Subgroups — C₂²×C₄×#11745689 #29495264

Order 15104.

12878 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11745689	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11745689	C₂	1	48	yes	maximal, direct product factor
3	C₂²×Q₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₂₃₆	C₂	1	3	yes	maximal, complemented
5	C₂²×#14273689	C₂	1	3	yes	maximal, complemented
6	C₂²×#14273685	C₂	1	3	yes	maximal, complemented
7	C₂²×C₄²×D₅₉	C₂	1	3	yes	maximal, complemented
8	C₂²×C₄×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11745689	C₄	1	32	yes	direct product factor
10	C₂²×#11745689	C₂²	1	24	yes	—
11	C₂³×D₂₃₆	C₂²	1	3	yes	—
12	C₂³×C₄×D₅₉	C₂²	1	3	yes	—
13	C₂³×C₄×D₅₉	C₂²	1	3	yes	—
14	C₂³×C₄×D₅₉	C₂²	1	3	yes	—
15	C₄×#11745689	C₂²	1	256	yes	direct product factor
16	C₂×Q₈×Dic₅₉	C₂²	1	24	yes	complemented
17	C₂×C₄×D₂₃₆	C₂²	1	72	yes	complemented
18	C₂×#14273689	C₂²	1	72	yes	complemented
19	C₂×#14273685	C₂²	1	72	yes	complemented
20	C₂²×C₄⋊Dic₅₉	C₂²	1	3	yes	—
21	C₂²×(C₂×D₅₉)⋊C₄	C₂²	1	6	yes	—
22	C₂²×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄×Dic₅₉	C₂²	1	3	yes	—
24	C₂²×C₄×Dic₅₉	C₂²	1	3	yes	—
25	C₂×C₄²×D₅₉	C₂²	1	72	yes	complemented
26	C₂³×Q₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₅₉	C₂²	1	24	yes	complemented
28	C₂²×C₄⋊C₄×C₅₉	C₂²	1	3	yes	—
29	C₂²×C₄²×C₅₉	C₂²	1	3	yes	—
30	C₂⁴×D₅₉	C₂³	1	3	yes	—
31	C₂×#11745689	C₂×C₄	1	384	yes	direct product factor
32	C₂×#11745689	C₂³	1	64	yes	—
33	C₂²×D₂₃₆	C₂×C₄	1	48	yes	complemented
34	C₂²×D₂₃₆	C₂³	1	36	yes	—
35	C₂³×Dic₅₉	C₂³	1	3	yes	—
36	C₂³×Dic₅₉	C₂³	1	1	yes	characteristic
37	C₂³×Dic₅₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×D₅₉	—	2	24	no	—
39	C₂²×C₄×D₅₉	—	2	24	no	—
40	C₂²×C₄×D₅₉	C₂×C₄	1	48	yes	complemented
41	C₂²×C₄×D₅₉	C₂³	1	36	yes	—
42	C₂²×C₄×D₅₉	C₂³	1	36	yes	—
43	C₂²×C₄×D₅₉	C₂³	1	36	yes	—
44	Q₈×Dic₅₉	C₂³	1	64	yes	complemented
45	C₄×D₂₃₆	C₂³	1	192	yes	complemented
46	#14273689	C₂³	1	192	yes	complemented
47	#14273685	C₂³	1	192	yes	complemented
48	C₂×C₄⋊Dic₅₉	C₂³	1	36	yes	—
49	C₂×(C₂×D₅₉)⋊C₄	C₂³	1	72	yes	—
50	C₂×C₄×Dic₅₉	C₂³	1	12	yes	—
51	C₂×C₄×Dic₅₉	C₂³	1	36	yes	—
52	C₂×C₄×Dic₅₉	C₂³	1	36	yes	—
53	C₄²×D₅₉	C₂³	1	192	yes	complemented
54	C₂²×Q₈×C₅₉	C₂×C₄	1	16	yes	complemented
55	C₂²×Q₈×C₅₉	C₂³	1	12	yes	—
56	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
