Subgroups — C₂×#21426301 #28743464

Order 15104.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837440	C₂	1	1	yes	maximal, characteristic
2	#21426301	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837229	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837227	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837206	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837206	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₉×#467	C₂	1	1	yes	maximal, characteristic
9	C₂×#14273686	—	2	1	no	—
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16837440	—	2	2	no	—
13	#16837440	C₂²	1	4	yes	—
14	C₂×#14273661	C₂²	1	1	yes	characteristic
15	C₂×#14273661	C₂²	1	1	yes	characteristic
16	C₂×#14273659	—	2	1	no	—
17	#16837229	C₂²	1	4	yes	complemented
18	#16837227	C₂²	1	4	yes	complemented
19	#16837206	C₂²	1	4	yes	complemented
20	#16837206	C₂²	1	4	yes	complemented
21	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	4	yes	—
22	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Q₈×C₅₉	—	2	1	no	—
25	C₅₉×#467	—	2	2	no	—
26	C₅₉×#467	C₂²	1	4	yes	—
27	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₅₉	—	2	1	no	—
29	#14273686	—	2	4	no	—
30	#14273686	—	4	2	no	—
31	C₂×C₄⋊Dic₅₉	—	2	1	no	—
32	C₂×C₄⋊Dic₅₉	—	2	1	no	—
33	C₂×C₂²⋊Dic₅₉	—	4	1	no	—
34	C₂×C₄×Dic₅₉	—	4	1	no	—
35	C₂×C₅₉⋊C₁₆	—	2	2	no	—
36	C₂×C₅₉⋊C₁₆	—	2	2	no	—
37	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
38	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
39	#14273661	—	2	1	no	—
40	#14273661	—	2	1	no	—
41	#14273661	C₂³	1	2	yes	—
42	#14273661	C₂³	1	2	yes	—
43	#14273659	—	2	4	no	—
44	C₂²×Q₈×C₅₉	—	2	1	no	—
45	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
46	C₂×Q₁₆×C₅₉	—	2	1	no	—
47	C₂×Q₁₆×C₅₉	—	2	2	no	—
48	C₂×Q₁₆×C₅₉	—	2	2	no	—
49	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₉	—	2	2	no	—
51	C₂²×C₈×C₅₉	—	2	2	no	—
52	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	D₄	1	2	yes	—
54	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
56	C₂²⋊Q₈×C₅₉	—	2	4	no	—
57	C₂²⋊Q₈×C₅₉	—	4	2	no	—
58	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
60	C₂×C₂²⋊C₄×C₅₉	—	4	1	no	—
61	#107×C₅₉	—	2	1	no	—
62	#107×C₅₉	C₂³	1	2	yes	—
63	#103×C₅₉	—	2	4	no	—
64	C₂⁴×C₅₉	D₈	1	1	yes	characteristic
65	C₂×Q₈×C₅₉	—	2	2	no	—
66	C₂×Q₈×C₅₉	—	4	1	no	—
67	C₂×Q₈×C₅₉	—	4	2	no	—
68	C₂×Q₈×C₅₉	—	4	2	no	—
69	C₂²×C₄×C₅₉	—	2	2	no	—
70	C₂²×C₄×C₅₉	—	2	2	no	—
71	C₂²×C₄×C₅₉	—	4	1	no	—
72	C₂²×C₄×C₅₉	—	4	1	no	—
73	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	D₈	1	2	yes	—
75	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₅₉	—	4	1	no	—
78	C₂²×Dic₅₉	—	4	1	no	—
79	Q₁₆×C₅₉	—	2	2	no	—
80	Q₁₆×C₅₉	—	2	2	no	—
81	Q₁₆×C₅₉	—	4	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	2	4	no	—
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	—	2	2	no	—
86	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	#41	1	2	yes	—
89	C₂×C₈×C₅₉	#41	1	2	yes	—
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⋊C₄×C₅₉	—	4	4	no	—
94	C₂²⋊C₄×C₅₉	—	4	4	no	—
