Subgroups — C₂×#21425885 #28741724

Order 15104.

718 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273667	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14273661	C₂	1	1	yes	maximal, characteristic
3	#21425885	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837229	C₂	1	2	yes	maximal, complemented
5	C₂×#16837227	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#427	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₅₉	—	2	1	no	—
8	C₂×#14273667	C₄	1	8	yes	complemented
9	C₂×#14273667	C₂²	1	4	yes	—
10	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
11	C₂×#14273661	—	2	1	no	—
12	C₂×#14273661	—	2	2	no	—
13	C₂×#14273661	C₂²	1	2	yes	—
14	C₂×#14273661	C₂²	1	4	yes	—
15	#16837229	C₂²	1	8	yes	complemented
16	#16837227	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×#117×C₅₉	—	2	1	no	—
19	C₅₉×#427	—	2	2	no	—
20	C₅₉×#427	C₂²	1	4	yes	—
21	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×#106×C₅₉	—	2	1	no	—
24	C₂³×Dic₅₉	—	4	1	no	—
25	C₂×C₄⋊Dic₅₉	—	2	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	4	no	—
27	C₂×C₄⋊Dic₅₉	—	4	1	no	—
28	C₂×C₄⋊Dic₅₉	—	4	2	no	—
29	#14273667	C₂×C₄	1	16	yes	complemented
30	C₂×C₅₉⋊C₁₆	C₂×C₄	1	8	yes	complemented
31	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
32	#14273661	—	2	4	no	—
33	#14273661	—	2	2	no	—
34	#14273661	C₂³	1	4	yes	—
35	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	D₄	1	2	yes	—
39	C₂²×C₈×C₅₉	Q₈	1	4	yes	—
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
42	#117×C₅₉	—	2	4	no	—
43	#117×C₅₉	—	4	2	no	—
44	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₅₉	—	4	1	no	—
47	C₂×C₄²×C₅₉	—	4	1	no	—
48	#107×C₅₉	—	2	1	no	—
49	#107×C₅₉	—	2	1	no	—
50	#107×C₅₉	C₂³	1	2	yes	—
51	#107×C₅₉	C₂³	1	2	yes	—
52	#106×C₅₉	—	2	4	no	—
53	C₂⁴×C₅₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	—	4	1	no	—
55	C₂²×C₄×C₅₉	—	4	1	no	—
56	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	D₈	1	2	yes	—
59	C₂²×C₄×C₅₉	Q₁₆	1	4	yes	—
60	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
62	C₂²×Dic₅₉	—	4	4	no	—
63	C₂²×Dic₅₉	—	4	2	no	—
64	C₂²×Dic₅₉	—	8	4	no	—
65	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
67	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	—	2	2	no	—
75	C₄⋊C₄×C₅₉	—	2	2	no	—
76	C₄⋊C₄×C₅₉	—	4	1	no	—
77	C₄⋊C₄×C₅₉	—	4	1	no	—
78	C₂²⋊C₄×C₅₉	—	4	4	no	—
79	C₄²×C₅₉	—	4	4	no	—
80	C₄⋊Dic₅₉	—	2	4	no	—
81	C₄⋊Dic₅₉	—	4	4	no	—
82	C₄⋊Dic₅₉	—	4	2	no	—
83	C₅₉⋊C₁₆	C₂²×C₄	1	8	yes	complemented
84	C₂³×C₅₉	—	2	2	no	—
85	C₂³×C₅₉	—	2	2	no	—
86	C₂³×C₅₉	#107	1	4	yes	—
87	C₂³×C₅₉	C₂×D₈	1	2	yes	—
88	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₉	—	4	2	no	—
90	C₂×C₄×C₅₉	—	4	2	no	—
91	C₂×C₄×C₅₉	—	8	2	no	—
92	C₂×C₄×C₅₉	—	8	2	no	—
93	C₂×C₄×C₅₉	#107	1	8	yes	—
94	C₂×C₄×C₅₉	#107	1	4	yes	—
95	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
100	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₅₉	—	4	4	no	—
103	C₂×Dic₅₉	—	8	8	no	—
104	C₂×Dic₅₉	—	8	4	no	—
105	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₈×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
107	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
109	C₂×#2159	—	59	1	no	maximal, Sylow
110	C₂²×C₅₉	—	2	2	no	—
111	C₂²×C₅₉	—	2	2	no	—
112	C₂²×C₅₉	—	2	8	no	—
113	C₂²×C₅₉	C₂×#107	1	4	yes	—
114	C₂²×C₅₉	C₂×#107	1	2	yes	—
115	C₂²×C₅₉	#509	1	2	yes	—
116	C₂²×C₅₉	#510	1	2	yes	—
117	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
118	C₄×C₅₉	—	8	2	no	—
119	C₄×C₅₉	—	8	2	no	—
120	C₄×C₅₉	C₂×#107	1	4	yes	—
121	C₄×C₅₉	C₂×#107	1	2	yes	—
122	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
124	Dic₅₉	—	8	4	no	—
125	C₂²×M₃₂	—	59	1	no	—
126	C₂×#427	D₅₉	1	1	yes	characteristic
127	C₂²×#107	—	59	1	no	—
128	#2159	—	59	8	no	—
129	C₂×#367	—	59	2	no	—
130	C₂×#366	—	59	2	no	—
131	C₂×C₅₉	—	2	4	no	—
132	C₂×C₅₉	#2159	1	4	yes	complemented
133	C₂×C₅₉	C₂²×#107	1	1	yes	characteristic
134	C₂×C₅₉	C₂×#509	1	1	yes	characteristic
135	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
136	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
137	C₂×#117	—	2	1	no	—
138	C₂²×C₄⋊C₄	—	118	1	no	—
139	C₂×M₃₂	—	59	8	no	—
140	C₂×M₃₂	—	59	4	no	—
141	C₂²×C₁₆	—	59	2	no	—
142	#427	—	2	2	no	—
143	#427	C₂×D₅₉	1	4	yes	—
