Subgroups — C₄×#16837204 #28503894

Order 15104.

534 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837204	C₂	1	2	yes	maximal
3	C₄×#14273661	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422188	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₂₃₆	—	2	1	no	—
8	C₄×D₄₇₂	—	2	2	no	—
9	C₄×D₄₇₂	C₄	1	4	yes	complemented
10	C₂×#14273661	C₂²	1	1	yes	characteristic
11	C₂×#14273661	C₂²	1	2	yes	—
12	C₂×#14273649	—	2	1	no	—
13	C₄×C₄⋊Dic₅₉	—	2	1	no	—
14	#16837204	C₄	1	8	yes	direct product factor
15	#16837165	—	2	1	no	—
16	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₅₉	—	2	2	no	—
19	C₂²×D₂₃₆	—	2	1	no	—
20	C₂²×C₄×D₅₉	—	4	1	no	—
21	C₂×D₄₇₂	—	2	1	no	—
22	C₂×D₄₇₂	—	2	2	no	—
23	C₂×D₄₇₂	C₂×C₄	1	4	yes	—
24	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
25	C₄×D₂₃₆	—	2	4	no	—
26	C₄×D₂₃₆	—	4	2	no	—
27	C₂×C₄⋊Dic₅₉	—	2	1	no	—
28	C₂×C₄⋊Dic₅₉	—	2	2	no	—
29	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
30	C₂×C₄×Dic₅₉	—	4	1	no	—
31	#14273661	—	2	1	no	—
32	#14273661	—	2	2	no	—
33	#14273661	C₂×C₄	1	2	yes	—
34	#14273661	C₂×C₄	1	4	yes	complemented
35	#14273649	—	2	4	no	—
36	C₅₉⋊#95	—	4	1	no	—
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₉	—	2	2	no	—
41	C₂×C₁₆×C₅₉	—	2	2	no	—
42	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
43	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	D₄	1	2	yes	—
45	C₂³×D₅₉	—	4	1	no	—
46	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₉	—	4	2	no	—
50	C₂²×Dic₅₉	—	4	1	no	—
51	C₂×D₂₃₆	—	2	2	no	—
52	C₂×D₂₃₆	—	2	4	no	—
53	C₂×D₂₃₆	—	4	1	no	—
54	C₂×D₂₃₆	—	4	2	no	—
55	C₂×C₄×D₅₉	—	4	4	no	—
56	C₂×C₄×D₅₉	—	8	2	no	—
57	C₂×C₄×D₅₉	—	8	2	no	—
58	C₂×C₈×C₅₉	—	2	2	no	—
59	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	#41	1	2	yes	—
64	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₉	SD₁₆	1	2	yes	—
66	(C₂×D₅₉)⋊C₄	—	4	4	no	—
67	C₄⋊Dic₅₉	—	2	2	no	—
68	C₄⋊Dic₅₉	—	2	4	no	—
69	C₄⋊Dic₅₉	—	4	1	no	—
70	C₄⋊Dic₅₉	—	4	2	no	—
71	C₄×Dic₅₉	—	8	2	no	—
72	D₄₇₂	—	2	2	no	—
73	D₄₇₂	—	2	4	no	—
74	D₄₇₂	C₄²	1	4	yes	complemented
75	C₁₆×C₅₉	—	2	4	no	—
76	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
77	C₂²×D₅₉	—	4	2	no	—
78	C₂²×D₅₉	—	4	4	no	—
79	C₂²×D₅₉	—	8	1	no	—
80	C₂²×D₅₉	—	8	1	no	—
81	C₂²×D₅₉	—	8	2	no	—
82	C₂×C₄×C₅₉	—	2	2	no	—
83	C₂×C₄×C₅₉	#102	1	2	yes	—
84	C₂×C₄×C₅₉	#102	1	2	yes	—
85	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
86	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
87	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#135	1	2	yes	—
91	C₂×Dic₅₉	—	4	2	no	—
92	C₂×Dic₅₉	—	4	4	no	—
93	C₂×Dic₅₉	—	8	2	no	—
94	C₂×Dic₅₉	—	8	1	no	—
95	C₂×Dic₅₉	—	8	1	no	—
96	C₂×Dic₅₉	—	8	2	no	—
97	D₂₃₆	—	2	4	no	—
98	D₂₃₆	—	4	2	no	—
99	D₂₃₆	—	4	4	no	—
100	C₄×D₅₉	—	8	8	no	—
