Subgroups — C₂×#21426270 #28743423

Order 15104.

526 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837802	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837519	C₂	1	1	yes	maximal, characteristic
3	#21426270	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837225	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837224	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837222	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16837212	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₉×#360	C₂	1	1	yes	maximal, characteristic
9	C₂×#14273734	—	2	1	no	—
10	C₂×#14273718	—	2	1	no	—
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	#16837802	—	2	2	no	—
13	#16837802	C₂²	1	4	yes	complemented
14	#16837519	—	2	2	no	—
15	#16837519	C₂²	1	4	yes	—
16	C₂×#14273663	C₂²	1	1	yes	characteristic
17	C₂×#14273660	—	2	1	no	—
18	C₂×#14273659	C₂²	1	1	yes	characteristic
19	C₂×#14273654	—	2	1	no	—
20	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
21	#16837225	C₂²	1	4	yes	complemented
22	#16837224	C₂²	1	4	yes	complemented
23	#16837222	C₂²	1	4	yes	complemented
24	#16837212	C₂²	1	4	yes	—
25	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#360	C₂²	1	4	yes	—
29	#14273734	—	2	4	no	—
30	#14273734	—	4	2	no	—
31	C₂×#11745664	—	2	1	no	—
32	C₂×#11745664	—	4	1	no	—
33	C₂²×C₅₉⋊C₈	—	4	1	no	—
34	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	#14273718	—	2	4	no	—
36	#14273718	—	4	2	no	—
37	C₂×C₄⋊Dic₅₉	—	2	1	no	—
38	C₂×C₄⋊Dic₅₉	—	4	1	no	—
39	C₂×Dic₅₉⋊C₄	—	4	1	no	—
40	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
41	#14273667	—	2	1	no	—
42	#14273667	C₂³	1	2	yes	—
43	C₂×C₅₉⋊C₁₆	—	2	1	no	—
44	#14273663	—	2	1	no	—
45	#14273663	C₂³	1	2	yes	—
46	#14273660	—	2	4	no	—
47	#14273659	—	2	1	no	—
48	#14273659	C₂³	1	2	yes	—
49	#14273654	—	2	4	no	—
50	C₈×Dic₅₉	D₄	1	2	yes	—
51	C₈×Dic₅₉	C₂³	1	2	yes	—
52	C₂×#41×C₅₉	—	2	1	no	—
53	#135×C₅₉	—	2	1	no	—
54	#135×C₅₉	—	2	1	no	—
55	#135×C₅₉	Q₈	1	2	yes	—
56	#135×C₅₉	C₂³	1	2	yes	—
57	C₂×Q₁₆×C₅₉	Q₈	1	1	yes	characteristic
58	C₂×SD₁₆×C₅₉	—	2	1	no	—
59	C₂×D₈×C₅₉	Q₈	1	1	yes	characteristic
60	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
62	M₃₂×C₅₉	—	2	1	no	—
63	M₃₂×C₅₉	C₂³	1	2	yes	—
64	C₂×C₁₆×C₅₉	—	2	1	no	—
65	#106×C₅₉	—	2	1	no	—
66	#106×C₅₉	C₂³	1	2	yes	—
67	#41×C₅₉	—	2	2	no	—
68	#41×C₅₉	—	2	2	no	—
69	#41×C₅₉	—	4	1	no	—
70	#41×C₅₉	—	4	1	no	—
71	C₂×Q₈×C₅₉	—	2	1	no	—
72	C₂×D₄×C₅₉	—	2	1	no	—
73	C₂×D₄×C₅₉	—	4	1	no	—
74	C₂²×C₄×C₅₉	—	4	1	no	—
75	C₂²×C₄×C₅₉	—	4	1	no	—
76	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₅₉	—	2	1	no	—
78	C₂²×Dic₅₉	—	4	1	no	—
79	Q₁₆×C₅₉	—	2	1	no	—
80	Q₁₆×C₅₉	C₂×Q₈	1	2	yes	—
81	SD₁₆×C₅₉	—	2	2	no	—
82	SD₁₆×C₅₉	—	2	2	no	—
83	D₈×C₅₉	—	2	1	no	—
84	D₈×C₅₉	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
89	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₉	—	2	2	no	—
92	C₄⋊C₄×C₅₉	—	4	1	no	—
93	C₄⋊Dic₅₉	—	2	2	no	—
94	C₄⋊Dic₅₉	—	4	4	no	—
95	C₄⋊Dic₅₉	—	4	1	no	—
96	Dic₅₉⋊C₄	—	4	4	no	—
97	C₄×Dic₅₉	—	2	1	no	—
98	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
99	#11745664	—	2	2	no	—
100	#11745664	—	4	1	no	—
101	#11745664	—	4	4	no	—
102	C₂×C₅₉⋊C₈	—	4	1	no	—
103	C₂×C₅₉⋊C₈	—	4	4	no	—
104	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	SD₁₆	1	2	yes	—
107	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
