Subgroups — C₅₉×#35674 #28447783

Order 15104.

2278 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×#519	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2484	C₂	1	8	yes	maximal
4	C₅₉×#2446	C₂	1	16	yes	maximal, complemented
5	C₅₉×#2310	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#2310	C₂	1	2	yes	maximal
7	C₅₉×#2296	C₂	1	2	yes	maximal
8	C₂³×Q₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₂³×Q₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₅₉	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₅₉	C₂²	1	8	yes	—
12	C₅₉×#519	—	2	8	no	—
13	C₂×C₄×Q₈×C₅₉	C₂²	1	8	yes	—
14	C₂×#117×C₅₉	C₂²	1	4	yes	—
15	C₂×#117×C₅₉	C₂²	1	2	yes	—
16	C₂×#117×C₅₉	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₅₉	C₂²	1	4	yes	—
18	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#400	C₂²	1	16	yes	—
20	C₅₉×#398	C₂²	1	16	yes	—
21	C₅₉×#391	C₂²	1	8	yes	—
22	C₅₉×#391	C₂²	1	8	yes	—
23	C₅₉×#391	C₂²	1	8	yes	—
24	C₅₉×#389	C₂²	1	8	yes	—
25	C₅₉×#387	C₂²	1	32	yes	—
26	C₅₉×#385	C₂²	1	8	yes	—
27	C₅₉×#381	C₂²	1	4	yes	—
28	C₅₉×#381	C₂²	1	8	yes	—
29	C₄×C₄⋊C₄×C₅₉	C₂²	1	8	yes	—
30	C₂²×Q₈×C₅₉	—	2	4	no	—
31	C₂²×Q₈×C₅₉	—	2	8	no	—
32	C₂²×Q₈×C₅₉	—	2	8	no	—
33	C₂²×Q₈×C₅₉	D₄	1	8	yes	—
34	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
35	C₂²×Q₈×C₅₉	C₂³	1	4	yes	—
36	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
37	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
38	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
39	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₉	—	2	32	no	—
41	C₄×Q₈×C₅₉	—	2	32	no	—
42	#117×C₅₉	—	2	16	no	—
43	#117×C₅₉	—	2	8	no	—
44	#117×C₅₉	—	2	8	no	—
45	C₂×C₄⋊C₄×C₅₉	—	2	16	no	—
46	C₂×C₄⋊C₄×C₅₉	C₂³	1	16	yes	—
47	C₂×C₄⋊C₄×C₅₉	C₂³	1	16	yes	—
48	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₅₉	C₂³	1	8	yes	—
51	C₂×C₂²⋊C₄×C₅₉	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₅₉	C₂³	1	4	yes	—
53	C₂×C₄²×C₅₉	—	2	4	no	—
54	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
55	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
56	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
57	C₂×C₄²×C₅₉	C₂³	1	8	yes	—
58	#95×C₅₉	C₂³	1	16	yes	—
59	#95×C₅₉	C₂³	1	16	yes	—
60	#95×C₅₉	C₂³	1	32	yes	—
61	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
62	C₂×Q₈×C₅₉	—	2	32	no	—
63	C₂×Q₈×C₅₉	—	2	8	no	—
64	C₂×Q₈×C₅₉	—	2	8	no	—
65	C₂×Q₈×C₅₉	—	2	8	no	—
66	C₂×Q₈×C₅₉	—	2	8	no	—
67	C₂×Q₈×C₅₉	—	2	8	no	—
68	C₂×Q₈×C₅₉	—	4	16	no	—
69	C₂×Q₈×C₅₉	C₂×D₄	1	16	yes	—
70	C₂²×C₄×C₅₉	—	2	4	no	—
71	C₂²×C₄×C₅₉	—	2	8	no	—
72	C₂²×C₄×C₅₉	—	2	8	no	—
73	C₂²×C₄×C₅₉	—	2	4	no	—
74	C₂²×C₄×C₅₉	—	2	8	no	—
75	C₂²×C₄×C₅₉	C₂×D₄	1	8	yes	—
76	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₅₉	#41	1	8	yes	—
78	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
83	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
84	C₄⋊C₄×C₅₉	—	2	32	no	—
85	C₄⋊C₄×C₅₉	—	2	16	no	—
86	C₄⋊C₄×C₅₉	—	2	8	no	—