57	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
58	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
59	C₄×Q₈×C₅₉	C₂³	1	64	yes	complemented
60	C₂×C₄⋊C₄×C₅₉	C₂³	1	36	yes	—
61	C₂×C₄²×C₅₉	C₂³	1	36	yes	—
62	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
63	C₂³×D₅₉	—	2	12	no	—
64	C₂³×D₅₉	—	2	12	no	—
65	C₂³×D₅₉	C₂²×C₄	1	24	yes	—
66	C₂³×D₅₉	C₂⁴	1	18	yes	—
67	C₂×Q₈×C₅₉	C₂²×C₄	1	96	yes	complemented
68	C₂×Q₈×C₅₉	C₂⁴	1	16	yes	—
69	C₂²×C₄×C₅₉	—	2	12	no	—
70	C₂²×C₄×C₅₉	C₂²×C₄	1	24	yes	—
71	C₂²×C₄×C₅₉	#41	1	8	yes	complemented
72	C₂²×C₄×C₅₉	C₂⁴	1	18	yes	—
73	C₂²×C₄×C₅₉	C₂⁴	1	18	yes	—
74	C₂²×C₄×C₅₉	C₂⁴	1	6	yes	—
75	C₂²×Dic₅₉	—	2	12	no	—
76	C₂²×Dic₅₉	C₂²×C₄	1	8	yes	—
77	C₂²×Dic₅₉	#41	1	8	yes	complemented
78	C₂²×Dic₅₉	C₂⁴	1	18	yes	—
79	C₂²×Dic₅₉	C₂⁴	1	6	yes	—
80	C₂²×Dic₅₉	C₂⁴	1	6	yes	—
81	#11745689	C₂²×C₄	1	512	yes	direct product factor
82	C₂×D₂₃₆	C₂²×C₄	1	288	yes	complemented
83	C₂×D₂₃₆	C₂⁴	1	48	yes	—
84	C₂×C₄×D₅₉	—	2	144	no	—
85	C₂×C₄×D₅₉	—	2	144	no	—
86	C₂×C₄×D₅₉	C₂²×C₄	1	288	yes	complemented
87	C₂×C₄×D₅₉	C₂⁴	1	48	yes	—
88	C₂×C₄×D₅₉	C₂⁴	1	48	yes	—
89	C₂×C₄×D₅₉	C₂⁴	1	48	yes	—
90	C₄⋊C₄×C₅₉	C₂⁴	1	48	yes	—
91	C₄²×C₅₉	C₂⁴	1	48	yes	—
92	(C₂×D₅₉)⋊C₄	C₂⁴	1	96	yes	—
93	C₄⋊Dic₅₉	C₂⁴	1	48	yes	—
94	C₄×Dic₅₉	C₂⁴	1	16	yes	—
95	C₄×Dic₅₉	C₂⁴	1	48	yes	—
96	C₄×Dic₅₉	C₂⁴	1	48	yes	—
97	C₂³×C₅₉	C₂³×C₄	1	4	yes	—
98	C₂³×C₅₉	C₂×#41	1	4	yes	—
99	C₂³×C₅₉	C₂×#41	1	4	yes	—
100	C₂³×C₅₉	C₂⁵	1	3	yes	—
101	C₂²×D₅₉	—	2	96	no	—
102	C₂²×D₅₉	—	2	36	no	—
103	C₂²×D₅₉	—	2	36	no	—
104	C₂²×D₅₉	C₂³×C₄	1	72	yes	—
105	C₂²×D₅₉	C₂⁵	1	12	yes	—
106	Q₈×C₅₉	C₂³×C₄	1	64	yes	complemented
107	C₂×C₄×C₅₉	—	2	36	no	—
108	C₂×C₄×C₅₉	C₂³×C₄	1	72	yes	—
109	C₂×C₄×C₅₉	C₂×#41	1	24	yes	complemented
110	C₂×C₄×C₅₉	C₂⁵	1	12	yes	—
111	C₂×C₄×C₅₉	C₂⁵	1	12	yes	—
112	C₂×C₄×C₅₉	C₂⁵	1	4	yes	—
113	C₂×Dic₅₉	—	2	36	no	—
114	C₂×Dic₅₉	C₂³×C₄	1	24	yes	—
115	C₂×Dic₅₉	C₂×#41	1	24	yes	complemented
116	C₂×Dic₅₉	C₂⁵	1	12	yes	—
117	C₂×Dic₅₉	C₂⁵	1	4	yes	—
118	C₂×Dic₅₉	C₂⁵	1	4	yes	—
119	D₂₃₆	C₂³×C₄	1	192	yes	complemented
120	C₄×D₅₉	—	2	96	no	—
121	C₄×D₅₉	—	2	96	no	—
122	C₄×D₅₉	C₂³×C₄	1	192	yes	complemented
123	C₂²×C₄×#41	—	59	1	no	maximal, Sylow
124	C₂²×C₅₉	C₄×#41	1	16	yes	complemented
125	C₂²×C₅₉	C₂⁴×C₄	1	6	yes	—
126	C₂²×C₅₉	C₂²×#41	1	6	yes	—
127	C₂²×C₅₉	C₂²×#41	1	6	yes	—
