Subgroups — #28745427

Order 15104.

696 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837834	C₂	1	2	yes	maximal, complemented
2	#21435856	C₂	1	2	yes	maximal, complemented
3	#21426346	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426182	C₂	1	2	yes	maximal, complemented
5	#21426137	C₂	1	2	yes	maximal, complemented
6	#21426021	C₂	1	4	yes	maximal, complemented
7	C₄×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₂²	1	4	yes	—
10	#16838073	—	2	2	no	—
11	#16837834	—	2	8	no	—
12	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
13	C₂×#14273672	C₂²	1	4	yes	—
14	C₂×D₉₄₄	C₂²	1	2	yes	—
15	C₂×D₉₄₄	C₂²	1	2	yes	—
16	#16837779	C₂²	1	2	yes	—
17	#16837638	C₂²	1	4	yes	—
18	#16837603	—	2	2	no	—
19	#16837549	—	2	4	no	—
20	C₄×Dic₂₃₆	C₂²	1	2	yes	—
21	#16837228	C₂²	1	1	yes	characteristic
22	#16837207	C₂²	1	2	yes	—
23	#16837204	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
25	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×#11745685	—	2	4	no	—
30	#14273749	—	2	8	no	—
31	#14273749	—	4	4	no	—
32	C₂×Dic₂₃₆	C₂³	1	2	yes	—
33	C₂×#11745660	—	2	4	no	—
34	C₂×D₄₇₂	—	2	2	no	—
35	C₂×D₄₇₂	C₂³	1	4	yes	—
36	#14273724	—	2	2	no	—
37	#14273723	—	4	2	no	—
38	#14273701	—	2	4	no	—
39	#14273700	—	4	4	no	—
40	C₄×Dic₁₁₈	—	2	2	no	—
41	#14273689	—	4	2	no	—
42	Dic₄₇₂	D₄	1	4	yes	complemented
43	#14273672	—	2	8	no	—
44	D₉₄₄	—	2	4	no	—
45	D₉₄₄	—	4	2	no	—
46	#14273661	C₂³	1	2	yes	—
47	#14273652	—	2	2	no	—
48	#14273649	—	2	4	no	—
49	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
50	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₅₉	—	2	4	no	—
52	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
54	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
56	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
57	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
60	C₂×C₂²⋊D₅₉	—	4	4	no	—
61	C₂×C₂²⋊D₅₉	—	4	4	no	—
62	#11745685	—	4	8	no	—
63	#11745685	—	8	4	no	—
64	C₂×D₂₃₆	—	2	4	no	—
65	C₂×D₂₃₆	—	4	2	no	—
66	C₂×D₂₃₆	—	4	2	no	—
67	C₂×C₄×D₅₉	—	4	4	no	—
68	C₂×Dic₁₁₈	—	2	2	no	—
69	C₂×C₈×C₅₉	—	2	2	no	—
70	C₂×C₈×C₅₉	—	2	2	no	—
71	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
75	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
76	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
77	(C₂×D₅₉)⋊C₄	—	4	4	no	—
78	C₄⋊Dic₅₉	—	2	2	no	—
79	Dic₅₉⋊C₄	—	4	2	no	—
80	C₄×Dic₅₉	—	4	2	no	—
81	Dic₂₃₆	—	2	2	no	—
82	Dic₂₃₆	C₂×D₄	1	4	yes	—
83	D₄₇₂	—	2	4	no	—
84	D₄₇₂	—	2	4	no	—
85	D₄₇₂	—	8	2	no	—
86	#11745660	—	4	8	no	—
87	C₁₆×C₅₉	—	2	2	no	—
88	C₁₆×C₅₉	—	2	1	no	—
89	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
90	C₂³×C₅₉	C₂×D₈	1	2	yes	—
91	C₂²×D₅₉	—	4	4	no	—
92	D₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
99	C₂²⋊D₅₉	—	8	8	no	—
100	C₂²⋊D₅₉	—	16	4	no	—
101	C₂×Dic₅₉	—	4	2	no	—
102	C₂×Dic₅₉	—	4	2	no	—
103	D₂₃₆	—	4	4	no	—
104	D₂₃₆	—	4	4	no	—
105	D₂₃₆	—	8	4	no	—
106	D₂₃₆	—	16	2	no	—
107	C₄×D₅₉	—	8	8	no	—
108	Dic₁₁₈	—	2	4	no	—
109	Dic₁₁₈	—	4	2	no	—
110	C₈×C₅₉	—	2	2	no	—
111	C₈×C₅₉	—	2	1	no	—
112	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
114	#33853	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	—	2	2	no	—
116	C₂²×C₅₉	—	2	2	no	—
117	C₂²×C₅₉	—	2	2	no	—
118	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
119	C₂×D₅₉	—	8	4	no	—
120	C₂×D₅₉	—	8	4	no	—
121	C₂×D₅₉	—	8	4	no	—
122	C₄×C₅₉	—	2	1	no	—
123	C₄×C₅₉	—	2	2	no	—
124	C₄×C₅₉	#508	1	2	yes	complemented
125	C₄×C₅₉	D₄²	1	1	yes	characteristic
126	C₄×C₅₉	#576	1	1	yes	characteristic
127	Dic₅₉	—	4	4	no	—
128	Dic₅₉	—	8	2	no	—
129	C₂×#508	—	59	2	no	—
130	#3287	—	59	2	no	—
131	#2246	—	59	1	no	—
132	#2215	—	59	2	no	—
133	#2207	—	59	2	no	—
134	#2186	—	59	4	no	—
135	C₄×Q₃₂	—	59	1	no	—
136	D₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
137	C₂×C₅₉	—	4	2	no	—
138	C₂×C₅₉	D₄×D₈	1	1	yes	characteristic
139	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
140	C₂×C₅₉	#3415	1	1	yes	characteristic
141	D₅₉	—	16	4	no	—
142	C₂×#135	—	59	4	no	—
143	#550	—	118	2	no	—
144	#508	—	118	8	no	—
145	C₂×Q₃₂	—	59	1	no	—
