Subgroups — #34285748

Order 20224.

1054 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	C₂×C₄×#11929608	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098409	C₂	1	1	yes	maximal, characteristic
3	#23763153	C₂	1	4	yes	maximal, complemented
4	#23763151	C₂	1	4	yes	maximal, complemented
5	#23763093	C₂	1	4	yes	maximal
6	C₂×C₄×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11929608	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₇₉	C₂²	1	2	yes	—
9	C₄×#11929608	C₄	1	8	yes	complemented
10	C₂×#14579072	C₂²	1	2	yes	—
11	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
12	#19098409	C₄	1	4	yes	—
13	#19098409	C₂²	1	2	yes	—
14	#19098409	C₂²	1	2	yes	—
15	C₂×C₇₉⋊#95	C₄	1	4	yes	—
16	#19098317	C₂²	1	16	yes	—
17	C₂²×M₁₆×C₇₉	C₂²	1	2	yes	—
18	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₇₉	C₄	1	8	yes	complemented
20	C₂×#97×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
22	C₂³×Dic₇₉	C₂×C₄	1	4	yes	—
23	C₂×#11929608	—	2	8	no	—
24	C₂×#11929608	C₂×C₄	1	8	yes	—
25	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
26	C₂×C₄×Dic₇₉	—	2	8	no	—
27	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
28	C₂×C₄×Dic₇₉	C₂³	1	4	yes	—
29	#14579072	C₂×C₄	1	8	yes	complemented
30	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	complemented
31	C₇₉⋊#95	C₂×C₄	1	8	yes	—
32	C₇₉⋊#95	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₇₉	—	2	8	no	—
36	C₂×M₁₆×C₇₉	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
38	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
43	#97×C₇₉	C₂×C₄	1	8	yes	complemented
44	C₄×C₈×C₇₉	C₂×C₄	1	8	yes	complemented
45	C₂²×Dic₇₉	—	2	8	no	—
46	C₂²×Dic₇₉	—	4	8	no	—
47	C₂²×Dic₇₉	C₂²×C₄	1	8	yes	—
48	#11929608	—	2	16	no	—
49	C₂×C₇₉⋊C₈	—	2	8	no	—
50	C₂×C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
51	C₄×Dic₇₉	—	2	8	no	—
52	C₄×Dic₇₉	—	2	8	no	—
53	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	—	4	4	no	—
55	C₂²×C₄×C₇₉	C₄²	1	8	yes	—
56	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₇₉	#41	1	8	yes	—
62	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₇₉	—	2	16	no	—
64	C₂×C₈×C₇₉	—	2	8	no	—
65	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
66	C₄²×C₇₉	C₄²	1	8	yes	—
67	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
68	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
69	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
70	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
71	C₂³×C₇₉	—	2	4	no	—
72	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
73	C₂³×C₇₉	#117	1	4	yes	—
74	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	—	2	4	no	—
76	C₂×C₄×C₇₉	—	2	4	no	—
77	C₂×C₄×C₇₉	—	4	8	no	—
78	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
79	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
80	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₇₉	#117	1	8	yes	—
82	C₂×C₄×C₇₉	#117	1	4	yes	—
83	C₂×C₄×C₇₉	#117	1	8	yes	—
84	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
87	C₂×Dic₇₉	—	2	8	no	—
88	C₂×Dic₇₉	—	2	16	no	—
89	C₂×Dic₇₉	—	4	16	no	—
90	C₈×C₇₉	—	2	8	no	—
91	C₇₉⋊C₈	—	2	8	no	—
92	C₂²×C₇₉	—	2	4	no	—
93	C₂²×C₇₉	—	2	8	no	—
94	C₂²×C₇₉	#376	1	4	yes	—
95	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
96	C₂²×C₇₉	C₂×#117	1	2	yes	—
97	C₂²×C₇₉	#568	1	4	yes	—
98	C₄×C₇₉	—	2	4	no	—
99	C₄×C₇₉	—	2	4	no	—
100	C₄×C₇₉	#376	1	4	yes	—
101	C₄×C₇₉	C₂×#117	1	2	yes	—
102	C₄×C₇₉	C₂×#117	1	2	yes	—
103	Dic₇₉	—	4	8	no	—
104	#20365	—	79	1	no	maximal, Sylow
105	C₂×C₇₉	—	2	4	no	—
106	C₂×C₇₉	C₂×#376	1	1	yes	characteristic
107	C₂×C₇₉	#2873	1	2	yes	—
108	C₂×C₇₉	#2921	1	4	yes	—
109	C₂×C₄×M₁₆	—	79	1	no	—
110	C₂×C₄×M₁₆	D₇₉	1	1	yes	characteristic
111	C₂×#376	—	79	1	no	—
112	#1746	—	79	4	no	—
113	#1746	—	79	4	no	—
114	#1728	—	79	4	no	—
115	C₇₉	#20365	1	1	yes	characteristic, complemented, Sylow
116	C₂²×M₁₆	—	79	2	no	—