87	C₄⋊C₄×C₅₉	—	2	8	no	—
88	C₄⋊C₄×C₅₉	—	2	32	no	—
89	C₂²⋊C₄×C₅₉	—	2	8	no	—
90	C₂²⋊C₄×C₅₉	—	2	8	no	—
91	C₄²×C₅₉	—	2	8	no	—
92	C₄²×C₅₉	—	2	16	no	—
93	C₄²×C₅₉	—	2	8	no	—
94	C₄²×C₅₉	—	2	8	no	—
95	C₂³×C₅₉	—	2	4	no	—
96	C₂³×C₅₉	C₂²×D₄	1	2	yes	—
97	C₂³×C₅₉	C₂×#41	1	4	yes	—
98	C₂³×C₅₉	C₂⁵	1	1	yes	characteristic
99	Q₈×C₅₉	—	2	32	no	—
100	Q₈×C₅₉	—	4	16	no	—
101	C₂×C₄×C₅₉	—	2	16	no	—
102	C₂×C₄×C₅₉	—	2	16	no	—
103	C₂×C₄×C₅₉	—	2	4	no	—
104	C₂×C₄×C₅₉	—	2	8	no	—
105	C₂×C₄×C₅₉	—	2	4	no	—
106	C₂×C₄×C₅₉	—	2	8	no	—
107	C₂×C₄×C₅₉	—	2	4	no	—
108	C₂×C₄×C₅₉	—	2	8	no	—
109	C₂×C₄×C₅₉	—	2	8	no	—
110	C₂×C₄×C₅₉	—	2	16	no	—
111	C₂×C₄×C₅₉	—	2	16	no	—
112	C₂×C₄×C₅₉	—	4	8	no	—
113	C₂×C₄×C₅₉	#123	1	16	yes	—
114	C₂×C₄×C₅₉	C₂²×D₄	1	8	yes	—
115	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
116	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
117	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
118	C₂×C₄×C₅₉	2₋¹⁺⁴	1	8	yes	—
119	#35674	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₅₉	—	2	4	no	—
121	C₂²×C₅₉	—	2	8	no	—
122	C₂²×C₅₉	C₂×#123	1	4	yes	—
123	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
124	C₂²×C₅₉	C₂²×#41	1	2	yes	—
125	C₂²×C₅₉	C₂×2₋¹⁺⁴	1	4	yes	—
126	C₄×C₅₉	—	2	8	no	—
127	C₄×C₅₉	—	2	8	no	—
128	C₄×C₅₉	—	4	4	no	—
129	C₄×C₅₉	—	4	8	no	—
130	C₄×C₅₉	—	4	8	no	—
131	C₄×C₅₉	C₂×#123	1	8	yes	—
132	C₂²×C₄⋊Q₈	C₂×C₅₉	1	1	yes	characteristic
133	C₂×#519	C₂×C₅₉	1	1	yes	characteristic
134	#2484	C₂×C₅₉	1	8	yes	—
135	#2446	C₂×C₅₉	1	16	yes	complemented
136	#2310	C₂×C₅₉	1	1	yes	characteristic
137	#2310	C₂×C₅₉	1	2	yes	—
138	#2296	C₂×C₅₉	1	2	yes	—
139	C₂×C₅₉	—	2	4	no	—
140	C₂×C₅₉	C₂²×#123	1	1	yes	characteristic
141	C₂×C₅₉	#3486	1	2	yes	—
142	C₂×C₅₉	#3509	1	4	yes	—
143	C₂³×Q₈	C₂²×C₅₉	1	1	yes	characteristic
144	C₂³×Q₈	C₂²×C₅₉	1	1	yes	characteristic
145	C₂×C₄⋊Q₈	—	2	8	no	—
146	C₂×C₄⋊Q₈	C₂²×C₅₉	1	8	yes	—
147	#519	—	2	8	no	—
148	C₂×C₄×Q₈	C₂²×C₅₉	1	8	yes	—
149	C₂×#117	C₂²×C₅₉	1	2	yes	—
150	C₂×#117	C₂²×C₅₉	1	2	yes	—
151	C₂×#117	C₂²×C₅₉	1	4	yes	—
152	C₂²×C₄⋊C₄	C₂²×C₅₉	1	4	yes	—
153	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
154	#400	C₂²×C₅₉	1	16	yes	—
155	#398	C₂²×C₅₉	1	16	yes	—
156	#391	C₂²×C₅₉	1	8	yes	—
157	#391	C₂²×C₅₉	1	8	yes	—
158	#391	C₂²×C₅₉	1	8	yes	—
159	#389	C₂²×C₅₉	1	8	yes	—
160	#387	C₂²×C₅₉	1	32	yes	—
161	#385	C₂²×C₅₉	1	8	yes	—
162	#381	C₂²×C₅₉	1	8	yes	—
163	#381	C₂²×C₅₉	1	4	yes	—
164	C₄×C₄⋊C₄	C₂²×C₅₉	1	8	yes	—
165	C₅₉	#35674	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×Q₈	—	2	4	no	—
167	C₂²×Q₈	—	2	8	no	—
168	C₂²×Q₈	—	2	8	no	—
169	C₂²×Q₈	D₄×C₅₉	1	8	yes	—
170	C₂²×Q₈	C₂³×C₅₉	1	4	yes	—
171	C₂²×Q₈	C₂³×C₅₉	1	4	yes	—
172	C₂³×C₄	C₂³×C₅₉	1	2	yes	—
173	C₂³×C₄	C₂³×C₅₉	1	2	yes	—
174	C₂³×C₄	C₂³×C₅₉	1	2	yes	—
175	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
176	C₄⋊Q₈	—	2	32	no	—
177	C₄×Q₈	—	2	32	no	—
178	#117	—	2	16	no	—
179	#117	—	2	8	no	—
180	#117	—	2	8	no	—
181	C₂×C₄⋊C₄	—	2	16	no	—
