Subgroups — #21432764

Order 7552.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11745685	C₂	1	1	yes	maximal, characteristic
2	#16837462	C₂	1	4	yes	maximal, complemented
3	#16837458	C₂	1	2	yes	maximal, complemented
4	#16837454	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊(C₅₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837370	C₂	1	2	yes	maximal
7	#16837360	C₂	1	2	yes	maximal, complemented
8	C₅₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11745685	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
11	C₄×Dic₁₁₈	C₄	1	1	yes	characteristic
12	C₄×Dic₁₁₈	C₄	1	1	yes	characteristic
13	C₄×D₂₃₆	C₄	1	1	yes	characteristic
14	C₄×D₂₃₆	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₅₉	C₄	1	2	yes	—
16	C₄×C₂²⋊D₅₉	C₄	1	2	yes	—
17	#14273686	C₂²	1	1	yes	characteristic
18	#14273684	C₂²	1	2	yes	—
19	C₄²×D₅₉	C₂²	1	2	yes	—
20	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
23	(C₂×D₅₉)⋊C₈	C₂²	1	4	yes	—
24	(C₂×D₅₉)⋊C₈	C₂²	1	4	yes	—
25	#14273637	C₂²	1	4	yes	—
26	C₈×Dic₅₉	C₂²	1	4	yes	—
27	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
28	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
30	C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
31	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
35	#11745685	—	2	2	no	—
36	#11745685	—	2	2	no	—
37	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₅₉	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₅₉	C₂³	1	2	yes	—
41	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
42	M₁₆×C₅₉	—	2	4	no	—
43	C₂×C₈×C₅₉	C₂³	1	4	yes	—
44	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₅₉	C₂³	1	1	yes	characteristic
47	C₄²×C₅₉	C₂³	1	1	yes	characteristic
48	C₂²⋊Dic₅₉	C₂×C₄	1	2	yes	—
49	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
50	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
53	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
56	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
57	C₄×Dic₅₉	C₂³	1	2	yes	—
58	C₂×C₅₉⋊C₈	—	2	4	no	—
59	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
60	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
61	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₅₉	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	4	1	no	—
65	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#41	1	2	yes	—
71	C₂×C₄×C₅₉	#41	1	2	yes	—
72	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₅₉	—	2	4	no	—
74	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
77	D₂₃₆	—	2	2	no	—
78	C₄×D₅₉	—	2	4	no	—
79	C₄×D₅₉	—	2	2	no	—
80	C₄×D₅₉	—	2	2	no	—
81	C₄×D₅₉	—	4	2	no	—
82	Dic₁₁₈	—	2	2	no	—
83	C₈×C₅₉	—	2	4	no	—
84	C₅₉⋊C₈	—	2	2	no	—
85	C₅₉⋊C₈	—	2	2	no	—
86	C₂²×C₅₉	—	2	1	no	—
87	C₂²×C₅₉	—	2	1	no	—
88	C₂²×C₅₉	#117	1	2	yes	—
89	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₅₉	—	2	2	no	—
91	C₂×D₅₉	—	2	2	no	—
92	C₂×D₅₉	—	2	2	no	—
93	C₄×C₅₉	—	2	1	no	—
94	C₄×C₅₉	—	2	1	no	—
95	C₄×C₅₉	—	2	1	no	—
96	C₄×C₅₉	#117	1	2	yes	—
97	C₄×C₅₉	#131	1	2	yes	—
98	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
99	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
100	Dic₅₉	—	2	2	no	—
101	Dic₅₉	—	2	2	no	—
102	Dic₅₉	—	2	2	no	—
103	#2925	—	59	1	no	maximal, Sylow
104	C₂×C₅₉	—	2	2	no	—
105	C₂×C₅₉	C₂×#117	1	1	yes	characteristic
