Subgroups — C₂×#21426138 #28742916

Order 15104.

766 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273670	C₂	1	2	yes	maximal
2	C₂×C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
3	#21426138	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837206	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₃₆	—	2	1	no	—
9	C₂×#14273670	—	2	8	no	—
10	C₂×#14273670	C₂²	1	8	yes	—
11	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
12	C₄×D₄₇₂	—	2	2	no	—
13	C₄×D₄₇₂	C₂²	1	4	yes	—
14	C₂×#14273661	C₂²	1	1	yes	characteristic
15	C₂×#14273649	—	2	1	no	—
16	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	complemented
17	#16837206	C₂²	1	8	yes	complemented
18	C₂²×Q₁₆×C₅₉	—	2	1	no	—
19	C₂²×Q₁₆×C₅₉	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₅₉	—	2	1	no	—
21	C₅₉×#494	—	2	2	no	—
22	C₅₉×#494	C₂²	1	4	yes	complemented
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×D₂₃₆	—	2	1	no	—
25	C₂²×C₄×D₅₉	—	4	1	no	—
26	C₂×D₄₇₂	—	2	1	no	—
27	C₂×D₄₇₂	—	2	2	no	—
28	C₂×D₄₇₂	D₄	1	4	yes	—
29	C₂×D₄₇₂	C₂³	1	2	yes	—
30	C₄×D₂₃₆	—	2	4	no	—
31	C₄×D₂₃₆	—	4	2	no	—
32	C₂×C₄⋊Dic₅₉	—	2	1	no	—
33	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
34	#14273670	—	2	16	no	—
35	C₂×C₅₉⋊C₁₆	—	2	4	no	—
36	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
37	#14273661	—	2	1	no	—
38	#14273661	C₂³	1	2	yes	—
39	#14273649	—	2	4	no	—
40	C₂²×Q₈×C₅₉	—	2	2	no	—
41	C₂×Q₁₆×C₅₉	—	2	4	no	—
42	C₂×Q₁₆×C₅₉	—	2	2	no	—
43	C₂×Q₁₆×C₅₉	—	2	4	no	—
44	C₂×Q₁₆×C₅₉	—	2	4	no	—
45	C₂×Q₁₆×C₅₉	—	4	4	no	—
46	C₂×Q₁₆×C₅₉	C₂³	1	4	yes	—
47	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₅₉	—	2	4	no	—
50	C₄⋊Q₈×C₅₉	—	4	2	no	—
51	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
52	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
53	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₉	D₄	1	2	yes	—
55	C₄×C₈×C₅₉	C₂³	1	2	yes	—
56	C₂³×D₅₉	—	4	1	no	—
57	C₂×Q₈×C₅₉	—	2	4	no	—
58	C₂×Q₈×C₅₉	—	4	2	no	—
59	C₂×Q₈×C₅₉	—	4	4	no	—
60	C₂×Q₈×C₅₉	—	4	4	no	—
61	C₂²×C₄×C₅₉	—	4	2	no	—
62	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅₉	—	4	1	no	—
66	C₂×D₂₃₆	—	2	4	no	—
67	C₂×D₂₃₆	—	2	2	no	—
68	C₂×D₂₃₆	—	4	1	no	—
69	C₂×D₂₃₆	—	4	2	no	—
70	C₂×C₄×D₅₉	—	4	4	no	—
71	C₂×C₄×D₅₉	—	8	4	no	—
72	Q₁₆×C₅₉	—	2	4	no	—
73	Q₁₆×C₅₉	—	2	4	no	—
74	Q₁₆×C₅₉	—	4	8	no	—
75	Q₁₆×C₅₉	—	4	4	no	—
76	C₂×C₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₅₉	#41	1	2	yes	—
81	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	—	4	4	no	—
83	C₄⋊C₄×C₅₉	—	4	4	no	—
84	C₄²×C₅₉	D₈	1	2	yes	—
85	C₄²×C₅₉	C₂×D₄	1	2	yes	—
86	(C₂×D₅₉)⋊C₄	—	4	4	no	—
87	C₄⋊Dic₅₉	—	2	2	no	—
88	C₄⋊Dic₅₉	—	4	1	no	—
89	D₄₇₂	—	2	4	no	—
90	D₄₇₂	—	2	2	no	—
91	D₄₇₂	C₂×D₄	1	4	yes	—
92	C₅₉⋊C₁₆	—	2	4	no	—
93	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
94	C₂²×D₅₉	—	4	2	no	—
95	C₂²×D₅₉	—	4	4	no	—
96	C₂²×D₅₉	—	8	2	no	—
97	C₂²×D₅₉	—	8	2	no	—
98	Q₈×C₅₉	—	4	4	no	—
99	Q₈×C₅₉	—	4	4	no	—
100	Q₈×C₅₉	—	8	4	no	—
101	C₂×C₄×C₅₉	—	2	2	no	—
102	C₂×C₄×C₅₉	—	4	4	no	—
