Subgroups — C₂×#21426136 #28742933

Order 15104.

806 conjugacy classes of subgroups.

The subgroup lattice has 288 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, complemented
2	C₂×#16837779	C₂	1	1	yes	maximal, characteristic
3	#21426136	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837202	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₇₂	—	2	1	no	—
8	C₂²×D₄₇₂	C₂²	1	2	yes	—
9	C₂×#14273724	—	2	1	no	—
10	C₂×#14273670	—	2	8	no	—
11	C₂×#14273670	C₂²	1	8	yes	complemented
12	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
13	#16837779	—	2	2	no	—
14	#16837779	C₂²	1	4	yes	—
15	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	complemented
16	#16837202	C₂²	1	8	yes	complemented
17	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₉	—	2	1	no	—
19	C₄×Q₁₆×C₅₉	—	2	2	no	—
20	C₄×Q₁₆×C₅₉	C₂²	1	4	yes	complemented
21	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#103×C₅₉	—	2	1	no	—
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×D₂₃₆	—	2	2	no	—
25	C₂²×D₂₃₆	—	4	1	no	—
26	C₂²×D₂₃₆	—	4	1	no	—
27	C₂×D₄₇₂	—	2	4	no	—
28	C₂×D₄₇₂	—	2	4	no	—
29	C₂×D₄₇₂	—	2	4	no	—
30	C₂×D₄₇₂	—	2	2	no	—
31	C₂×D₄₇₂	—	4	4	no	—
32	C₂×D₄₇₂	C₂³	1	4	yes	—
33	#14273724	—	2	4	no	—
34	#14273724	—	4	2	no	—
35	#14273670	—	2	16	no	—
36	C₂×C₅₉⋊C₁₆	—	2	4	no	—
37	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
38	C₂²×Q₈×C₅₉	—	2	1	no	—
39	C₂×Q₁₆×C₅₉	—	2	1	no	—
40	C₂×Q₁₆×C₅₉	—	2	2	no	—
41	C₂×Q₁₆×C₅₉	D₄	1	4	yes	complemented
42	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₅₉	—	2	4	no	—
46	C₄×Q₈×C₅₉	—	4	2	no	—
47	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
49	C₂×C₄²×C₅₉	—	4	1	no	—
50	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
51	#107×C₅₉	—	2	1	no	—
52	#107×C₅₉	C₂³	1	2	yes	—
53	#103×C₅₉	—	2	4	no	—
54	C₄×C₈×C₅₉	D₄	1	2	yes	—
55	C₄×C₈×C₅₉	C₂³	1	2	yes	—
56	C₂³×D₅₉	—	4	2	no	—
57	C₂×Q₈×C₅₉	—	2	4	no	—
58	C₂×Q₈×C₅₉	—	2	2	no	—
59	C₂×Q₈×C₅₉	—	4	1	no	—
60	C₂×Q₈×C₅₉	—	4	2	no	—
61	C₂²×C₄×C₅₉	—	4	1	no	—
62	C₂²×C₄×C₅₉	—	4	1	no	—
63	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×D₂₃₆	—	2	4	no	—
67	C₂×D₂₃₆	—	4	4	no	—
68	C₂×D₂₃₆	—	4	4	no	—
69	C₂×D₂₃₆	—	4	4	no	—
70	C₂×D₂₃₆	—	4	4	no	—
71	C₂×D₂₃₆	—	4	2	no	—
72	C₂×D₂₃₆	—	8	4	no	—
73	C₂×D₂₃₆	—	8	4	no	—
74	Q₁₆×C₅₉	—	2	4	no	—
75	Q₁₆×C₅₉	—	2	2	no	—
76	Q₁₆×C₅₉	C₂×D₄	1	4	yes	complemented
77	C₂×C₈×C₅₉	—	2	2	no	—
78	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₉	#41	1	2	yes	—
82	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₅₉	—	2	2	no	—
84	C₄⋊C₄×C₅₉	—	4	1	no	—
85	C₄⋊C₄×C₅₉	—	4	4	no	—
86	C₄²×C₅₉	—	4	4	no	—
87	C₄²×C₅₉	D₈	1	2	yes	—
88	C₄²×C₅₉	C₂×D₄	1	2	yes	—
89	D₄₇₂	—	2	4	no	—
90	D₄₇₂	—	2	4	no	—
91	D₄₇₂	—	4	8	no	—
92	D₄₇₂	—	4	4	no	—
93	C₅₉⋊C₁₆	—	2	4	no	—
94	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
95	C₂²×D₅₉	—	4	4	no	—
96	C₂²×D₅₉	—	8	4	no	—
97	C₂²×D₅₉	—	8	4	no	—
98	C₂²×D₅₉	—	8	4	no	—
99	Q₈×C₅₉	—	2	4	no	—
100	Q₈×C₅₉	—	4	2	no	—
101	Q₈×C₅₉	—	4	4	no	—
102	C₂×C₄×C₅₉	—	2	2	no	—
103	C₂×C₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	—	4	4	no	—
105	C₂×C₄×C₅₉	—	4	2	no	—
