈	—	118	4	no	—
226	C₈	—	236	2	no	—
227	C₈	C₂×C₈×D₅₉	1	4	yes	—
228	C₂²	#16837467	1	2	yes	—
229	C₂²	#16837508	1	2	yes	—
230	C₂²	#16837523	1	2	yes	—
231	C₂²	C₂×#14273689	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	118	8	no	—
235	C₄	—	236	4	no	—
236	C₄	#16837467	1	2	yes	—
237	C₄	#16837508	1	2	yes	—
238	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
239	C₄	C₂×#14273735	1	1	yes	characteristic
240	C₄	C₂×#14273756	1	2	yes	—
241	C₂	#21422914	1	4	yes	direct product factor
242	C₂	C₂×#16837467	1	1	yes	characteristic
243	C₂	C₂×#16837508	1	1	yes	characteristic
244	C₂	C₂×#16837523	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₈
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#334	C₂
socle, prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂×#334	D₅₉
hypocenter	C₅₉	C₂×#1579
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #21422914	C₂	direct (N × H)
25. C₂×Dic₂₃₆	C₈	semidirect
81. Dic₂₃₆	C₂×C₈	semidirect
125. C₂×C₅₉	#1579	semidirect
146. C₅₉	C₂×#1579	semidirect
241. C₂	#21422914	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×#16837526	2
C₂×C₄×Dic₂₃₆	2
#21422914	2
C₂×#16837132	2
C₂×#16837123	2
C₂×C₅₉×#334	2
C₂×#1579	59

Subgroups — C2×#21422914 #28558081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422914 #28558081