101	C₈×C₅₉	—	2	2	no	—
102	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
103	C₈×C₅₉	#117	1	1	yes	characteristic
104	C₈×C₅₉	C₄×D₄	1	2	yes	—
105	C₄×#357	—	59	1	no	maximal, Sylow
106	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
107	C₂²×C₅₉	C₄×D₈	1	2	yes	—
108	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
109	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
110	C₂²×C₅₉	#508	1	2	yes	—
111	C₂×D₅₉	—	4	4	no	—
112	C₂×D₅₉	—	8	2	no	—
113	C₂×D₅₉	—	8	2	no	—
114	C₂×D₅₉	—	8	4	no	—
115	C₂×D₅₉	—	8	4	no	—
116	C₄×C₅₉	—	2	2	no	—
117	C₄×C₅₉	#357	1	4	yes	complemented
118	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₅₉	#416	1	1	yes	characteristic
120	C₄×C₅₉	C₄×SD₁₆	1	2	yes	—
121	Dic₅₉	—	8	2	no	—
122	Dic₅₉	—	8	4	no	—
123	C₂×C₄×D₈	—	59	1	no	—
124	C₂×#357	—	59	2	no	—
125	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
126	C₄×#107	—	59	1	no	—
127	#1380	—	59	2	no	—
128	C₂×C₅₉	C₄×#102	1	1	yes	characteristic
129	C₂×C₅₉	C₂×#357	1	1	yes	characteristic
130	C₂×C₅₉	#2138	1	1	yes	characteristic
131	C₂×C₅₉	C₄×D₁₆	1	2	yes	—
132	C₂×C₅₉	C₄×SD₃₂	1	2	yes	—
133	D₅₉	—	8	4	no	—
134	C₂²×D₈	—	59	1	no	—
135	C₂×C₄×D₄	—	118	1	no	—
136	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
137	C₄×D₈	—	59	4	no	—
138	C₄×D₈	—	118	2	no	—
139	C₂×#107	—	59	2	no	—
140	C₂×#107	—	59	1	no	—
141	C₂×#102	—	118	1	no	—
142	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
143	C₄×C₄⋊C₄	—	118	1	no	—
144	#357	—	59	8	no	—
145	C₄×C₁₆	—	2	2	no	—
146	#340	—	118	1	no	—
147	C₅₉	C₄×#357	1	1	yes	characteristic, complemented, Sylow
148	C₂²×D₄	—	118	1	no	—
149	C₂³×C₄	—	236	1	no	—
150	C₂×D₈	—	59	4	no	—
151	C₂×D₈	—	59	2	no	—
152	C₂×D₈	—	118	2	no	—
153	C₂×D₈	—	118	1	no	—
154	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
156	C₄×D₄	—	118	4	no	—
157	C₄×D₄	—	236	2	no	—
158	C₂×C₄⋊C₄	—	118	1	no	—
159	C₂×C₄⋊C₄	—	118	2	no	—
160	C₂×C₂²⋊C₄	—	236	1	no	—
161	C₂×C₄²	—	236	1	no	—
162	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	C₄×D₅₉	1	4	yes	complemented
166	#107	—	59	4	no	—
167	#107	—	59	2	no	—
168	#107	—	118	2	no	—
169	#107	—	118	1	no	—
170	#102	—	118	4	no	—
171	C₄×C₈	C₄×D₅₉	1	2	yes	—
172	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
173	#95	—	236	1	no	—
174	C₂⁴	—	236	1	no	—
175	C₂×D₄	—	118	2	no	—
176	C₂×D₄	—	118	4	no	—
177	C₂×D₄	—	236	1	no	—
178	C₂×D₄	—	236	2	no	—
179	C₂²×C₄	—	472	2	no	—
180	C₂²×C₄	—	472	2	no	—
181	C₂²×C₄	—	236	2	no	—
182	C₂²×C₄	—	236	1	no	—
183	C₂²×C₄	—	236	4	no	—
184	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
185	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
187	D₈	—	59	4	no	—
188	D₈	—	118	2	no	—
189	D₈	—	118	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
194	C₂×C₈	#11745685	1	2	yes	—