106	C₂×C₄×C₅₉	—	8	2	no	—
107	C₂×C₄×C₅₉	SD₃₂	1	4	yes	complemented
108	C₂×C₄×C₅₉	#121	1	2	yes	—
109	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
114	D₂₃₆	—	4	4	no	—
115	D₂₃₆	—	4	4	no	—
116	D₂₃₆	—	8	8	no	—
117	D₂₃₆	—	8	4	no	—
118	D₂₃₆	—	8	8	no	—
119	C₈×C₅₉	—	2	2	no	—
120	C₈×C₅₉	#121	1	2	yes	—
121	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
123	C₂×#2207	—	59	1	no	maximal, Sylow
124	C₂²×C₅₉	#459	1	2	yes	—
125	C₂²×C₅₉	C₂×SD₃₂	1	2	yes	—
126	C₂²×C₅₉	#509	1	2	yes	—
127	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
128	C₂×D₅₉	—	8	4	no	—
129	C₂×D₅₉	—	8	4	no	—
130	C₂×D₅₉	—	8	4	no	—
131	C₂×D₅₉	—	16	8	no	—
132	C₄×C₅₉	—	2	2	no	—
133	C₄×C₅₉	—	4	4	no	—
134	C₄×C₅₉	—	8	2	no	—
135	C₄×C₅₉	#459	1	2	yes	—
136	C₄×C₅₉	C₂×SD₃₂	1	4	yes	complemented
137	C₄×C₅₉	C₂×#121	1	1	yes	characteristic
138	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂²×SD₃₂	—	59	2	no	—
140	C₂×#493	—	59	1	no	—
141	C₂×C₄×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
142	#2207	—	59	8	no	—
143	C₂×C₄⋊C₁₆	—	59	1	no	—
144	C₂×#357	—	59	2	no	—
145	C₂×C₅₉	#2207	1	4	yes	complemented
146	C₂×C₅₉	C₂×#459	1	1	yes	characteristic
147	C₂×C₅₉	C₂²×SD₃₂	1	1	yes	characteristic
148	C₂×C₅₉	C₂×#509	1	1	yes	characteristic
149	D₅₉	—	16	4	no	—
150	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
151	C₂²×D₈	—	59	2	no	—
152	C₂²×D₈	—	118	1	no	—
153	C₂×#127	—	118	1	no	—
154	C₂×C₄×Q₈	—	2	1	no	—
155	C₂×SD₃₂	—	59	8	no	—
156	C₂×SD₃₂	—	118	8	no	—
157	C₂²×C₁₆	—	59	2	no	—
158	#493	—	59	4	no	—
159	#493	—	118	2	no	—
160	C₄×Q₁₆	—	2	2	no	—
161	C₄×Q₁₆	C₂×D₅₉	1	4	yes	complemented
162	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
163	C₂×#103	—	2	1	no	—
164	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
165	C₄⋊C₁₆	—	59	4	no	—
166	#357	—	59	8	no	—
167	C₅₉	C₂×#2207	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×D₄	—	118	2	no	—
170	C₂²×D₄	—	236	1	no	—
171	C₂²×D₄	—	236	1	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	2	1	no	—
174	C₂×Q₁₆	D₂₃₆	1	4	yes	complemented
175	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
176	C₂×D₈	—	59	4	no	—
177	C₂×D₈	—	118	4	no	—
178	C₂×D₈	—	118	4	no	—
179	C₂×D₈	—	118	2	no	—
180	C₂×D₈	—	118	4	no	—
181	C₂×D₈	—	236	4	no	—
182	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
184	#127	—	118	4	no	—
185	#127	—	236	2	no	—
186	C₄×Q₈	—	2	4	no	—
187	C₄×Q₈	—	4	2	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	4	1	no	—
190	C₂×C₄²	—	4	1	no	—
191	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
192	SD₃₂	—	118	16	no	—
193	C₂×C₁₆	—	59	4	no	—
194	C₂×C₁₆	—	118	4	no	—
195	#107	—	2	1	no	—
196	#107	C₂²×D₅₉	1	2	yes	—
197	#103	—	2	4	no	—
198	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
199	C₄×C₈	C₂²×D₅₉	1	2	yes	—
200	C₂⁴	—	236	2	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×D₄	—	118	4	no	—
206	C₂×D₄	—	472	4	no	—
207	C₂×D₄	—	472	4	no	—
208	C₂×D₄	—	236	4	no	—
209	C₂×D₄	—	236	4	no	—
210	C₂×D₄	—	236	4	no	—
211	C₂×D₄	—	236	2	no	—
212	C₂×D₄	—	236	4	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	#11745669	1	1	yes	characteristic
