Subgroups — C₂×#21426267 #28743419

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837800	C₂	1	1	yes	maximal, characteristic, complemented
2	#21426267	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837226	C₂	1	2	yes	maximal, complemented
4	C₂×#16837223	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837212	C₂	1	2	yes	maximal
6	C₂×C₅₉×#443	C₂	1	1	yes	maximal, characteristic
7	C₂×#14273727	—	2	1	no	—
8	C₂×#14273667	C₂²	1	2	yes	—
9	#16837800	—	2	2	no	—
10	#16837800	C₂²	1	4	yes	complemented
11	C₂×#14273659	—	2	1	no	—
12	C₂×#14273659	C₂²	1	2	yes	—
13	#16837226	C₂²	1	8	yes	complemented
14	#16837223	C₂²	1	4	yes	complemented
15	#16837212	C₂²	1	8	yes	—
16	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#131×C₅₉	—	2	1	no	—
18	C₅₉×#443	—	2	2	no	—
19	C₅₉×#443	C₂²	1	4	yes	—
20	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#104×C₅₉	—	2	1	no	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₁₈	—	4	1	no	—
24	#14273727	—	2	4	no	—
25	#14273727	—	4	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	—	4	1	no	—
28	#14273667	—	2	2	no	—
29	#14273667	C₂³	1	4	yes	—
30	C₂×C₅₉⋊C₁₆	—	2	2	no	—
31	#14273659	—	2	4	no	—
32	#14273659	—	2	2	no	—
33	#14273659	C₂³	1	4	yes	—
34	C₂×#41×C₅₉	—	2	1	no	—
35	#135×C₅₉	—	2	1	no	—
36	#135×C₅₉	—	2	1	no	—
37	#135×C₅₉	Q₈	1	2	yes	complemented
38	#135×C₅₉	C₂³	1	2	yes	—
39	C₂×Q₁₆×C₅₉	Q₈	1	1	yes	characteristic, complemented
40	C₂×SD₁₆×C₅₉	—	2	1	no	—
41	C₂×D₈×C₅₉	Q₈	1	1	yes	characteristic, complemented
42	#131×C₅₉	—	2	4	no	—
43	#131×C₅₉	—	4	2	no	—
44	C₂×M₁₆×C₅₉	—	2	1	no	—
45	C₂×M₁₆×C₅₉	—	4	1	no	—
46	C₂²×C₈×C₅₉	—	4	1	no	—
47	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
50	#108×C₅₉	—	2	1	no	—
51	#108×C₅₉	C₂³	1	2	yes	—
52	#104×C₅₉	—	2	4	no	—
53	C₄×C₈×C₅₉	D₄	1	2	yes	—
54	C₄×C₈×C₅₉	C₂³	1	2	yes	—
55	#41×C₅₉	—	2	2	no	—
56	#41×C₅₉	—	2	2	no	—
57	#41×C₅₉	—	4	1	no	—
58	#41×C₅₉	—	4	1	no	—
59	C₂×Q₈×C₅₉	—	2	1	no	—
60	C₂×D₄×C₅₉	—	2	1	no	—
61	C₂×D₄×C₅₉	—	4	1	no	—
62	C₂²×C₄×C₅₉	—	2	1	no	—
63	C₂²×C₄×C₅₉	—	4	1	no	—
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅₉	—	4	2	no	—
66	C₂×Dic₁₁₈	—	4	4	no	—
67	C₂×Dic₁₁₈	—	8	4	no	—
68	Q₁₆×C₅₉	—	2	1	no	—
69	Q₁₆×C₅₉	C₂×Q₈	1	2	yes	complemented
70	SD₁₆×C₅₉	—	2	2	no	—
71	SD₁₆×C₅₉	—	2	2	no	—
72	D₈×C₅₉	—	2	1	no	—
73	D₈×C₅₉	C₂×Q₈	1	2	yes	complemented
74	M₁₆×C₅₉	—	2	2	no	—
75	M₁₆×C₅₉	—	4	1	no	—
76	M₁₆×C₅₉	—	4	4	no	—
77	C₂×C₈×C₅₉	—	4	1	no	—
78	C₂×C₈×C₅₉	—	4	4	no	—
79	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
80	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
81	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
82	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
89	C₄²×C₅₉	—	2	1	no	—
90	C₄²×C₅₉	C₂×D₄	1	2	yes	—
91	C₄⋊Dic₅₉	—	2	4	no	—
92	C₄⋊Dic₅₉	—	4	4	no	—
93	C₄⋊Dic₅₉	—	4	2	no	—
94	C₅₉⋊C₁₆	—	2	4	no	—
95	C₂³×C₅₉	—	4	1	no	—
96	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	Q₈×C₅₉	—	2	2	no	—
98	Q₈×C₅₉	—	4	1	no	—
99	D₄×C₅₉	—	2	2	no	—
100	D₄×C₅₉	—	4	2	no	—
101	D₄×C₅₉	—	4	2	no	—
102	D₄×C₅₉	—	4	1	no	—
103	C₂×C₄×C₅₉	—	2	4	no	—
104	C₂×C₄×C₅₉	—	2	2	no	—
105	C₂×C₄×C₅₉	—	4	1	no	—
