Subgroups — C₂×#21426213 #28743243

Order 15104.

678 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837818	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837669	C₂	1	2	yes	maximal, complemented
3	#21426213	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837232	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837215	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14273753	—	2	2	no	—
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	#16837818	—	2	2	no	—
10	#16837818	C₂²	1	4	yes	complemented
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	C₂×#14273666	—	2	1	no	—
13	C₂×C₁₆×D₅₉	—	2	1	no	—
14	#16837669	—	2	4	no	—
15	#16837669	C₂²	1	8	yes	complemented
16	C₂×#14273663	C₂²	1	2	yes	—
17	C₂×#14273644	—	2	2	no	—
18	#16837232	C₂²	1	4	yes	complemented
19	#16837215	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₅₉	C₂²	1	2	yes	—
21	C₂×D₁₆×C₅₉	—	2	1	no	—
22	C₂×D₁₆×C₅₉	—	2	1	no	—
23	C₂×D₁₆×C₅₉	—	2	2	no	—
24	C₂×D₁₆×C₅₉	C₂²	1	4	yes	complemented
25	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#11745685	D₄	1	1	yes	characteristic
27	#14273753	—	2	8	no	—
28	#14273753	—	4	4	no	—
29	C₂×#11745670	—	4	2	no	—
30	C₂×#11745669	—	4	2	no	—
31	#14273733	—	2	1	no	—
32	#14273733	—	2	1	no	—
33	#14273733	D₄	1	2	yes	—
34	#14273733	C₂³	1	2	yes	—
35	C₂×#11745664	—	2	2	no	—
36	C₂×#11745664	D₄	1	1	yes	characteristic
37	C₂×#11745659	—	2	1	no	—
38	C₂×C₈×D₅₉	—	2	1	no	—
39	#14273667	—	2	1	no	—
40	#14273667	C₂³	1	2	yes	—
41	#14273666	—	2	4	no	—
42	C₂×C₅₉⋊C₁₆	—	2	1	no	—
43	C₁₆×D₅₉	—	2	4	no	—
44	#14273663	—	2	2	no	—
45	#14273663	C₂³	1	4	yes	—
46	#14273644	—	2	8	no	—
47	C₂²×D₄×C₅₉	—	2	2	no	—
48	C₂×D₈×C₅₉	—	2	2	no	—
49	C₂×D₈×C₅₉	—	2	2	no	—
50	C₂×D₈×C₅₉	—	2	2	no	—
51	C₂×D₈×C₅₉	—	2	2	no	—
52	C₂×D₈×C₅₉	—	2	2	no	—
53	C₂×D₈×C₅₉	C₂³	1	4	yes	—
54	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	D₁₆×C₅₉	—	2	4	no	—
56	D₁₆×C₅₉	—	2	4	no	—
57	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
60	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
61	C₂⁴×C₅₉	—	4	2	no	—
62	C₂×D₄×C₅₉	—	2	4	no	—
63	C₂×D₄×C₅₉	—	4	2	no	—
64	C₂×D₄×C₅₉	—	4	2	no	—
65	C₂×D₄×C₅₉	—	4	2	no	—
66	C₂×D₄×C₅₉	—	4	2	no	—
67	C₂×D₄×C₅₉	—	4	2	no	—
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₉	—	2	1	no	—
70	#11745685	—	2	1	no	—
71	#11745685	—	2	1	no	—
72	#11745685	D₈	1	2	yes	—
73	#11745685	C₂×D₄	1	2	yes	—
74	C₂×D₂₃₆	D₈	1	1	yes	characteristic
75	C₂×C₄×D₅₉	—	2	1	no	—
76	C₂×Dic₁₁₈	D₈	1	1	yes	characteristic
77	D₈×C₅₉	—	2	4	no	—
78	D₈×C₅₉	—	2	4	no	—
79	D₈×C₅₉	—	4	2	no	—
80	D₈×C₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
87	#11745670	—	4	8	no	—
88	#11745669	—	4	8	no	—
89	#11745664	—	2	1	no	—
90	#11745664	—	2	4	no	—
91	#11745664	—	4	2	no	—
92	#11745664	C₂×D₄	1	2	yes	—
93	C₂×C₅₉⋊C₈	—	2	1	no	—
94	C₂×C₅₉⋊C₈	—	4	2	no	—
95	#11745659	—	2	2	no	—
96	#11745659	—	2	2	no	—
97	C₈×D₅₉	—	2	2	no	—
98	C₈×D₅₉	—	2	2	no	—
99	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
100	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
101	C₅₉⋊C₁₆	—	2	2	no	—
102	C₂³×C₅₉	—	4	4	no	—
103	C₂³×C₅₉	—	8	2	no	—
104	C₂³×C₅₉	—	8	2	no	—
105	C₂³×C₅₉	—	16	4	no	—
106	C₂³×C₅₉	#135	1	1	yes	characteristic
107	C₂²×D₅₉	—	2	1	no	—