103	C₂×C₄×C₅₉	—	8	4	no	—
104	C₂×C₄×C₅₉	SD₃₂	1	4	yes	complemented
105	C₂×C₄×C₅₉	#121	1	2	yes	—
106	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₅₉	—	4	2	no	—
112	C₂×Dic₅₉	—	8	2	no	—
113	D₂₃₆	—	2	4	no	—
114	D₂₃₆	—	4	2	no	—
115	D₂₃₆	—	4	4	no	—
116	C₄×D₅₉	—	8	8	no	—
117	C₈×C₅₉	—	2	2	no	—
118	C₈×C₅₉	#121	1	2	yes	—
119	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
121	C₂×#2208	—	59	1	no	maximal, Sylow
122	C₂²×C₅₉	#459	1	2	yes	—
123	C₂²×C₅₉	C₂×SD₃₂	1	2	yes	—
124	C₂²×C₅₉	#510	1	2	yes	—
125	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
126	C₂×D₅₉	—	4	4	no	—
127	C₂×D₅₉	—	8	2	no	—
128	C₂×D₅₉	—	8	2	no	—
129	C₂×D₅₉	—	8	8	no	—
130	C₄×C₅₉	—	2	2	no	—
131	C₄×C₅₉	—	8	4	no	—
132	C₄×C₅₉	#459	1	2	yes	—
133	C₄×C₅₉	C₂×SD₃₂	1	4	yes	complemented
134	C₄×C₅₉	C₂×#121	1	1	yes	characteristic
135	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
136	Dic₅₉	—	8	2	no	—
137	C₂²×SD₃₂	—	59	2	no	—
138	C₂×#494	D₅₉	1	1	yes	characteristic, complemented
139	C₂×C₄×D₈	—	59	1	no	—
140	#2208	—	59	8	no	—
141	C₂×C₄⋊C₁₆	—	59	1	no	—
142	C₂×#358	—	59	2	no	—
143	C₂×C₅₉	#2208	1	4	yes	complemented
144	C₂×C₅₉	C₂×#459	1	1	yes	characteristic
145	C₂×C₅₉	C₂²×SD₃₂	1	1	yes	characteristic
146	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
147	D₅₉	—	8	4	no	—
148	C₂²×Q₁₆	—	2	1	no	—
149	C₂²×Q₁₆	C₂×D₅₉	1	2	yes	—
150	C₂²×D₈	—	59	1	no	—
151	C₂×C₄⋊Q₈	—	2	1	no	—
152	C₂×C₄×D₄	—	118	1	no	—
153	C₂×SD₃₂	—	59	8	no	—
154	C₂×SD₃₂	—	118	8	no	—
155	C₂²×C₁₆	—	59	2	no	—
156	#494	—	2	2	no	—
157	#494	C₂×D₅₉	1	4	yes	complemented
158	C₄×D₈	—	59	4	no	—
159	C₄×D₈	—	118	2	no	—
160	C₂×#107	—	59	1	no	—
161	C₂×#102	—	118	1	no	—
162	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
163	C₄⋊C₁₆	—	59	4	no	—
164	#358	—	59	8	no	—
165	C₅₉	C₂×#2208	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×D₄	—	118	1	no	—
168	C₂³×C₄	—	236	1	no	—
169	C₂×Q₁₆	—	2	4	no	—
170	C₂×Q₁₆	—	2	4	no	—
171	C₂×Q₁₆	—	2	4	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	4	4	no	—
174	C₂×Q₁₆	C₂²×D₅₉	1	4	yes	—
175	C₂×D₈	—	59	4	no	—
176	C₂×D₈	—	59	2	no	—
177	C₂×D₈	—	118	2	no	—
178	C₂×D₈	—	118	1	no	—
179	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
181	C₄⋊Q₈	—	2	4	no	—
182	C₄⋊Q₈	—	4	2	no	—
183	C₄×D₄	—	118	4	no	—
184	C₄×D₄	—	236	2	no	—
185	C₂×C₄⋊C₄	—	4	1	no	—
186	C₂×C₄⋊C₄	—	4	1	no	—
187	C₂×C₄⋊C₄	—	118	1	no	—
188	C₂×C₂²⋊C₄	—	236	1	no	—
189	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
190	SD₃₂	—	118	16	no	—
191	C₂×C₁₆	—	59	4	no	—
192	C₂×C₁₆	—	118	4	no	—
193	#107	—	59	2	no	—
194	#107	—	118	1	no	—
195	#102	—	118	4	no	—
196	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
197	C₄×C₈	C₂²×D₅₉	1	2	yes	—
198	C₂⁴	—	236	1	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×D₄	—	118	4	no	—
204	C₂×D₄	—	118	2	no	—
205	C₂×D₄	—	236	1	no	—
206	C₂×D₄	—	236	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	472	4	no	—
209	C₂²×C₄	—	236	1	no	—
210	C₂²×C₄	—	236	4	no	—
211	C₂²×C₄	#11745669	1	1	yes	characteristic
