Subgroups — C₂×C₅₉×#1633 #28430616

Order 15104.

1010 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×#494	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×#477	C₂	1	2	yes	maximal
4	C₅₉×#1633	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₉×#333	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₉×Q₈⋊C₈	C₂	1	2	yes	maximal
7	C₂²×Q₁₆×C₅₉	—	2	2	no	—
8	Q₈²×C₅₉	—	2	2	no	—
9	Q₈²×C₅₉	—	2	4	no	—
10	Q₈²×C₅₉	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₅₉	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₅₉	—	2	2	no	—
13	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₅₉	—	2	2	no	—
16	C₂×C₄×Q₈×C₅₉	C₂²	1	2	yes	—
17	C₅₉×#494	—	2	2	no	—
18	C₅₉×#494	C₂²	1	4	yes	—
19	C₅₉×#477	—	2	4	no	—
20	C₅₉×#477	C₂²	1	8	yes	—
21	C₂×#107×C₅₉	—	2	2	no	—
22	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
23	C₂×#103×C₅₉	—	2	2	no	—
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#333	C₂²	1	4	yes	—
26	C₅₉×Q₈⋊C₈	C₂²	1	8	yes	—
27	C₂²×Q₈×C₅₉	—	2	2	no	—
28	C₂²×Q₈×C₅₉	—	2	1	no	—
29	C₂²×Q₈×C₅₉	D₄	1	2	yes	—
30	C₂²×Q₈×C₅₉	D₄	1	2	yes	—
31	C₂×Q₁₆×C₅₉	—	2	8	no	—
32	C₂×Q₁₆×C₅₉	—	4	8	no	—
33	C₂²×C₈×C₅₉	—	2	1	no	—
34	C₂²×C₈×C₅₉	—	2	2	no	—
35	C₄⋊Q₈×C₅₉	—	2	8	no	—
36	C₄⋊Q₈×C₅₉	—	2	8	no	—
37	C₄⋊Q₈×C₅₉	—	2	8	no	—
38	C₄⋊Q₈×C₅₉	—	2	8	no	—
39	C₄⋊Q₈×C₅₉	—	2	2	no	—
40	C₄⋊Q₈×C₅₉	—	2	2	no	—
41	C₄⋊Q₈×C₅₉	—	2	1	no	—
42	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
43	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	—
45	C₄×Q₈×C₅₉	—	2	8	no	—
46	C₄×Q₈×C₅₉	—	2	8	no	—
47	C₄×Q₈×C₅₉	—	2	2	no	—
48	C₄×Q₈×C₅₉	—	2	2	no	—
49	C₄×Q₈×C₅₉	D₄	1	4	yes	—
50	C₄×Q₈×C₅₉	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
52	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
56	C₂×C₄²×C₅₉	—	2	2	no	—
57	C₂×C₄²×C₅₉	—	2	2	no	—
58	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
59	#107×C₅₉	—	2	8	no	—
60	C₄⋊C₈×C₅₉	—	2	2	no	—
61	C₄⋊C₈×C₅₉	C₂³	1	4	yes	—
62	#103×C₅₉	—	2	8	no	—
63	C₄×C₈×C₅₉	—	2	1	no	—
64	C₄×C₈×C₅₉	C₂³	1	2	yes	—
65	C₂×Q₈×C₅₉	—	2	8	no	—
66	C₂×Q₈×C₅₉	—	2	2	no	—
67	C₂×Q₈×C₅₉	—	2	2	no	—
68	C₂×Q₈×C₅₉	—	2	4	no	—
69	C₂×Q₈×C₅₉	—	2	2	no	—
70	C₂×Q₈×C₅₉	—	4	4	no	—
71	C₂×Q₈×C₅₉	—	4	4	no	—
72	C₂×Q₈×C₅₉	—	4	1	no	—
73	C₂×Q₈×C₅₉	—	8	4	no	—
74	C₂×Q₈×C₅₉	Q₁₆	1	8	yes	complemented
75	C₂×Q₈×C₅₉	C₂×D₄	1	4	yes	—
76	C₂×Q₈×C₅₉	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₅₉	—	2	2	no	—
78	C₂²×C₄×C₅₉	—	2	2	no	—
79	C₂²×C₄×C₅₉	—	2	2	no	—
80	C₂²×C₄×C₅₉	—	4	1	no	—
81	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
83	Q₁₆×C₅₉	—	4	16	no	—
84	C₂×C₈×C₅₉	—	2	4	no	—
85	C₂×C₈×C₅₉	—	2	2	no	—
86	C₂×C₈×C₅₉	—	2	4	no	—
87	C₂×C₈×C₅₉	—	4	4	no	—
88	C₄⋊C₄×C₅₉	—	2	16	no	—
89	C₄⋊C₄×C₅₉	—	2	4	no	—
90	C₄⋊C₄×C₅₉	—	2	4	no	—
91	C₄⋊C₄×C₅₉	—	2	2	no	—
92	C₄⋊C₄×C₅₉	—	2	4	no	—
