Subgroups — #34285744

Order 20224.

1286 conjugacy classes of subgroups.

The subgroup lattice has 230 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×C₄²×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23763148	C₂	1	4	yes	maximal, complemented
4	#23763145	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	C₄	1	4	yes	—
9	C₂²×C₄×Dic₇₉	C₂²	1	2	yes	—
10	C₄×#11929608	C₄	1	8	yes	complemented
11	C₂×#14579072	C₂²	1	2	yes	—
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
13	C₄²×Dic₇₉	C₄	1	4	yes	—
14	C₄²×Dic₇₉	C₂²	1	2	yes	—
15	C₄²×Dic₇₉	C₂²	1	2	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	—	2	16	no	—
28	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₇₉	C₂×C₄	1	8	yes	—
30	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₇₉	C₂³	1	4	yes	—
33	#14579072	C₂×C₄	1	8	yes	complemented
34	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	complemented
35	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₇₉	—	2	8	no	—
38	C₂×M₁₆×C₇₉	C₂×C₄	1	8	yes	—
39	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
40	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
45	#97×C₇₉	C₂×C₄	1	8	yes	complemented
46	C₄×C₈×C₇₉	C₂×C₄	1	8	yes	complemented
47	C₂²×Dic₇₉	—	2	16	no	—
48	C₂²×Dic₇₉	—	2	8	no	—
49	C₂²×Dic₇₉	C₂²×C₄	1	8	yes	—
50	#11929608	—	2	16	no	—
51	C₂×C₇₉⋊C₈	—	2	8	no	—
52	C₂×C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
53	C₄×Dic₇₉	—	2	16	no	—
54	C₄×Dic₇₉	—	2	8	no	—
55	C₄×Dic₇₉	—	2	8	no	—
56	C₄×Dic₇₉	—	2	8	no	—
57	C₄×Dic₇₉	M₁₆	1	16	yes	complemented
58	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	—	2	8	no	—
60	C₂²×C₄×C₇₉	C₄²	1	8	yes	—
61	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₇₉	#41	1	8	yes	—
67	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₇₉	—	2	16	no	—
69	C₂×C₈×C₇₉	—	2	8	no	—
70	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
71	C₄²×C₇₉	C₄²	1	8	yes	—
72	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
73	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
74	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
75	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
76	C₂³×C₇₉	—	2	4	no	—
77	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
78	C₂³×C₇₉	#117	1	4	yes	—
79	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₉	—	2	4	no	—
81	C₂×C₄×C₇₉	—	2	16	no	—
82	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
85	C₂×C₄×C₇₉	#117	1	8	yes	—
86	C₂×C₄×C₇₉	#117	1	4	yes	—
87	C₂×C₄×C₇₉	#117	1	8	yes	—
88	C₂×C₄×C₇₉	C₂×M₁₆	1	8	yes	—
89	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
92	C₂×Dic₇₉	—	2	32	no	—
93	C₂×Dic₇₉	—	2	8	no	—
94	C₂×Dic₇₉	—	2	8	no	—
95	C₂×Dic₇₉	C₂×M₁₆	1	16	yes	—
96	C₈×C₇₉	—	2	8	no	—
97	C₇₉⋊C₈	—	2	8	no	—
98	C₂²×C₇₉	—	2	4	no	—
99	C₂²×C₇₉	—	2	8	no	—
100	C₂²×C₇₉	#376	1	4	yes	—
101	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
102	C₂²×C₇₉	C₂×#117	1	2	yes	—
103	C₂²×C₇₉	C₂²×M₁₆	1	4	yes	—
104	C₄×C₇₉	—	2	4	no	—
105	C₄×C₇₉	#376	1	4	yes	—
106	C₄×C₇₉	C₄×M₁₆	1	8	yes	—
107	C₄×C₇₉	C₂×#117	1	2	yes	—
108	C₄×C₇₉	C₂×#117	1	2	yes	—
109	Dic₇₉	—	2	16	no	—
110	#20364	—	79	1	no	maximal, Sylow
111	C₂×C₇₉	—	2	4	no	—
112	C₂×C₇₉	C₂×#376	1	1	yes	characteristic
113	C₂×C₇₉	C₂×C₄×M₁₆	1	2	yes	—
114	C₂×C₇₉	#2920	1	4	yes	—
115	C₂×C₄×M₁₆	—	79	1	no	—
116	C₂×C₄×M₁₆	D₇₉	1	1	yes	characteristic
117	C₂×C₄³	—	79	1	no	—
118	#1745	—	79	4	no	—
119	#1744	—	79	4	no	—
120	#1728	—	79	4	no	—
121	C₇₉	#20364	1	1	yes	characteristic, complemented, Sylow
122	C₂²×M₁₆	—	79	2	no	—
123	C₂²×M₁₆	C₂×D₇₉	1	2	yes	—
124	C₂²×C₄²	—	79	2	no	—
125	C₂²×C₄²	—	79	4	no	—
126	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
127	C₄×M₁₆	—	79	8	no	—