212	C₂²×C₄	#11745669	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	2	4	no	—
216	Q₁₆	—	4	8	no	—
217	Q₁₆	—	4	4	no	—
218	D₈	—	59	4	no	—
219	D₈	—	118	4	no	—
220	D₈	—	118	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	D₄×D₅₉	1	2	yes	—
223	C₂×C₈	#11745687	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
226	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	118	2	no	—
230	C₄⋊C₄	—	236	1	no	—
231	C₂²⋊C₄	—	236	4	no	—
232	C₄²	#11745669	1	2	yes	—
233	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
234	C₁₆	—	118	4	no	—
235	C₂³	—	472	2	no	—
236	C₂³	—	472	2	no	—
237	C₂³	—	236	2	no	—
238	C₂³	—	236	4	no	—
239	C₂³	C₂×#11745669	1	1	yes	characteristic
240	Q₈	—	4	4	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	8	4	no	—
243	D₄	—	118	4	no	—
244	D₄	—	236	2	no	—
245	D₄	—	236	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	472	2	no	—
250	C₂×C₄	—	472	8	no	—
251	C₂×C₄	—	236	2	no	—
252	C₂×C₄	#14273670	1	4	yes	complemented
253	C₂×C₄	#14273698	1	2	yes	—
254	C₂×C₄	C₂×#11745669	1	2	yes	—
255	C₂×C₄	C₂×#11745669	1	2	yes	—
256	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
257	C₂×C₄	#14273753	1	2	yes	—
258	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	#14273698	1	2	yes	—
261	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
262	C₈	C₂×#11745687	1	1	yes	characteristic
263	C₂²	—	472	2	no	—
264	C₂²	—	472	2	no	—
265	C₂²	—	472	8	no	—
266	C₂²	—	236	4	no	—
267	C₂²	#16837594	1	2	yes	—
268	C₂²	C₂×#14273670	1	2	yes	—
269	C₂²	#16837843	1	2	yes	—
270	C₂²	C₂²×#11745669	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	8	4	no	—
273	C₄	—	472	2	no	—
274	C₄	#16837594	1	2	yes	—
275	C₄	C₂×#14273670	1	4	yes	complemented
276	C₄	C₂×#14273698	1	1	yes	characteristic
277	C₄	C₂×#14273753	1	1	yes	characteristic
278	C₂	—	472	4	no	—
279	C₂	#21426138	1	4	yes	direct product factor
280	C₂	C₂×#16837594	1	1	yes	characteristic
281	C₂	C₂²×#14273670	1	1	yes	characteristic
282	C₂	C₂×#16837843	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#494	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂²×D₄₇₂	C₂²
hypercenter	C₂×#494	D₅₉
hypocenter	C₅₉	C₂×#2208
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #21426138	C₂	direct (N × H)
4. C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
5. C₂×#16837206	C₂	semidirect
6. C₂×C₅₉×#494	C₂	semidirect
16. C₄⋊(C₅₉⋊C₁₆)	C₂²	semidirect
17. #16837206	C₂²	semidirect
22. C₅₉×#494	C₂²	semidirect
104. C₂×C₄×C₅₉	SD₃₂	semidirect
133. C₄×C₅₉	C₂×SD₃₂	semidirect
138. C₂×#494	D₅₉	semidirect
143. C₂×C₅₉	#2208	semidirect
157. #494	C₂×D₅₉	semidirect
165. C₅₉	C₂×#2208	semidirect
252. C₂×C₄	#14273670	semidirect
275. C₄	C₂×#14273670	semidirect
279. C₂	#21426138	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×#14273670	2
C₂×C₄×D₄₇₂	2
#21426138	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×#16837206	2
C₂×C₅₉×#494	2
C₂×#2208	59

Subgroups — C2×#21426138 #28742916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426138 #28742916