Subgroups — C₂×#21426268 #28743420

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	C₂×#16837520	C₂	1	1	yes	maximal, characteristic
3	#21426268	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837225	C₂	1	2	yes	maximal, complemented
5	C₂×#16837213	C₂	1	2	yes	maximal
6	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14273749	C₂²	1	1	yes	characteristic
8	C₂×#14273733	—	2	1	no	—
9	C₂×#14273727	—	2	1	no	—
10	#16837800	—	2	2	no	—
11	#16837800	C₂²	1	4	yes	complemented
12	#16837520	—	2	2	no	—
13	#16837520	C₂²	1	4	yes	—
14	C₂×#14273663	C₂²	1	1	yes	characteristic
15	C₂×#14273659	—	2	1	no	—
16	C₂×#14273659	C₂²	1	2	yes	—
17	C₂×#14273655	—	2	1	no	—
18	#16837225	C₂²	1	8	yes	complemented
19	#16837213	C₂²	1	8	yes	—
20	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#364	C₂²	1	4	yes	complemented
23	C₂×#11745685	—	2	1	no	—
24	C₂²×Dic₁₁₈	—	4	1	no	—
25	#14273749	—	2	1	no	—
26	#14273749	—	2	1	no	—
27	#14273749	Q₈	1	2	yes	complemented
28	#14273749	C₂³	1	2	yes	—
29	C₂×Dic₂₃₆	Q₈	1	1	yes	characteristic, complemented
30	C₂×#11745660	—	2	1	no	—
31	C₂×D₄₇₂	Q₈	1	1	yes	characteristic, complemented
32	#14273733	—	2	4	no	—
33	#14273733	—	4	2	no	—
34	C₂×#11745664	—	2	1	no	—
35	C₂×#11745659	—	4	1	no	—
36	C₂×C₈×D₅₉	—	4	1	no	—
37	#14273727	—	2	4	no	—
38	#14273727	—	4	2	no	—
39	C₂×C₄⋊Dic₅₉	—	2	2	no	—
40	C₂×C₄⋊Dic₅₉	—	4	1	no	—
41	#14273663	—	2	1	no	—
42	#14273663	C₂³	1	2	yes	—
43	#14273659	—	2	4	no	—
44	#14273659	—	2	2	no	—
45	#14273659	C₂³	1	4	yes	—
46	#14273655	—	2	4	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	M₃₂×C₅₉	—	2	2	no	—
51	M₃₂×C₅₉	C₂³	1	4	yes	—
52	C₂×C₁₆×C₅₉	—	2	2	no	—
53	C₄×C₈×C₅₉	D₄	1	2	yes	—
54	C₄×C₈×C₅₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₅₉	—	2	1	no	—
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₅₉	—	4	1	no	—
58	C₂²×Dic₅₉	—	4	2	no	—
59	#11745685	—	2	2	no	—
60	#11745685	—	2	2	no	—
61	#11745685	—	4	1	no	—
62	#11745685	—	4	1	no	—
63	C₂×D₂₃₆	—	2	1	no	—
64	C₂×C₄×D₅₉	—	4	1	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	—	4	4	no	—
67	C₂×Dic₁₁₈	—	8	4	no	—
68	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
69	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
70	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
71	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
76	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
78	C₄²×C₅₉	—	2	1	no	—
79	C₄²×C₅₉	C₂×D₄	1	2	yes	—
80	C₄⋊Dic₅₉	—	2	4	no	—
81	C₄⋊Dic₅₉	—	4	4	no	—
82	C₄⋊Dic₅₉	—	4	2	no	—
83	#11745664	—	2	2	no	—
84	#11745664	—	4	1	no	—
85	C₂×C₅₉⋊C₈	—	4	1	no	—
86	Dic₂₃₆	—	2	1	no	—
87	Dic₂₃₆	C₂×Q₈	1	2	yes	complemented
88	D₄₇₂	—	2	1	no	—
89	D₄₇₂	C₂×Q₈	1	2	yes	complemented
90	#11745660	—	2	2	no	—
91	#11745660	—	2	2	no	—
92	#11745659	—	4	4	no	—
93	C₈×D₅₉	—	4	4	no	—
94	C₁₆×C₅₉	—	2	4	no	—
95	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂²×D₅₉	—	4	1	no	—
97	C₂×C₄×C₅₉	—	2	4	no	—
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
105	C₂²⋊D₅₉	—	4	2	no	—
106	C₂²⋊D₅₉	—	4	2	no	—
107	C₂×Dic₅₉	—	4	1	no	—
108	C₂×Dic₅₉	—	4	4	no	—
