Subgroups — #28607512

Order 15104.

838 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#21427008	C₂	1	2	yes	maximal
3	#21424857	C₂	1	2	yes	maximal
4	#21424856	C₂	1	4	yes	maximal
5	#21424251	C₂	1	2	yes	maximal
6	#106×Dic₅₉	C₂	1	2	yes	maximal
7	C₄×#14273659	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#1844	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
10	C₂×#14273692	—	2	2	no	—
11	C₈×Dic₁₁₈	C₄	1	8	yes	complemented
12	C₂×#14273659	C₂²	1	1	yes	characteristic
13	C₂×#14273659	C₂²	1	1	yes	characteristic
14	C₂×#14273658	C₂²	1	4	yes	—
15	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₉⋊C₈	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
18	#16837298	—	2	2	no	—
19	#16837297	—	2	4	no	—
20	#16837271	C₂²	1	4	yes	—
21	C₄⋊C₄×Dic₅₉	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₅₉	C₂²	1	2	yes	—
23	#16837165	C₂²	1	2	yes	—
24	#16837164	C₂²	1	4	yes	—
25	C₂×#106×C₅₉	C₂²	1	2	yes	—
26	C₂×#106×C₅₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#384	C₂²	1	2	yes	—
29	C₂²×Dic₁₁₈	D₄	1	1	yes	characteristic
30	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₁₁₈	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₁₈	Q₈	1	4	yes	—
33	#14273692	—	4	8	no	—
34	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅₉	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₅₉	C₂³	1	2	yes	—
37	C₂×Dic₅₉⋊C₄	—	2	4	no	—
38	C₂×Dic₅₉⋊C₄	—	2	4	no	—
39	C₂×Dic₅₉⋊C₄	D₄	1	2	yes	—
40	C₂×C₄×Dic₅₉	—	2	2	no	—
41	C₂×C₄×Dic₅₉	—	2	4	no	—
42	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
43	#14273659	—	2	2	no	—
44	#14273659	Q₈	1	4	yes	complemented
45	#14273658	—	4	4	no	—
46	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
47	Dic₅₉⋊C₈	C₂×C₄	1	8	yes	—
48	C₈×Dic₅₉	C₂×C₄	1	8	yes	—
49	C₅₉⋊#95	—	2	2	no	—
50	C₅₉⋊#95	—	2	4	no	—
51	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
56	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
57	#106×C₅₉	—	2	4	no	—
58	#106×C₅₉	—	4	2	no	—
59	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
60	C₂²×C₄×C₅₉	—	2	4	no	—
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₅₉	—	2	2	no	—
65	C₂²×Dic₅₉	—	2	2	no	—
66	C₂²×Dic₅₉	C₂×D₄	1	2	yes	—
67	C₂²×Dic₅₉	C₂×D₄	1	2	yes	—
68	C₂×Dic₁₁₈	C₄⋊C₄	1	4	yes	—
69	C₂×Dic₁₁₈	SD₁₆	1	8	yes	—
70	C₂×C₈×C₅₉	—	2	1	no	—
71	C₂×C₈×C₅₉	—	2	1	no	—
72	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₅₉	#41	1	2	yes	—
76	C₄⋊C₄×C₅₉	—	2	4	no	—
77	C₄⋊C₄×C₅₉	—	2	4	no	—
78	C₄⋊C₄×C₅₉	—	4	4	no	—
79	C₄⋊C₄×C₅₉	—	8	2	no	—
80	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
81	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
82	C₄⋊Dic₅₉	—	2	2	no	—
83	C₄⋊Dic₅₉	—	2	2	no	—
84	C₄⋊Dic₅₉	—	2	2	no	—
85	C₄⋊Dic₅₉	C₄⋊C₄	1	4	yes	—
86	C₄⋊Dic₅₉	C₂×Q₈	1	4	yes	—
87	Dic₅₉⋊C₄	—	4	8	no	—
88	Dic₅₉⋊C₄	—	8	4	no	—
89	Dic₅₉⋊C₄	C₄⋊C₄	1	8	yes	—
90	C₄×Dic₅₉	—	4	4	no	—
91	C₄×Dic₅₉	—	4	8	no	—
92	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
93	C₄×Dic₅₉	C₂×Q₈	1	4	yes	—