216	C₂²×C₄	#11745669	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	2	no	—
220	Q₁₆	C₂×D₂₃₆	1	4	yes	complemented
221	D₈	—	118	4	no	—
222	D₈	—	118	4	no	—
223	D₈	—	236	8	no	—
224	D₈	—	236	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
227	C₂×C₈	#11745685	1	2	yes	—
228	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
229	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
230	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄²	—	4	4	no	—
235	C₄²	#11745669	1	2	yes	—
236	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
237	C₁₆	—	118	4	no	—
238	C₂³	—	472	4	no	—
239	C₂³	—	472	4	no	—
240	C₂³	—	472	4	no	—
241	C₂³	—	236	4	no	—
242	C₂³	C₂×#11745669	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	4	4	no	—
246	D₄	—	472	8	no	—
247	D₄	—	472	8	no	—
248	D₄	—	472	4	no	—
249	D₄	—	236	4	no	—
250	D₄	—	236	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	#14273670	1	4	yes	complemented
257	C₂×C₄	#14273701	1	2	yes	—
258	C₂×C₄	C₂×#11745669	1	2	yes	—
259	C₂×C₄	C₂×#11745669	1	2	yes	—
260	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
261	C₂×C₄	#14273752	1	2	yes	—
262	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	#14273701	1	2	yes	—
265	C₈	C₂²×D₂₃₆	1	1	yes	characteristic
266	C₈	C₂×#11745685	1	1	yes	characteristic
267	C₂²	—	944	8	no	—
268	C₂²	—	472	4	no	—
269	C₂²	—	472	4	no	—
270	C₂²	—	472	4	no	—
271	C₂²	#16837597	1	2	yes	—
272	C₂²	C₂×#14273670	1	2	yes	—
273	C₂²	#16837837	1	2	yes	—
274	C₂²	C₂²×#11745669	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	8	2	no	—
278	C₄	#16837597	1	2	yes	—
279	C₄	C₂×#14273670	1	4	yes	complemented
280	C₄	C₂×#14273701	1	1	yes	characteristic
281	C₄	C₂×#14273752	1	1	yes	characteristic
282	C₂	—	944	4	no	—
283	C₂	#21426136	1	4	yes	direct product factor
284	C₂	C₂×#16837597	1	1	yes	characteristic
285	C₂	C₂²×#14273670	1	1	yes	characteristic
286	C₂	C₂×#16837837	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×Q₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂×#16837779	C₂
hypercenter	C₂×C₄×Q₁₆	D₅₉
hypocenter	C₅₉	C₂×#2207
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#14273670	C₂	semidirect
3. #21426136	C₂	direct (N × H)
4. C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
5. C₂×#16837202	C₂	semidirect
6. C₂×C₄×Q₁₆×C₅₉	C₂	semidirect
11. C₂×#14273670	C₂²	semidirect
15. C₄⋊(C₅₉⋊C₁₆)	C₂²	semidirect
16. #16837202	C₂²	semidirect
20. C₄×Q₁₆×C₅₉	C₂²	semidirect
41. C₂×Q₁₆×C₅₉	D₄	semidirect
76. Q₁₆×C₅₉	C₂×D₄	semidirect
107. C₂×C₄×C₅₉	SD₃₂	semidirect
136. C₄×C₅₉	C₂×SD₃₂	semidirect
141. C₂×C₄×Q₁₆	D₅₉	semidirect
145. C₂×C₅₉	#2207	semidirect
161. C₄×Q₁₆	C₂×D₅₉	semidirect
167. C₅₉	C₂×#2207	semidirect
174. C₂×Q₁₆	D₂₃₆	semidirect
220. Q₁₆	C₂×D₂₃₆	semidirect
256. C₂×C₄	#14273670	semidirect
279. C₄	C₂×#14273670	semidirect
283. C₂	#21426136	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×#16837779	2
#21426136	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×#16837202	2
C₂×C₄×Q₁₆×C₅₉	2
C₂×#2207	59

Subgroups — C2×#21426136 #28742933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426136 #28742933