95	C₂²⋊Dic₅₉	—	4	4	no	—
96	C₄⋊Dic₅₉	—	2	2	no	—
97	C₄⋊Dic₅₉	—	2	2	no	—
98	C₄⋊Dic₅₉	—	4	1	no	—
99	C₄⋊Dic₅₉	—	4	1	no	—
100	C₄×Dic₅₉	—	4	4	no	—
101	C₅₉⋊C₁₆	—	2	2	no	—
102	C₅₉⋊C₁₆	—	2	2	no	—
103	C₂³×C₅₉	—	2	2	no	—
104	C₂³×C₅₉	—	2	2	no	—
105	C₂³×C₅₉	—	2	2	no	—
106	C₂³×C₅₉	C₂×D₈	1	2	yes	—
107	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
108	Q₈×C₅₉	—	4	2	no	—
109	Q₈×C₅₉	—	4	2	no	—
110	Q₈×C₅₉	—	8	2	no	—
111	C₂×C₄×C₅₉	—	2	2	no	—
112	C₂×C₄×C₅₉	—	2	2	no	—
113	C₂×C₄×C₅₉	—	2	2	no	—
114	C₂×C₄×C₅₉	—	2	4	no	—
115	C₂×C₄×C₅₉	—	4	2	no	—
116	C₂×C₄×C₅₉	—	4	2	no	—
117	C₂×C₄×C₅₉	—	8	2	no	—
118	C₂×C₄×C₅₉	—	8	2	no	—
119	C₂×C₄×C₅₉	#123	1	2	yes	—
120	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	#137	1	2	yes	—
123	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₅₉	—	4	2	no	—
125	C₂×Dic₅₉	—	4	2	no	—
126	C₂×Dic₅₉	—	8	2	no	—
127	C₂×Dic₅₉	—	8	2	no	—
128	C₈×C₅₉	—	2	2	no	—
129	C₈×C₅₉	#123	1	2	yes	—
130	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
131	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
132	C₂×#2237	—	59	1	no	maximal, Sylow
133	C₂²×C₅₉	—	2	2	no	—
134	C₂²×C₅₉	—	2	2	no	—
135	C₂²×C₅₉	—	2	2	no	—
136	C₂²×C₅₉	—	4	4	no	—
137	C₂²×C₅₉	#480	1	2	yes	—
138	C₂²×C₅₉	#508	1	2	yes	—
139	C₂²×C₅₉	#510	1	2	yes	—
140	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₅₉	—	2	2	no	—
142	C₄×C₅₉	—	8	2	no	—
143	C₄×C₅₉	—	8	2	no	—
144	C₄×C₅₉	#480	1	2	yes	—
145	C₄×C₅₉	C₂×#123	1	1	yes	characteristic
146	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
147	Dic₅₉	—	8	2	no	—
148	Dic₅₉	—	8	2	no	—
149	C₂×#467	D₅₉	1	1	yes	characteristic
150	C₂×#427	—	59	1	no	—
151	#2237	—	59	8	no	—
152	C₂×#367	—	59	1	no	—
153	C₂×#366	—	59	1	no	—
154	C₂×#358	—	59	1	no	—
155	C₂×#358	—	59	1	no	—
156	C₂×C₂²⋊C₁₆	—	59	1	no	—
157	C₂×C₅₉	—	4	2	no	—
158	C₂×C₅₉	#2237	1	4	yes	complemented
159	C₂×C₅₉	C₂×#480	1	1	yes	characteristic
160	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
161	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
162	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
163	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
164	C₂×C₂²⋊Q₈	—	2	1	no	—
165	C₂×#117	—	118	1	no	—
166	C₂²×C₁₆	—	59	1	no	—
167	C₂²×C₁₆	—	59	1	no	—
168	#467	—	2	2	no	—
169	#467	C₂×D₅₉	1	4	yes	—
170	#427	—	59	4	no	—
171	#427	—	118	2	no	—
172	C₂×#107	—	59	1	no	—
173	C₂×#107	—	59	1	no	—
174	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
175	C₂×#106	—	118	1	no	—
176	C₂×#103	—	2	1	no	—
177	#367	—	59	4	no	—
178	#366	—	59	4	no	—
179	#358	—	59	4	no	—
180	#358	—	59	4	no	—
181	C₂²⋊C₁₆	—	59	4	no	—
182	C₅₉	C₂×#2237	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	2	1	no	—
184	C₂³×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₅₉	1	2	yes	—
193	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
194	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
195	C₂²⋊Q₈	—	2	4	no	—
196	C₂²⋊Q₈	—	4	2	no	—
197	#117	—	118	4	no	—
198	#117	—	236	2	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	4	1	no	—