108	D₄×C₅₉	—	4	4	no	—
109	D₄×C₅₉	—	4	4	no	—
110	D₄×C₅₉	—	8	2	no	—
111	D₄×C₅₉	—	8	2	no	—
112	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₅₉	—	2	2	no	—
119	C₂²⋊D₅₉	—	2	2	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	D₂₃₆	—	2	1	no	—
122	D₂₃₆	C₂×D₈	1	2	yes	—
123	C₄×D₅₉	—	2	2	no	—
124	C₄×D₅₉	—	2	2	no	—
125	Dic₁₁₈	—	2	1	no	—
126	Dic₁₁₈	C₂×D₈	1	2	yes	—
127	C₈×C₅₉	#121	1	1	yes	characteristic
128	C₈×C₅₉	#121	1	1	yes	characteristic
129	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
131	C₅₉⋊C₈	—	2	2	no	—
132	C₅₉⋊C₈	—	4	4	no	—
133	C₂×#2221	—	59	1	no	maximal, Sylow
134	C₂²×C₅₉	—	2	2	no	—
135	C₂²×C₅₉	—	8	4	no	—
136	C₂²×C₅₉	—	16	4	no	—
137	C₂²×C₅₉	—	16	4	no	—
138	C₂²×C₅₉	—	16	4	no	—
139	C₂²×C₅₉	#466	1	1	yes	characteristic
140	C₂²×C₅₉	#466	1	1	yes	characteristic
141	C₂²×C₅₉	C₂×#135	1	1	yes	characteristic
142	C₂×D₅₉	—	2	2	no	—
143	C₂×D₅₉	—	4	2	no	—
144	C₄×C₅₉	#466	1	1	yes	characteristic
145	C₄×C₅₉	#466	1	1	yes	characteristic
146	C₄×C₅₉	C₂×#121	1	1	yes	characteristic
147	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
148	Dic₅₉	—	2	2	no	—
149	C₂²×D₁₆	D₅₉	1	1	yes	characteristic, complemented
150	C₂×#504	—	59	1	no	—
151	C₂×#472	—	59	2	no	—
152	#2221	—	59	8	no	—
153	C₂×#368	—	59	1	no	—
154	C₂×#361	—	59	2	no	—
155	C₂×C₅₉	—	2	1	no	—
156	C₂×C₅₉	—	2	1	no	—
157	C₂×C₅₉	—	16	4	no	—
158	C₂×C₅₉	#2221	1	2	yes	complemented
159	C₂×C₅₉	C₂×#466	1	1	yes	characteristic
160	D₅₉	—	4	2	no	—
161	C₂×AGL(1,ℤ/8)	—	118	2	no	—
162	C₂²×D₈	C₂×D₅₉	1	2	yes	—
163	C₂×#131	—	59	1	no	—
164	C₂×D₁₆	—	2	1	no	—
165	C₂×D₁₆	—	2	1	no	—
166	C₂×D₁₆	—	2	2	no	—
167	C₂×D₁₆	C₂×D₅₉	1	4	yes	complemented
168	#504	—	59	4	no	—
169	#504	—	118	2	no	—
170	C₂×M₃₂	—	59	1	no	—
171	C₂×M₃₂	—	118	1	no	—
172	C₂²×C₁₆	—	118	1	no	—
173	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
174	#472	—	59	8	no	—
175	#472	—	118	4	no	—
176	C₂×#108	—	59	2	no	—
177	C₂×#100	—	118	2	no	—
178	#368	—	59	4	no	—
179	#361	—	59	8	no	—
180	C₅₉	C₂×#2221	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	—	59	1	no	—
182	C₂²×D₄	—	2	2	no	—
183	AGL(1,ℤ/8)	—	118	8	no	—
184	AGL(1,ℤ/8)	—	236	4	no	—
185	C₂×SD₁₆	—	236	2	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	236	2	no	—
192	C₂×D₈	C₂²×D₅₉	1	4	yes	—
193	#131	—	59	2	no	—
194	#131	—	59	2	no	—
195	#131	—	118	1	no	—
196	#131	—	118	1	no	—
197	C₂×M₁₆	—	59	1	no	—
198	C₂×M₁₆	—	118	1	no	—
199	C₂×M₁₆	—	118	2	no	—
200	C₂²×C₈	—	118	1	no	—
201	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
202	D₁₆	—	2	4	no	—
203	D₁₆	—	2	4	no	—
204	M₃₂	—	59	2	no	—
205	M₃₂	—	118	1	no	—
206	M₃₂	—	118	4	no	—
207	C₂×C₁₆	—	118	1	no	—
208	C₂×C₁₆	—	118	4	no	—
209	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
210	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
211	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
212	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
213	#108	—	59	4	no	—
214	#108	—	118	2	no	—
215	#100	—	118	8	no	—
216	C₂⁴	—	4	2	no	—
217	#41	—	59	2	no	—
218	#41	—	59	2	no	—
219	#41	—	118	1	no	—
220	#41	—	118	1	no	—
221	C₂×Q₈	—	59	1	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	59	1	no	—
229	C₂×D₄	—	118	1	no	—
230	C₂²×C₄	—	118	1	no	—
