Subgroups — #23777098

Order 10112.

1052 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19099246	C₂	1	8	yes	maximal, complemented
2	#19099237	C₂	1	8	yes	maximal, complemented
3	#19099189	C₂	1	1	yes	maximal, characteristic, complemented
4	#19099160	C₂	1	4	yes	maximal, complemented
5	#19099125	C₂	1	1	yes	maximal, characteristic
6	#19099097	C₂	1	4	yes	maximal, complemented
7	C₂×#14579144	C₂	1	2	yes	maximal, complemented
8	#19099036	C₂	1	2	yes	maximal, complemented
9	C₂×#117×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₂²⋊D₇₉	C₂²	1	2	yes	—
11	#14579156	C₂²	1	8	yes	complemented
12	#14579155	C₂²	1	8	yes	complemented
13	#14579153	C₂²	1	4	yes	complemented
14	#14579151	C₂²	1	8	yes	complemented
15	#14579149	C₂²	1	4	yes	complemented
16	#14579146	C₂²	1	8	yes	complemented
17	#14579145	C₂²	1	4	yes	—
18	#14579144	C₂²	1	8	yes	—
19	#14579144	C₂²	1	4	yes	complemented
20	#14579144	C₂²	1	4	yes	complemented
21	#14579140	C₂²	1	4	yes	complemented
22	#14579139	C₂²	1	8	yes	complemented
23	#14579138	C₂²	1	8	yes	complemented
24	#14579135	C₂²	1	4	yes	—
25	#14579131	C₂²	1	4	yes	—
26	C₄×Dic₁₅₈	C₂²	1	4	yes	complemented
27	C₄×D₃₁₆	C₂²	1	4	yes	complemented
28	C₄×C₂²⋊D₇₉	C₂²	1	8	yes	complemented
29	(C₂²⋊D₇₉)⋊C₄	C₂²	1	8	yes	—
30	#14579118	C₂²	1	8	yes	complemented
31	#14579117	C₂²	1	4	yes	complemented
32	C₂×Dic₇₉⋊C₄	C₂²	1	4	yes	complemented
33	C₂×C₂²⋊Dic₇₉	C₂²	1	2	yes	—
34	C₂×(C₂×D₇₉)⋊C₄	C₂²	1	4	yes	complemented
35	C₂²⋊C₄×D₇₉	C₂²	1	4	yes	complemented
36	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
37	#117×C₇₉	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	C₂²	1	2	yes	complemented
42	C₂³×D₇₉	C₂³	1	2	yes	—
43	C₂×Dic₁₅₈	C₂³	1	4	yes	—
44	C₂×D₃₁₆	C₂³	1	4	yes	—
45	C₂×C₂²⋊D₇₉	—	2	8	no	—
46	C₂×C₂²⋊D₇₉	C₂³	1	4	yes	—
47	C₂×C₂²⋊D₇₉	C₂³	1	4	yes	—
48	C₂²×Dic₇₉	C₂³	1	2	yes	—
49	C₂²×Dic₇₉	C₂³	1	4	yes	—
50	C₂×C₄×D₇₉	C₂³	1	8	yes	—
51	C₄⋊Dic₇₉	C₂³	1	4	yes	—
52	C₄⋊Dic₇₉	C₂³	1	4	yes	—
53	Dic₇₉⋊C₄	C₂³	1	8	yes	—
54	Dic₇₉⋊C₄	C₂³	1	8	yes	—
55	Dic₇₉⋊C₄	C₂³	1	8	yes	—
56	Dic₇₉⋊C₄	C₂³	1	8	yes	—
57	C₂²⋊Dic₇₉	C₂³	1	4	yes	—
58	C₂²⋊Dic₇₉	C₂³	1	4	yes	—
59	(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
60	(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
61	(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
62	(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
63	C₄×Dic₇₉	C₂³	1	8	yes	—
64	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	—	2	4	no	—
66	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂³	1	4	yes	—
69	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
70	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
71	C₄⋊C₄×C₇₉	C₂³	1	8	yes	—
72	C₂²⋊C₄×C₇₉	C₂³	1	4	yes	—
73	C₂²⋊C₄×C₇₉	C₂³	1	4	yes	—
74	C₄²×C₇₉	C₂³	1	4	yes	—
75	C₄²×C₇₉	C₂³	1	4	yes	—
76	C₂³×C₇₉	—	4	2	no	—
