Subgroups — #23776803

Order 10112.

996 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19099193	C₂	1	4	yes	maximal, complemented
2	#19099192	C₂	1	4	yes	maximal, complemented
3	#19099190	C₂	1	4	yes	maximal, complemented
4	#19099188	C₂	1	4	yes	maximal
5	#19099120	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099117	C₂	1	8	yes	maximal, complemented
7	#19099116	C₂	1	2	yes	maximal, complemented
8	C₂×#14579118	C₂	1	1	yes	maximal, characteristic
9	#117×D₇₉	C₂	1	2	yes	maximal
10	#533×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₄×D₇₉	C₂²	1	1	yes	characteristic
12	#14579161	C₂²	1	2	yes	complemented
13	#14579160	C₂²	1	4	yes	complemented
14	#14579160	C₂²	1	8	yes	complemented
15	#14579159	C₂²	1	8	yes	complemented
16	#14579153	C₂²	1	2	yes	complemented
17	#14579152	C₂²	1	4	yes	complemented
18	#14579151	C₂²	1	8	yes	complemented
19	#14579151	C₂²	1	8	yes	complemented
20	#14579141	C₂²	1	8	yes	—
21	#14579140	C₂²	1	4	yes	—
22	#14579139	C₂²	1	8	yes	—
23	#14579139	C₂²	1	8	yes	—
24	#14579138	C₂²	1	8	yes	—
25	#14579137	C₂²	1	8	yes	—
26	C₄×Dic₁₅₈	C₂²	1	8	yes	complemented
27	C₄×Dic₁₅₈	C₂²	1	4	yes	complemented
28	#14579120	C₂²	1	2	yes	—
29	#14579119	C₂²	1	4	yes	—
30	#14579118	C₂²	1	2	yes	—
31	#14579118	C₂²	1	2	yes	—
32	#14579118	C₂²	1	4	yes	—
33	#14579118	C₂²	1	4	yes	—
34	#14579117	C₂²	1	4	yes	—
35	C₂×Dic₇₉⋊C₄	C₂²	1	2	yes	—
36	C₄⋊C₄×D₇₉	C₂²	1	4	yes	—
37	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
38	C₂²⋊C₄×D₇₉	C₂²	1	4	yes	—
39	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
40	C₄²×D₇₉	C₂²	1	4	yes	—
41	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	complemented
42	#125×C₇₉	C₂²	1	2	yes	complemented
43	C₂²⋊Q₈×C₇₉	C₂²	1	4	yes	—
44	C₄×Q₈×C₇₉	C₂²	1	4	yes	complemented
45	#117×C₇₉	C₂²	1	2	yes	—
46	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
47	C₂³×D₇₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₅₈	C₂³	1	8	yes	—
49	C₂×Dic₁₅₈	C₂³	1	4	yes	—
50	C₂²×Dic₇₉	C₂³	1	2	yes	—
51	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₇₉	Q₈	1	8	yes	complemented
53	C₂×C₄×D₇₉	C₂³	1	8	yes	—
54	C₂×C₄×D₇₉	C₂³	1	2	yes	—
55	C₂×C₄×D₇₉	C₂³	1	2	yes	—
56	C₄⋊Dic₇₉	C₂³	1	4	yes	—
57	C₄⋊Dic₇₉	C₂³	1	8	yes	—
58	C₄⋊Dic₇₉	C₂³	1	4	yes	—
59	Dic₇₉⋊C₄	C₂³	1	8	yes	—
60	Dic₇₉⋊C₄	C₂³	1	4	yes	—
61	Dic₇₉⋊C₄	C₂³	1	4	yes	—
62	Dic₇₉⋊C₄	C₂³	1	8	yes	—
63	Dic₇₉⋊C₄	C₂³	1	8	yes	—
64	Dic₇₉⋊C₄	C₂³	1	8	yes	—
65	C₂²⋊Dic₇₉	C₂³	1	4	yes	—
66	(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
67	(C₂×D₇₉)⋊C₄	C₂³	1	4	yes	—
68	(C₂×D₇₉)⋊C₄	C₂³	1	4	yes	—
69	C₄×Dic₇₉	C₂³	1	8	yes	—
70	C₄×Dic₇₉	C₂³	1	4	yes	—
71	C₄×Dic₇₉	C₂³	1	2	yes	—
72	C₄×Dic₇₉	C₂³	1	2	yes	—
73	C₂×Q₈×C₇₉	C₂³	1	4	yes	—
74	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
77	C₄⋊C₄×C₇₉	C₂³	1	8	yes	—
78	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₇₉	C₂³	1	4	yes	—