201	C₂×C₄⋊C₄	—	118	1	no	—
202	C₂×C₄⋊C₄	—	118	1	no	—
203	C₂×C₂²⋊C₄	—	4	1	no	—
204	C₂×C₂²⋊C₄	—	236	1	no	—
205	C₂×C₄²	—	236	1	no	—
206	C₂×C₁₆	—	59	2	no	—
207	C₂×C₁₆	—	59	2	no	—
208	C₂×C₁₆	—	118	2	no	—
209	C₂×C₁₆	—	118	2	no	—
210	#107	—	2	1	no	—
211	#107	—	59	2	no	—
212	#107	—	59	2	no	—
213	#107	—	118	1	no	—
214	#107	—	118	1	no	—
215	#107	C₂²×D₅₉	1	2	yes	—
216	#106	—	118	4	no	—
217	#103	—	2	4	no	—
218	C₂⁴	#11745669	1	1	yes	characteristic
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	236	1	no	—
228	C₂²×C₄	—	236	1	no	—
229	C₂²×C₄	#11745669	1	2	yes	—
230	C₂²×C₄	#11745669	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₅₉	1	2	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	#11745687	1	2	yes	—
241	C₂×C₈	#11745687	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
244	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	118	2	no	—
249	C₄⋊C₄	—	118	2	no	—
250	C₄⋊C₄	—	236	1	no	—
251	C₄⋊C₄	—	236	1	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₂²⋊C₄	—	236	4	no	—
254	C₄²	—	236	4	no	—
255	C₁₆	—	118	2	no	—
256	C₁₆	—	118	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×#11745669	1	2	yes	—
261	C₂³	C₂×#11745669	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	472	2	no	—
274	C₂×C₄	—	472	2	no	—
275	C₂×C₄	—	236	2	no	—
276	C₂×C₄	—	236	2	no	—
277	C₂×C₄	#14273708	1	2	yes	—
278	C₂×C₄	C₂×#11745669	1	2	yes	—
279	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
280	C₂×C₄	#14273757	1	2	yes	—
281	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	#14273708	1	2	yes	—
284	C₈	C₂×#11745687	1	1	yes	characteristic
285	C₈	C₂×#11745687	1	1	yes	characteristic
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	#16837713	1	2	yes	—
291	C₂²	#16837832	1	2	yes	—
292	C₂²	#16837843	1	2	yes	—
293	C₂²	C₂²×#11745669	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	472	2	no	—
298	C₄	—	472	2	no	—
299	C₄	#16837713	1	2	yes	—
300	C₄	C₂×#14273708	1	1	yes	characteristic
301	C₄	C₂×#14273757	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	#21426301	1	4	yes	direct product factor
304	C₂	C₂×#16837713	1	1	yes	characteristic
305	C₂	C₂×#16837832	1	1	yes	characteristic
306	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×#467	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂⁴×C₅₉	D₈
hypercenter	C₂×#467	D₅₉
hypocenter	C₅₉	C₂×#2237
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. #21426301	C₂	direct (N × H)
3. C₂×#16837229	C₂	semidirect
4. C₂×#16837227	C₂	semidirect
5. C₂×#16837206	C₂	semidirect
6. C₂×#16837206	C₂	semidirect
17. #16837229	C₂²	semidirect
18. #16837227	C₂²	semidirect
19. #16837206	C₂²	semidirect
20. #16837206	C₂²	semidirect
158. C₂×C₅₉	#2237	semidirect
182. C₅₉	C₂×#2237	semidirect
303. C₂	#21426301	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×#16837440	2
#21426301	2
C₂×#16837229	2
C₂×#16837227	2
C₂×#16837206	2
C₂×C₂²⋊(C₅₉⋊C₁₆)	2
C₂×C₅₉×#467	2
C₂×#2237	59

Subgroups — C2×#21426301 #28743464

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426301 #28743464