Subgroups — #28504746

Order 15104.

430 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837207	C₂	1	2	yes	maximal
3	#21424260	C₂	1	1	yes	maximal, characteristic
4	#21422188	C₂	1	2	yes	maximal
5	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₁₈	—	2	1	no	—
8	C₄×Dic₂₃₆	—	2	2	no	—
9	C₄×Dic₂₃₆	C₄	1	4	yes	complemented
10	C₂×#14273661	C₂²	1	2	yes	—
11	C₂×#14273661	C₂²	1	1	yes	characteristic
12	C₂×#14273659	—	2	1	no	—
13	C₂×#14273652	—	2	1	no	—
14	#16837268	—	2	1	no	—
15	#16837207	—	2	4	no	—
16	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×C₄⋊C₁₆	—	2	2	no	—
19	C₂²×Dic₁₁₈	—	2	1	no	—
20	C₂×Dic₂₃₆	—	2	1	no	—
21	C₂×Dic₂₃₆	—	2	1	no	—
22	C₂×Dic₂₃₆	—	2	1	no	—
23	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
24	C₂×Dic₂₃₆	D₄	1	2	yes	—
25	C₂×Dic₂₃₆	Q₈	1	2	yes	—
26	C₄×Dic₁₁₈	—	2	4	no	—
27	C₄×Dic₁₁₈	—	4	2	no	—
28	C₂×C₄⋊Dic₅₉	—	2	1	no	—
29	C₂×C₄⋊Dic₅₉	—	2	2	no	—
30	C₂×Dic₅₉⋊C₄	—	4	1	no	—
31	C₂×C₄×Dic₅₉	—	4	1	no	—
32	#14273661	—	2	1	no	—
33	#14273661	—	2	2	no	—
34	#14273661	—	2	2	no	—
35	#14273661	C₂×C₄	1	2	yes	—
36	#14273659	—	4	2	no	—
37	#14273652	—	2	4	no	—
38	C₅₉⋊#95	—	4	1	no	—
39	C₅₉⋊#95	—	4	1	no	—
40	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₉	—	2	2	no	—
44	C₂×C₁₆×C₅₉	—	2	2	no	—
45	C₂×C₁₆×C₅₉	—	2	2	no	—
46	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₉	D₄	1	2	yes	—
48	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₉	—	4	1	no	—
52	C₂²×Dic₅₉	—	4	1	no	—
53	C₂²×Dic₅₉	—	4	2	no	—
54	C₂×Dic₁₁₈	—	2	2	no	—
55	C₂×Dic₁₁₈	—	2	2	no	—
56	C₂×Dic₁₁₈	—	2	2	no	—
57	C₂×Dic₁₁₈	—	4	1	no	—
58	C₂×Dic₁₁₈	—	4	1	no	—
59	C₂×Dic₁₁₈	—	4	1	no	—
60	C₂×C₈×C₅₉	—	2	1	no	—
61	C₂×C₈×C₅₉	—	2	1	no	—
62	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
68	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
69	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₉	SD₁₆	1	2	yes	—
71	C₄⋊Dic₅₉	—	2	2	no	—
72	C₄⋊Dic₅₉	—	4	1	no	—
73	C₄⋊Dic₅₉	—	4	2	no	—
74	C₄⋊Dic₅₉	—	4	2	no	—
75	Dic₅₉⋊C₄	—	4	4	no	—
76	C₄×Dic₅₉	—	4	4	no	—
77	Dic₂₃₆	—	2	2	no	—
78	Dic₂₃₆	—	2	2	no	—
79	Dic₂₃₆	—	2	2	no	—
80	Dic₂₃₆	C₄⋊C₄	1	4	yes	—
81	C₁₆×C₅₉	—	4	2	no	—
82	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	—	2	2	no	—
85	C₂×C₄×C₅₉	#102	1	2	yes	—
86	C₂×C₄×C₅₉	#102	1	2	yes	—
87	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
93	C₂×Dic₅₉	—	4	2	no	—
94	C₂×Dic₅₉	—	4	2	no	—
95	C₂×Dic₅₉	—	4	2	no	—
96	C₂×Dic₅₉	—	4	2	no	—
97	C₂×Dic₅₉	—	8	2	no	—
98	C₂×Dic₅₉	—	8	2	no	—
99	C₂×Dic₅₉	—	8	2	no	—
100	C₂×Dic₅₉	—	8	2	no	—
101	Dic₁₁₈	—	2	4	no	—
102	Dic₁₁₈	—	4	2	no	—
103	Dic₁₁₈	—	4	2	no	—
104	Dic₁₁₈	—	4	2	no	—
105	C₈×C₅₉	—	2	2	no	—
106	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₈×C₅₉	#117	1	1	yes	characteristic
108	C₈×C₅₉	#121	1	1	yes	characteristic
109	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
110	#12814	—	59	1	no	maximal, Sylow
111	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
112	C₂²×C₅₉	#459	1	1	yes	characteristic
113	C₂²×C₅₉	#474	1	1	yes	characteristic
114	C₂²×C₅₉	#508	1	2	yes	—
115	C₂²×C₅₉	#510	1	2	yes	—
116	C₄×C₅₉	—	2	2	no	—
117	C₄×C₅₉	—	2	2	no	—
118	C₄×C₅₉	#380	1	1	yes	characteristic
119	C₄×C₅₉	#417	1	1	yes	characteristic
120	C₄×C₅₉	#461	1	1	yes	characteristic
121	C₄×C₅₉	#476	1	1	yes	characteristic
122	Dic₅₉	—	4	4	no	—
123	Dic₅₉	—	8	2	no	—
124	Dic₅₉	—	16	2	no	—
125	C₂×C₄×Q₁₆	—	59	1	no	—
126	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
127	C₂×#358	—	59	2	no	—
128	#1846	—	59	1	no	—
129	#1380	—	59	2	no	—
130	C₂×C₅₉	#1802	1	1	yes	characteristic
131	C₂×C₅₉	#2138	1	1	yes	characteristic
132	C₂×C₅₉	#2141	1	1	yes	characteristic
133	C₂×C₅₉	#2211	1	2	yes	—
134	C₂×C₅₉	#2229	1	2	yes	—
135	C₂²×Q₁₆	—	59	1	no	—
136	C₂×C₄×Q₈	—	118	1	no	—
137	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
138	C₄×Q₁₆	—	59	4	no	—
