Subgroups — #29458234

Order 15104.

3386 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838207	C₂	1	4	yes	maximal, complemented
2	#21437874	C₂	1	2	yes	maximal, complemented
3	#21434582	C₂	1	12	yes	maximal, complemented
4	#21434548	C₂	1	6	yes	maximal, complemented
5	#21434524	C₂	1	3	yes	maximal, complemented
6	C₂×#16837779	C₂	1	3	yes	maximal, complemented
7	C₄×#131×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273773	C₂²	1	4	yes	—
9	#16838207	—	2	32	no	—
10	C₂×#14273755	C₂²	1	24	yes	complemented
11	C₂×#14273749	C₂²	1	12	yes	complemented
12	C₂²×D₄₇₂	C₂²	1	12	yes	complemented
13	#16838074	—	2	4	no	—
14	D₄×D₂₃₆	—	2	12	no	—
15	#16837971	C₂²	1	12	yes	complemented
16	#16837952	C₂²	1	6	yes	complemented
17	C₂×#14273724	C₂²	1	6	yes	complemented
18	#16837791	C₂²	1	24	yes	complemented
19	#16837786	C₂²	1	12	yes	complemented
20	#16837782	C₂²	1	1	yes	characteristic, complemented
21	#16837779	C₂²	1	3	yes	complemented
22	#16837779	C₂²	1	6	yes	complemented
23	#16837779	C₂²	1	12	yes	complemented
24	#16837775	C₂²	1	6	yes	complemented
25	C₂×#131×C₅₉	C₂²	1	2	yes	—
26	C₄×#41×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#405	C₂²	1	6	yes	complemented
28	C₄×M₁₆×C₅₉	C₂²	1	3	yes	complemented
29	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	complemented
30	#14273773	—	2	32	no	—
31	C₂×#11745689	—	2	4	no	—
32	C₂×#11745685	C₂³	1	12	yes	—
33	C₂²×D₂₃₆	—	2	12	no	—
34	C₂²×D₂₃₆	C₂³	1	12	yes	—
35	C₂×D₄×D₅₉	—	2	12	no	—
36	#14273755	—	2	96	no	—
37	#14273749	—	2	48	no	—
38	C₂×Dic₂₃₆	C₂³	1	4	yes	—
39	C₂×#11745660	C₂³	1	24	yes	—
40	C₂×D₄₇₂	—	2	48	no	—
41	C₂×D₄₇₂	C₂³	1	12	yes	—
42	C₂×D₄₇₂	C₂³	1	24	yes	—
43	#14273727	C₂³	1	2	yes	—
44	#14273724	—	2	12	no	—
45	#14273724	C₂³	1	6	yes	—
46	#14273724	C₂³	1	12	yes	—
47	#14273715	C₂³	1	12	yes	—
48	#14273702	—	2	24	no	—
49	#14273701	—	2	12	no	—
50	#14273697	—	2	24	no	—
51	C₄×D₂₃₆	—	2	12	no	—
52	C₂×#41×C₅₉	C₂³	1	1	yes	characteristic
53	#131×C₅₉	D₄	1	16	yes	complemented
54	C₂×M₁₆×C₅₉	C₂³	1	6	yes	—
55	C₂²×C₈×C₅₉	C₂³	1	6	yes	—
56	C₄×Q₈×C₅₉	D₄	1	2	yes	—
57	C₄×D₄×C₅₉	D₄	1	6	yes	—
58	#117×C₅₉	C₂³	1	3	yes	—
59	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
60	#97×C₅₉	C₂³	1	6	yes	—
61	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₉	C₂³	1	3	yes	—
63	C₄×C₈×C₅₉	C₂³	1	6	yes	—
64	C₂³×D₅₉	—	2	12	no	—
65	#41×C₅₉	C₂×D₄	1	8	yes	—