146	C₂×SD₃₂	—	59	4	no	—
147	C₂×D₁₆	—	59	2	no	—
148	C₂×D₁₆	—	59	2	no	—
149	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
150	#493	—	59	2	no	—
151	#466	—	59	4	no	—
152	#460	—	118	2	no	—
153	#449	—	118	4	no	—
154	C₄×Q₁₆	—	59	2	no	—
155	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
156	#366	—	59	1	no	—
157	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
158	#358	—	59	2	no	—
159	#357	—	59	4	no	—
160	C₂²⋊C₁₆	C₂×D₅₉	1	2	yes	—
161	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
162	C₅₉	#33853	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	118	4	no	—
164	#135	—	118	8	no	—
165	#135	—	236	4	no	—
166	C₂×Q₁₆	—	59	2	no	—
167	C₂×SD₁₆	—	118	4	no	—
168	C₂×D₈	—	59	4	no	—
169	C₂×D₈	—	118	2	no	—
170	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
171	#127	—	118	2	no	—
172	#127	—	236	2	no	—
173	#121	—	118	4	no	—
174	#121	—	236	4	no	—
175	C₄×Q₈	—	118	2	no	—
176	C₄×D₄	—	236	2	no	—
177	C₄×D₄	D₂₃₆	1	1	yes	characteristic
178	Q₃₂	—	59	4	no	—
179	SD₃₂	—	118	8	no	—
180	D₁₆	—	118	4	no	—
181	D₁₆	—	236	2	no	—
182	C₂×C₁₆	—	2	4	no	—
183	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
185	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
186	#107	—	59	2	no	—
187	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
188	#103	—	118	2	no	—
189	#102	—	118	4	no	—
190	C₂²⋊C₈	D₂₃₆	1	2	yes	—
191	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
192	#41	—	472	4	no	—
193	#41	—	236	8	no	—
194	C₂×Q₈	—	118	2	no	—
195	C₂×D₄	—	118	4	no	—
196	C₂×D₄	—	236	4	no	—
197	C₂×D₄	—	236	4	no	—
198	C₂×D₄	—	236	2	no	—
199	C₂×D₄	—	236	2	no	—
200	C₂×D₄	D₄₇₂	1	1	yes	characteristic
201	C₂²×C₄	—	236	4	no	—
202	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
203	Q₁₆	—	59	4	no	—
204	Q₁₆	—	118	2	no	—
205	SD₁₆	—	236	8	no	—
206	D₈	—	118	4	no	—
207	D₈	—	118	4	no	—
208	D₈	—	472	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
212	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
213	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
214	C₄⋊C₄	—	118	2	no	—
215	C₄⋊C₄	—	236	2	no	—
216	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
217	C₂²⋊C₄	—	236	4	no	—
218	C₂²⋊C₄	D₄₇₂	1	2	yes	—
219	C₄²	—	236	2	no	—
220	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
221	C₁₆	—	2	2	no	—
222	C₁₆	—	2	1	no	—
223	C₁₆	D₄×D₅₉	1	2	yes	—
224	C₂³	—	236	4	no	—
225	C₂³	C₂×D₄₇₂	1	2	yes	—
226	Q₈	—	118	4	no	—
227	Q₈	—	236	2	no	—
228	D₄	—	2	2	no	—
229	D₄	—	944	2	no	—
230	D₄	—	944	4	no	—
231	D₄	—	472	8	no	—
232	D₄	—	472	4	no	—
233	D₄	—	236	4	no	—
234	D₄	—	236	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	472	8	no	—
238	C₂×C₄	—	236	2	no	—
239	C₂×C₄	—	236	2	no	—
240	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
242	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	2	no	—
246	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
247	C₈	#14273773	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	472	4	no	—
252	C₂²	—	472	4	no	—
253	C₂²	—	472	4	no	—
254	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	472	2	no	—
258	C₄	—	236	4	no	—
259	C₄	#16837834	1	2	yes	complemented
260	C₄	D₄×D₂₃₆	1	1	yes	characteristic
261	C₄	#16838207	1	1	yes	characteristic
262	C₂	—	4	2	no	—
263	C₂	—	944	4	no	—
264	C₂	D₄×D₄₇₂	1	1	yes	characteristic
265	C₂	C₂×#16837834	1	1	yes	characteristic
266	C₂	#21437237	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	D₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	D₄×C₁₆	D₅₉
hypocenter	C₅₉	#33853
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16837834	C₂	semidirect
2. #21435856	C₂	semidirect
3. #21426346	C₂	semidirect
4. #21426182	C₂	semidirect
5. #21426137	C₂	semidirect
6. #21426021	C₂	semidirect
7. C₄×Dic₄₇₂	C₂	semidirect
8. D₄×C₁₆×C₅₉	C₂	semidirect
42. Dic₄₇₂	D₄	semidirect
124. C₄×C₅₉	#508	semidirect
136. D₄×C₁₆	D₅₉	semidirect
162. C₅₉	#33853	semidirect
259. C₄	#16837834	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837834	2
#21435856	2
#21426346	2
#21426182	2
#21426137	2
#21426021	2
C₄×Dic₄₇₂	2
D₄×C₁₆×C₅₉	2
#33853	59