93	C₄⋊C₄×C₅₉	—	2	4	no	—
94	C₄⋊C₄×C₅₉	—	2	4	no	—
95	C₄⋊C₄×C₅₉	—	4	2	no	—
96	C₄⋊C₄×C₅₉	C₂×D₄	1	4	yes	—
97	C₄²×C₅₉	—	2	8	no	—
98	C₄²×C₅₉	—	2	4	no	—
99	C₄²×C₅₉	—	2	4	no	—
100	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅₉	C₂×D₄	1	2	yes	—
102	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
103	C₂³×C₅₉	#120	1	1	yes	characteristic
104	Q₈×C₅₉	—	2	4	no	—
105	Q₈×C₅₉	—	2	8	no	—
106	Q₈×C₅₉	—	4	4	no	—
107	Q₈×C₅₉	—	4	4	no	—
108	Q₈×C₅₉	—	8	2	no	—
109	Q₈×C₅₉	—	8	2	no	—
110	Q₈×C₅₉	—	8	8	no	—
111	Q₈×C₅₉	C₂×Q₁₆	1	8	yes	complemented
112	C₂×C₄×C₅₉	—	2	4	no	—
113	C₂×C₄×C₅₉	—	2	8	no	—
114	C₂×C₄×C₅₉	—	2	2	no	—
115	C₂×C₄×C₅₉	—	2	4	no	—
116	C₂×C₄×C₅₉	—	2	4	no	—
117	C₂×C₄×C₅₉	—	4	4	no	—
118	C₂×C₄×C₅₉	—	4	4	no	—
119	C₂×C₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	8	2	no	—
121	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
122	C₂×C₄×C₅₉	#120	1	2	yes	—
123	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
124	C₂×C₄×C₅₉	#137	1	4	yes	—
125	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
127	C₈×C₅₉	—	2	4	no	—
128	C₈×C₅₉	—	4	4	no	—
129	C₂×#1633	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₅₉	#451	1	4	yes	—
131	C₂²×C₅₉	#453	1	2	yes	—
132	C₂²×C₅₉	C₂×#120	1	1	yes	characteristic
133	C₄×C₅₉	—	2	8	no	—
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	4	4	no	—
137	C₄×C₅₉	—	4	4	no	—
138	C₄×C₅₉	—	8	2	no	—
139	C₄×C₅₉	#451	1	4	yes	—
140	C₄×C₅₉	C₂²×Q₁₆	1	2	yes	—
141	C₄×C₅₉	C₂×#137	1	2	yes	—
142	C₂×Q₈²	C₂×C₅₉	1	1	yes	characteristic
143	C₂×#494	C₂×C₅₉	1	1	yes	characteristic
144	C₂×#477	C₂×C₅₉	1	2	yes	—
145	#1633	C₂×C₅₉	1	8	yes	direct product factor
146	C₂×#333	C₂×C₅₉	1	1	yes	characteristic
147	C₂×Q₈⋊C₈	C₂×C₅₉	1	2	yes	—
148	C₂×C₅₉	#1633	1	4	yes	direct product factor
149	C₂×C₅₉	C₂×#451	1	2	yes	—
150	C₂×C₅₉	C₂×#453	1	1	yes	characteristic
151	C₂²×Q₁₆	—	2	2	no	—
152	Q₈²	—	2	2	no	—
153	Q₈²	—	2	4	no	—
154	Q₈²	C₂²×C₅₉	1	4	yes	—
155	C₂×C₄⋊Q₈	—	2	2	no	—
156	C₂×C₄⋊Q₈	—	2	2	no	—
157	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
158	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
159	C₂×C₄×Q₈	—	2	2	no	—
160	C₂×C₄×Q₈	C₂²×C₅₉	1	2	yes	—
161	#494	—	2	2	no	—
162	#494	C₂²×C₅₉	1	4	yes	—
163	#477	—	2	4	no	—
164	#477	C₂²×C₅₉	1	8	yes	—
165	C₂×#107	—	2	2	no	—
166	C₂×C₄⋊C₈	C₂²×C₅₉	1	2	yes	—
167	C₂×#103	—	2	2	no	—
168	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
169	#333	C₂²×C₅₉	1	4	yes	—
170	Q₈⋊C₈	C₂²×C₅₉	1	8	yes	—
171	C₅₉	C₂×#1633	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×Q₈	—	2	2	no	—
173	C₂²×Q₈	—	2	1	no	—
174	C₂²×Q₈	D₄×C₅₉	1	2	yes	—
175	C₂²×Q₈	D₄×C₅₉	1	2	yes	—
176	C₂×Q₁₆	—	2	8	no	—
177	C₂×Q₁₆	—	4	8	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	8	no	—
181	C₄⋊Q₈	—	2	8	no	—
