Subgroups — C₂×#21423690 #28576335

Order 15104.

814 conjugacy classes of subgroups.

The subgroup lattice has 244 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×#16837798	C₂	1	2	yes	maximal
3	C₂×#16837754	C₂	1	2	yes	maximal
4	#21423690	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic
7	C₂×#14273727	C₂²	1	2	yes	—
8	#16837807	—	2	2	no	—
9	#16837807	C₂²	1	4	yes	—
10	#16837798	—	2	4	no	—
11	#16837798	C₂²	1	8	yes	—
12	#16837754	—	2	4	no	—
13	#16837754	C₂²	1	8	yes	—
14	C₂×#14273661	—	2	2	no	—
15	C₂×#14273658	—	2	4	no	—
16	C₂×#14273650	—	2	4	no	—
17	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×C₅₉⋊C₈	C₂²	1	4	yes	—
20	C₂²×Q₁₆×C₅₉	—	2	2	no	—
21	C₂×C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
22	C₅₉×#494	—	2	2	no	—
23	C₅₉×#494	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₁₈	—	2	2	no	—
26	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
27	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
28	#14273727	—	2	2	no	—
29	#14273727	—	2	2	no	—
30	#14273727	—	2	2	no	—
31	#14273727	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₅₉	—	2	2	no	—
33	C₂×C₄⋊Dic₅₉	—	2	2	no	—
34	#14273661	—	2	8	no	—
35	#14273658	—	2	16	no	—
36	#14273650	—	2	16	no	—
37	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
38	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
39	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
40	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
41	C₂²×Q₈×C₅₉	—	2	2	no	—
42	C₂×Q₁₆×C₅₉	—	2	8	no	—
43	C₂×Q₁₆×C₅₉	—	4	8	no	—
44	C₂²×C₈×C₅₉	D₄	1	2	yes	—
45	C₄⋊Q₈×C₅₉	—	2	2	no	—
46	C₄⋊Q₈×C₅₉	—	2	2	no	—
47	C₄⋊Q₈×C₅₉	—	2	2	no	—
48	C₄⋊Q₈×C₅₉	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
51	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₅₉	D₄	1	2	yes	—
53	C₄×C₈×C₅₉	C₂³	1	2	yes	—
54	C₂×Q₈×C₅₉	—	2	4	no	—
55	C₂×Q₈×C₅₉	—	4	2	no	—
56	C₂×Q₈×C₅₉	—	4	8	no	—
57	C₂²×C₄×C₅₉	—	4	2	no	—
58	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₅₉	—	4	2	no	—
62	C₂×Dic₁₁₈	—	2	4	no	—
63	C₂×Dic₁₁₈	—	4	8	no	—
64	C₂×Dic₁₁₈	—	4	2	no	—
65	Q₁₆×C₅₉	—	4	16	no	—
66	C₂×C₈×C₅₉	SD₁₆	1	8	yes	—
67	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
68	C₄⋊C₄×C₅₉	—	2	4	no	—
69	C₄⋊C₄×C₅₉	—	2	4	no	—
70	C₄⋊C₄×C₅₉	—	4	2	no	—
71	C₄⋊C₄×C₅₉	—	4	2	no	—
72	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
76	C₄⋊Dic₅₉	—	2	4	no	—
77	C₄⋊Dic₅₉	—	2	4	no	—
78	C₄⋊Dic₅₉	—	4	2	no	—
79	C₄⋊Dic₅₉	—	4	2	no	—
80	C₂×C₅₉⋊C₈	—	2	4	no	—
81	C₂×C₅₉⋊C₈	Q₁₆	1	8	yes	complemented
82	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
83	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
84	C₂³×C₅₉	#127	1	1	yes	characteristic
85	Q₈×C₅₉	—	4	8	no	—
86	Q₈×C₅₉	—	8	4	no	—
87	C₂×C₄×C₅₉	—	4	4	no	—
88	C₂×C₄×C₅₉	—	8	4	no	—
89	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
93	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₅₉	#135	1	4	yes	—
95	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₅₉	—	4	4	no	—
99	C₂×Dic₅₉	—	8	4	no	—
100	Dic₁₁₈	—	4	8	no	—
101	Dic₁₁₈	—	8	4	no	—
102	C₈×C₅₉	C₂×SD₁₆	1	8	yes	—
103	C₅₉⋊C₈	—	2	4	no	—
104	C₅₉⋊C₈	C₂×Q₁₆	1	8	yes	complemented
105	C₂×#1707	—	59	1	no	maximal, Sylow
106	C₂²×C₅₉	#492	1	2	yes	—
107	C₂²×C₅₉	#494	1	2	yes	—
108	C₂²×C₅₉	#495	1	2	yes	—
109	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
110	C₄×C₅₉	—	8	4	no	—
111	C₄×C₅₉	#492	1	2	yes	—
112	C₄×C₅₉	#494	1	2	yes	—
113	C₄×C₅₉	#495	1	2	yes	—
114	C₄×C₅₉	C₂²×SD₁₆	1	2	yes	—
115	C₄×C₅₉	C₂²×Q₁₆	1	2	yes	—
116	C₄×C₅₉	C₂×#135	1	2	yes	—
117	Dic₅₉	—	8	4	no	—
118	C₂×#500	—	59	1	no	—
119	C₂×#498	—	59	2	no	—
120	C₂×#494	D₅₉	1	1	yes	characteristic
121	C₂×#487	—	59	2	no	—
122	#1707	—	59	8	no	—
123	C₂×C₈²	—	59	1	no	—
124	C₂×C₅₉	#1707	1	4	yes	complemented
125	C₂×C₅₉	C₂×#492	1	1	yes	characteristic