80	C₄²×C₇₉	C₂³	1	2	yes	—
81	C₄²×C₇₉	C₂³	1	1	yes	characteristic
82	C₄²×C₇₉	C₂³	1	1	yes	characteristic
83	C₄²×C₇₉	C₂³	1	4	yes	—
84	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
85	C₂²×D₇₉	—	4	2	no	—
86	C₂²×D₇₉	C₂×Q₈	1	4	yes	—
87	C₂²×D₇₉	C₂⁴	1	2	yes	—
88	Q₈×C₇₉	—	4	4	no	—
89	C₂×C₄×C₇₉	—	4	2	no	—
90	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
91	C₂×C₄×C₇₉	C₂⁴	1	8	yes	—
92	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
93	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
94	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
96	C₂×Dic₇₉	—	4	2	no	—
97	C₂×Dic₇₉	C₂×Q₈	1	4	yes	—
98	C₂×Dic₇₉	C₂⁴	1	4	yes	—
99	C₂×Dic₇₉	C₂⁴	1	8	yes	—
100	C₂×Dic₇₉	C₂⁴	1	2	yes	—
101	C₄×D₇₉	—	4	8	no	—
102	C₄×D₇₉	C₂×Q₈	1	8	yes	—
103	C₄×D₇₉	C₂×Q₈	1	8	yes	—
104	Dic₁₅₈	—	4	4	no	—
105	Dic₁₅₈	—	4	8	no	—
106	C₂²×C₇₉	—	2	2	no	—
107	C₂²×C₇₉	C₂²×Q₈	1	2	yes	—
108	C₂²×C₇₉	C₂⁵	1	1	yes	characteristic
109	C₂×D₇₉	—	2	4	no	—
110	C₂×D₇₉	—	4	4	no	—
111	C₂×D₇₉	C₂²×Q₈	1	4	yes	—
112	C₄×C₇₉	—	2	2	no	—
113	C₄×C₇₉	—	2	2	no	—
114	C₄×C₇₉	—	2	8	no	—
115	C₄×C₇₉	C₂²×Q₈	1	2	yes	—
116	C₄×C₇₉	C₂²×Q₈	1	2	yes	—
117	Dic₇₉	—	2	8	no	—
118	Dic₇₉	—	2	4	no	—
119	Dic₇₉	C₂²×Q₈	1	4	yes	—
120	C₂×C₇₉	—	2	2	no	—
121	C₂×C₇₉	C₂³×Q₈	1	1	yes	characteristic
122	C₂×C₇₉	#585	1	2	yes	—
123	D₇₉	—	2	4	no	—
124	#3458	—	79	1	no	maximal, Sylow
125	C₇₉	#3458	1	1	yes	characteristic, complemented, Sylow
126	#544	—	79	4	no	—
127	#544	—	79	4	no	—
128	#544	—	79	4	no	—
129	#543	—	79	4	no	—
130	#533	—	79	8	no	—
131	#533	—	79	2	no	—
132	#533	—	79	1	no	—
133	#533	D₇₉	1	1	yes	characteristic, complemented
134	C₂×#117	—	79	2	no	—
135	C₂×#117	—	79	1	no	—
136	C₂³×C₄	—	79	1	no	—
137	C₄⋊Q₈	—	79	8	no	—
138	C₄⋊Q₈	—	79	8	no	—
139	C₄⋊Q₈	—	79	4	no	—
140	C₄⋊Q₈	—	79	2	no	—
141	C₄⋊Q₈	C₂×D₇₉	1	2	yes	complemented
142	#125	—	79	8	no	—
143	#125	—	79	8	no	—
144	#125	—	79	4	no	—
145	#125	—	79	2	no	—
146	#125	C₂×D₇₉	1	2	yes	complemented
147	C₂²⋊Q₈	—	79	8	no	—
148	C₂²⋊Q₈	—	79	4	no	—
149	C₂²⋊Q₈	—	79	8	no	—
150	C₂²⋊Q₈	—	79	8	no	—
151	C₂²⋊Q₈	—	79	8	no	—
152	C₂²⋊Q₈	—	79	8	no	—
153	C₂²⋊Q₈	C₂×D₇₉	1	4	yes	—
154	C₄×Q₈	—	79	8	no	—
155	C₄×Q₈	—	79	4	no	—
156	C₄×Q₈	C₂×D₇₉	1	4	yes	complemented
157	#117	—	79	2	no	—
158	#117	—	79	4	no	—
159	#117	—	79	4	no	—
160	#117	—	79	4	no	—
161	#117	—	79	4	no	—
162	#117	—	79	2	no	—
163	#117	—	79	2	no	—
164	#117	C₂×D₇₉	1	2	yes	—
165	C₂×C₄⋊C₄	—	79	2	no	—
166	C₂×C₄⋊C₄	—	79	4	no	—
167	C₂×C₂²⋊C₄	—	79	2	no	—
168	C₂×C₂²⋊C₄	—	79	4	no	—
169	C₂×C₄²	—	79	1	no	—
170	C₂×C₄²	—	79	4	no	—
171	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
172	C₂⁴	—	79	1	no	—
173	C₂×Q₈	—	79	8	no	—
174	C₂×Q₈	—	79	4	no	—
175	C₂×Q₈	C₂²×D₇₉	1	4	yes	—
176	C₂²×C₄	—	79	2	no	—
177	C₂²×C₄	—	79	8	no	—
178	C₂²×C₄	—	79	1	no	—
179	C₂²×C₄	—	79	8	no	—
180	C₂²×C₄	—	79	2	no	—
181	C₂²×C₄	—	79	2	no	—
