Subgroups — C₂×#21426269 #28743422

Order 15104.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837520	C₂	1	1	yes	maximal, characteristic
2	#21426269	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837224	C₂	1	2	yes	maximal, complemented
4	C₂×#16837223	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837211	C₂	1	2	yes	maximal
6	C₂×C₅₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14273749	C₂²	1	1	yes	characteristic
8	C₂×#14273733	—	2	1	no	—
9	C₂×#14273667	C₂²	1	2	yes	—
10	#16837520	—	2	2	no	—
11	#16837520	C₂²	1	4	yes	—
12	C₂×#14273663	C₂²	1	1	yes	characteristic
13	C₂×#14273655	—	2	1	no	—
14	#16837224	C₂²	1	8	yes	complemented
15	#16837223	C₂²	1	4	yes	complemented
16	#16837211	C₂²	1	8	yes	—
17	C₂×#125×C₅₉	—	2	1	no	—
18	C₅₉×#499	—	2	2	no	—
19	C₅₉×#499	C₂²	1	4	yes	complemented
20	C₂×#107×C₅₉	—	2	1	no	—
21	C₂×#106×C₅₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×#11745685	—	2	1	no	—
24	#14273749	—	2	1	no	—
25	#14273749	—	2	1	no	—
26	#14273749	Q₈	1	2	yes	—
27	#14273749	C₂³	1	2	yes	—
28	C₂×Dic₂₃₆	Q₈	1	1	yes	characteristic
29	C₂×#11745660	—	2	1	no	—
30	C₂×D₄₇₂	Q₈	1	1	yes	characteristic
31	#14273733	—	2	4	no	—
32	#14273733	—	4	2	no	—
33	C₂×#11745664	—	2	1	no	—
34	C₂×#11745659	—	4	1	no	—
35	C₂×C₈×D₅₉	—	4	1	no	—
36	#14273667	—	2	2	no	—
37	#14273667	C₂³	1	4	yes	—
38	C₂×C₅₉⋊C₁₆	—	2	2	no	—
39	#14273663	—	2	1	no	—
40	#14273663	C₂³	1	2	yes	—
41	#14273655	—	2	4	no	—
42	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	#125×C₅₉	—	2	4	no	—
45	#125×C₅₉	—	4	2	no	—
46	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
48	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
49	#107×C₅₉	—	2	4	no	—
50	#106×C₅₉	—	2	2	no	—
51	#106×C₅₉	C₂³	1	4	yes	—
52	C₄×C₈×C₅₉	D₄	1	2	yes	—
53	C₄×C₈×C₅₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₅₉	—	2	1	no	—
55	C₂²×C₄×C₅₉	—	4	2	no	—
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₅₉	—	4	1	no	—
58	#11745685	—	2	2	no	—
59	#11745685	—	2	2	no	—
60	#11745685	—	4	1	no	—
61	#11745685	—	4	1	no	—
62	C₂×D₂₃₆	—	2	1	no	—
63	C₂×C₄×D₅₉	—	4	1	no	—
64	C₂×Dic₁₁₈	—	2	1	no	—
65	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
66	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
67	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
68	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₅₉	—	2	4	no	—
76	C₄⋊C₄×C₅₉	—	4	8	no	—
77	C₄⋊C₄×C₅₉	—	4	2	no	—
78	C₄²×C₅₉	—	2	1	no	—
79	C₄²×C₅₉	C₂×D₄	1	2	yes	—
80	#11745664	—	2	2	no	—
81	#11745664	—	4	1	no	—
82	C₂×C₅₉⋊C₈	—	4	1	no	—
83	Dic₂₃₆	—	2	1	no	—
84	Dic₂₃₆	C₂×Q₈	1	2	yes	—
85	D₄₇₂	—	2	1	no	—
86	D₄₇₂	C₂×Q₈	1	2	yes	—
87	#11745660	—	2	2	no	—
88	#11745660	—	2	2	no	—
89	#11745659	—	4	4	no	—
90	C₈×D₅₉	—	4	4	no	—
91	C₅₉⋊C₁₆	—	2	4	no	—
92	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂²×D₅₉	—	4	1	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	4	2	no	—
96	C₂×C₄×C₅₉	—	4	4	no	—
97	C₂×C₄×C₅₉	—	8	4	no	—
