Subgroups — #28746092

Order 15104.

484 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21436489	C₂	1	2	yes	maximal
2	#21426371	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426299	C₂	1	4	yes	maximal, complemented
4	#21426249	C₂	1	2	yes	maximal, complemented
5	#21426187	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16837228	C₂	1	2	yes	maximal, complemented
8	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
9	#16838109	—	2	2	no	—
10	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
11	#16837807	C₂²	1	2	yes	—
12	#16837738	—	2	4	no	—
13	#16837698	—	2	2	no	—
14	#16837643	C₂²	1	4	yes	—
15	C₄×Dic₂₃₆	C₂²	1	2	yes	—
16	C₂×#14273661	C₂²	1	4	yes	—
17	#16837228	C₂²	1	2	yes	—
18	#16837228	C₂²	1	1	yes	characteristic
19	#16837228	C₂²	1	4	yes	—
20	#16837228	C₂²	1	2	yes	—
21	#16837207	C₂²	1	4	yes	—
22	#16837207	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
24	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₂₃₆	C₂³	1	2	yes	—
29	#14273727	—	2	2	no	—
30	#14273718	—	4	2	no	—
31	#14273706	—	2	4	no	—
32	#14273704	—	4	4	no	—
33	C₄×Dic₁₁₈	—	2	2	no	—
34	D₄×Dic₅₉	—	4	2	no	—
35	C₂×C₄⋊Dic₅₉	—	2	4	no	—
36	Dic₄₇₂	—	2	2	no	—
37	#14273661	—	2	4	no	—
38	#14273661	—	2	2	no	—
39	#14273661	C₂³	1	4	yes	—
40	#14273661	C₂³	1	2	yes	—
41	#14273652	—	2	4	no	—
42	#14273652	—	2	2	no	—
43	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
44	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₉	—	2	4	no	—
46	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
48	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
50	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
51	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
54	C₂²×Dic₅₉	—	4	4	no	—
55	C₂×Dic₁₁₈	—	2	2	no	—
56	C₂×Dic₁₁₈	—	4	2	no	—
57	C₂×C₈×C₅₉	—	2	2	no	—
58	C₂×C₈×C₅₉	—	2	2	no	—
59	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
63	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
64	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²⋊Dic₅₉	—	4	4	no	—
66	C₄⋊Dic₅₉	—	2	2	no	—
67	C₄⋊Dic₅₉	—	2	4	no	—
68	C₄⋊Dic₅₉	—	4	4	no	—
69	C₄⋊Dic₅₉	—	4	2	no	—
70	Dic₅₉⋊C₄	—	4	2	no	—
71	Dic₅₉⋊C₄	—	4	4	no	—
72	C₄×Dic₅₉	—	4	2	no	—
73	Dic₂₃₆	—	2	2	no	—
74	Dic₂₃₆	—	2	2	no	—
75	C₁₆×C₅₉	—	2	2	no	—
76	C₁₆×C₅₉	—	2	1	no	—
77	C₁₆×C₅₉	#41	1	2	yes	—
78	C₂³×C₅₉	C₂×D₈	1	2	yes	—
79	D₄×C₅₉	Q₃₂	1	4	yes	complemented
80	C₂×C₄×C₅₉	—	2	2	no	—
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₅₉	—	4	2	no	—
87	C₂×Dic₅₉	—	4	2	no	—
88	C₂×Dic₅₉	—	4	4	no	—
89	C₂×Dic₅₉	—	16	4	no	—
90	Dic₁₁₈	—	4	2	no	—
91	Dic₁₁₈	—	4	2	no	—
92	Dic₁₁₈	—	16	2	no	—
93	C₈×C₅₉	—	2	2	no	—
94	C₈×C₅₉	—	2	1	no	—
95	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
96	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
97	#33905	—	59	1	no	maximal, Sylow
98	C₂²×C₅₉	—	2	2	no	—
99	C₂²×C₅₉	—	2	2	no	—
100	C₂²×C₅₉	C₂×Q₃₂	1	4	yes	—
101	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₅₉	—	2	2	no	—
103	C₄×C₅₉	—	2	1	no	—
104	C₄×C₅₉	C₂×Q₃₂	1	2	yes	—
105	C₄×C₅₉	#547	1	1	yes	characteristic
106	C₄×C₅₉	#578	1	1	yes	characteristic
107	Dic₅₉	—	8	2	no	—
108	Dic₅₉	—	8	4	no	—
109	Dic₅₉	—	8	2	no	—
110	#3338	—	59	2	no	—
111	#2252	—	59	1	no	—
112	#2236	—	59	4	no	—
113	#2227	—	59	2	no	—
114	#2216	—	59	2	no	—
115	C₄×Q₃₂	—	59	1	no	—
116	D₄×C₁₆	D₅₉	1	1	yes	characteristic
117	C₂×#366	—	59	2	no	—
118	C₂×C₅₉	—	2	4	no	—
119	C₂×C₅₉	#3334	1	1	yes	characteristic
120	C₂×C₅₉	C₂²×Q₃₂	1	1	yes	characteristic
121	C₂×C₅₉	#3415	1	1	yes	characteristic
122	#554	—	118	2	no	—
123	C₂×Q₃₂	—	59	1	no	—
124	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
125	#500	—	59	2	no	—
