Subgroups — C₄×C₅₉×#329 #28420331

Order 15104.

616 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#124×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₄×#97×C₅₉	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₉×#329	C₂	1	2	yes	maximal
4	C₅₉×#1295	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₅₉	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₅₉	C₄	1	1	yes	characteristic
8	C₂×#97×C₅₉	—	2	2	no	—
9	C₂×#97×C₅₉	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₅₉	—	2	2	no	—
11	C₅₉×#391	C₄	1	1	yes	characteristic
12	C₅₉×#390	C₄	1	1	yes	characteristic
13	C₅₉×#388	—	2	2	no	—
14	C₅₉×#382	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₅₉	—	2	2	no	—
16	C₄³×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#329	C₄	1	8	yes	direct product factor
18	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅₉	—	2	1	no	—
21	C₂²×C₈×C₅₉	—	2	4	no	—
22	#124×C₅₉	—	2	2	no	—
23	#124×C₅₉	C₂×C₄	1	4	yes	—
24	C₄×Q₈×C₅₉	—	4	2	no	—
25	C₄×D₄×C₅₉	—	4	2	no	—
26	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
32	C₂×C₄²×C₅₉	—	2	1	no	—
33	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	D₄	1	2	yes	—
36	C₂×C₄²×C₅₉	D₄	1	2	yes	—
37	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
38	#97×C₅₉	—	2	8	no	—
39	#97×C₅₉	—	2	4	no	—
40	#97×C₅₉	C₂×C₄	1	8	yes	complemented
41	C₄×C₈×C₅₉	—	2	8	no	—
42	#95×C₅₉	—	2	2	no	—
43	#95×C₅₉	—	2	2	no	—
44	C₂⁴×C₅₉	—	2	1	no	—
45	C₂×Q₈×C₅₉	—	2	1	no	—
46	C₂×Q₈×C₅₉	—	4	1	no	—
47	C₂×Q₈×C₅₉	—	4	1	no	—
48	C₂×Q₈×C₅₉	C₄²	1	2	yes	—
49	C₂×D₄×C₅₉	—	2	1	no	—
50	C₂×D₄×C₅₉	—	4	1	no	—
51	C₂×D₄×C₅₉	—	4	1	no	—
52	C₂×D₄×C₅₉	C₄²	1	2	yes	—
53	C₂²×C₄×C₅₉	—	2	1	no	—
54	C₂²×C₄×C₅₉	—	2	1	no	—
55	C₂²×C₄×C₅₉	—	2	1	no	—
56	C₂²×C₄×C₅₉	—	4	2	no	—
57	C₂²×C₄×C₅₉	—	4	4	no	—
58	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₉	—	2	8	no	—
64	C₂×C₈×C₅₉	—	2	8	no	—
65	C₂×C₈×C₅₉	—	2	8	no	—
66	C₄⋊C₄×C₅₉	—	4	2	no	—
67	C₄⋊C₄×C₅₉	—	4	1	no	—
68	C₄⋊C₄×C₅₉	—	4	1	no	—
69	C₂²⋊C₄×C₅₉	—	2	8	no	—
70	C₂²⋊C₄×C₅₉	—	4	2	no	—
71	C₂²⋊C₄×C₅₉	—	4	4	no	—
72	C₄²×C₅₉	—	2	2	no	—
73	C₄²×C₅₉	—	2	4	no	—
74	C₄²×C₅₉	—	2	4	no	—
75	C₄²×C₅₉	—	4	1	no	—
76	C₄²×C₅₉	—	4	2	no	—
77	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
78	C₄²×C₅₉	#41	1	2	yes	—
79	C₂³×C₅₉	—	2	2	no	—
80	C₂³×C₅₉	—	4	1	no	—
81	C₂³×C₅₉	—	4	1	no	—
82	C₂³×C₅₉	—	4	2	no	—
83	C₂³×C₅₉	—	4	2	no	—
84	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₅₉	—	4	2	no	—
86	Q₈×C₅₉	—	4	2	no	—
87	D₄×C₅₉	—	4	2	no	—
88	D₄×C₅₉	—	4	2	no	—
89	C₂×C₄×C₅₉	—	2	1	no	—
90	C₂×C₄×C₅₉	—	2	2	no	—
91	C₂×C₄×C₅₉	—	2	1	no	—
92	C₂×C₄×C₅₉	—	2	2	no	—
93	C₂×C₄×C₅₉	—	2	2	no	—
