Subgroups — C₂×#21426265 #28743417

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837799	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837519	C₂	1	1	yes	maximal, characteristic
3	#21426265	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×#14273726	—	2	1	no	—
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	#16837799	—	2	2	no	—
13	#16837799	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×#14273659	C₂²	1	1	yes	characteristic
18	C₂×#14273658	—	2	1	no	—
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	C₂²×Dic₁₁₈	—	4	1	no	—
30	#14273734	—	2	4	no	—
31	#14273734	—	4	2	no	—
32	C₂×#11745664	—	2	1	no	—
33	C₂×#11745664	—	4	1	no	—
34	C₂²×C₅₉⋊C₈	—	4	1	no	—
35	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
36	#14273726	—	2	4	no	—
37	#14273726	—	4	2	no	—
38	C₂×C₄⋊Dic₅₉	—	2	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	#14273659	—	2	1	no	—
47	#14273659	C₂³	1	2	yes	—
48	#14273658	—	2	4	no	—
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	complemented
56	#135×C₅₉	C₂³	1	2	yes	—
57	C₂×Q₁₆×C₅₉	Q₈	1	1	yes	characteristic, complemented
58	C₂×SD₁₆×C₅₉	—	2	1	no	—
59	C₂×D₈×C₅₉	Q₈	1	1	yes	characteristic, complemented
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	C₂×Dic₁₁₈	—	4	4	no	—
80	C₂×Dic₁₁₈	—	8	4	no	—
81	Q₁₆×C₅₉	—	2	1	no	—
82	Q₁₆×C₅₉	C₂×Q₈	1	2	yes	complemented
83	SD₁₆×C₅₉	—	2	2	no	—
84	SD₁₆×C₅₉	—	2	2	no	—
85	D₈×C₅₉	—	2	1	no	—
86	D₈×C₅₉	C₂×Q₈	1	2	yes	complemented
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
91	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₉	—	2	2	no	—
94	C₄⋊C₄×C₅₉	—	4	1	no	—
95	C₄⋊Dic₅₉	—	2	2	no	—
96	C₄⋊Dic₅₉	—	4	1	no	—
97	Dic₅₉⋊C₄	—	4	4	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
100	#11745664	—	2	2	no	—
101	#11745664	—	4	1	no	—
102	#11745664	—	4	4	no	—
103	C₂×C₅₉⋊C₈	—	4	1	no	—
104	C₂×C₅₉⋊C₈	—	4	4	no	—
105	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	SD₁₆	1	2	yes	—
108	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
109	C₁₆×C₅₉	—	2	2	no	—
110	C₅₉⋊C₁₆	—	2	2	no	—
111	C₂³×C₅₉	—	4	1	no	—
112	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	Q₈×C₅₉	—	2	2	no	—
114	Q₈×C₅₉	—	4	1	no	—
115	D₄×C₅₉	—	2	2	no	—
116	D₄×C₅₉	—	4	2	no	—
117	D₄×C₅₉	—	4	2	no	—
118	D₄×C₅₉	—	4	1	no	—
119	C₂×C₄×C₅₉	—	4	1	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	—	4	2	no	—
122	C₂×C₄×C₅₉	—	4	2	no	—
123	C₂×C₄×C₅₉	—	8	2	no	—
124	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₅₉	—	2	4	no	—
131	C₂×Dic₅₉	—	2	2	no	—
132	C₂×Dic₅₉	—	4	2	no	—
133	C₂×Dic₅₉	—	8	2	no	—
134	Dic₁₁₈	—	8	8	no	—
135	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
139	C₅₉⋊C₈	—	4	2	no	—
140	C₅₉⋊C₈	C₂×SD₁₆	1	2	yes	—
141	C₅₉⋊C₈	C₂×SD₁₆	1	2	yes	—
142	C₂×#2230	—	59	1	no	maximal, Sylow
143	C₂²×C₅₉	—	2	2	no	—
144	C₂²×C₅₉	—	4	2	no	—
145	C₂²×C₅₉	—	8	2	no	—
146	C₂²×C₅₉	#476	1	1	yes	characteristic
147	C₂²×C₅₉	#476	1	1	yes	characteristic