109	C₂×Dic₅₉	—	8	4	no	—
110	D₂₃₆	—	2	2	no	—
111	D₂₃₆	—	4	1	no	—
112	C₄×D₅₉	—	4	2	no	—
113	C₄×D₅₉	—	4	2	no	—
114	Dic₁₁₈	—	2	2	no	—
115	Dic₁₁₈	—	4	1	no	—
116	Dic₁₁₈	—	8	8	no	—
117	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
120	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
121	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
122	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
123	C₅₉⋊C₈	—	4	2	no	—
124	C₂×#2230	—	59	1	no	maximal, Sylow
125	C₂²×C₅₉	—	2	2	no	—
126	C₂²×C₅₉	#476	1	1	yes	characteristic
127	C₂²×C₅₉	#476	1	1	yes	characteristic
128	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×D₅₉	—	4	2	no	—
130	C₂×D₅₉	—	8	2	no	—
131	C₄×C₅₉	—	2	4	no	—
132	C₄×C₅₉	#476	1	1	yes	characteristic
133	C₄×C₅₉	#476	1	1	yes	characteristic
134	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
135	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
136	Dic₅₉	—	4	2	no	—
137	Dic₅₉	—	8	4	no	—
138	C₂×#498	—	59	1	no	—
139	C₂×#443	—	59	1	no	—
140	#2230	—	59	8	no	—
141	C₂×#365	—	59	2	no	—
142	C₂×#364	D₅₉	1	1	yes	characteristic, complemented
143	C₂×#360	—	59	2	no	—
144	C₂×C₅₉	—	2	1	no	—
145	C₂×C₅₉	—	2	1	no	—
146	C₂×C₅₉	#2230	1	2	yes	complemented
147	C₂×C₅₉	C₂×#476	1	1	yes	characteristic
148	D₅₉	—	8	2	no	—
149	C₂×#135	—	59	1	no	—
150	C₂×#131	—	118	1	no	—
151	C₂×C₄⋊Q₈	—	118	1	no	—
152	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
153	#498	—	59	4	no	—
154	#498	—	118	2	no	—
155	#443	—	59	4	no	—
156	#443	—	118	2	no	—
157	C₂×#108	—	59	1	no	—
158	C₂×#106	—	59	2	no	—
159	C₂×#106	—	118	1	no	—
160	C₂×#104	—	118	1	no	—
161	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
162	#365	—	59	8	no	—
163	#364	C₂×D₅₉	1	4	yes	complemented
164	#360	—	59	8	no	—
165	C₅₉	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
166	C₂×#41	—	118	1	no	—
167	C₂²×Q₈	—	236	1	no	—
168	#135	—	59	2	no	—
169	#135	—	59	2	no	—
170	#135	—	118	1	no	—
171	#135	—	118	1	no	—
172	C₂×Q₁₆	—	59	1	no	—
173	C₂×SD₁₆	—	118	1	no	—
174	C₂×D₈	—	59	1	no	—
175	#131	—	118	4	no	—
176	#131	—	236	2	no	—
177	C₂×M₁₆	—	118	1	no	—
178	C₂×M₁₆	—	236	1	no	—
179	C₂²×C₈	—	236	1	no	—
180	C₂²×C₈	D₂₃₆	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	D₂₃₆	1	1	yes	characteristic
187	M₃₂	—	2	2	no	—
188	M₃₂	C₂²×D₅₉	1	4	yes	—
189	C₂×C₁₆	—	2	2	no	—
190	#108	—	59	2	no	—
191	#108	—	118	1	no	—
192	#106	—	59	4	no	—
193	#106	—	118	2	no	—
194	#106	—	118	4	no	—
195	#104	—	118	4	no	—
196	C₄×C₈	D₂₃₆	1	2	yes	—
197	C₄×C₈	C₂²×D₅₉	1	2	yes	—
198	#41	—	118	2	no	—
199	#41	—	118	2	no	—
200	#41	—	236	1	no	—
201	#41	—	236	1	no	—
202	C₂×Q₈	—	118	1	no	—
203	C₂×Q₈	—	472	4	no	—
204	C₂×Q₈	—	236	4	no	—
205	C₂×D₄	—	118	1	no	—
206	C₂×D₄	—	236	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	236	2	no	—
209	C₂²×C₄	—	236	1	no	—
210	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
211	Q₁₆	—	59	2	no	—
212	Q₁₆	—	118	1	no	—
213	SD₁₆	—	118	2	no	—
214	SD₁₆	—	118	2	no	—
215	D₈	—	59	2	no	—
216	D₈	—	118	1	no	—
217	M₁₆	—	118	2	no	—
218	M₁₆	—	236	1	no	—
219	M₁₆	—	236	4	no	—