106	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
107	C₂×C₅₉	#568	1	1	yes	characteristic
108	D₅₉	—	4	2	no	—
109	C₄×#41	—	59	1	no	—
110	#433	—	59	4	no	—
111	#432	—	59	2	no	—
112	#431	—	59	2	no	—
113	#423	D₅₉	1	1	yes	characteristic, complemented
114	C₂×C₄⋊C₈	—	59	1	no	—
115	#408	—	59	2	no	—
116	#405	—	59	2	no	—
117	C₅₉	#2925	1	1	yes	characteristic, complemented, Sylow
118	C₂×#41	—	59	1	no	—
119	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
120	C₂²×C₈	—	59	2	no	—
121	C₄×Q₈	—	59	1	no	—
122	C₄×Q₈	—	59	1	no	—
123	C₄×D₄	—	59	2	no	—
124	C₄×D₄	—	59	2	no	—
125	C₄×D₄	—	59	1	no	—
126	C₄×D₄	—	59	1	no	—
127	#117	—	59	1	no	—
128	#117	—	59	2	no	—
129	C₂×C₄²	—	59	2	no	—
130	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
131	C₄⋊C₈	—	59	1	no	—
132	C₄⋊C₈	—	59	1	no	—
133	C₄⋊C₈	—	59	2	no	—
134	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
135	C₄⋊C₈	C₂×D₅₉	1	2	yes	—
136	C₂²⋊C₈	—	59	4	no	—
137	C₂²⋊C₈	—	59	4	no	—
138	#97	—	59	4	no	—
139	C₄×C₈	—	59	4	no	—
140	#41	—	118	2	no	—
141	#41	—	118	2	no	—
142	C₂×Q₈	—	59	1	no	—
143	C₂×D₄	—	59	2	no	—
144	C₂×D₄	—	59	1	no	—
145	C₂²×C₄	—	59	2	no	—
146	C₂²×C₄	—	59	2	no	—
147	C₂²×C₄	—	59	2	no	—
148	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
151	M₁₆	—	2	4	no	—
152	C₂×C₈	—	59	2	no	—
153	C₂×C₈	—	59	2	no	—
154	C₂×C₈	—	118	4	no	—
155	C₂×C₈	C₂²×D₅₉	1	4	yes	—
156	C₄⋊C₄	—	59	2	no	—
157	C₄⋊C₄	—	59	2	no	—
158	C₄⋊C₄	—	59	1	no	—
159	C₄⋊C₄	—	59	1	no	—
160	C₂²⋊C₄	—	59	2	no	—
161	C₂²⋊C₄	—	59	2	no	—
162	C₂²⋊C₄	—	59	2	no	—
163	C₄²	—	59	2	no	—
164	C₄²	—	59	2	no	—
165	C₄²	—	59	2	no	—
166	C₄²	C₄×D₅₉	1	1	yes	characteristic
167	C₄²	C₄×D₅₉	1	1	yes	characteristic
168	C₄²	C₂²×D₅₉	1	1	yes	characteristic
169	C₄²	C₂²×D₅₉	1	1	yes	characteristic
170	C₂³	—	59	2	no	—
171	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
172	Q₈	—	118	2	no	—
173	D₄	—	118	4	no	—
174	D₄	—	118	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	59	2	no	—
178	C₂×C₄	—	59	2	no	—
179	C₂×C₄	—	59	2	no	—
180	C₂×C₄	—	118	4	no	—
181	C₂×C₄	—	118	2	no	—
182	C₂×C₄	—	118	2	no	—
183	C₂×C₄	—	236	2	no	—
184	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
189	C₂×C₄	#11745687	1	2	yes	—
190	C₂×C₄	#11745689	1	2	yes	—
191	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
192	C₈	—	2	4	no	—
193	C₈	—	118	2	no	—
194	C₈	—	118	2	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	118	2	no	—
198	C₂²	—	118	2	no	—
199	C₂²	—	118	2	no	—
200	C₂²	#14273685	1	2	yes	—
201	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	118	2	no	—
206	C₄	—	118	2	no	—
207	C₄	—	118	2	no	—
208	C₄	#14273685	1	2	yes	—
209	C₄	#14273735	1	2	yes	—
210	C₄	C₂×#11745687	1	1	yes	characteristic
211	C₄	C₂×#11745689	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	236	2	no	—
214	C₂	C₂×#14273685	1	1	yes	characteristic
215	C₂	C₂×#14273735	1	1	yes	characteristic
216	C₂	#16838168	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#423	C₂
hypercenter	#423	D₅₉
hypocenter	C₅₉	#2925
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #16837462	C₂	semidirect
3. #16837458	C₂	semidirect
4. #16837454	C₂	semidirect
5. C₂×C₄⋊(C₅₉⋊C₈)	C₂	semidirect
7. #16837360	C₂	semidirect
8. C₅₉×#423	C₂	semidirect
113. #423	D₅₉	semidirect
117. C₅₉	#2925	semidirect