66	C₂×Q₈×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₅₉	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
70	C₂×C₂²⋊D₅₉	—	2	12	no	—
71	#11745689	—	4	16	no	—
72	D₄×D₅₉	—	4	48	no	—
73	#11745685	—	2	48	no	—
74	C₂×D₂₃₆	—	2	48	no	—
75	C₂×D₂₃₆	—	2	12	no	—
76	C₂×D₂₃₆	—	2	12	no	—
77	C₂×D₂₃₆	—	2	24	no	—
78	C₂×D₂₃₆	—	8	24	no	—
79	C₂×D₂₃₆	C₂⁴	1	12	yes	—
80	C₂×C₄×D₅₉	—	2	12	no	—
81	C₂×Dic₁₁₈	C₂⁴	1	4	yes	—
82	M₁₆×C₅₉	C₂×D₄	1	24	yes	complemented
83	C₂×C₈×C₅₉	C₂×D₄	1	24	yes	complemented
84	C₂×C₈×C₅₉	C₂⁴	1	2	yes	—
85	C₂×C₈×C₅₉	C₂⁴	1	6	yes	—
86	C₄⋊C₄×C₅₉	C₂×D₄	1	6	yes	—
87	C₂²⋊C₄×C₅₉	C₂×D₄	1	6	yes	—
88	C₄²×C₅₉	C₂×D₄	1	6	yes	—
89	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
90	C₄²×C₅₉	C₂⁴	1	3	yes	—
91	(C₂×D₅₉)⋊C₄	—	2	24	no	—
92	C₄⋊Dic₅₉	—	2	4	no	—
93	Dic₂₃₆	—	2	8	no	—
94	D₄₇₂	—	2	24	no	—
95	D₄₇₂	—	2	48	no	—
96	#11745660	—	2	48	no	—
97	C₂³×C₅₉	C₂²×D₄	1	3	yes	—
98	C₂²×D₅₉	—	2	12	no	—
99	C₂²×D₅₉	—	4	24	no	—
100	C₂²×D₅₉	—	8	24	no	—
101	Q₈×C₅₉	#127	1	4	yes	—
102	D₄×C₅₉	#127	1	12	yes	—
103	C₂×C₄×C₅₉	—	2	12	no	—
104	C₂×C₄×C₅₉	C₂²×D₄	1	6	yes	—
105	C₂×C₄×C₅₉	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₅₉	C₂²×D₄	1	12	yes	—
107	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
109	C₂²⋊D₅₉	—	4	24	no	—
110	C₂×Dic₅₉	—	2	4	no	—
111	D₂₃₆	—	2	24	no	—
112	D₂₃₆	—	4	24	no	—
113	D₂₃₆	—	4	24	no	—
114	D₂₃₆	—	8	24	no	—
115	C₄×D₅₉	—	4	24	no	—
116	Dic₁₁₈	—	2	8	no	—
117	C₈×C₅₉	C₂²×D₄	1	12	yes	—
118	C₈×C₅₉	C₂²×D₄	1	4	yes	—
119	#60647	—	59	1	no	maximal, Sylow
120	C₂²×C₅₉	—	2	6	no	—
121	C₂²×C₅₉	C₂×#127	1	6	yes	—
122	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
123	C₂×D₅₉	—	4	12	no	—
124	C₂×D₅₉	—	4	24	no	—
125	C₂×D₅₉	—	16	24	no	—
126	C₄×C₅₉	—	2	6	no	—
127	C₄×C₅₉	C₂×#127	1	6	yes	—
128	C₄×C₅₉	#576	1	4	yes	complemented
129	C₄×C₅₉	C₂³×D₄	1	2	yes	—
130	Dic₅₉	—	4	4	no	—
131	C₂×#576	—	59	4	no	—
132	#3498	—	59	2	no	—
133	#3155	—	59	12	no	—
134	#3151	—	59	6	no	—
135	#3147	—	59	3	no	—
136	C₂×#493	—	59	3	no	—
137	C₄×#131	D₅₉	1	1	yes	characteristic, complemented
138	C₂×C₅₉	—	2	6	no	—
139	C₂×C₅₉	C₂²×#127	1	1	yes	characteristic
140	C₂×C₅₉	C₂×#576	1	2	yes	—
141	D₅₉	—	8	12	no	—
142	C₂×H(5,𝔽₂)	—	59	4	no	—
143	#576	—	118	32	no	—
144	C₂×AGL(1,ℤ/8)	—	59	24	no	—
145	C₂×#135	—	59	12	no	—
146	C₂²×D₈	—	59	12	no	—
147	C₂×#131	C₂×D₅₉	1	2	yes	—
148	#550	—	118	4	no	—
149	D₄²	—	118	12	no	—
150	#535	—	59	12	no	—