94	C₂×C₅₉⋊C₈	—	2	2	no	—
95	C₂×C₅₉⋊C₈	—	2	2	no	—
96	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
97	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	—	4	4	no	—
100	C₂×C₄×C₅₉	—	4	4	no	—
101	C₂×C₄×C₅₉	—	4	4	no	—
102	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
105	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₅₉	—	2	4	no	—
112	C₂×Dic₅₉	—	2	4	no	—
113	C₂×Dic₅₉	—	4	2	no	—
114	C₂×Dic₅₉	—	4	2	no	—
115	C₂×Dic₅₉	—	4	2	no	—
116	C₂×Dic₅₉	—	4	2	no	—
117	C₂×Dic₅₉	—	4	2	no	—
118	C₂×Dic₅₉	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×Dic₅₉	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×Dic₅₉	C₂×SD₁₆	1	8	yes	—
121	Dic₁₁₈	#106	1	16	yes	complemented
122	C₈×C₅₉	—	2	2	no	—
123	C₈×C₅₉	C₄×Q₈	1	4	yes	complemented
124	C₅₉⋊C₈	—	2	4	no	—
125	Q₈×#106	—	59	1	no	maximal, Sylow
126	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₅₉	D₄×Q₈	1	1	yes	characteristic
128	C₂²×C₅₉	#550	1	1	yes	characteristic
129	C₂²×C₅₉	C₂²×SD₁₆	1	2	yes	—
130	C₂²×C₅₉	#577	1	2	yes	—
131	C₄×C₅₉	—	2	2	no	—
132	C₄×C₅₉	—	8	4	no	—
133	C₄×C₅₉	C₂×#106	1	4	yes	—
134	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₄×C₅₉	#520	1	1	yes	characteristic
136	C₄×C₅₉	#557	1	1	yes	characteristic
137	C₄×C₅₉	Q₈²	1	1	yes	characteristic
138	Dic₅₉	—	2	4	no	—
139	Dic₅₉	—	4	4	no	—
140	Dic₅₉	—	8	2	no	—
141	Dic₅₉	C₂×#106	1	8	yes	—
142	C₂×C₈×Q₈	—	59	1	no	—
143	Q₈×C₄⋊C₄	—	59	2	no	—
144	#1948	—	59	4	no	—
145	#1948	—	59	2	no	—
146	#1844	—	59	2	no	—
147	#1844	D₅₉	1	1	yes	characteristic
148	C₄×#106	—	59	2	no	—
149	C₄×#106	—	59	1	no	—
150	C₂×C₅₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₅₉	C₂²×#106	1	1	yes	characteristic
152	C₂×C₅₉	#2912	1	1	yes	characteristic
153	C₂×C₅₉	Q₈×SD₁₆	1	2	yes	—
154	C₂×C₅₉	#3392	1	2	yes	—
155	C₂×C₄×Q₈	—	59	1	no	—
156	C₂×C₄×Q₈	—	118	2	no	—
157	C₈×Q₈	—	59	8	no	—
158	C₂×#106	—	59	1	no	—
159	C₂×#106	—	59	1	no	—
160	C₂×#106	—	59	4	no	—
161	C₂×#106	C₂×D₅₉	1	2	yes	—
162	C₂×#106	C₂×D₅₉	1	2	yes	—
163	C₂×C₄⋊C₈	—	59	2	no	—
164	C₂×C₄⋊C₈	—	59	1	no	—
165	C₂×C₄×C₈	—	59	2	no	—
166	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
167	#389	—	118	4	no	—
168	#389	—	118	2	no	—
169	#384	—	59	4	no	—
170	#384	C₂×D₅₉	1	2	yes	—
171	C₄×C₄⋊C₄	—	59	4	no	—
172	C₄×C₄⋊C₄	—	59	2	no	—
173	#340	—	59	2	no	—
174	#340	—	59	4	no	—
175	C₅₉	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	59	1	no	—
177	C₂²×C₈	—	59	2	no	—
178	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
180	C₄×Q₈	—	59	4	no	—
181	C₄×Q₈	—	59	4	no	—
182	C₄×Q₈	—	236	8	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	59	2	no	—
186	C₂×C₄⋊C₄	—	59	1	no	—
187	C₂×C₄⋊C₄	—	59	2	no	—
188	C₂×C₄⋊C₄	—	59	2	no	—
189	C₂×C₄⋊C₄	—	118	4	no	—
190	C₂×C₄⋊C₄	—	118	4	no	—
191	C₂×C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
192	C₂×C₄²	—	59	2	no	—
193	C₂×C₄²	—	118	4	no	—
194	C₂×C₄²	—	118	2	no	—
195	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
196	#106	—	2	4	no	—
197	#106	—	4	2	no	—