108	C₁₆×C₅₉	—	2	2	no	—
109	C₅₉⋊C₁₆	—	2	2	no	—
110	C₂³×C₅₉	—	4	1	no	—
111	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
112	Q₈×C₅₉	—	2	2	no	—
113	Q₈×C₅₉	—	4	1	no	—
114	D₄×C₅₉	—	2	2	no	—
115	D₄×C₅₉	—	4	2	no	—
116	D₄×C₅₉	—	4	2	no	—
117	D₄×C₅₉	—	4	1	no	—
118	C₂×C₄×C₅₉	—	4	1	no	—
119	C₂×C₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	—	4	2	no	—
122	C₂×C₄×C₅₉	—	8	2	no	—
123	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₅₉	—	2	2	no	—
130	C₂×Dic₅₉	—	4	2	no	—
131	C₂×Dic₅₉	—	4	2	no	—
132	C₂×Dic₅₉	—	8	2	no	—
133	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
134	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
136	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
137	C₅₉⋊C₈	—	4	2	no	—
138	C₅₉⋊C₈	C₂×SD₁₆	1	2	yes	—
139	C₅₉⋊C₈	C₂×SD₁₆	1	2	yes	—
140	C₂×#2231	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	—	2	2	no	—
142	C₂²×C₅₉	—	4	2	no	—
143	C₂²×C₅₉	—	8	2	no	—
144	C₂²×C₅₉	#476	1	1	yes	characteristic
145	C₂²×C₅₉	#476	1	1	yes	characteristic
146	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₄×C₅₉	—	4	2	no	—
148	C₄×C₅₉	—	8	2	no	—
149	C₄×C₅₉	#476	1	1	yes	characteristic
150	C₄×C₅₉	#476	1	1	yes	characteristic
151	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
153	Dic₅₉	—	4	2	no	—
154	Dic₅₉	—	8	2	no	—
155	C₂×#499	—	59	1	no	—
156	C₂×#443	—	59	1	no	—
157	#2231	—	59	8	no	—
158	C₂×#365	—	59	1	no	—
159	C₂×#365	—	59	1	no	—
160	C₂×#364	—	59	1	no	—
161	C₂×#360	—	59	1	no	—
162	C₂×#360	D₅₉	1	1	yes	characteristic
163	C₂×C₅₉	—	2	1	no	—
164	C₂×C₅₉	—	2	1	no	—
165	C₂×C₅₉	—	8	2	no	—
166	C₂×C₅₉	#2231	1	2	yes	complemented
167	C₂×C₅₉	C₂×#476	1	1	yes	characteristic
168	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
169	C₂×#131	—	118	1	no	—
170	C₂×#125	—	118	1	no	—
171	C₂×M₃₂	—	59	1	no	—
172	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
173	#499	—	59	4	no	—
174	#499	—	118	2	no	—
175	#443	—	59	4	no	—
176	#443	—	118	2	no	—
177	C₂×#108	—	59	1	no	—
178	C₂×#107	—	118	1	no	—
179	C₂×#106	—	59	1	no	—
180	C₂×#106	C₂×D₅₉	1	1	yes	characteristic
181	C₂×#104	—	118	1	no	—
182	C₂×C₄×C₈	—	59	1	no	—
183	#365	—	59	4	no	—
184	#365	—	59	4	no	—
185	#364	—	59	4	no	—
186	#360	—	59	4	no	—
187	#360	C₂×D₅₉	1	4	yes	—
188	C₅₉	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
189	C₂×#41	—	2	1	no	—
190	#135	—	2	1	no	—
191	#135	—	2	1	no	—
192	#135	Dic₁₁₈	1	2	yes	—
193	#135	C₂²×D₅₉	1	2	yes	—
194	C₂×Q₁₆	Dic₁₁₈	1	1	yes	characteristic
195	C₂×SD₁₆	—	2	1	no	—
196	C₂×D₈	Dic₁₁₈	1	1	yes	characteristic
197	#131	—	118	4	no	—
198	#131	—	236	2	no	—
199	C₂×M₁₆	—	118	1	no	—
200	C₂×M₁₆	—	236	1	no	—
201	C₂²×C₈	—	59	1	no	—
202	C₂²×C₈	—	236	1	no	—
203	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
204	#125	—	118	4	no	—
205	#125	—	236	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	118	1	no	—
208	C₂×C₄⋊C₄	—	236	1	no	—
209	C₂×C₄⋊C₄	—	236	1	no	—
210	C₂×C₄²	—	59	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	59	2	no	—
213	M₃₂	—	118	1	no	—
214	M₃₂	C₂²×D₅₉	1	2	yes	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	118	1	no	—
217	#108	—	59	2	no	—
218	#108	—	118	1	no	—
219	#107	—	118	4	no	—
220	#106	—	2	1	no	—
221	#106	—	59	2	no	—
222	#106	—	118	1	no	—
223	#106	C₂²×D₅₉	1	2	yes	—
224	#104	—	118	4	no	—
225	C₄×C₈	—	59	2	no	—
