Subgroups — #21433178

Order 7552.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₅₉	C₂	1	1	yes	maximal, characteristic
2	#16837533	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837532	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837531	C₂	1	2	yes	maximal, complemented
5	#16837530	C₂	1	2	yes	maximal, complemented
6	#16837529	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837458	C₂	1	2	yes	maximal
8	#16837457	C₂	1	2	yes	maximal
9	#16837454	C₂	1	1	yes	maximal, characteristic
10	#16837453	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₅₉	C₄	1	1	yes	characteristic
13	Q₈×Dic₅₉	C₄	1	1	yes	characteristic
14	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₁₈	C₂²	1	2	yes	—
16	#14273692	C₄	1	1	yes	characteristic
17	#14273692	C₄	1	2	yes	—
18	C₄⋊C₄×D₅₉	C₄	1	2	yes	—
19	C₄⋊C₄×D₅₉	C₄	1	1	yes	characteristic
20	C₄²×D₅₉	C₂²	1	1	yes	characteristic
21	C₄²×D₅₉	C₂²	1	2	yes	—
22	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
24	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₅₉⋊C₈	C₂²	1	2	yes	—
27	Dic₅₉⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
31	#14273638	C₂²	1	2	yes	—
32	#14273637	C₂²	1	2	yes	—
33	#14273637	C₂²	1	2	yes	—
34	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₅₉	C₂²	1	2	yes	—
36	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
39	#97×C₅₉	C₂²	1	2	yes	—
40	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	Q₈×D₅₉	—	2	4	no	—
43	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
49	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
50	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₅₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄²×C₅₉	C₂³	1	2	yes	—
57	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
59	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
62	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
63	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
65	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
67	C₄×Dic₅₉	C₂³	1	2	yes	—
68	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
69	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
71	C₂²×D₅₉	C₂²×C₄	1	1	yes	characteristic
72	Q₈×C₅₉	—	2	2	no	—
73	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
81	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
82	C₂×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₄×D₅₉	—	2	4	no	—
84	C₄×D₅₉	—	2	1	no	—
85	C₄×D₅₉	—	2	1	no	—
86	C₄×D₅₉	—	4	1	no	—
87	C₄×D₅₉	—	4	2	no	—
88	C₄×D₅₉	M₁₆	1	4	yes	—
89	Dic₁₁₈	—	2	2	no	—
90	Dic₁₁₈	—	2	4	no	—
91	C₈×C₅₉	—	2	1	no	—
92	C₈×C₅₉	—	2	2	no	—
93	C₈×C₅₉	—	2	1	no	—
94	C₅₉⋊C₈	—	2	1	no	—
95	C₅₉⋊C₈	—	2	2	no	—
96	C₅₉⋊C₈	—	2	1	no	—
97	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₅₉	—	2	1	no	—
99	C₂×D₅₉	—	2	1	no	—
100	C₂×D₅₉	C₂×M₁₆	1	2	yes	—
101	C₄×C₅₉	—	2	2	no	—
102	C₄×C₅₉	—	2	1	no	—
103	C₄×C₅₉	—	2	2	no	—
104	C₄×C₅₉	C₂×M₁₆	1	2	yes	—
105	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
106	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
107	Dic₅₉	—	2	1	no	—
108	Dic₅₉	—	2	2	no	—
109	Dic₅₉	—	2	1	no	—
110	Dic₅₉	—	2	2	no	—
111	Dic₅₉	C₂×M₁₆	1	2	yes	—
112	#2983	—	59	1	no	maximal, Sylow
113	C₂×C₅₉	#519	1	1	yes	characteristic
114	C₂×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