231	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
232	SD₁₆	—	236	8	no	—
233	D₈	—	2	4	no	—
234	D₈	—	2	4	no	—
235	D₈	—	4	2	no	—
236	D₈	—	4	2	no	—
237	D₈	—	236	8	no	—
238	M₁₆	—	59	2	no	—
239	M₁₆	—	118	1	no	—
240	M₁₆	—	118	2	no	—
241	M₁₆	—	118	2	no	—
242	M₁₆	—	118	4	no	—
243	M₁₆	—	236	2	no	—
244	C₂×C₈	—	118	1	no	—
245	C₂×C₈	—	118	2	no	—
246	C₂×C₈	—	118	2	no	—
247	C₂×C₈	—	236	2	no	—
248	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
249	C₂×C₈	#11745687	1	1	yes	characteristic
250	C₂×C₈	#11745687	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
252	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
254	C₁₆	—	118	2	no	—
255	C₁₆	C₂×C₂²⋊D₅₉	1	2	yes	—
256	C₁₆	C₂×C₂²⋊D₅₉	1	2	yes	—
257	C₂³	—	4	4	no	—
258	C₂³	—	8	2	no	—
259	C₂³	—	8	2	no	—
260	C₂³	—	16	4	no	—
261	C₂³	—	118	1	no	—
262	C₂³	#14273746	1	1	yes	characteristic
263	Q₈	—	59	2	no	—
264	Q₈	—	118	1	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	2	no	—
268	D₄	—	8	2	no	—
269	D₄	—	59	2	no	—
270	D₄	—	118	2	no	—
271	D₄	—	118	2	no	—
272	D₄	—	118	1	no	—
273	C₂×C₄	—	118	1	no	—
274	C₂×C₄	—	118	2	no	—
275	C₂×C₄	—	118	2	no	—
276	C₂×C₄	#14273698	1	1	yes	characteristic
277	C₂×C₄	#14273698	1	1	yes	characteristic
278	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
279	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
280	C₂×C₄	#14273746	1	1	yes	characteristic
281	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
282	C₈	—	118	2	no	—
283	C₈	—	236	4	no	—
284	C₈	#14273698	1	1	yes	characteristic
285	C₈	#14273698	1	1	yes	characteristic
286	C₈	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
287	C₈	C₂×#11745687	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	16	4	no	—
291	C₂²	—	16	4	no	—
292	C₂²	—	16	4	no	—
293	C₂²	—	118	2	no	—
294	C₂²	—	236	2	no	—
295	C₂²	#16837635	1	1	yes	characteristic
296	C₂²	#16837635	1	1	yes	characteristic
297	C₂²	C₂×#14273746	1	1	yes	characteristic
298	C₄	—	118	2	no	—
299	C₄	#16837635	1	1	yes	characteristic
300	C₄	#16837635	1	1	yes	characteristic
301	C₄	C₂×#14273698	1	1	yes	characteristic
302	C₄	C₂×D₈×D₅₉	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	16	4	no	—
306	C₂	—	236	2	no	—
307	C₂	#21426213	1	2	yes	direct product factor
308	C₂	C₂×#16837635	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837635
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂²×D₁₆×C₅₉	C₂
socle	C₂²×C₅₉	#466
hypercenter	C₂²×D₁₆	D₅₉
hypocenter	C₅₉	C₂×#2221
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16837818	C₂	semidirect
2. C₂×#16837669	C₂	semidirect
3. #21426213	C₂	direct (N × H)
4. C₂×#16837232	C₂	semidirect
5. C₂×#16837215	C₂	semidirect
6. C₂²×D₁₆×C₅₉	C₂	semidirect
10. #16837818	C₂²	semidirect
15. #16837669	C₂²	semidirect
18. #16837232	C₂²	semidirect
19. #16837215	C₂²	semidirect
24. C₂×D₁₆×C₅₉	C₂²	semidirect
149. C₂²×D₁₆	D₅₉	semidirect
158. C₂×C₅₉	#2221	semidirect
167. C₂×D₁₆	C₂×D₅₉	semidirect
180. C₅₉	C₂×#2221	semidirect
307. C₂	#21426213	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×#16837818	2
C₂×#16837669	2
#21426213	2
C₂×#16837232	2
C₂×#16837215	2
C₂²×D₁₆×C₅₉	2
C₂×#2221	59

Subgroups — C2×#21426213 #28743243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426213 #28743243