151	#532	—	59	6	no	—
152	C₂×#127	—	59	6	no	—
153	C₄×#41	C₂×D₅₉	1	1	yes	characteristic
154	#496	—	59	24	no	—
155	#495	—	59	12	no	—
156	#494	—	59	1	no	—
157	#493	—	59	3	no	—
158	#493	—	59	6	no	—
159	#493	—	59	12	no	—
160	#492	—	59	6	no	—
161	#405	C₂×D₅₉	1	6	yes	complemented
162	C₄×M₁₆	C₂×D₅₉	1	3	yes	complemented
163	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	complemented
164	C₅₉	#60647	1	1	yes	characteristic, complemented, Sylow
165	H(5,𝔽₂)	—	118	32	no	—
166	C₂×#41	—	59	12	no	—
167	C₂×#41	—	118	4	no	—
168	C₂×#41	C₂²×D₅₉	1	1	yes	characteristic
169	C₂²×D₄	—	59	12	no	—
170	C₂²×D₄	—	118	12	no	—
171	C₂²×D₄	—	118	12	no	—
172	AGL(1,ℤ/8)	—	118	96	no	—
173	#135	—	118	48	no	—
174	C₂×Q₁₆	—	59	4	no	—
175	C₂×SD₁₆	—	59	24	no	—
176	C₂×D₈	—	59	12	no	—
177	C₂×D₈	—	59	24	no	—
178	C₂×D₈	—	118	48	no	—
179	#131	D₂₃₆	1	16	yes	complemented
180	C₂×M₁₆	C₂²×D₅₉	1	6	yes	—
181	C₂²×C₈	C₂²×D₅₉	1	6	yes	—
182	C₄⋊Q₈	—	59	2	no	—
183	#127	—	59	6	no	—
184	#127	—	59	12	no	—
185	#127	—	118	12	no	—
186	#124	—	59	12	no	—
187	#121	—	118	12	no	—
188	#121	—	118	24	no	—
189	#120	—	118	24	no	—
190	C₄×Q₈	D₂₃₆	1	2	yes	—
191	C₄×D₄	—	118	12	no	—
192	C₄×D₄	D₂₃₆	1	6	yes	—
193	#117	C₂²×D₅₉	1	3	yes	—
194	C₂×C₄²	C₂²×D₅₉	1	3	yes	—
195	#97	C₂²×D₅₉	1	6	yes	—
196	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₅₉	1	3	yes	—
198	C₄×C₈	C₂²×D₅₉	1	6	yes	—
199	C₂⁴	—	118	12	no	—
200	#41	—	118	48	no	—
201	#41	—	236	16	no	—
202	#41	C₂×D₂₃₆	1	8	yes	—
203	C₂×Q₈	—	59	4	no	—
204	C₂×Q₈	C₂×D₂₃₆	1	1	yes	characteristic
205	C₂×D₄	—	59	12	no	—
206	C₂×D₄	—	118	12	no	—
207	C₂×D₄	—	118	48	no	—
208	C₂×D₄	—	118	24	no	—
209	C₂×D₄	—	118	12	no	—
210	C₂×D₄	—	118	12	no	—
211	C₂×D₄	—	472	24	no	—
212	C₂×D₄	—	236	48	no	—
213	C₂×D₄	C₂×D₂₃₆	1	3	yes	—
214	C₂²×C₄	—	118	12	no	—
215	C₂²×C₄	C₂×D₂₃₆	1	6	yes	—
216	C₂²×C₄	C₂³×D₅₉	1	3	yes	—
217	Q₁₆	—	118	8	no	—
218	SD₁₆	—	118	48	no	—
219	D₈	—	118	24	no	—
220	D₈	—	118	48	no	—
221	M₁₆	C₂×D₂₃₆	1	24	yes	complemented
222	C₂×C₈	C₂×D₂₃₆	1	24	yes	complemented
223	C₂×C₈	C₂³×D₅₉	1	2	yes	—
224	C₂×C₈	C₂³×D₅₉	1	6	yes	—
225	C₄⋊C₄	—	118	4	no	—
226	C₄⋊C₄	C₂×D₂₃₆	1	6	yes	—
227	C₂²⋊C₄	—	118	24	no	—
228	C₂²⋊C₄	C₂×D₂₃₆	1	6	yes	—
229	C₄²	C₂×D₂₃₆	1	6	yes	—
230	C₄²	C₂³×D₅₉	1	1	yes	characteristic
231	C₄²	C₂³×D₅₉	1	3	yes	—
232	C₂³	—	118	12	no	—
233	C₂³	—	472	24	no	—
234	C₂³	—	236	24	no	—
235	C₂³	C₂²×D₂₃₆	1	3	yes	—
236	Q₈	—	118	8	no	—