128	C₂²×C₅₉	C₂⁶	1	1	yes	characteristic
129	C₂×D₅₉	—	2	12	no	—
130	C₂×D₅₉	—	2	12	no	—
131	C₂×D₅₉	—	2	144	no	—
132	C₂×D₅₉	C₂⁴×C₄	1	24	yes	—
133	C₄×C₅₉	—	2	12	no	—
134	C₄×C₅₉	C₂⁴×C₄	1	24	yes	—
135	C₄×C₅₉	C₂²×#41	1	8	yes	complemented
136	Dic₅₉	—	2	12	no	—
137	Dic₅₉	C₂⁴×C₄	1	8	yes	—
138	Dic₅₉	C₂²×#41	1	8	yes	complemented
139	C₂³×#41	—	59	1	no	—
140	C₂×C₄×#41	—	59	48	no	—
141	C₂²×C₄×Q₈	—	59	1	no	—
142	C₂²×C₄×Q₈	D₅₉	1	1	yes	characteristic, complemented
143	C₂²×C₄×D₄	—	59	3	no	—
144	C₂²×C₄×D₄	—	59	3	no	—
145	C₂²×#117	—	59	3	no	—
146	C₂³×C₄²	—	59	3	no	—
147	C₂×C₅₉	C₂×C₄×#41	1	12	yes	complemented
148	C₂×C₅₉	C₂⁵×C₄	1	1	yes	characteristic
149	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
150	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
151	D₅₉	—	2	24	no	—
152	C₂²×#41	—	59	32	no	—
153	C₂²×#41	—	59	24	no	—
154	C₂³×Q₈	C₂×D₅₉	1	1	yes	characteristic
155	C₂³×D₄	—	59	3	no	—
156	C₂⁴×C₄	—	59	3	no	—
157	C₂⁴×C₄	—	59	3	no	—
158	C₂⁴×C₄	—	59	3	no	—
159	C₄×#41	—	59	256	no	—
160	C₂×C₄×Q₈	—	59	24	no	—
161	C₂×C₄×Q₈	C₂×D₅₉	1	24	yes	complemented
162	C₂×C₄×D₄	—	59	72	no	—
163	C₂×C₄×D₄	—	59	72	no	—
164	C₂×#117	—	59	72	no	—
165	C₂²×C₄⋊C₄	—	59	3	no	—
166	C₂²×C₄⋊C₄	C₂×D₅₉	1	3	yes	—
167	C₂²×C₂²⋊C₄	—	59	6	no	—
168	C₂²×C₄²	—	59	3	no	—
169	C₂²×C₄²	—	59	3	no	—
170	C₂²×C₄²	—	59	1	no	—
171	C₂²×C₄²	—	59	72	no	—
172	C₂²×C₄²	C₂×D₅₉	1	3	yes	—
173	C₅₉	C₂²×C₄×#41	1	1	yes	characteristic, complemented, Sylow
174	C₂⁵	—	59	3	no	—
175	C₂×#41	—	59	384	no	—
176	C₂×#41	—	59	64	no	—
177	C₂²×Q₈	C₄×D₅₉	1	16	yes	complemented
178	C₂²×Q₈	C₂²×D₅₉	1	12	yes	—
179	C₂²×D₄	—	59	48	no	—
180	C₂²×D₄	—	59	36	no	—
181	C₂³×C₄	—	59	1	no	—
182	C₂³×C₄	—	59	1	no	—
183	C₂³×C₄	—	59	48	no	—
184	C₂³×C₄	—	59	36	no	—
185	C₂³×C₄	—	59	3	no	—
186	C₂³×C₄	—	59	36	no	—
187	C₂³×C₄	—	59	36	no	—
188	C₂³×C₄	—	118	24	no	—
189	C₂³×C₄	—	118	24	no	—
190	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
191	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
192	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
193	C₄×Q₈	—	59	64	no	—
194	C₄×Q₈	C₂²×D₅₉	1	64	yes	complemented
195	C₄×D₄	—	59	192	no	—
196	C₄×D₄	—	59	192	no	—
197	#117	—	59	192	no	—
198	C₂×C₄⋊C₄	—	59	36	no	—
199	C₂×C₄⋊C₄	C₂²×D₅₉	1	36	yes	—
200	C₂×C₂²⋊C₄	—	59	72	no	—
201	C₂×C₄²	—	59	36	no	—
202	C₂×C₄²	—	59	36	no	—
203	C₂×C₄²	—	59	12	no	—