106	C₂×C₄×C₅₉	—	4	2	no	—
107	C₂×C₄×C₅₉	—	4	2	no	—
108	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₅₉	—	4	4	no	—
115	C₂×Dic₅₉	—	8	4	no	—
116	Dic₁₁₈	—	8	8	no	—
117	C₈×C₅₉	—	4	2	no	—
118	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
121	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
122	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
124	C₂×#2230	—	59	1	no	maximal, Sylow
125	C₂²×C₅₉	—	2	2	no	—
126	C₂²×C₅₉	—	4	2	no	—
127	C₂²×C₅₉	—	8	2	no	—
128	C₂²×C₅₉	#476	1	1	yes	characteristic
129	C₂²×C₅₉	#476	1	1	yes	characteristic
130	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₄×C₅₉	—	2	4	no	—
132	C₄×C₅₉	—	4	2	no	—
133	C₄×C₅₉	#476	1	1	yes	characteristic
134	C₄×C₅₉	#476	1	1	yes	characteristic
135	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
137	Dic₅₉	—	8	4	no	—
138	C₂×#498	—	59	1	no	—
139	C₂×#443	D₅₉	1	1	yes	characteristic
140	#2230	—	59	8	no	—
141	C₂×#365	—	59	2	no	—
142	C₂×#364	—	59	1	no	—
143	C₂×#360	—	59	2	no	—
144	C₂×C₅₉	—	2	1	no	—
145	C₂×C₅₉	—	2	1	no	—
146	C₂×C₅₉	—	8	2	no	—
147	C₂×C₅₉	#2230	1	2	yes	complemented
148	C₂×C₅₉	C₂×#476	1	1	yes	characteristic
149	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
150	C₂×#131	—	2	1	no	—
151	C₂×C₄⋊Q₈	—	118	1	no	—
152	C₂×M₃₂	—	59	2	no	—
153	#498	—	59	4	no	—
154	#498	—	118	2	no	—
155	#443	—	2	2	no	—
156	#443	C₂×D₅₉	1	4	yes	—
157	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
158	C₂×#106	—	59	2	no	—
159	C₂×#106	—	118	1	no	—
160	C₂×#104	—	2	1	no	—
161	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
162	#365	—	59	8	no	—
163	#364	—	59	4	no	—
164	#360	—	59	8	no	—
165	C₅₉	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
166	C₂×#41	—	2	1	no	—
167	C₂²×Q₈	—	236	1	no	—
168	#135	—	2	1	no	—
169	#135	—	2	1	no	—
170	#135	Dic₁₁₈	1	2	yes	complemented
171	#135	C₂²×D₅₉	1	2	yes	—
172	C₂×Q₁₆	Dic₁₁₈	1	1	yes	characteristic, complemented
173	C₂×SD₁₆	—	2	1	no	—
174	C₂×D₈	Dic₁₁₈	1	1	yes	characteristic, complemented
175	#131	—	2	4	no	—
176	#131	—	4	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	4	1	no	—
179	C₂²×C₈	—	4	1	no	—
180	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
182	C₄⋊Q₈	—	118	4	no	—
183	C₄⋊Q₈	—	236	2	no	—
184	C₂×C₄⋊C₄	—	118	2	no	—
185	C₂×C₄⋊C₄	—	236	1	no	—
186	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
187	M₃₂	—	59	4	no	—
188	M₃₂	—	118	2	no	—
189	C₂×C₁₆	—	118	2	no	—
190	#108	—	2	1	no	—
191	#108	C₂²×D₅₉	1	2	yes	—
192	#106	—	59	4	no	—
193	#106	—	118	2	no	—
194	#106	—	118	4	no	—
195	#104	—	2	4	no	—
196	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
197	C₄×C₈	C₂²×D₅₉	1	2	yes	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	4	1	no	—
201	#41	—	4	1	no	—
202	C₂×Q₈	—	2	1	no	—
203	C₂×Q₈	—	472	4	no	—
204	C₂×Q₈	—	236	4	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	236	2	no	—
210	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
211	Q₁₆	—	2	1	no	—
212	Q₁₆	C₂×Dic₁₁₈	1	2	yes	complemented
213	SD₁₆	—	2	2	no	—
214	SD₁₆	—	2	2	no	—
215	D₈	—	2	1	no	—
216	D₈	C₂×Dic₁₁₈	1	2	yes	complemented