139	C₄×Q₁₆	—	118	2	no	—
140	C₂×#107	—	59	1	no	—
141	C₂×#107	—	59	2	no	—
142	C₂×#106	—	118	1	no	—
143	C₂×#103	—	118	1	no	—
144	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
145	#382	—	118	1	no	—
146	C₄⋊C₁₆	—	2	2	no	—
147	#358	—	118	4	no	—
148	C₅₉	#12814	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	118	1	no	—
150	C₂×Q₁₆	—	59	2	no	—
151	C₂×Q₁₆	—	59	2	no	—
152	C₂×Q₁₆	—	59	2	no	—
153	C₂×Q₁₆	—	118	1	no	—
154	C₂×Q₁₆	—	118	1	no	—
155	C₂×Q₁₆	—	118	1	no	—
156	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
158	C₄×Q₈	—	118	4	no	—
159	C₄×Q₈	—	236	2	no	—
160	C₂×C₄⋊C₄	—	118	1	no	—
161	C₂×C₄⋊C₄	—	118	2	no	—
162	C₂×C₄⋊C₄	—	236	1	no	—
163	C₂×C₄²	—	236	1	no	—
164	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	#107	—	59	2	no	—
169	#107	—	118	1	no	—
170	#107	—	118	2	no	—
171	#107	—	118	2	no	—
172	#106	—	236	2	no	—
173	#103	—	118	4	no	—
174	C₄×C₈	C₄×D₅₉	1	2	yes	—
175	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
176	#95	—	236	1	no	—
177	#95	—	236	1	no	—
178	C₂×Q₈	—	118	2	no	—
179	C₂×Q₈	—	118	2	no	—
180	C₂×Q₈	—	118	2	no	—
181	C₂×Q₈	—	236	1	no	—
182	C₂×Q₈	—	236	1	no	—
183	C₂×Q₈	—	236	1	no	—
184	C₂²×C₄	—	236	2	no	—
185	C₂²×C₄	—	236	1	no	—
186	C₂²×C₄	—	236	1	no	—
187	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
188	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
190	Q₁₆	—	59	4	no	—
191	Q₁₆	—	118	2	no	—
192	Q₁₆	—	118	2	no	—
193	Q₁₆	—	118	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
197	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
198	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
199	C₂×C₈	#11745687	1	1	yes	characteristic
200	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
201	C₂×C₈	#11745689	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
203	C₄⋊C₄	—	118	2	no	—
204	C₄⋊C₄	—	236	2	no	—
205	C₄⋊C₄	—	236	2	no	—
206	C₄⋊C₄	—	236	4	no	—
207	C₄⋊C₄	—	236	1	no	—
208	C₄²	—	236	4	no	—
209	C₄²	#11745660	1	2	yes	—
210	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
211	C₁₆	—	4	2	no	—
212	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
213	Q₈	—	118	4	no	—
214	Q₈	—	236	2	no	—
215	Q₈	—	236	2	no	—
216	Q₈	—	236	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	472	2	no	—
220	C₂×C₄	—	472	2	no	—
221	C₂×C₄	—	472	2	no	—
222	C₂×C₄	—	472	2	no	—
223	C₂×C₄	—	236	2	no	—
224	C₂×C₄	—	236	2	no	—
225	C₂×C₄	—	236	2	no	—
226	C₂×C₄	—	236	2	no	—
227	C₂×C₄	#14273649	1	2	yes	—
228	C₂×C₄	#14273649	1	2	yes	—
229	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	#14273702	1	1	yes	characteristic
231	C₂×C₄	#14273707	1	1	yes	characteristic
232	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
233	C₂×C₄	#14273755	1	1	yes	characteristic
234	C₂×C₄	#14273760	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
237	C₈	#14273685	1	1	yes	characteristic
238	C₈	#14273698	1	1	yes	characteristic
239	C₈	#14273703	1	1	yes	characteristic
240	C₂²	C₂×#14273649	1	1	yes	characteristic
241	C₂²	#16837598	1	1	yes	characteristic
242	C₂²	#16837683	1	1	yes	characteristic
243	C₂²	#16837834	1	2	yes	—
244	C₂²	#16837845	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	944	2	no	—
248	C₄	—	472	2	no	—
249	C₄	—	236	4	no	—
250	C₄	#16837259	1	1	yes	characteristic
251	C₄	#16837404	1	1	yes	characteristic
252	C₄	#16837608	1	1	yes	characteristic
253	C₄	#16837693	1	1	yes	characteristic
254	C₂	#21424097	1	1	yes	characteristic
255	C₂	#21425804	1	1	yes	characteristic
256	C₂	#21425817	1	1	yes	characteristic
257	C₂	#21426159	1	2	yes	—
258	C₂	#21426261	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#12814
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×Dic₂₃₆	C₄	semidirect
148. C₅₉	#12814	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₃₆	2
C₂×#16837207	2
#21424260	2
#21422188	2
C₂×C₅₉×C₄⋊C₁₆	2
#12814	59