182	C₂×C₄⋊C₄	C₂³×C₅₉	1	16	yes	—
183	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
184	C₂×C₄⋊C₄	C₂³×C₅₉	1	8	yes	—
185	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
186	C₂×C₄⋊C₄	C₂³×C₅₉	1	16	yes	—
187	C₂×C₂²⋊C₄	C₂³×C₅₉	1	4	yes	—
188	C₂×C₂²⋊C₄	C₂³×C₅₉	1	4	yes	—
189	C₂×C₄²	—	2	4	no	—
190	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
191	C₂×C₄²	C₂³×C₅₉	1	8	yes	—
192	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
193	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
194	#95	C₂³×C₅₉	1	16	yes	—
195	#95	C₂³×C₅₉	1	16	yes	—
196	#95	C₂³×C₅₉	1	32	yes	—
197	C₂⁴	C₂⁴×C₅₉	1	1	yes	characteristic
198	C₂×Q₈	—	2	32	no	—
199	C₂×Q₈	—	2	8	no	—
200	C₂×Q₈	—	2	8	no	—
201	C₂×Q₈	—	2	8	no	—
202	C₂×Q₈	—	2	8	no	—
203	C₂×Q₈	—	2	8	no	—
204	C₂×Q₈	—	4	16	no	—
205	C₂×Q₈	C₂×D₄×C₅₉	1	16	yes	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	C₂×D₄×C₅₉	1	8	yes	—
212	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
213	C₂²×C₄	#41×C₅₉	1	8	yes	—
214	C₂²×C₄	C₂⁴×C₅₉	1	8	yes	—
215	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
216	C₂²×C₄	C₂⁴×C₅₉	1	8	yes	—
217	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
218	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
219	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
220	C₄⋊C₄	—	2	32	no	—
221	C₄⋊C₄	—	2	16	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	2	32	no	—
225	C₂²⋊C₄	—	2	8	no	—
226	C₂²⋊C₄	—	2	8	no	—
227	C₄²	—	2	8	no	—
228	C₄²	—	2	16	no	—
229	C₄²	—	2	8	no	—
230	C₄²	—	2	8	no	—
231	C₂³	—	2	4	no	—
232	C₂³	C₂²×D₄×C₅₉	1	2	yes	—
233	C₂³	C₂×#41×C₅₉	1	4	yes	—
234	C₂³	C₂⁵×C₅₉	1	1	yes	characteristic
235	Q₈	—	2	32	no	—
236	Q₈	—	4	16	no	—
237	C₂×C₄	—	2	16	no	—
238	C₂×C₄	—	2	16	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	16	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	#123×C₅₉	1	16	yes	—
250	C₂×C₄	C₂²×D₄×C₅₉	1	8	yes	—
251	C₂×C₄	C₂²×D₄×C₅₉	1	4	yes	—
252	C₂×C₄	C₂×#41×C₅₉	1	4	yes	—
253	C₂×C₄	C₂×#41×C₅₉	1	4	yes	—
254	C₂×C₄	2₋¹⁺⁴×C₅₉	1	8	yes	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	4	no	—
257	C₂²	C₂×#123×C₅₉	1	4	yes	—
258	C₂²	C₂³×D₄×C₅₉	1	1	yes	characteristic
259	C₂²	C₂²×#41×C₅₉	1	2	yes	—
260	C₂²	C₂×2₋¹⁺⁴×C₅₉	1	4	yes	—
261	C₄	—	2	8	no	—
262	C₄	—	2	8	no	—
263	C₄	—	4	8	no	—
264	C₄	—	4	4	no	—
265	C₄	—	4	8	no	—
266	C₄	C₂×#123×C₅₉	1	8	yes	—
267	C₂	—	2	4	no	—
268	C₂	C₂²×#123×C₅₉	1	1	yes	characteristic
269	C₂	C₅₉×#3486	1	2	yes	—
270	C₂	C₅₉×#3509	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₉
prime radical	C₂⁴×C₅₉	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₉×#2446	C₂	semidirect
119. #35674	C₅₉	direct (N × H)
135. #2446	C₂×C₅₉	semidirect
165. C₅₉	#35674	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₅₉	2
C₂×C₅₉×#519	2
C₅₉×#2484	2
C₅₉×#2446	2
C₅₉×#2310	2
C₅₉×#2296	2
#35674	59

Subgroups — C59×#35674 #28447783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#35674 #28447783