204	C₂×C₄²	—	59	192	no	—
205	C₂×C₄²	C₂²×D₅₉	1	36	yes	—
206	C₂⁴	—	59	24	no	—
207	C₂⁴	—	59	18	no	—
208	C₂⁴	—	118	12	no	—
209	C₂⁴	—	118	12	no	—
210	C₂⁴	C₂³×D₅₉	1	1	yes	characteristic
211	#41	—	59	512	no	—
212	C₂×Q₈	C₂×C₄×D₅₉	1	96	yes	complemented
213	C₂×Q₈	C₂³×D₅₉	1	16	yes	—
214	C₂×D₄	—	59	288	no	—
215	C₂×D₄	—	59	48	no	—
216	C₂²×C₄	—	2	12	no	—
217	C₂²×C₄	—	59	8	no	—
218	C₂²×C₄	—	59	8	no	—
219	C₂²×C₄	—	59	6	no	—
220	C₂²×C₄	—	59	6	no	—
221	C₂²×C₄	—	59	288	no	—
222	C₂²×C₄	—	59	48	no	—
223	C₂²×C₄	—	59	18	no	—
224	C₂²×C₄	—	59	48	no	—
225	C₂²×C₄	—	59	48	no	—
226	C₂²×C₄	—	118	144	no	—
227	C₂²×C₄	—	118	12	no	—
228	C₂²×C₄	—	118	144	no	—
229	C₂²×C₄	C₂×C₄×D₅₉	1	24	yes	—
230	C₂²×C₄	#11745689	1	8	yes	direct product factor
231	C₂²×C₄	C₂³×D₅₉	1	18	yes	—
232	C₂²×C₄	C₂³×D₅₉	1	18	yes	—
233	C₂²×C₄	C₂³×D₅₉	1	6	yes	—
234	C₄⋊C₄	—	59	48	no	—
235	C₄⋊C₄	C₂³×D₅₉	1	48	yes	—
236	C₂²⋊C₄	—	59	96	no	—
237	C₄²	—	59	16	no	—
238	C₄²	—	59	48	no	—
239	C₄²	—	59	48	no	—
240	C₄²	C₂³×D₅₉	1	48	yes	—
241	C₂³	—	59	72	no	—
242	C₂³	—	59	12	no	—
243	C₂³	—	118	36	no	—
244	C₂³	—	118	36	no	—
245	C₂³	—	118	96	no	—
246	C₂³	C₂²×C₄×D₅₉	1	4	yes	—
247	C₂³	#41×D₅₉	1	4	yes	—
248	C₂³	C₂×#11745689	1	4	yes	—
249	C₂³	C₂⁴×D₅₉	1	3	yes	—
250	Q₈	C₂²×C₄×D₅₉	1	64	yes	complemented
251	D₄	—	59	192	no	—
252	C₂×C₄	—	2	36	no	—
253	C₂×C₄	—	59	24	no	—
254	C₂×C₄	—	59	24	no	—
255	C₂×C₄	—	59	4	no	—
256	C₂×C₄	—	59	4	no	—
257	C₂×C₄	—	59	192	no	—
258	C₂×C₄	—	59	12	no	—
259	C₂×C₄	—	118	96	no	—
260	C₂×C₄	—	118	36	no	—
261	C₂×C₄	—	118	96	no	—
262	C₂×C₄	C₂²×C₄×D₅₉	1	72	yes	—
263	C₂×C₄	C₂×#11745689	1	24	yes	direct product factor
264	C₂×C₄	C₂⁴×D₅₉	1	12	yes	—
265	C₂×C₄	C₂⁴×D₅₉	1	12	yes	—
266	C₂×C₄	C₂⁴×D₅₉	1	4	yes	—
267	C₂²	—	59	24	no	—
268	C₂²	—	118	144	no	—
269	C₂²	—	118	12	no	—
270	C₂²	—	118	12	no	—
271	C₂²	C₄×#11745689	1	16	yes	direct product factor
272	C₂²	C₂³×C₄×D₅₉	1	6	yes	—
273	C₂²	C₂×#41×D₅₉	1	6	yes	—
274	C₂²	C₂²×#11745689	1	6	yes	—
275	C₂²	C₂⁵×D₅₉	1	1	yes	characteristic
276	C₄	—	2	12	no	—
277	C₄	—	59	8	no	—
278	C₄	—	59	8	no	—
279	C₄	—	118	12	no	—
280	C₄	C₂³×C₄×D₅₉	1	24	yes	—
281	C₄	C₂²×#11745689	1	8	yes	direct product factor
282	C₂	—	118	24	no	—
283	C₂	C₂×C₄×#11745689	1	12	yes	direct product factor
284	C₂	C₂⁴×C₄×D₅₉	1	1	yes	characteristic
285	C₂	C₂²×#41×D₅₉	1	1	yes	characteristic