126	#484	—	118	4	no	—
127	#477	—	118	2	no	—
128	#467	—	59	4	no	—
129	C₄×Q₁₆	—	59	2	no	—
130	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
131	C₂×#107	—	59	4	no	—
132	#366	—	59	1	no	—
133	#366	—	59	2	no	—
134	#366	—	59	4	no	—
135	#366	—	59	2	no	—
136	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
137	#358	—	59	2	no	—
138	#358	—	59	4	no	—
139	C₂²⋊C₁₆	C₂×D₅₉	1	2	yes	—
140	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
141	C₅₉	#33905	1	1	yes	characteristic, complemented, Sylow
142	C₂×Q₁₆	—	59	2	no	—
143	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
144	C₄⋊Q₈	—	118	2	no	—
145	#125	—	236	2	no	—
146	C₂²⋊Q₈	—	118	4	no	—
147	C₂²⋊Q₈	—	236	4	no	—
148	C₄×Q₈	—	118	2	no	—
149	C₄×D₄	—	236	2	no	—
150	C₄×D₄	D₂₃₆	1	1	yes	characteristic
151	C₂×C₄⋊C₄	—	118	4	no	—
152	Q₃₂	—	118	2	no	—
153	C₂×C₁₆	—	2	4	no	—
154	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
156	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
157	#107	—	59	2	no	—
158	#107	—	59	4	no	—
159	#107	—	118	4	no	—
160	#107	—	118	2	no	—
161	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
162	#103	—	118	2	no	—
163	#103	—	118	4	no	—
164	C₂²⋊C₈	D₂₃₆	1	2	yes	—
165	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
166	C₂×Q₈	—	118	2	no	—
167	C₂×Q₈	—	236	2	no	—
168	C₂×D₄	D₄₇₂	1	1	yes	characteristic
169	C₂²×C₄	—	236	4	no	—
170	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
171	Q₁₆	—	118	2	no	—
172	Q₁₆	—	118	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
176	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
177	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
178	C₄⋊C₄	—	118	2	no	—
179	C₄⋊C₄	—	118	4	no	—
180	C₄⋊C₄	—	236	4	no	—
181	C₄⋊C₄	—	236	2	no	—
182	C₄⋊C₄	—	236	2	no	—
183	C₄⋊C₄	—	236	4	no	—
184	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
185	C₂²⋊C₄	—	236	4	no	—
186	C₂²⋊C₄	D₄₇₂	1	2	yes	—
187	C₄²	—	236	2	no	—
188	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
189	C₁₆	—	2	2	no	—
190	C₁₆	—	2	1	no	—
191	C₁₆	#11745687	1	2	yes	—
192	C₂³	C₂×D₄₇₂	1	2	yes	—
193	Q₈	—	944	2	no	—
194	Q₈	—	236	2	no	—
195	Q₈	—	236	2	no	—
196	D₄	Dic₄₇₂	1	4	yes	complemented
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	944	4	no	—
200	C₂×C₄	—	236	2	no	—
201	C₂×C₄	—	236	2	no	—
202	C₂×C₄	—	236	4	no	—
203	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
204	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
205	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	2	no	—
209	C₈	C₂×#11745687	1	1	yes	characteristic
210	C₈	#14273774	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	C₂×Dic₄₇₂	1	4	yes	—
214	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	472	2	no	—
218	C₄	—	472	2	no	—
219	C₄	—	472	4	no	—
220	C₄	C₂×Dic₄₇₂	1	2	yes	—
221	C₄	#16838056	1	1	yes	characteristic
222	C₄	#16838219	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	#21436444	1	1	yes	characteristic
225	C₂	C₂²×Dic₄₇₂	1	1	yes	characteristic
226	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
prime radical	C₂×D₄×C₅₉	D₈
hypercenter	D₄×C₁₆	D₅₉
hypocenter	C₅₉	#33905
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
2. #21426371	C₂	semidirect
3. #21426299	C₂	semidirect
4. #21426249	C₂	semidirect
5. #21426187	C₂	semidirect
6. C₄×Dic₄₇₂	C₂	semidirect
7. C₂×#16837228	C₂	semidirect
79. D₄×C₅₉	Q₃₂	semidirect
141. C₅₉	#33905	semidirect
196. D₄	Dic₄₇₂	semidirect

Maximal subgroups

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

subgroup	index
#21436489	2
#21426371	2
#21426299	2
#21426249	2
#21426187	2
C₄×Dic₄₇₂	2
C₂×#16837228	2
D₄×C₁₆×C₅₉	2
#33905	59