226	C₄×C₈	—	59	2	no	—
227	#41	—	2	2	no	—
228	#41	—	2	2	no	—
229	#41	—	4	1	no	—
230	#41	—	4	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	118	1	no	—
237	C₂²×C₄	—	236	1	no	—
238	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
239	Q₁₆	—	2	1	no	—
240	Q₁₆	C₂×Dic₁₁₈	1	2	yes	—
241	SD₁₆	—	2	2	no	—
242	SD₁₆	—	2	2	no	—
243	D₈	—	2	1	no	—
244	D₈	C₂×Dic₁₁₈	1	2	yes	—
245	M₁₆	—	118	2	no	—
246	M₁₆	—	236	4	no	—
247	M₁₆	—	236	1	no	—
248	C₂×C₈	—	59	1	no	—
249	C₂×C₈	—	59	1	no	—
250	C₂×C₈	—	59	2	no	—
251	C₂×C₈	—	59	2	no	—
252	C₂×C₈	—	236	4	no	—
253	C₂×C₈	—	236	1	no	—
254	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
255	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
256	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
257	C₂×C₈	#11745687	1	1	yes	characteristic
258	C₂×C₈	#11745687	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	118	2	no	—
263	C₄⋊C₄	—	236	4	no	—
264	C₄⋊C₄	—	236	4	no	—
265	C₄⋊C₄	—	236	1	no	—
266	C₄²	—	59	2	no	—
267	C₄²	—	118	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	118	2	no	—
270	C₂³	—	4	1	no	—
271	C₂³	#14273751	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	4	1	no	—
274	D₄	—	2	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	118	2	no	—
284	C₂×C₄	—	472	2	no	—
285	C₂×C₄	—	236	2	no	—
286	C₂×C₄	—	236	2	no	—
287	C₂×C₄	#14273704	1	1	yes	characteristic
288	C₂×C₄	#14273704	1	1	yes	characteristic
289	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
290	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
291	C₂×C₄	#14273751	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
293	C₈	—	59	2	no	—
294	C₈	—	59	2	no	—
295	C₈	—	236	2	no	—
296	C₈	#14273704	1	1	yes	characteristic
297	C₈	#14273704	1	1	yes	characteristic
298	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
299	C₈	C₂×#11745687	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	#16837690	1	1	yes	characteristic
304	C₂²	#16837690	1	1	yes	characteristic
305	C₂²	C₂×#14273751	1	1	yes	characteristic
306	C₄	—	4	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	472	2	no	—
309	C₄	—	236	2	no	—
310	C₄	#16837690	1	1	yes	characteristic
311	C₄	#16837690	1	1	yes	characteristic
312	C₄	C₂×#14273704	1	1	yes	characteristic
313	C₄	C₂×SD₁₆×D₅₉	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	8	2	no	—
317	C₂	#21426270	1	2	yes	direct product factor
318	C₂	C₂×#16837690	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837690
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#360	C₂
socle	C₂²×C₅₉	#476
prime radical	C₂×#135×C₅₉	C₂²
hypercenter	C₂×#360	D₅₉
hypocenter	C₅₉	C₂×#2231
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
1. C₂×#16837802	C₂	semidirect
3. #21426270	C₂	direct (N × H)
4. C₂×#16837225	C₂	semidirect
5. C₂×#16837224	C₂	semidirect
6. C₂×#16837222	C₂	semidirect
13. #16837802	C₂²	semidirect
21. #16837225	C₂²	semidirect
22. #16837224	C₂²	semidirect
23. #16837222	C₂²	semidirect
166. C₂×C₅₉	#2231	semidirect
188. C₅₉	C₂×#2231	semidirect
317. C₂	#21426270	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×#16837802	2
C₂×#16837519	2
#21426270	2
C₂×#16837225	2
C₂×#16837224	2
C₂×#16837222	2
C₂×#16837212	2
C₂×C₅₉×#360	2
C₂×#2231	59

Subgroups — C2×#21426270 #28743423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426270 #28743423