182	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
183	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
184	C₄⋊C₄	—	79	8	no	—
185	C₄⋊C₄	—	79	4	no	—
186	C₄⋊C₄	—	79	4	no	—
187	C₄⋊C₄	—	79	8	no	—
188	C₄⋊C₄	—	79	8	no	—
189	C₄⋊C₄	—	79	8	no	—
190	C₄⋊C₄	—	79	4	no	—
191	C₄⋊C₄	—	79	8	no	—
192	C₄⋊C₄	—	79	4	no	—
193	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
194	C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
195	C₄⋊C₄	C₂²×D₇₉	1	8	yes	—
196	C₂²⋊C₄	—	79	4	no	—
197	C₂²⋊C₄	—	79	8	no	—
198	C₂²⋊C₄	—	79	4	no	—
199	C₂²⋊C₄	—	79	4	no	—
200	C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
201	C₄²	—	79	8	no	—
202	C₄²	—	79	2	no	—
203	C₄²	—	79	4	no	—
204	C₄²	—	79	2	no	—
205	C₄²	C₂²×D₇₉	1	1	yes	characteristic
206	C₄²	C₂²×D₇₉	1	1	yes	characteristic
207	C₄²	C₂²×D₇₉	1	4	yes	—
208	C₄²	C₂²×D₇₉	1	2	yes	—
209	C₂³	—	316	2	no	—
210	C₂³	—	79	4	no	—
211	C₂³	—	79	2	no	—
212	C₂³	C₂³×D₇₉	1	1	yes	characteristic
213	Q₈	—	4	4	no	—
214	Q₈	—	316	4	no	—
215	Q₈	—	316	8	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	316	8	no	—
218	C₂×C₄	—	316	2	no	—
219	C₂×C₄	—	79	4	no	—
220	C₂×C₄	—	79	2	no	—
221	C₂×C₄	—	79	8	no	—
222	C₂×C₄	—	79	8	no	—
223	C₂×C₄	—	79	8	no	—
224	C₂×C₄	—	79	4	no	—
225	C₂×C₄	Q₈×D₇₉	1	4	yes	—
226	C₂×C₄	C₂³×D₇₉	1	8	yes	—
227	C₂×C₄	C₂³×D₇₉	1	2	yes	—
228	C₂×C₄	C₂³×D₇₉	1	2	yes	—
229	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
230	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	316	4	no	—
233	C₂²	—	79	4	no	—
234	C₂²	—	158	4	no	—
235	C₂²	C₂×Q₈×D₇₉	1	2	yes	—
236	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	8	no	—
240	C₄	—	79	4	no	—
241	C₄	—	158	8	no	—
242	C₄	—	158	4	no	—
243	C₄	C₂×Q₈×D₇₉	1	2	yes	—
244	C₄	C₂×Q₈×D₇₉	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	—	158	4	no	—
247	C₂	C₂²×Q₈×D₇₉	1	1	yes	characteristic
248	C₂	#19099399	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁵
Frattini	C₂²	C₂⁴×D₇₉
Fitting	#533×C₇₉	C₂
prime radical	C₂³×D₇₉	C₂³
hypercenter	#533	D₇₉
hypocenter	C₇₉	#3458
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #19099193	C₂	semidirect
2. #19099192	C₂	semidirect
3. #19099190	C₂	semidirect
5. #19099120	C₂	semidirect
6. #19099117	C₂	semidirect
7. #19099116	C₂	semidirect
10. #533×C₇₉	C₂	semidirect
12. #14579161	C₂²	semidirect
13. #14579160	C₂²	semidirect
14. #14579160	C₂²	semidirect
15. #14579159	C₂²	semidirect
16. #14579153	C₂²	semidirect
17. #14579152	C₂²	semidirect
18. #14579151	C₂²	semidirect
19. #14579151	C₂²	semidirect
26. C₄×Dic₁₅₈	C₂²	semidirect
27. C₄×Dic₁₅₈	C₂²	semidirect
41. C₄⋊Q₈×C₇₉	C₂²	semidirect
42. #125×C₇₉	C₂²	semidirect
44. C₄×Q₈×C₇₉	C₂²	semidirect
52. C₂×C₄×D₇₉	Q₈	semidirect
125. C₇₉	#3458	semidirect
133. #533	D₇₉	semidirect
141. C₄⋊Q₈	C₂×D₇₉	semidirect
146. #125	C₂×D₇₉	semidirect
156. C₄×Q₈	C₂×D₇₉	semidirect