217	M₁₆	—	2	2	no	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	4	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	#11745670	1	1	yes	characteristic
223	C₂×C₈	#11745670	1	1	yes	characteristic
224	C₂×C₈	#11745670	1	2	yes	—
225	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
226	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
227	C₂×C₈	#11745685	1	1	yes	characteristic
228	C₂×C₈	#11745685	1	1	yes	characteristic
229	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
231	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
232	C₄⋊C₄	—	118	4	no	—
233	C₄⋊C₄	—	236	4	no	—
234	C₄⋊C₄	—	236	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
237	C₁₆	—	118	4	no	—
238	C₂³	—	4	1	no	—
239	C₂³	#14273752	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	4	1	no	—
242	Q₈	—	472	8	no	—
243	D₄	—	2	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	4	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	472	4	no	—
253	C₂×C₄	—	236	4	no	—
254	C₂×C₄	#14273706	1	1	yes	characteristic
255	C₂×C₄	#14273706	1	1	yes	characteristic
256	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
257	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
258	C₂×C₄	#14273752	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	#14273706	1	1	yes	characteristic
262	C₈	#14273706	1	1	yes	characteristic
263	C₈	C₂×#11745670	1	2	yes	—
264	C₈	C₂×#11745670	1	2	yes	—
265	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
266	C₈	C₂×#11745685	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	#16837692	1	1	yes	characteristic
271	C₂²	#16837692	1	1	yes	characteristic
272	C₂²	C₂×#14273752	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	4	2	no	—
275	C₄	—	472	4	no	—
276	C₄	#16837692	1	1	yes	characteristic
277	C₄	#16837692	1	1	yes	characteristic
278	C₄	C₂×#14273706	1	1	yes	characteristic
279	C₄	C₂²×#11745670	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	8	2	no	—
283	C₂	#21426267	1	2	yes	direct product factor
284	C₂	C₂×#16837692	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837692
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#443	C₂
socle	C₂²×C₅₉	#476
prime radical	C₂×#135×C₅₉	C₂²
hypercenter	C₂×#443	D₅₉
hypocenter	C₅₉	C₂×#2230
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16837800	C₂	semidirect
2. #21426267	C₂	direct (N × H)
3. C₂×#16837226	C₂	semidirect
4. C₂×#16837223	C₂	semidirect
10. #16837800	C₂²	semidirect
13. #16837226	C₂²	semidirect
14. #16837223	C₂²	semidirect
37. #135×C₅₉	Q₈	semidirect
39. C₂×Q₁₆×C₅₉	Q₈	semidirect
41. C₂×D₈×C₅₉	Q₈	semidirect
69. Q₁₆×C₅₉	C₂×Q₈	semidirect
73. D₈×C₅₉	C₂×Q₈	semidirect
147. C₂×C₅₉	#2230	semidirect
165. C₅₉	C₂×#2230	semidirect
170. #135	Dic₁₁₈	semidirect
172. C₂×Q₁₆	Dic₁₁₈	semidirect
174. C₂×D₈	Dic₁₁₈	semidirect
212. Q₁₆	C₂×Dic₁₁₈	semidirect
216. D₈	C₂×Dic₁₁₈	semidirect
283. C₂	#21426267	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×#16837800	2
#21426267	2
C₂×#16837226	2
C₂×#16837223	2
C₂×#16837212	2
C₂×C₅₉×#443	2
C₂×#2230	59

Subgroups — C2×#21426267 #28743419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426267 #28743419