148	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₅₉	—	4	2	no	—
150	C₄×C₅₉	—	8	2	no	—
151	C₄×C₅₉	#476	1	1	yes	characteristic
152	C₄×C₅₉	#476	1	1	yes	characteristic
153	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
154	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
155	Dic₅₉	—	2	4	no	—
156	Dic₅₉	—	8	2	no	—
157	C₂×#498	—	59	1	no	—
158	C₂×#443	—	59	1	no	—
159	#2230	—	59	8	no	—
160	C₂×#365	—	59	1	no	—
161	C₂×#365	—	59	1	no	—
162	C₂×#364	—	59	1	no	—
163	C₂×#360	—	59	1	no	—
164	C₂×#360	D₅₉	1	1	yes	characteristic
165	C₂×C₅₉	—	2	1	no	—
166	C₂×C₅₉	—	2	1	no	—
167	C₂×C₅₉	—	8	2	no	—
168	C₂×C₅₉	#2230	1	2	yes	complemented
169	C₂×C₅₉	C₂×#476	1	1	yes	characteristic
170	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
171	C₂×#131	—	118	1	no	—
172	C₂×C₄⋊Q₈	—	118	1	no	—
173	C₂×M₃₂	—	59	1	no	—
174	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
175	#498	—	59	4	no	—
176	#498	—	118	2	no	—
177	#443	—	59	4	no	—
178	#443	—	118	2	no	—
179	C₂×#108	—	59	1	no	—
180	C₂×#106	—	59	1	no	—
181	C₂×#106	—	118	1	no	—
182	C₂×#106	C₂×D₅₉	1	1	yes	characteristic
183	C₂×#104	—	118	1	no	—
184	C₂×C₄×C₈	—	59	1	no	—
185	#365	—	59	4	no	—
186	#365	—	59	4	no	—
187	#364	—	59	4	no	—
188	#360	—	59	4	no	—
189	#360	C₂×D₅₉	1	4	yes	—
190	C₅₉	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	2	1	no	—
192	C₂²×Q₈	—	236	1	no	—
193	#135	—	2	1	no	—
194	#135	—	2	1	no	—
195	#135	Dic₁₁₈	1	2	yes	complemented
196	#135	C₂²×D₅₉	1	2	yes	—
197	C₂×Q₁₆	Dic₁₁₈	1	1	yes	characteristic, complemented
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	Dic₁₁₈	1	1	yes	characteristic, complemented
200	#131	—	118	4	no	—
201	#131	—	236	2	no	—
202	C₂×M₁₆	—	118	1	no	—
203	C₂×M₁₆	—	236	1	no	—
204	C₂²×C₈	—	59	1	no	—
205	C₂²×C₈	—	236	1	no	—
206	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
207	C₄⋊Q₈	—	118	4	no	—
208	C₄⋊Q₈	—	236	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	118	1	no	—
211	C₂×C₄⋊C₄	—	236	1	no	—
212	C₂×C₄²	—	59	1	no	—
213	M₃₂	—	2	1	no	—
214	M₃₂	—	59	2	no	—
215	M₃₂	—	118	1	no	—
216	M₃₂	C₂²×D₅₉	1	2	yes	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	118	1	no	—
219	#108	—	59	2	no	—
220	#108	—	118	1	no	—
221	#106	—	2	1	no	—
222	#106	—	59	2	no	—
223	#106	—	118	4	no	—
224	#106	—	118	1	no	—
225	#106	C₂²×D₅₉	1	2	yes	—
226	#104	—	118	4	no	—
227	C₄×C₈	—	59	2	no	—
228	C₄×C₈	—	59	2	no	—
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	#41	—	4	1	no	—
232	#41	—	4	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	472	4	no	—
235	C₂×Q₈	—	236	4	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	118	1	no	—
241	C₂²×C₄	—	236	1	no	—
242	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
243	Q₁₆	—	2	1	no	—
244	Q₁₆	C₂×Dic₁₁₈	1	2	yes	complemented
245	SD₁₆	—	2	2	no	—
246	SD₁₆	—	2	2	no	—
247	D₈	—	2	1	no	—
248	D₈	C₂×Dic₁₁₈	1	2	yes	complemented
249	M₁₆	—	118	2	no	—
250	M₁₆	—	236	4	no	—
251	M₁₆	—	236	1	no	—
252	C₂×C₈	—	59	1	no	—
