Subgroups — #21433081

Order 7552.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837936	C₂	1	1	yes	maximal, characteristic
2	#16837530	C₂	1	2	yes	maximal, complemented
3	#16837478	C₂	1	2	yes	maximal, complemented
4	#16837475	C₂	1	2	yes	maximal, complemented
5	#16837471	C₂	1	2	yes	maximal, complemented
6	#16837461	C₂	1	2	yes	maximal, complemented
7	#16837459	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837356	C₂	1	2	yes	maximal, complemented
9	C₄×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11745659	C₂²	1	2	yes	—
11	C₂×C₈×D₅₉	C₂²	1	2	yes	—
12	#14273721	C₄	1	2	yes	—
13	#14273719	C₄	1	1	yes	characteristic
14	#14273715	C₄	1	1	yes	characteristic
15	#14273710	C₄	1	2	yes	—
16	#14273706	C₄	1	1	yes	characteristic
17	#14273701	C₄	1	1	yes	characteristic
18	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
19	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₅₉	C₂²	1	2	yes	—
21	#14273684	C₂²	1	2	yes	—
22	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
23	Dic₅₉⋊C₈	C₂²	1	2	yes	—
24	Dic₅₉⋊C₈	C₂²	1	2	yes	—
25	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
26	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
27	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₅₉)⋊C₈	C₂²	1	2	yes	—
29	#14273638	C₂²	1	2	yes	—
30	#14273637	C₂²	1	2	yes	—
31	C₈×Dic₅₉	C₂²	1	2	yes	—
32	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
34	#97×C₅₉	C₂²	1	2	yes	—
35	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
41	C₂×D₂₃₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₅₉	C₂³	1	2	yes	—
43	C₂×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
44	M₁₆×C₅₉	—	2	4	no	—
45	C₂×C₈×C₅₉	C₂³	1	2	yes	—
46	C₂×C₈×C₅₉	C₂³	1	2	yes	—
47	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₅₉	C₂³	1	1	yes	characteristic
50	C₄²×C₅₉	C₂³	1	1	yes	characteristic
51	C₂²⋊Dic₅₉	C₂×C₄	1	2	yes	—
52	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
58	Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
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	C₂³	1	2	yes	—
63	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
64	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
65	#11745659	—	2	4	no	—
66	C₈×D₅₉	—	2	4	no	—
67	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₅₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₅₉	—	2	2	no	—
70	C₂×C₄×C₅₉	—	4	1	no	—
71	C₂×C₄×C₅₉	—	4	1	no	—
72	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₅₉	—	4	2	no	—
79	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
80	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
81	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
82	D₂₃₆	—	4	1	no	—
83	C₄×D₅₉	—	2	2	no	—
84	C₄×D₅₉	—	2	2	no	—
85	Dic₁₁₈	—	4	1	no	—
86	C₈×C₅₉	—	2	2	no	—
87	C₈×C₅₉	#41	1	4	yes	—
88	C₅₉⋊C₈	—	2	2	no	—
89	C₅₉⋊C₈	—	2	2	no	—
90	C₂²×C₅₉	—	2	1	no	—
91	C₂²×C₅₉	—	2	1	no	—
92	C₂²×C₅₉	—	2	1	no	—
93	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₅₉	—	2	2	no	—
95	C₂×D₅₉	—	2	2	no	—
96	C₂×D₅₉	—	2	2	no	—
97	C₄×C₅₉	—	2	1	no	—
98	C₄×C₅₉	—	2	1	no	—
99	C₄×C₅₉	—	2	1	no	—
100	C₄×C₅₉	—	2	1	no	—
101	C₄×C₅₉	—	2	1	no	—
102	C₄×C₅₉	C₂×#41	1	2	yes	—
103	Dic₅₉	—	2	2	no	—
104	Dic₅₉	—	2	2	no	—
105	Dic₅₉	—	2	2	no	—