94	C₂×C₄×C₅₉	—	2	2	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	2	1	no	—
97	C₂×C₄×C₅₉	—	2	2	no	—
98	C₂×C₄×C₅₉	—	4	1	no	—
99	C₂×C₄×C₅₉	—	4	1	no	—
100	C₂×C₄×C₅₉	—	4	1	no	—
101	C₂×C₄×C₅₉	—	4	1	no	—
102	C₂×C₄×C₅₉	—	4	2	no	—
103	C₂×C₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	—	8	4	no	—
105	C₂×C₄×C₅₉	#100	1	2	yes	—
106	C₂×C₄×C₅₉	#104	1	4	yes	—
107	C₂×C₄×C₅₉	#104	1	4	yes	—
108	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₄×D₄	1	4	yes	—
111	C₈×C₅₉	—	2	16	no	—
112	C₄×#329	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₉	—	4	2	no	—
114	C₂²×C₅₉	—	4	4	no	—
115	C₂²×C₅₉	—	8	1	no	—
116	C₂²×C₅₉	—	8	2	no	—
117	C₂²×C₅₉	—	8	1	no	—
118	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₅₉	C₂×#100	1	1	yes	characteristic
120	C₂²×C₅₉	#413	1	1	yes	characteristic
121	C₂²×C₅₉	C₂×#104	1	2	yes	—
122	C₂²×C₅₉	C₂×#104	1	2	yes	—
123	C₄×C₅₉	—	2	1	no	—
124	C₄×C₅₉	—	2	1	no	—
125	C₄×C₅₉	—	2	4	no	—
126	C₄×C₅₉	—	2	4	no	—
127	C₄×C₅₉	—	4	2	no	—
128	C₄×C₅₉	—	4	1	no	—
129	C₄×C₅₉	—	8	1	no	—
130	C₄×C₅₉	—	8	1	no	—
131	C₄×C₅₉	#329	1	4	yes	direct product factor
132	C₄×#124	C₂×C₅₉	1	1	yes	characteristic
133	C₄×#97	C₂×C₅₉	1	2	yes	complemented
134	C₂×#329	C₂×C₅₉	1	2	yes	—
135	#1295	C₂×C₅₉	1	2	yes	complemented
136	C₂×C₅₉	—	8	2	no	—
137	C₂×C₅₉	C₂×#329	1	1	yes	characteristic
138	C₂×C₅₉	#1524	1	2	yes	—
139	C₂×C₅₉	#1525	1	1	yes	characteristic
140	C₂×C₅₉	C₄×#100	1	1	yes	characteristic
141	C₂×C₅₉	C₄×#104	1	2	yes	—
142	C₂×#124	C₂²×C₅₉	1	1	yes	characteristic
143	C₂×C₄×Q₈	C₄×C₅₉	1	1	yes	characteristic
144	C₂×C₄×D₄	C₄×C₅₉	1	1	yes	characteristic
145	C₂×#97	—	2	2	no	—
146	C₂×#97	C₂²×C₅₉	1	4	yes	—
147	C₂×C₄×C₈	—	2	2	no	—
148	#391	C₄×C₅₉	1	1	yes	characteristic
149	#390	C₄×C₅₉	1	1	yes	characteristic
150	#388	—	2	2	no	—
151	#382	C₂²×C₅₉	1	1	yes	characteristic
152	C₄×C₂²⋊C₄	—	2	2	no	—
153	C₄³	C₂²×C₅₉	1	1	yes	characteristic
154	#329	C₄×C₅₉	1	8	yes	direct product factor
155	C₅₉	C₄×#329	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
157	C₂²×D₄	C₂×C₄×C₅₉	1	1	yes	characteristic
158	C₂³×C₄	—	2	1	no	—
159	C₂²×C₈	—	2	4	no	—
160	#124	—	2	2	no	—
161	#124	C₂×C₄×C₅₉	1	4	yes	—
162	C₄×Q₈	—	4	2	no	—
163	C₄×D₄	—	4	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	—	2	2	no	—
168	C₂×C₂²⋊C₄	—	2	1	no	—
169	C₂×C₂²⋊C₄	—	2	2	no	—
170	C₂×C₄²	—	2	1	no	—
171	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
172	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
173	C₂×C₄²	D₄×C₅₉	1	2	yes	—
174	C₂×C₄²	D₄×C₅₉	1	2	yes	—
175	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
176	#97	—	2	8	no	—
177	#97	—	2	4	no	—
178	#97	C₂×C₄×C₅₉	1	8	yes	complemented