98	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
104	C₂²⋊D₅₉	—	4	2	no	—
105	C₂²⋊D₅₉	—	4	2	no	—
106	C₂×Dic₅₉	—	4	1	no	—
107	D₂₃₆	—	2	2	no	—
108	D₂₃₆	—	4	1	no	—
109	C₄×D₅₉	—	4	2	no	—
110	C₄×D₅₉	—	4	2	no	—
111	Dic₁₁₈	—	2	2	no	—
112	Dic₁₁₈	—	4	1	no	—
113	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
116	C₈×C₅₉	C₂×SD₁₆	1	2	yes	—
117	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
118	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
119	C₅₉⋊C₈	—	4	2	no	—
120	C₂×#2231	—	59	1	no	maximal, Sylow
121	C₂²×C₅₉	—	2	2	no	—
122	C₂²×C₅₉	#476	1	1	yes	characteristic
123	C₂²×C₅₉	#476	1	1	yes	characteristic
124	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×D₅₉	—	4	2	no	—
126	C₂×D₅₉	—	8	2	no	—
127	C₄×C₅₉	—	4	2	no	—
128	C₄×C₅₉	—	8	4	no	—
129	C₄×C₅₉	#476	1	1	yes	characteristic
130	C₄×C₅₉	#476	1	1	yes	characteristic
131	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
133	Dic₅₉	—	4	2	no	—
134	C₂×#499	D₅₉	1	1	yes	characteristic, complemented
135	C₂×#443	—	59	1	no	—
136	#2231	—	59	8	no	—
137	C₂×#365	—	59	2	no	—
138	C₂×#364	—	59	1	no	—
139	C₂×#360	—	59	2	no	—
140	C₂×C₅₉	—	2	1	no	—
141	C₂×C₅₉	—	2	1	no	—
142	C₂×C₅₉	#2231	1	2	yes	complemented
143	C₂×C₅₉	C₂×#476	1	1	yes	characteristic
144	D₅₉	—	8	2	no	—
145	C₂×#135	—	59	1	no	—
146	C₂×#131	—	118	1	no	—
147	C₂×#125	—	2	1	no	—
148	C₂×M₃₂	—	59	2	no	—
149	#499	—	2	2	no	—
150	#499	C₂×D₅₉	1	4	yes	complemented
151	#443	—	59	4	no	—
152	#443	—	118	2	no	—
153	C₂×#108	—	59	1	no	—
154	C₂×#107	—	2	1	no	—
155	C₂×#106	C₂×D₅₉	1	2	yes	—
156	C₂×#104	—	118	1	no	—
157	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
158	#365	—	59	8	no	—
159	#364	—	59	4	no	—
160	#360	—	59	8	no	—
161	C₅₉	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	118	1	no	—
163	#135	—	59	2	no	—
164	#135	—	59	2	no	—
165	#135	—	118	1	no	—
166	#135	—	118	1	no	—
167	C₂×Q₁₆	—	59	1	no	—
168	C₂×SD₁₆	—	118	1	no	—
169	C₂×D₈	—	59	1	no	—
170	#131	—	118	4	no	—
171	#131	—	236	2	no	—
172	C₂×M₁₆	—	118	1	no	—
173	C₂×M₁₆	—	236	1	no	—
174	C₂²×C₈	—	236	1	no	—
175	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
177	#125	—	2	4	no	—
178	#125	—	4	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	4	2	no	—
181	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
182	M₃₂	—	59	4	no	—
183	M₃₂	—	118	2	no	—
184	C₂×C₁₆	—	118	2	no	—
185	#108	—	59	2	no	—
186	#108	—	118	1	no	—
187	#107	—	2	4	no	—
188	#106	—	2	2	no	—
189	#106	C₂²×D₅₉	1	4	yes	—
190	#104	—	118	4	no	—
191	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
192	C₄×C₈	C₂²×D₅₉	1	2	yes	—
193	#41	—	118	2	no	—
194	#41	—	118	2	no	—
195	#41	—	236	1	no	—
196	#41	—	236	1	no	—
197	C₂×Q₈	—	118	1	no	—
198	C₂×D₄	—	118	1	no	—
199	C₂×D₄	—	236	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	236	1	no	—
203	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
204	Q₁₆	—	59	2	no	—
205	Q₁₆	—	118	1	no	—
206	SD₁₆	—	118	2	no	—
207	SD₁₆	—	118	2	no	—
208	D₈	—	59	2	no	—
209	D₈	—	118	1	no	—
210	M₁₆	—	118	2	no	—