106	#2967	—	59	1	no	maximal, Sylow
107	C₂×C₅₉	—	4	1	no	—
108	C₂×C₅₉	C₄×#41	1	1	yes	characteristic
109	C₂×C₅₉	#568	1	1	yes	characteristic
110	C₂×C₅₉	#568	1	1	yes	characteristic
111	D₅₉	—	4	2	no	—
112	#529	—	59	1	no	—
113	#446	—	59	2	no	—
114	#436	—	59	2	no	—
115	#435	—	59	2	no	—
116	#435	—	59	2	no	—
117	#433	—	59	2	no	—
118	#432	—	59	1	no	—
119	#405	—	59	2	no	—
120	C₄×M₁₆	D₅₉	1	1	yes	characteristic, complemented
121	C₅₉	#2967	1	1	yes	characteristic, complemented, Sylow
122	C₂×M₁₆	—	59	2	no	—
123	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
124	C₂²×C₈	—	59	2	no	—
125	#126	—	59	2	no	—
126	#125	—	59	1	no	—
127	#124	—	59	1	no	—
128	#123	—	59	2	no	—
129	C₂²⋊Q₈	—	59	1	no	—
130	#121	—	59	1	no	—
131	C₄×Q₈	—	59	1	no	—
132	C₄×D₄	—	59	2	no	—
133	C₄×D₄	—	59	1	no	—
134	#117	—	59	2	no	—
135	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
136	C₄⋊C₈	—	59	2	no	—
137	C₄⋊C₈	—	59	2	no	—
138	C₄⋊C₈	—	59	2	no	—
139	C₂²⋊C₈	—	59	1	no	—
140	C₂²⋊C₈	—	59	1	no	—
141	C₂²⋊C₈	—	59	2	no	—
142	C₂²⋊C₈	—	59	2	no	—
143	#97	—	59	2	no	—
144	#97	—	59	2	no	—
145	#97	C₂×D₅₉	1	2	yes	—
146	C₄×C₈	—	59	2	no	—
147	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
148	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
149	C₂×Q₈	—	59	1	no	—
150	C₂×D₄	—	59	2	no	—
151	C₂×D₄	—	59	1	no	—
152	C₂²×C₄	—	59	2	no	—
153	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
156	M₁₆	—	2	4	no	—
157	M₁₆	—	118	4	no	—
158	C₂×C₈	—	59	2	no	—
159	C₂×C₈	—	59	2	no	—
160	C₂×C₈	—	118	4	no	—
161	C₂×C₈	C₂²×D₅₉	1	2	yes	—
162	C₂×C₈	C₂²×D₅₉	1	2	yes	—
163	C₄⋊C₄	—	59	2	no	—
164	C₄⋊C₄	—	59	2	no	—
165	C₄⋊C₄	—	59	1	no	—
166	C₄⋊C₄	—	59	2	no	—
167	C₄⋊C₄	—	59	1	no	—
168	C₄⋊C₄	—	59	2	no	—
169	C₂²⋊C₄	—	59	2	no	—
170	C₂²⋊C₄	—	59	2	no	—
171	C₂²⋊C₄	—	59	2	no	—
172	C₂²⋊C₄	—	59	2	no	—
173	C₂²⋊C₄	—	59	2	no	—
174	C₄²	—	59	2	no	—
175	C₄²	C₄×D₅₉	1	1	yes	characteristic
176	C₄²	C₄×D₅₉	1	1	yes	characteristic
177	C₄²	C₂²×D₅₉	1	1	yes	characteristic
178	C₄²	C₂²×D₅₉	1	1	yes	characteristic
179	C₂³	—	59	2	no	—
180	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
181	Q₈	—	236	1	no	—
182	D₄	—	236	1	no	—
183	D₄	—	236	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	59	2	no	—
188	C₂×C₄	—	59	2	no	—
189	C₂×C₄	—	59	2	no	—
190	C₂×C₄	—	118	2	no	—
191	C₂×C₄	—	118	2	no	—
192	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	118	2	no	—
200	C₈	—	118	2	no	—
201	C₈	#11745685	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	118	2	no	—
206	C₂²	—	118	2	no	—
207	C₂²	—	118	2	no	—
208	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	118	2	no	—
215	C₄	—	118	2	no	—
216	C₄	—	118	2	no	—
217	C₄	C₂×#11745685	1	2	yes	—
218	C₂	—	4	1	no	—
219	C₂	—	236	2	no	—
220	C₂	C₄×#11745685	1	1	yes	characteristic
221	C₂	#16838168	1	1	yes	characteristic
222	C₂	#16838168	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₄×M₁₆×C₅₉	C₂
hypercenter	C₄×M₁₆	D₅₉
hypocenter	C₅₉	#2967
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #16837530	C₂	semidirect
3. #16837478	C₂	semidirect
4. #16837475	C₂	semidirect
5. #16837471	C₂	semidirect
6. #16837461	C₂	semidirect
7. #16837459	C₂	semidirect
8. #16837356	C₂	semidirect
9. C₄×M₁₆×C₅₉	C₂	semidirect
120. C₄×M₁₆	D₅₉	semidirect
121. C₅₉	#2967	semidirect