237	Q₈	#14273724	1	4	yes	—
238	D₄	—	118	24	no	—
239	D₄	—	472	24	no	—
240	D₄	—	236	24	no	—
241	D₄	—	236	24	no	—
242	D₄	—	236	24	no	—
243	D₄	#14273724	1	12	yes	—
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	118	4	no	—
246	C₂×C₄	—	236	24	no	—
247	C₂×C₄	C₂²×D₂₃₆	1	6	yes	—
248	C₂×C₄	C₂²×D₂₃₆	1	3	yes	—
249	C₂×C₄	C₂²×D₂₃₆	1	12	yes	—
250	C₂×C₄	C₂²×D₂₃₆	1	2	yes	—
251	C₂×C₄	C₂⁴×D₅₉	1	1	yes	characteristic
252	C₈	C₂²×D₂₃₆	1	4	yes	—
253	C₈	C₂²×D₂₃₆	1	12	yes	—
254	C₂²	—	2	6	no	—
255	C₂²	—	944	24	no	—
256	C₂²	—	236	24	no	—
257	C₂²	—	236	12	no	—
258	C₂²	C₂×#14273724	1	6	yes	—
259	C₂²	C₂³×D₂₃₆	1	1	yes	characteristic
260	C₄	—	2	6	no	—
261	C₄	—	236	4	no	—
262	C₄	C₂×#14273724	1	6	yes	—
263	C₄	#16838207	1	4	yes	complemented
264	C₄	C₂³×D₂₃₆	1	2	yes	—
265	C₂	—	2	6	no	—
266	C₂	—	472	12	no	—
267	C₂	C₂²×#14273724	1	1	yes	characteristic
268	C₂	C₂×#16838207	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₂₃₆
commutator, cosocle	C₂×C₄×C₅₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₉
Fitting	C₄×#131×C₅₉	C₂
socle	C₂²×C₅₉	C₂³×D₄
hypercenter	C₄×#131	D₅₉
hypocenter	C₅₉	#60647
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#16838207	C₂	semidirect
2. #21437874	C₂	semidirect
3. #21434582	C₂	semidirect
4. #21434548	C₂	semidirect
5. #21434524	C₂	semidirect
6. C₂×#16837779	C₂	semidirect
7. C₄×#131×C₅₉	C₂	semidirect
10. C₂×#14273755	C₂²	semidirect
11. C₂×#14273749	C₂²	semidirect
12. C₂²×D₄₇₂	C₂²	semidirect
15. #16837971	C₂²	semidirect
16. #16837952	C₂²	semidirect
17. C₂×#14273724	C₂²	semidirect
18. #16837791	C₂²	semidirect
19. #16837786	C₂²	semidirect
20. #16837782	C₂²	semidirect
21. #16837779	C₂²	semidirect
22. #16837779	C₂²	semidirect
23. #16837779	C₂²	semidirect
24. #16837775	C₂²	semidirect
27. C₅₉×#405	C₂²	semidirect
28. C₄×M₁₆×C₅₉	C₂²	semidirect
29. C₂×C₄×C₈×C₅₉	C₂²	semidirect
53. #131×C₅₉	D₄	semidirect
82. M₁₆×C₅₉	C₂×D₄	semidirect
83. C₂×C₈×C₅₉	C₂×D₄	semidirect
128. C₄×C₅₉	#576	semidirect
137. C₄×#131	D₅₉	semidirect
161. #405	C₂×D₅₉	semidirect
162. C₄×M₁₆	C₂×D₅₉	semidirect
163. C₂×C₄×C₈	C₂×D₅₉	semidirect
164. C₅₉	#60647	semidirect
179. #131	D₂₃₆	semidirect
221. M₁₆	C₂×D₂₃₆	semidirect
222. C₂×C₈	C₂×D₂₃₆	semidirect
263. C₄	#16838207	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16838207	2
#21437874	2
#21434582	2
#21434548	2
#21434524	2
C₂×#16837779	2
C₄×#131×C₅₉	2
#60647	59