198	#106	—	59	4	no	—
199	#106	—	118	2	no	—
200	#106	—	236	4	no	—
201	C₄⋊C₈	—	59	8	no	—
202	C₄⋊C₈	—	59	4	no	—
203	C₄×C₈	—	59	8	no	—
204	C₄×C₈	C₄×D₅₉	1	4	yes	—
205	#95	—	118	4	no	—
206	#95	—	118	2	no	—
207	C₂×Q₈	—	59	4	no	—
208	C₂×Q₈	—	59	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	59	2	no	—
211	C₂²×C₄	—	59	2	no	—
212	C₂²×C₄	—	118	2	no	—
213	C₂²×C₄	—	118	2	no	—
214	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	59	4	no	—
220	C₂×C₈	—	118	2	no	—
221	C₂×C₈	—	118	2	no	—
222	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
224	C₂×C₈	#11745687	1	2	yes	—
225	C₂×C₈	Q₈×D₅₉	1	2	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	8	2	no	—
230	C₄⋊C₄	—	59	8	no	—
231	C₄⋊C₄	—	59	4	no	—
232	C₄⋊C₄	—	59	4	no	—
233	C₄⋊C₄	—	118	2	no	—
234	C₄⋊C₄	—	118	2	no	—
235	C₄⋊C₄	—	118	2	no	—
236	C₄⋊C₄	—	472	4	no	—
237	C₄⋊C₄	—	236	8	no	—
238	C₄²	—	59	4	no	—
239	C₄²	—	59	4	no	—
240	C₄²	—	236	8	no	—
241	C₄²	—	236	4	no	—
242	C₄²	C₂×C₄×D₅₉	1	2	yes	—
243	C₄²	Q₈×D₅₉	1	2	yes	—
244	C₂³	C₂×D₄×D₅₉	1	1	yes	characteristic
245	Q₈	—	59	16	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	59	8	no	—
251	C₂×C₄	—	59	4	no	—
252	C₂×C₄	—	59	4	no	—
253	C₂×C₄	—	118	4	no	—
254	C₂×C₄	—	118	4	no	—
255	C₂×C₄	—	236	2	no	—
256	C₂×C₄	—	236	2	no	—
257	C₂×C₄	—	236	2	no	—
258	C₂×C₄	—	236	2	no	—
259	C₂×C₄	—	236	2	no	—
260	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
262	C₂×C₄	SD₁₆×D₅₉	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
264	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
267	C₂×C₄	#14273772	1	1	yes	characteristic
268	C₂×C₄	#14273775	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	118	4	no	—
271	C₈	#14273692	1	4	yes	complemented
272	C₂²	C₂×C₄⋊C₄×D₅₉	1	1	yes	characteristic
273	C₂²	#16838067	1	1	yes	characteristic
274	C₂²	#16838072	1	1	yes	characteristic
275	C₂²	C₂×SD₁₆×D₅₉	1	2	yes	—
276	C₂²	#16838208	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	8	4	no	—
279	C₄	—	59	8	no	—
280	C₄	—	118	4	no	—
281	C₄	—	472	2	no	—
282	C₄	—	236	4	no	—
283	C₄	#106×D₅₉	1	4	yes	—
284	C₄	C₂×#14273692	1	1	yes	characteristic
285	C₄	#16837883	1	1	yes	characteristic
286	C₄	#16838127	1	1	yes	characteristic
287	C₄	#16838132	1	1	yes	characteristic
288	C₂	#21427008	1	1	yes	characteristic
289	C₂	C₂×#106×D₅₉	1	1	yes	characteristic
290	C₂	#21432687	1	1	yes	characteristic
291	C₂	#21436953	1	2	yes	—
292	C₂	#21437074	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#1844	C₂
socle, prime radical	C₂³×C₅₉	C₂²×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	#1844	D₅₉
hypocenter	C₅₉	Q₈×#106
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₁₁₈	C₄	semidirect
44. #14273659	Q₈	semidirect
121. Dic₁₁₈	#106	semidirect
123. C₈×C₅₉	C₄×Q₈	semidirect
175. C₅₉	Q₈×#106	semidirect
271. C₈	#14273692	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₁₈	2
#21427008	2
#21424857	2
#21424856	2
#21424251	2
#106×Dic₅₉	2
C₄×#14273659	2
C₅₉×#1844	2
Q₈×#106	59