211	M₁₆	—	236	1	no	—
212	M₁₆	—	236	4	no	—
213	C₂×C₈	—	236	1	no	—
214	C₂×C₈	—	236	4	no	—
215	C₂×C₈	#11745671	1	1	yes	characteristic
216	C₂×C₈	#11745671	1	1	yes	characteristic
217	C₂×C₈	#11745671	1	2	yes	—
218	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
219	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
220	C₂×C₈	#11745689	1	1	yes	characteristic
221	C₂×C₈	#11745689	1	1	yes	characteristic
222	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
223	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
224	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	8	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
230	C₁₆	—	118	4	no	—
231	C₂³	—	236	1	no	—
232	C₂³	#14273753	1	1	yes	characteristic
233	Q₈	—	118	2	no	—
234	Q₈	—	236	1	no	—
235	D₄	—	118	2	no	—
236	D₄	—	236	2	no	—
237	D₄	—	236	2	no	—
238	D₄	—	236	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	236	1	no	—
244	C₂×C₄	—	236	2	no	—
245	C₂×C₄	—	236	2	no	—
246	C₂×C₄	#14273703	1	1	yes	characteristic
247	C₂×C₄	#14273703	1	1	yes	characteristic
248	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
249	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
250	C₂×C₄	#14273753	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
252	C₈	—	236	2	no	—
253	C₈	#14273703	1	1	yes	characteristic
254	C₈	#14273703	1	1	yes	characteristic
255	C₈	C₂×#11745671	1	2	yes	—
256	C₈	C₂×#11745671	1	2	yes	—
257	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
258	C₈	C₂×#11745689	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	472	2	no	—
261	C₂²	—	236	2	no	—
262	C₂²	#16837689	1	1	yes	characteristic
263	C₂²	#16837689	1	1	yes	characteristic
264	C₂²	C₂×#14273753	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	8	4	no	—
267	C₄	—	236	2	no	—
268	C₄	#16837689	1	1	yes	characteristic
269	C₄	#16837689	1	1	yes	characteristic
270	C₄	C₂×#14273703	1	1	yes	characteristic
271	C₄	C₂²×#11745671	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	472	2	no	—
275	C₂	#21426269	1	2	yes	direct product factor
276	C₂	C₂×#16837689	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837689
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#499	C₂
socle	C₂²×C₅₉	#476
prime radical	C₂×#14273749	C₂²
hypercenter	C₂×#499	D₅₉
hypocenter	C₅₉	C₂×#2231
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #21426269	C₂	direct (N × H)
3. C₂×#16837224	C₂	semidirect
4. C₂×#16837223	C₂	semidirect
6. C₂×C₅₉×#499	C₂	semidirect
14. #16837224	C₂²	semidirect
15. #16837223	C₂²	semidirect
19. C₅₉×#499	C₂²	semidirect
134. C₂×#499	D₅₉	semidirect
142. C₂×C₅₉	#2231	semidirect
150. #499	C₂×D₅₉	semidirect
161. C₅₉	C₂×#2231	semidirect
275. C₂	#21426269	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×#16837520	2
#21426269	2
C₂×#16837224	2
C₂×#16837223	2
C₂×#16837211	2
C₂×C₅₉×#499	2
C₂×#2231	59

Subgroups — C2×#21426269 #28743422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426269 #28743422