182	C₄⋊Q₈	—	2	8	no	—
183	C₄⋊Q₈	—	2	8	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	D₄×C₅₉	1	2	yes	—
188	C₄⋊Q₈	C₂³×C₅₉	1	2	yes	—
189	C₄⋊Q₈	C₂³×C₅₉	1	2	yes	—
190	C₄×Q₈	—	2	8	no	—
191	C₄×Q₈	—	2	8	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×Q₈	D₄×C₅₉	1	4	yes	—
195	C₄×Q₈	C₂³×C₅₉	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	D₄×C₅₉	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
204	#107	—	2	8	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	C₂³×C₅₉	1	4	yes	—
207	#103	—	2	8	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	C₂³×C₅₉	1	2	yes	—
210	C₂×Q₈	—	2	8	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	4	4	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	8	4	no	—
219	C₂×Q₈	Q₁₆×C₅₉	1	8	yes	complemented
220	C₂×Q₈	C₂×D₄×C₅₉	1	4	yes	—
221	C₂×Q₈	C₂×D₄×C₅₉	1	4	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
228	Q₁₆	—	4	16	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	4	no	—
233	C₄⋊C₄	—	2	16	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	C₂×D₄×C₅₉	1	4	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₅₉	1	2	yes	—
247	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
248	C₂³	#120×C₅₉	1	1	yes	characteristic
249	Q₈	—	2	8	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	4	no	—
252	Q₈	—	4	4	no	—
253	Q₈	—	8	2	no	—
254	Q₈	—	8	2	no	—
255	Q₈	—	8	8	no	—
256	Q₈	C₂×Q₁₆×C₅₉	1	8	yes	complemented
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
267	C₂×C₄	#120×C₅₉	1	2	yes	—
268	C₂×C₄	C₂×Q₁₆×C₅₉	1	4	yes	—
269	C₂×C₄	#137×C₅₉	1	4	yes	—
270	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
272	C₈	—	2	4	no	—
273	C₈	—	4	4	no	—
274	C₂²	C₅₉×#451	1	4	yes	—
275	C₂²	C₅₉×#453	1	2	yes	—
276	C₂²	C₂×#120×C₅₉	1	1	yes	characteristic
277	C₄	—	2	8	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	4	4	no	—
281	C₄	—	4	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₅₉×#451	1	4	yes	—
284	C₄	C₂²×Q₁₆×C₅₉	1	2	yes	—
285	C₄	C₂×#137×C₅₉	1	2	yes	—
286	C₂	C₅₉×#1633	1	4	yes	direct product factor
287	C₂	C₂×C₅₉×#451	1	2	yes	—
288	C₂	C₂×C₅₉×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₉	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₉×#1633	C₂	direct (N × H)
74. C₂×Q₈×C₅₉	Q₁₆	semidirect
111. Q₈×C₅₉	C₂×Q₁₆	semidirect
129. C₂×#1633	C₅₉	direct (N × H)
145. #1633	C₂×C₅₉	direct (N × H)
148. C₂×C₅₉	#1633	direct (N × H)
171. C₅₉	C₂×#1633	direct (N × H)
219. C₂×Q₈	Q₁₆×C₅₉	semidirect
256. Q₈	C₂×Q₁₆×C₅₉	semidirect
286. C₂	C₅₉×#1633	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×Q₈²×C₅₉	2
C₂×C₅₉×#494	2
C₂×C₅₉×#477	2
C₅₉×#1633	2
C₂×C₅₉×#333	2
C₂×C₅₉×Q₈⋊C₈	2
C₂×#1633	59

Subgroups — C2×C59×#1633 #28430616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₉×#1633 #28430616