77	C₂³×C₇₉	#41	1	4	yes	—
78	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
79	C₂³×C₇₉	C₂⁴	1	2	yes	—
80	C₂²×D₇₉	—	2	4	no	—
81	C₂²×D₇₉	—	4	4	no	—
82	C₂²×D₇₉	C₂⁴	1	4	yes	—
83	C₂×C₄×C₇₉	—	2	2	no	—
84	C₂×C₄×C₇₉	—	2	2	no	—
85	C₂×C₄×C₇₉	—	2	8	no	—
86	C₂×C₄×C₇₉	—	4	2	no	—
87	C₂×C₄×C₇₉	—	4	2	no	—
88	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
89	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
90	C₂×C₄×C₇₉	C₂⁴	1	2	yes	—
91	C₂×C₄×C₇₉	C₂⁴	1	4	yes	—
92	C₂²⋊D₇₉	—	2	8	no	—
93	C₂²⋊D₇₉	—	2	8	no	—
94	C₂×Dic₇₉	—	2	4	no	—
95	C₂×Dic₇₉	—	4	4	no	—
96	C₂×Dic₇₉	C₂⁴	1	4	yes	—
97	C₂×Dic₇₉	C₂⁴	1	8	yes	—
98	D₃₁₆	—	4	4	no	—
99	C₄×D₇₉	—	4	8	no	—
100	Dic₁₅₈	—	4	4	no	—
101	C₂²×C₇₉	—	2	2	no	—
102	C₂²×C₇₉	—	2	4	no	—
103	C₂²×C₇₉	—	4	4	no	—
104	C₂²×C₇₉	C₂×#41	1	2	yes	—
105	C₂²×C₇₉	C₂×#41	1	4	yes	—
106	C₂²×C₇₉	C₂⁵	1	1	yes	characteristic
107	C₂×D₇₉	—	2	4	no	—
108	C₂×D₇₉	—	2	8	no	—
109	C₂×D₇₉	—	8	4	no	—
110	C₄×C₇₉	—	2	2	no	—
111	C₄×C₇₉	—	2	2	no	—
112	C₄×C₇₉	—	2	4	no	—
113	C₄×C₇₉	—	2	4	no	—
114	Dic₇₉	—	2	4	no	—
115	Dic₇₉	—	2	8	no	—
116	C₂×C₇₉	—	2	4	no	—
117	C₂×C₇₉	—	2	2	no	—
118	C₂×C₇₉	C₂²×#41	1	1	yes	characteristic
119	C₂×C₇₉	#585	1	2	yes	—
120	D₇₉	—	4	4	no	—
121	#3491	—	79	1	no	maximal, Sylow
122	C₇₉	#3491	1	1	yes	characteristic, complemented, Sylow
123	#552	—	79	8	no	—
124	#551	—	79	8	no	—
125	#543	—	79	1	no	—
126	#539	—	79	4	no	—
127	#534	—	79	1	no	—
128	#529	—	79	4	no	—
129	C₂×#123	—	79	2	no	—
130	#518	—	79	2	no	—
131	C₂×#117	D₇₉	1	1	yes	characteristic, complemented
132	C₂²×D₄	—	79	2	no	—
133	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
134	#126	—	79	8	no	—
135	#126	—	79	8	no	—
136	#125	—	79	8	no	—
137	#125	—	79	4	no	—
138	#124	—	79	4	no	—
139	#123	—	79	4	no	—
140	#123	—	79	4	no	—
141	#123	—	79	8	no	—
142	#123	—	79	8	no	—
143	#123	—	79	4	no	—
144	C₂²⋊Q₈	—	79	4	no	—
145	C₂²⋊Q₈	—	79	8	no	—
146	C₂²⋊Q₈	—	79	8	no	—
147	#121	—	79	4	no	—
148	#120	—	79	4	no	—
149	C₄×Q₈	—	79	4	no	—
150	C₄×D₄	—	79	8	no	—
151	C₄×D₄	—	79	8	no	—
152	C₄×D₄	—	79	4	no	—
153	#117	—	79	4	no	—
154	#117	—	79	8	no	—
155	#117	C₂×D₇₉	1	8	yes	complemented
156	C₂×C₄⋊C₄	—	79	4	no	—
157	C₂×C₄⋊C₄	C₂×D₇₉	1	2	yes	complemented
158	C₂×C₂²⋊C₄	—	79	2	no	—
159	C₂×C₂²⋊C₄	—	79	4	no	—
160	C₂×C₂²⋊C₄	—	79	4	no	—
161	C₂×C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
163	C₂×C₄²	C₂×D₇₉	1	2	yes	complemented
164	C₂⁴	—	79	2	no	—
165	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
166	C₂×Q₈	—	79	4	no	—
167	C₂×D₄	—	79	4	no	—
168	C₂×D₄	—	79	4	no	—
169	C₂×D₄	—	79	4	no	—
170	C₂×D₄	—	158	8	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	79	4	no	—
173	C₂²×C₄	—	79	8	no	—
174	C₂²×C₄	—	79	2	no	—
175	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₇₉	1	4	yes	—
177	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