286	C₂	C₂³×#11745689	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₉
commutator	C₂×C₅₉	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₉
Fitting	C₂²×C₄×Q₈×C₅₉	C₂
socle	C₂⁴×C₅₉	C₂⁴
prime radical	C₂³×#11745689	C₂
cosocle	C₂²×C₅₉	C₂⁶
hypercenter	C₂²×C₄×Q₈	D₅₉
hypocenter	C₅₉	C₂²×C₄×#41
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11745689	C₂	direct (N × H)
3. C₂²×Q₈×Dic₅₉	C₂	semidirect
4. C₂²×C₄×D₂₃₆	C₂	semidirect
5. C₂²×#14273689	C₂	semidirect
6. C₂²×#14273685	C₂	semidirect
7. C₂²×C₄²×D₅₉	C₂	semidirect
8. C₂²×C₄×Q₈×C₅₉	C₂	semidirect
9. C₂²×#11745689	C₄	direct (N × H)
15. C₄×#11745689	C₂²	direct (N × H)
16. C₂×Q₈×Dic₅₉	C₂²	semidirect
17. C₂×C₄×D₂₃₆	C₂²	semidirect
18. C₂×#14273689	C₂²	semidirect
19. C₂×#14273685	C₂²	semidirect
25. C₂×C₄²×D₅₉	C₂²	semidirect
27. C₂×C₄×Q₈×C₅₉	C₂²	semidirect
31. C₂×#11745689	C₂×C₄	direct (N × H)
33. C₂²×D₂₃₆	C₂×C₄	semidirect
40. C₂²×C₄×D₅₉	C₂×C₄	semidirect
44. Q₈×Dic₅₉	C₂³	semidirect
45. C₄×D₂₃₆	C₂³	semidirect
46. #14273689	C₂³	semidirect
47. #14273685	C₂³	semidirect
53. C₄²×D₅₉	C₂³	semidirect
54. C₂²×Q₈×C₅₉	C₂×C₄	semidirect
59. C₄×Q₈×C₅₉	C₂³	semidirect
67. C₂×Q₈×C₅₉	C₂²×C₄	semidirect
71. C₂²×C₄×C₅₉	#41	semidirect
77. C₂²×Dic₅₉	#41	semidirect
81. #11745689	C₂²×C₄	direct (N × H)
82. C₂×D₂₃₆	C₂²×C₄	semidirect
86. C₂×C₄×D₅₉	C₂²×C₄	semidirect
106. Q₈×C₅₉	C₂³×C₄	semidirect
109. C₂×C₄×C₅₉	C₂×#41	semidirect
115. C₂×Dic₅₉	C₂×#41	semidirect
119. D₂₃₆	C₂³×C₄	semidirect
122. C₄×D₅₉	C₂³×C₄	semidirect
124. C₂²×C₅₉	C₄×#41	semidirect
135. C₄×C₅₉	C₂²×#41	semidirect
138. Dic₅₉	C₂²×#41	semidirect
142. C₂²×C₄×Q₈	D₅₉	semidirect
147. C₂×C₅₉	C₂×C₄×#41	semidirect
161. C₂×C₄×Q₈	C₂×D₅₉	semidirect
173. C₅₉	C₂²×C₄×#41	semidirect
177. C₂²×Q₈	C₄×D₅₉	semidirect
194. C₄×Q₈	C₂²×D₅₉	semidirect
212. C₂×Q₈	C₂×C₄×D₅₉	semidirect
230. C₂²×C₄	#11745689	direct (N × H)
250. Q₈	C₂²×C₄×D₅₉	semidirect
263. C₂×C₄	C₂×#11745689	direct (N × H)
271. C₂²	C₄×#11745689	direct (N × H)
281. C₄	C₂²×#11745689	direct (N × H)
283. C₂	C₂×C₄×#11745689	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11745689	2
C₂×C₄×#11745689	2
C₂²×Q₈×Dic₅₉	2
C₂²×C₄×D₂₃₆	2
C₂²×#14273689	2
C₂²×#14273685	2
C₂²×C₄²×D₅₉	2
C₂²×C₄×Q₈×C₅₉	2
C₂²×C₄×#41	59

Subgroups — C22×C4×#11745689 #29495264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11745689 #29495264