179	C₄×C₈	—	2	8	no	—
180	#95	—	2	2	no	—
181	#95	—	2	2	no	—
182	C₂⁴	—	2	1	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	—	4	1	no	—
185	C₂×Q₈	—	4	1	no	—
186	C₂×Q₈	C₄²×C₅₉	1	2	yes	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	C₄²×C₅₉	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	2	8	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	2	8	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	2	no	—
215	C₄²	C₂²×C₄×C₅₉	1	2	yes	—
216	C₄²	#41×C₅₉	1	2	yes	—
217	C₂³	—	2	2	no	—
218	C₂³	—	4	1	no	—
219	C₂³	—	4	1	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	2	no	—
222	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
223	Q₈	—	4	2	no	—
224	Q₈	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	#100×C₅₉	1	2	yes	—
244	C₂×C₄	#104×C₅₉	1	4	yes	—
245	C₂×C₄	#104×C₅₉	1	4	yes	—
246	C₂×C₄	C₂×C₄²×C₅₉	1	1	yes	characteristic
247	C₂×C₄	#117×C₅₉	1	1	yes	characteristic
248	C₂×C₄	C₄×D₄×C₅₉	1	4	yes	—
249	C₈	—	2	16	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	8	1	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	1	no	—
255	C₂²	C₄×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
256	C₂²	C₂×#100×C₅₉	1	1	yes	characteristic
257	C₂²	C₅₉×#413	1	1	yes	characteristic
258	C₂²	C₂×#104×C₅₉	1	2	yes	—
259	C₂²	C₂×#104×C₅₉	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	4	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	8	1	no	—
267	C₄	—	8	1	no	—
268	C₄	C₅₉×#329	1	4	yes	direct product factor
269	C₂	—	8	2	no	—
270	C₂	C₂×C₅₉×#329	1	1	yes	characteristic
271	C₂	C₅₉×#1524	1	2	yes	—
272	C₂	C₅₉×#1525	1	1	yes	characteristic
273	C₂	C₄×#100×C₅₉	1	1	yes	characteristic
274	C₂	C₄×#104×C₅₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₉	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₉
Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
socle	C₂³×C₅₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₅₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₉ ◁ C₄³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#97×C₅₉	C₂	semidirect
4. C₅₉×#1295	C₂	semidirect
17. C₅₉×#329	C₄	direct (N × H)
40. #97×C₅₉	C₂×C₄	semidirect
112. C₄×#329	C₅₉	direct (N × H)
131. C₄×C₅₉	#329	direct (N × H)
133. C₄×#97	C₂×C₅₉	semidirect
135. #1295	C₂×C₅₉	semidirect
154. #329	C₄×C₅₉	direct (N × H)
155. C₅₉	C₄×#329	direct (N × H)
178. #97	C₂×C₄×C₅₉	semidirect
268. C₄	C₅₉×#329	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#124×C₅₉	2
C₄×#97×C₅₉	2
C₂×C₅₉×#329	2
C₅₉×#1295	2
C₄×#329	59

Subgroups — C4×C59×#329 #28420331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₉×#329 #28420331