253	C₂×C₈	—	59	1	no	—
254	C₂×C₈	—	59	2	no	—
255	C₂×C₈	—	59	2	no	—
256	C₂×C₈	—	236	4	no	—
257	C₂×C₈	—	236	1	no	—
258	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
259	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
260	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
261	C₂×C₈	#11745687	1	1	yes	characteristic
262	C₂×C₈	#11745687	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	118	2	no	—
267	C₄⋊C₄	—	236	4	no	—
268	C₄⋊C₄	—	236	1	no	—
269	C₄²	—	59	2	no	—
270	C₄²	—	118	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	118	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	#14273751	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	Q₈	—	4	1	no	—
277	Q₈	—	472	8	no	—
278	D₄	—	2	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	118	4	no	—
288	C₂×C₄	—	118	2	no	—
289	C₂×C₄	—	472	2	no	—
290	C₂×C₄	—	236	2	no	—
291	C₂×C₄	#14273704	1	1	yes	characteristic
292	C₂×C₄	#14273704	1	1	yes	characteristic
293	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
294	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
295	C₂×C₄	#14273751	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
297	C₈	—	59	2	no	—
298	C₈	—	59	2	no	—
299	C₈	—	236	2	no	—
300	C₈	#14273704	1	1	yes	characteristic
301	C₈	#14273704	1	1	yes	characteristic
302	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
303	C₈	C₂×#11745687	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	#16837690	1	1	yes	characteristic
308	C₂²	#16837690	1	1	yes	characteristic
309	C₂²	C₂×#14273751	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	118	4	no	—
313	C₄	—	472	2	no	—
314	C₄	#16837690	1	1	yes	characteristic
315	C₄	#16837690	1	1	yes	characteristic
316	C₄	C₂×#14273704	1	1	yes	characteristic
317	C₄	C₂×SD₁₆×D₅₉	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	8	2	no	—
321	C₂	#21426265	1	2	yes	direct product factor
322	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×#2230
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×#16837799	C₂	semidirect
3. #21426265	C₂	direct (N × H)
4. C₂×#16837225	C₂	semidirect
5. C₂×#16837224	C₂	semidirect
6. C₂×#16837222	C₂	semidirect
13. #16837799	C₂²	semidirect
21. #16837225	C₂²	semidirect
22. #16837224	C₂²	semidirect
23. #16837222	C₂²	semidirect
55. #135×C₅₉	Q₈	semidirect
57. C₂×Q₁₆×C₅₉	Q₈	semidirect
59. C₂×D₈×C₅₉	Q₈	semidirect
82. Q₁₆×C₅₉	C₂×Q₈	semidirect
86. D₈×C₅₉	C₂×Q₈	semidirect
168. C₂×C₅₉	#2230	semidirect
190. C₅₉	C₂×#2230	semidirect
195. #135	Dic₁₁₈	semidirect
197. C₂×Q₁₆	Dic₁₁₈	semidirect
199. C₂×D₈	Dic₁₁₈	semidirect
244. Q₁₆	C₂×Dic₁₁₈	semidirect
248. D₈	C₂×Dic₁₁₈	semidirect
321. C₂	#21426265	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×#16837799	2
C₂×#16837519	2
#21426265	2
C₂×#16837225	2
C₂×#16837224	2
C₂×#16837222	2
C₂×#16837212	2
C₂×C₅₉×#360	2
C₂×#2230	59

Subgroups — C2×#21426265 #28743417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426265 #28743417