117	C₂²×M₁₆	C₂×D₇₉	1	2	yes	—
118	C₂²×C₄²	—	79	2	no	—
119	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
120	C₄×M₁₆	—	79	8	no	—
121	C₄×M₁₆	Dic₇₉	1	8	yes	complemented
122	C₂×#97	—	79	2	no	—
123	C₂×#97	C₂×D₇₉	1	2	yes	—
124	C₂×C₄×C₈	—	79	2	no	—
125	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
126	#376	—	79	2	no	—
127	#376	—	79	2	no	—
128	#376	—	79	4	no	—
129	C₂×#95	—	79	4	no	—
130	#336	—	79	16	no	—
131	C₂³×C₄	—	79	4	no	—
132	C₂³×C₄	C₄×D₇₉	1	2	yes	—
133	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
134	C₂×M₁₆	—	2	8	no	—
135	C₂×M₁₆	—	79	8	no	—
136	C₂×M₁₆	—	158	8	no	—
137	C₂×M₁₆	C₂×Dic₇₉	1	8	yes	—
138	C₂²×C₈	—	79	4	no	—
139	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
140	C₂×C₄²	—	79	4	no	—
141	C₂×C₄²	—	79	4	no	—
142	C₂×C₄²	—	158	8	no	—
143	C₂×C₄²	C₄×D₇₉	1	2	yes	—
144	C₂×C₄²	C₄×D₇₉	1	4	yes	—
145	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
146	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
148	#97	—	79	8	no	—
149	#97	C₂×Dic₇₉	1	8	yes	complemented
150	C₄×C₈	—	79	8	no	—
151	C₄×C₈	C₂×Dic₇₉	1	8	yes	complemented
152	#95	—	79	8	no	—
153	#95	—	79	8	no	—
154	C₂⁴	C₂×C₄×D₇₉	1	1	yes	characteristic
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	316	8	no	—
157	C₂²×C₄	—	79	8	no	—
158	C₂²×C₄	—	158	8	no	—
159	C₂²×C₄	C₄×Dic₇₉	1	8	yes	—
160	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
161	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
162	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
164	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
165	C₂²×C₄	#11929624	1	8	yes	—
166	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
167	M₁₆	—	2	16	no	—
168	M₁₆	—	158	16	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	79	8	no	—
171	C₂×C₈	—	158	8	no	—
172	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
173	C₄²	—	158	8	no	—
174	C₄²	—	158	8	no	—
175	C₄²	C₄×Dic₇₉	1	8	yes	—
176	C₄²	C₂×C₄×D₇₉	1	2	yes	—
177	C₄²	C₂×C₄×D₇₉	1	2	yes	—
178	C₄²	C₂²×Dic₇₉	1	2	yes	—
179	C₄²	C₂²×Dic₇₉	1	2	yes	—
180	C₂³	—	2	4	no	—
181	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
182	C₂³	#14579118	1	4	yes	—
183	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	—	316	16	no	—
188	C₂×C₄	—	158	8	no	—
189	C₂×C₄	—	158	16	no	—
190	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
191	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
192	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
193	C₂×C₄	#14579118	1	4	yes	—
194	C₂×C₄	#14579118	1	8	yes	—
195	C₂×C₄	#14579120	1	8	yes	—
196	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
197	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
198	C₂×C₄	C₂×#11929624	1	4	yes	—
199	C₈	—	2	8	no	—
200	C₈	—	158	8	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	#19098409	1	4	yes	—
204	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
205	C₂²	C₂×#14579118	1	2	yes	—
206	C₂²	#19099329	1	4	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	316	8	no	—
210	C₄	#19098409	1	4	yes	—
211	C₄	C₂×#14579118	1	2	yes	—
212	C₄	C₂×#14579120	1	2	yes	—
213	C₂	—	2	4	no	—
214	C₂	C₂×#19098409	1	1	yes	characteristic
215	C₂	#23771818	1	2	yes	—
216	C₂	#23772036	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Fitting	C₂×C₄×M₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₇₉
hypocenter	C₇₉	#20365
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #23763153	C₂	semidirect
4. #23763151	C₂	semidirect
9. C₄×#11929608	C₄	semidirect
19. C₄×M₁₆×C₇₉	C₄	semidirect
29. #14579072	C₂×C₄	semidirect
30. C₄×C₇₉⋊C₈	C₂×C₄	semidirect
43. #97×C₇₉	C₂×C₄	semidirect
44. C₄×C₈×C₇₉	C₂×C₄	semidirect
115. C₇₉	#20365	semidirect
121. C₄×M₁₆	Dic₇₉	semidirect
149. #97	C₂×Dic₇₉	semidirect
151. C₄×C₈	C₂×Dic₇₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×#11929608	2
C₂×#19098409	2
#23763153	2
#23763151	2
#23763093	2
C₂×C₄×M₁₆×C₇₉	2
#20365	79