179	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
180	C₄⋊C₄	—	79	8	no	—
181	C₄⋊C₄	—	79	8	no	—
182	C₄⋊C₄	—	79	8	no	—
183	C₄⋊C₄	—	79	8	no	—
184	C₄⋊C₄	—	79	4	no	—
185	C₄⋊C₄	—	79	4	no	—
186	C₄⋊C₄	C₂²×D₇₉	1	8	yes	—
187	C₂²⋊C₄	—	79	4	no	—
188	C₂²⋊C₄	—	79	4	no	—
189	C₂²⋊C₄	—	79	8	no	—
190	C₂²⋊C₄	—	79	8	no	—
191	C₂²⋊C₄	—	79	8	no	—
192	C₂²⋊C₄	—	79	8	no	—
193	C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
194	C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
195	C₄²	—	79	8	no	—
196	C₄²	C₂²×D₇₉	1	4	yes	—
197	C₄²	C₂²×D₇₉	1	4	yes	—
198	C₂³	—	4	2	no	—
199	C₂³	—	316	4	no	—
200	C₂³	—	79	4	no	—
201	C₂³	—	158	4	no	—
202	C₂³	#11929624	1	4	yes	—
203	C₂³	C₂³×D₇₉	1	1	yes	characteristic
204	C₂³	C₂³×D₇₉	1	2	yes	—
205	Q₈	—	316	4	no	—
206	D₄	—	316	4	no	—
207	D₄	—	158	8	no	—
208	D₄	—	158	8	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	316	8	no	—
215	C₂×C₄	—	316	4	no	—
216	C₂×C₄	—	79	4	no	—
217	C₂×C₄	—	79	8	no	—
218	C₂×C₄	—	158	4	no	—
219	C₂×C₄	C₂³×D₇₉	1	4	yes	—
220	C₂×C₄	C₂³×D₇₉	1	2	yes	—
221	C₂×C₄	C₂³×D₇₉	1	4	yes	—
222	C₂×C₄	C₂³×D₇₉	1	2	yes	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	4	4	no	—
226	C₂²	—	632	4	no	—
227	C₂²	—	158	8	no	—
228	C₂²	—	158	4	no	—
229	C₂²	C₂×#11929624	1	4	yes	—
230	C₂²	C₂×#11929624	1	2	yes	—
231	C₂²	C₂⁴×D₇₉	1	1	yes	characteristic
232	C₄	—	2	4	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	158	8	no	—
237	C₄	—	158	4	no	—
238	C₂	—	2	4	no	—
239	C₂	—	2	2	no	—
240	C₂	—	316	4	no	—
241	C₂	C₂²×#11929624	1	1	yes	characteristic
242	C₂	#19099399	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁵
Fitting	C₂×#117×C₇₉	C₂
prime radical	#19099125	C₂
hypercenter	C₂×#117	D₇₉
hypocenter	C₇₉	#3491
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #19099246	C₂	semidirect
2. #19099237	C₂	semidirect
3. #19099189	C₂	semidirect
4. #19099160	C₂	semidirect
6. #19099097	C₂	semidirect
7. C₂×#14579144	C₂	semidirect
8. #19099036	C₂	semidirect
9. C₂×#117×C₇₉	C₂	semidirect
11. #14579156	C₂²	semidirect
12. #14579155	C₂²	semidirect
13. #14579153	C₂²	semidirect
14. #14579151	C₂²	semidirect
15. #14579149	C₂²	semidirect
16. #14579146	C₂²	semidirect
19. #14579144	C₂²	semidirect
20. #14579144	C₂²	semidirect
21. #14579140	C₂²	semidirect
22. #14579139	C₂²	semidirect
23. #14579138	C₂²	semidirect
26. C₄×Dic₁₅₈	C₂²	semidirect
27. C₄×D₃₁₆	C₂²	semidirect
28. C₄×C₂²⋊D₇₉	C₂²	semidirect
30. #14579118	C₂²	semidirect
31. #14579117	C₂²	semidirect
32. C₂×Dic₇₉⋊C₄	C₂²	semidirect
34. C₂×(C₂×D₇₉)⋊C₄	C₂²	semidirect
35. C₂²⋊C₄×D₇₉	C₂²	semidirect
37. #117×C₇₉	C₂²	semidirect
38. C₂×C₄⋊C₄×C₇₉	C₂²	semidirect
41. C₂×C₄²×C₇₉	C₂²	semidirect
122. C₇₉	#3491	semidirect
131. C₂×#117	D₇₉	semidirect
155. #117	C₂×D₇₉	semidirect
157. C₂×C₄⋊C₄	C₂×D₇₉	semidirect
163. C₂×C₄²	C₂×D₇₉	semidirect