220	C₂×C₈	—	236	1	no	—
221	C₂×C₈	—	236	4	no	—
222	C₂×C₈	#11745660	1	1	yes	characteristic
223	C₂×C₈	#11745660	1	1	yes	characteristic
224	C₂×C₈	#11745660	1	2	yes	—
225	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
226	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
227	C₂×C₈	#11745687	1	1	yes	characteristic
228	C₂×C₈	#11745687	1	1	yes	characteristic
229	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
230	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
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×D₂₃₆	1	2	yes	—
237	C₁₆	—	2	4	no	—
238	C₂³	—	236	1	no	—
239	C₂³	#14273755	1	1	yes	characteristic
240	Q₈	—	118	2	no	—
241	Q₈	—	472	8	no	—
242	Q₈	—	236	1	no	—
243	D₄	—	118	2	no	—
244	D₄	—	236	2	no	—
245	D₄	—	236	2	no	—
246	D₄	—	236	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	472	4	no	—
250	C₂×C₄	—	236	4	no	—
251	C₂×C₄	—	236	1	no	—
252	C₂×C₄	—	236	2	no	—
253	C₂×C₄	—	236	2	no	—
254	C₂×C₄	#14273707	1	1	yes	characteristic
255	C₂×C₄	#14273707	1	1	yes	characteristic
256	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
257	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
258	C₂×C₄	#14273755	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
260	C₈	—	236	2	no	—
261	C₈	#14273707	1	1	yes	characteristic
262	C₈	#14273707	1	1	yes	characteristic
263	C₈	C₂×#11745660	1	2	yes	—
264	C₈	C₂×#11745660	1	2	yes	—
265	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
266	C₈	C₂×#11745687	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	472	2	no	—
269	C₂²	—	236	2	no	—
270	C₂²	#16837693	1	1	yes	characteristic
271	C₂²	#16837693	1	1	yes	characteristic
272	C₂²	C₂×#14273755	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	472	4	no	—
275	C₄	—	236	2	no	—
276	C₄	#16837693	1	1	yes	characteristic
277	C₄	#16837693	1	1	yes	characteristic
278	C₄	C₂×#14273707	1	1	yes	characteristic
279	C₄	C₂²×#11745660	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	472	2	no	—
283	C₂	#21426268	1	2	yes	direct product factor
284	C₂	C₂×#16837693	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837693
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂²×C₅₉	#476
prime radical	C₂×#14273749	C₂²
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	C₂×#2230
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#16837800	C₂	semidirect
3. #21426268	C₂	direct (N × H)
4. C₂×#16837225	C₂	semidirect
6. C₂×C₅₉×#364	C₂	semidirect
11. #16837800	C₂²	semidirect
18. #16837225	C₂²	semidirect
22. C₅₉×#364	C₂²	semidirect
27. #14273749	Q₈	semidirect
29. C₂×Dic₂₃₆	Q₈	semidirect
31. C₂×D₄₇₂	Q₈	semidirect
87. Dic₂₃₆	C₂×Q₈	semidirect
89. D₄₇₂	C₂×Q₈	semidirect
142. C₂×#364	D₅₉	semidirect
146. C₂×C₅₉	#2230	semidirect
163. #364	C₂×D₅₉	semidirect
165. C₅₉	C₂×#2230	semidirect
283. C₂	#21426268	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
C₂×#16837520	2
#21426268	2
C₂×#16837225	2
C₂×#16837213	2
C₂×C₅₉×#364	2
C₂×#2230	59

Subgroups — C2×#21426268 #28743420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426268 #28743420