144	C₂×#107	—	59	2	no	—
145	C₂×#107	—	59	4	no	—
146	C₂×#107	—	118	1	no	—
147	C₂×#107	—	118	2	no	—
148	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
149	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
150	C₂×#106	—	2	1	no	—
151	#367	—	59	8	no	—
152	#366	—	59	8	no	—
153	C₅₉	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	—	236	1	no	—
155	C₂³×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
156	C₂²×C₈	Dic₁₁₈	1	4	yes	—
157	C₂²×C₈	C₄×D₅₉	1	4	yes	—
158	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
160	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
162	#117	—	2	4	no	—
163	#117	—	4	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	118	4	no	—
167	C₂×C₄⋊C₄	—	118	2	no	—
168	C₂×C₄⋊C₄	—	236	2	no	—
169	C₂×C₄⋊C₄	—	236	1	no	—
170	C₂×C₂²⋊C₄	—	4	1	no	—
171	C₂×C₄²	—	4	1	no	—
172	M₃₂	—	59	16	no	—
173	C₂×C₁₆	—	59	8	no	—
174	C₂×C₁₆	—	59	4	no	—
175	#107	—	2	1	no	—
176	#107	—	2	1	no	—
177	#107	—	59	4	no	—
178	#107	—	118	4	no	—
179	#107	—	118	2	no	—
180	#107	C₂²×D₅₉	1	2	yes	—
181	#107	C₂²×D₅₉	1	2	yes	—
182	#106	—	2	4	no	—
183	C₂⁴	#11745669	1	1	yes	characteristic
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	472	4	no	—
187	C₂²×C₄	—	236	4	no	—
188	C₂²×C₄	—	236	2	no	—
189	C₂²×C₄	Dic₅₉⋊C₄	1	4	yes	—
190	C₂²×C₄	#11745669	1	2	yes	—
191	C₂²×C₄	#11745669	1	1	yes	characteristic
192	C₂²×C₄	#11745672	1	4	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
195	C₂×C₈	Dic₅₉⋊C₄	1	4	yes	—
196	C₂×C₈	Dic₅₉⋊C₄	1	8	yes	—
197	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
198	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
199	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
200	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
201	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
203	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	118	4	no	—
209	C₄⋊C₄	—	236	4	no	—
210	C₄⋊C₄	—	236	2	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	4	4	no	—
213	C₁₆	—	59	8	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#14273660	1	4	yes	—
217	C₂³	C₂×#11745669	1	2	yes	—
218	C₂³	C₂×#11745669	1	1	yes	characteristic
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	472	8	no	—
224	C₂×C₄	—	472	4	no	—
225	C₂×C₄	—	236	4	no	—
226	C₂×C₄	#14273660	1	8	yes	—
227	C₂×C₄	#14273660	1	4	yes	—
228	C₂×C₄	C₂×Dic₅₉⋊C₄	1	4	yes	—
229	C₂×C₄	C₂×Dic₅₉⋊C₄	1	2	yes	—
230	C₂×C₄	C₂×#11745669	1	2	yes	—
231	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
232	C₂×C₄	C₂×#11745672	1	4	yes	—
233	C₂×C₄	C₂×#11745672	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
235	C₈	C₂×Dic₅₉⋊C₄	1	2	yes	—
236	C₈	C₂×Dic₅₉⋊C₄	1	4	yes	—
237	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
238	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	8	no	—
242	C₂²	C₂×#14273660	1	4	yes	—
243	C₂²	C₂×#14273660	1	2	yes	—
244	C₂²	#16837837	1	2	yes	—
245	C₂²	#16837842	1	2	yes	—
246	C₂²	C₂²×#11745669	1	1	yes	characteristic
247	C₄	—	8	2	no	—
248	C₄	—	8	2	no	—
249	C₄	—	472	4	no	—
250	C₄	C₂×#14273660	1	4	yes	—
251	C₄	C₂×#14273660	1	2	yes	—
252	C₄	C₂²×Dic₅₉⋊C₄	1	1	yes	characteristic
253	C₄	C₂²×#11745672	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#21425885	1	4	yes	direct product factor
256	C₂	C₂²×#14273660	1	1	yes	characteristic
257	C₂	C₂×#16837837	1	1	yes	characteristic
258	C₂	C₂×#16837842	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#427	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂⁴×C₅₉	D₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#427	D₅₉
hypocenter	C₅₉	C₂×#2159
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #21425885	C₂	direct (N × H)
4. C₂×#16837229	C₂	semidirect
5. C₂×#16837227	C₂	semidirect
8. C₂×#14273667	C₄	semidirect
15. #16837229	C₂²	semidirect
16. #16837227	C₂²	semidirect
29. #14273667	C₂×C₄	semidirect
30. C₂×C₅₉⋊C₁₆	C₂×C₄	semidirect
83. C₅₉⋊C₁₆	C₂²×C₄	semidirect
132. C₂×C₅₉	#2159	semidirect
153. C₅₉	C₂×#2159	semidirect
255. C₂	#21425885	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14273667	2
C₂²×#14273661	2
#21425885	2
C₂×#16837229	2
C₂×#16837227	2
C₂×C₅₉×#427	2
C₂×#2159	59

Subgroups — C2×#21425885 #28741724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425885 #28741724