195	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
196	C₄⋊C₄	—	118	2	no	—
197	C₄⋊C₄	—	118	4	no	—
198	C₄⋊C₄	—	236	1	no	—
199	C₄⋊C₄	—	236	2	no	—
200	C₂²⋊C₄	—	236	4	no	—
201	C₄²	—	472	2	no	—
202	C₄²	#11745660	1	2	yes	—
203	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
204	C₁₆	—	2	4	no	—
205	C₂³	—	472	1	no	—
206	C₂³	—	472	1	no	—
207	C₂³	—	472	2	no	—
208	C₂³	—	236	2	no	—
209	C₂³	—	236	4	no	—
210	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
211	D₄	—	118	4	no	—
212	D₄	—	236	2	no	—
213	D₄	—	236	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	472	2	no	—
216	C₂×C₄	—	472	1	no	—
217	C₂×C₄	—	472	1	no	—
218	C₂×C₄	—	472	2	no	—
219	C₂×C₄	—	472	8	no	—
220	C₂×C₄	—	236	2	no	—
221	C₂×C₄	—	236	4	no	—
222	C₂×C₄	#14273649	1	2	yes	—
223	C₂×C₄	#14273649	1	2	yes	—
224	C₂×C₄	D₉₄₄	1	2	yes	—
225	C₂×C₄	#14273672	1	2	yes	—
226	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
228	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
229	C₂×C₄	#14273749	1	2	yes	—
230	C₈	—	2	2	no	—
231	C₈	C₄²×D₅₉	1	1	yes	characteristic
232	C₈	#14273684	1	1	yes	characteristic
233	C₈	C₄×C₂²⋊D₅₉	1	2	yes	—
234	C₂²	—	472	4	no	—
235	C₂²	—	472	4	no	—
236	C₂²	—	472	2	no	—
237	C₂²	—	472	2	no	—
238	C₂²	—	236	4	no	—
239	C₂²	C₂×#14273649	1	1	yes	characteristic
240	C₂²	C₄×D₄₇₂	1	2	yes	—
241	C₂²	C₂×D₉₄₄	1	1	yes	characteristic
242	C₂²	C₂×#14273672	1	1	yes	characteristic
243	C₂²	#16837834	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	472	4	no	—
246	C₄	—	472	2	no	—
247	C₄	#16837204	1	4	yes	direct product factor
248	C₄	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
249	C₄	#16837399	1	1	yes	characteristic
250	C₄	C₄×#11745660	1	2	yes	—
251	C₂	—	472	4	no	—
252	C₂	C₄×#14273649	1	1	yes	characteristic
253	C₂	C₂×#16837204	1	1	yes	characteristic
254	C₂	#21425804	1	1	yes	characteristic
255	C₂	C₄×D₉₄₄	1	2	yes	—
256	C₂	C₄×#14273672	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄₇₂
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂²×D₄₇₂	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	C₄×#357
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#14273661	C₂	semidirect
4. #21422188	C₂	semidirect
5. C₂×C₄×C₁₆×C₅₉	C₂	semidirect
9. C₄×D₄₇₂	C₄	semidirect
14. #16837204	C₄	direct (N × H)
34. #14273661	C₂×C₄	semidirect
42. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
74. D₄₇₂	C₄²	semidirect
117. C₄×C₅₉	#357	semidirect
125. C₂×C₄×C₁₆	D₅₉	semidirect
147. C₅₉	C₄×#357	semidirect
165. C₂×C₁₆	C₄×D₅₉	semidirect
247. C₄	#16837204	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×C₄×D₄₇₂	2
C₂×#16837204	2
C₄×#14273661	2
#21422188	2
C₂×C₄×C₁₆×C₅₉	2
C₄×#357	59

Subgroups — C4×#16837204 #28503894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837204 #28503894