126	C₂×C₅₉	C₂×#494	1	1	yes	characteristic
127	C₂×C₅₉	C₂×#495	1	1	yes	characteristic
128	C₂²×Q₁₆	—	2	2	no	—
129	C₂×C₄⋊Q₈	—	59	2	no	—
130	C₂×C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
131	#500	—	59	4	no	—
132	#500	—	118	2	no	—
133	#498	—	59	8	no	—
134	#498	—	118	4	no	—
135	#494	—	2	2	no	—
136	#494	C₂×D₅₉	1	4	yes	—
137	#487	—	59	8	no	—
138	#487	—	118	4	no	—
139	C₂×#107	—	118	2	no	—
140	C₂×#106	—	118	4	no	—
141	C₂×#103	—	118	4	no	—
142	C₂×C₄×C₈	—	59	1	no	—
143	C₂×C₄×C₈	—	59	1	no	—
144	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
145	C₈²	—	59	4	no	—
146	C₅₉	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	—	118	2	no	—
149	C₂×Q₁₆	—	2	8	no	—
150	C₂×Q₁₆	—	4	8	no	—
151	C₂²×C₈	—	59	2	no	—
152	C₂²×C₈	—	59	2	no	—
153	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	59	4	no	—
158	C₄⋊Q₈	—	118	2	no	—
159	C₄⋊Q₈	—	118	2	no	—
160	C₄⋊Q₈	—	118	2	no	—
161	C₄⋊Q₈	C₂²×D₅₉	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	118	2	no	—
165	C₂×C₄⋊C₄	—	118	2	no	—
166	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
167	#107	—	118	8	no	—
168	#106	—	118	16	no	—
169	#103	—	118	16	no	—
170	C₄×C₈	—	59	2	no	—
171	C₄×C₈	—	59	2	no	—
172	C₄×C₈	—	59	2	no	—
173	C₄×C₈	—	59	2	no	—
174	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
175	C₄×C₈	C₂²×D₅₉	1	2	yes	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	8	no	—
179	C₂×Q₈	—	118	4	no	—
180	C₂×Q₈	—	236	8	no	—
181	C₂×Q₈	—	236	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	236	2	no	—
184	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
187	Q₁₆	—	4	16	no	—
188	C₂×C₈	—	59	8	no	—
189	C₂×C₈	—	59	4	no	—
190	C₂×C₈	—	59	4	no	—
191	C₂×C₈	—	118	4	no	—
192	C₂×C₈	#11745670	1	8	yes	—
193	C₂×C₈	C₂×C₂²⋊D₅₉	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	118	4	no	—
199	C₄⋊C₄	—	118	4	no	—
200	C₄⋊C₄	—	236	2	no	—
201	C₄⋊C₄	—	236	2	no	—
202	C₄²	D₄×D₅₉	1	1	yes	characteristic
203	C₄²	D₄×D₅₉	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
205	C₄²	C₂³×D₅₉	1	1	yes	characteristic
206	C₂³	#14273723	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	Q₈	—	8	4	no	—
209	Q₈	—	472	4	no	—
210	Q₈	—	236	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	472	4	no	—
214	C₂×C₄	—	236	4	no	—
215	C₂×C₄	#14273723	1	1	yes	characteristic
216	C₂×C₄	#14273723	1	1	yes	characteristic
217	C₂×C₄	#14273723	1	1	yes	characteristic
218	C₂×C₄	C₂×#11745670	1	4	yes	—
219	C₂×C₄	Q₁₆×D₅₉	1	4	yes	—
220	C₂×C₄	#14273750	1	4	yes	—
221	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
224	C₈	—	59	8	no	—
225	C₈	—	118	4	no	—
226	C₈	C₂×#11745670	1	8	yes	—
227	C₂²	#16837774	1	2	yes	—
228	C₂²	#16837780	1	2	yes	—
229	C₂²	#16837787	1	2	yes	—
230	C₂²	C₂×#14273723	1	1	yes	characteristic
231	C₄	—	8	4	no	—
232	C₄	—	472	4	no	—
233	C₄	#16837774	1	2	yes	—
234	C₄	#16837780	1	2	yes	—
235	C₄	#16837787	1	2	yes	—
236	C₄	C₂²×#11745670	1	2	yes	—
237	C₄	C₂×Q₁₆×D₅₉	1	2	yes	—
238	C₄	C₂×#14273750	1	2	yes	—
239	C₂	#21423690	1	4	yes	direct product factor
240	C₂	C₂×#16837774	1	1	yes	characteristic
241	C₂	C₂×#16837780	1	1	yes	characteristic
242	C₂	C₂×#16837787	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273723
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#494	C₂
socle, prime radical	C₂³×C₅₉	#127
hypercenter	C₂×#494	D₅₉
hypocenter	C₅₉	C₂×#1707
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
4. #21423690	C₂	direct (N × H)
81. C₂×C₅₉⋊C₈	Q₁₆	semidirect
104. C₅₉⋊C₈	C₂×Q₁₆	semidirect
124. C₂×C₅₉	#1707	semidirect
146. C₅₉	C₂×#1707	semidirect
239. C₂	#21423690	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×#16837798	2
C₂×#16837754	2
#21423690	2
C₂×C₈×C₅₉⋊C₈	2
C₂×C₅₉×#494	2
C₂×#1707	59

Subgroups — C2×#21423690 #28576335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21423690 #28576335