115	C₂×C₅₉	#568	1	1	yes	characteristic
116	D₅₉	—	2	2	no	—
117	C₂×C₄×Q₈	—	59	1	no	—
118	#446	—	59	1	no	—
119	#446	—	59	1	no	—
120	#446	—	59	2	no	—
121	#446	—	59	2	no	—
122	#446	—	59	1	no	—
123	#446	D₅₉	1	1	yes	characteristic, complemented
124	#432	—	59	2	no	—
125	#432	—	59	2	no	—
126	#431	—	59	1	no	—
127	#431	—	59	1	no	—
128	C₅₉	#2983	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	59	1	no	—
130	C₄×Q₈	—	59	1	no	—
131	C₄×Q₈	—	59	2	no	—
132	C₄×Q₈	—	59	1	no	—
133	C₄×Q₈	—	59	2	no	—
134	C₄×Q₈	—	59	1	no	—
135	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	59	1	no	—
137	C₂×C₄⋊C₄	—	59	2	no	—
138	C₂×C₄²	—	59	1	no	—
139	C₂×C₄²	—	59	2	no	—
140	C₄⋊C₈	—	59	1	no	—
141	C₄⋊C₈	—	59	1	no	—
142	C₄⋊C₈	—	59	2	no	—
143	C₄⋊C₈	—	59	2	no	—
144	C₄⋊C₈	—	59	1	no	—
145	C₄⋊C₈	—	59	2	no	—
146	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
147	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
148	C₂²⋊C₈	—	59	2	no	—
149	C₂²⋊C₈	—	59	1	no	—
150	C₂²⋊C₈	—	59	1	no	—
151	#97	—	59	2	no	—
152	#97	—	59	2	no	—
153	#97	—	59	2	no	—
154	#97	C₂×D₅₉	1	2	yes	—
155	C₄×C₈	—	59	2	no	—
156	C₄×C₈	—	59	1	no	—
157	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
158	C₂×Q₈	—	59	2	no	—
159	C₂×Q₈	—	59	1	no	—
160	C₂×Q₈	—	118	4	no	—
161	C₂×Q₈	C₄×D₅₉	1	1	yes	characteristic
162	C₂²×C₄	—	59	1	no	—
163	C₂²×C₄	—	59	2	no	—
164	C₂²×C₄	—	59	1	no	—
165	C₂²×C₄	—	59	1	no	—
166	C₂²×C₄	—	59	2	no	—
167	C₂×C₈	—	59	1	no	—
168	C₂×C₈	—	59	2	no	—
169	C₂×C₈	—	59	1	no	—
170	C₂×C₈	C₂²×D₅₉	1	2	yes	—
171	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
173	C₄⋊C₄	—	59	2	no	—
174	C₄⋊C₄	—	59	2	no	—
175	C₄⋊C₄	—	59	2	no	—
176	C₄⋊C₄	—	59	2	no	—
177	C₄⋊C₄	—	59	1	no	—
178	C₄⋊C₄	C₄×D₅₉	1	2	yes	—
179	C₄⋊C₄	C₄×D₅₉	1	1	yes	characteristic
180	C₄²	—	59	1	no	—
181	C₄²	—	59	2	no	—
182	C₄²	—	59	1	no	—
183	C₄²	—	59	2	no	—
184	C₄²	—	59	1	no	—
185	C₄²	—	59	2	no	—
186	C₄²	C₂²×D₅₉	1	1	yes	characteristic
187	C₄²	C₂²×D₅₉	1	2	yes	—
188	C₂³	—	59	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	118	4	no	—
191	Q₈	—	118	2	no	—
192	C₂×C₄	—	59	1	no	—
193	C₂×C₄	—	59	4	no	—
194	C₂×C₄	—	59	1	no	—
195	C₂×C₄	—	59	1	no	—
196	C₂×C₄	—	59	2	no	—
197	C₂×C₄	—	59	2	no	—
198	C₂×C₄	—	118	4	no	—
199	C₂×C₄	—	118	1	no	—
200	C₂×C₄	—	118	1	no	—
201	C₂×C₄	—	236	1	no	—
202	C₂×C₄	—	236	2	no	—
203	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	118	1	no	—
212	C₈	—	118	2	no	—
213	C₈	—	118	1	no	—
214	C₂²	—	59	2	no	—
215	C₂²	—	118	1	no	—
216	C₂²	—	118	1	no	—
217	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	59	2	no	—
222	C₄	—	118	2	no	—
223	C₄	—	118	1	no	—
224	C₄	—	118	2	no	—
225	C₄	—	118	1	no	—
226	C₄	M₁₆×D₅₉	1	2	yes	—
227	C₄	#14273774	1	1	yes	characteristic
228	C₄	#14273775	1	1	yes	characteristic
229	C₂	—	118	2	no	—
230	C₂	#16837878	1	1	yes	characteristic
231	C₂	C₂×M₁₆×D₅₉	1	1	yes	characteristic
232	C₂	#16838167	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#446	C₂
hypercenter	#446	D₅₉
hypocenter	C₅₉	#2983
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #16837533	C₂	semidirect
3. #16837532	C₂	semidirect
4. #16837531	C₂	semidirect
5. #16837530	C₂	semidirect
6. #16837529	C₂	semidirect
11. C₅₉×#446	C₂	semidirect
123. #446	D₅₉	semidirect
128. C₅₉	#2983	semidirect