128	C₄×M₁₆	Dic₇₉	1	8	yes	complemented
129	C₂×#97	—	79	2	no	—
130	C₂×#97	C₂×D₇₉	1	2	yes	—
131	C₂×C₄×C₈	—	79	2	no	—
132	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
133	C₄³	—	79	2	no	—
134	C₄³	—	79	2	no	—
135	C₄³	—	79	4	no	—
136	#336	—	79	16	no	—
137	C₂³×C₄	—	79	4	no	—
138	C₂³×C₄	C₄×D₇₉	1	2	yes	—
139	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
140	C₂×M₁₆	—	2	8	no	—
141	C₂×M₁₆	—	79	8	no	—
142	C₂×M₁₆	—	158	8	no	—
143	C₂×M₁₆	C₂×Dic₇₉	1	8	yes	—
144	C₂²×C₈	—	79	4	no	—
145	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
146	C₂×C₄²	—	79	4	no	—
147	C₂×C₄²	—	79	8	no	—
148	C₂×C₄²	—	79	4	no	—
149	C₂×C₄²	—	79	4	no	—
150	C₂×C₄²	—	79	4	no	—
151	C₂×C₄²	—	158	16	no	—
152	C₂×C₄²	—	158	8	no	—
153	C₂×C₄²	C₄×D₇₉	1	2	yes	—
154	C₂×C₄²	C₄×D₇₉	1	4	yes	—
155	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
156	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
158	#97	—	79	8	no	—
159	#97	C₂×Dic₇₉	1	8	yes	complemented
160	C₄×C₈	—	79	8	no	—
161	C₄×C₈	C₂×Dic₇₉	1	8	yes	complemented
162	C₂⁴	C₂×C₄×D₇₉	1	1	yes	characteristic
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	79	8	no	—
165	C₂²×C₄	—	158	16	no	—
166	C₂²×C₄	—	158	8	no	—
167	C₂²×C₄	C₄×Dic₇₉	1	8	yes	—
168	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
173	C₂²×C₄	#11929624	1	8	yes	—
174	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
175	M₁₆	—	2	16	no	—
176	M₁₆	—	158	16	no	—
177	C₂×C₈	—	2	8	no	—
178	C₂×C₈	—	79	8	no	—
179	C₂×C₈	—	158	8	no	—
180	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
181	C₄²	—	79	16	no	—
182	C₄²	—	158	16	no	—
183	C₄²	—	158	8	no	—
184	C₄²	—	158	8	no	—
185	C₄²	—	158	8	no	—
186	C₄²	C₄×Dic₇₉	1	8	yes	—
187	C₄²	C₂×C₄×D₇₉	1	2	yes	—
188	C₄²	C₂×C₄×D₇₉	1	2	yes	—
189	C₄²	C₂²×Dic₇₉	1	2	yes	—
190	C₄²	C₂²×Dic₇₉	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
193	C₂³	#14579118	1	4	yes	—
194	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	79	16	no	—
198	C₂×C₄	—	158	32	no	—
199	C₂×C₄	—	158	8	no	—
200	C₂×C₄	—	158	8	no	—
201	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
202	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
203	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
204	C₂×C₄	#14579118	1	4	yes	—
205	C₂×C₄	#14579118	1	8	yes	—
206	C₂×C₄	#14579120	1	8	yes	—
207	C₂×C₄	M₁₆×D₇₉	1	8	yes	—
208	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
209	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
210	C₂×C₄	C₂×#11929624	1	4	yes	—
211	C₈	—	2	8	no	—
212	C₈	—	158	8	no	—
213	C₂²	—	2	8	no	—
214	C₂²	—	2	4	no	—
215	C₂²	#19098409	1	4	yes	—
216	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
217	C₂²	C₂×#14579118	1	2	yes	—
218	C₂²	C₂×M₁₆×D₇₉	1	4	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	158	16	no	—
221	C₄	#19098409	1	4	yes	—
222	C₄	M₁₆×Dic₇₉	1	8	yes	—
223	C₄	C₂×#14579118	1	2	yes	—
224	C₄	C₂×#14579120	1	2	yes	—
225	C₂	—	2	4	no	—
226	C₂	C₂×#19098409	1	1	yes	characteristic
227	C₂	C₂×M₁₆×Dic₇₉	1	2	yes	—
228	C₂	#23772031	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
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	#20364
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #23763148	C₂	semidirect
4. #23763145	C₂	semidirect
10. C₄×#11929608	C₄	semidirect
19. C₄×M₁₆×C₇₉	C₄	semidirect
33. #14579072	C₂×C₄	semidirect
34. C₄×C₇₉⋊C₈	C₂×C₄	semidirect
45. #97×C₇₉	C₂×C₄	semidirect
46. C₄×C₈×C₇₉	C₂×C₄	semidirect
57. C₄×Dic₇₉	M₁₆	semidirect
121. C₇₉	#20364	semidirect
128. C₄×M₁₆	Dic₇₉	semidirect
159. #97	C₂×Dic₇₉	semidirect
161. 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×C₄²×Dic₇₉	2
#23763148	2
#23763145	2
#23763093	2
C₂×C₄×M₁₆×C₇₉	2
#20364	79