Subgroups — #34285745

Order 20224.

1286 conjugacy classes of subgroups.

The subgroup lattice has 188 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	2	yes	maximal
2	#23763149	C₂	1	4	yes	maximal, complemented
3	#23763146	C₂	1	4	yes	maximal, complemented
4	#23763094	C₂	1	4	yes	maximal
5	C₂×C₄³×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	4	yes	—
7	C₄×#11929608	C₄	1	16	yes	complemented
8	C₂×#14579072	C₂²	1	4	yes	—
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
10	#19098318	C₂²	1	16	yes	—
11	C₂²×C₄²×C₇₉	C₄	1	4	yes	—
12	C₂²×C₄²×C₇₉	C₂²	1	2	yes	—
13	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
14	C₄³×C₇₉	C₄	1	4	yes	—
15	C₄³×C₇₉	C₂²	1	2	yes	—
16	C₄³×C₇₉	C₂²	1	2	yes	—
17	C₂×#11929608	—	2	16	no	—
18	C₂×#11929608	C₂×C₄	1	16	yes	—
19	C₂²×C₇₉⋊C₈	C₂³	1	8	yes	—
20	#14579072	C₂×C₄	1	16	yes	complemented
21	C₄×C₇₉⋊C₈	C₂×C₄	1	16	yes	complemented
22	C₂³×C₄×C₇₉	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇₉	—	2	8	no	—
26	C₂×C₄²×C₇₉	—	2	16	no	—
27	C₂×C₄²×C₇₉	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₇₉	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
34	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
36	#11929608	—	2	32	no	—
37	C₂×C₇₉⋊C₈	—	2	16	no	—
38	C₂×C₇₉⋊C₈	C₂²×C₄	1	16	yes	—
39	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	—	2	8	no	—
41	C₂²×C₄×C₇₉	—	2	16	no	—
42	C₂²×C₄×C₇₉	—	2	8	no	—
43	C₂²×C₄×C₇₉	C₄²	1	8	yes	—
44	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
46	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₇₉	#41	1	8	yes	—
50	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
51	C₄²×C₇₉	—	2	8	no	—
52	C₄²×C₇₉	—	2	16	no	—
53	C₄²×C₇₉	—	2	8	no	—
54	C₄²×C₇₉	—	2	8	no	—
55	C₄²×C₇₉	C₄²	1	8	yes	—
56	C₄²×C₇₉	M₁₆	1	16	yes	complemented
57	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
58	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
59	C₂³×C₇₉	—	2	4	no	—
60	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
61	C₂³×C₇₉	#117	1	4	yes	—
62	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	32	no	—
64	C₂×C₄×C₇₉	—	2	8	no	—
65	C₂×C₄×C₇₉	—	2	4	no	—
66	C₂×C₄×C₇₉	—	2	8	no	—
67	C₂×C₄×C₇₉	—	2	16	no	—
68	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
69	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
70	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
71	C₂×C₄×C₇₉	#117	1	16	yes	—
72	C₂×C₄×C₇₉	#117	1	4	yes	—
73	C₂×C₄×C₇₉	C₂×M₁₆	1	16	yes	—
74	C₂×C₄×C₇₉	C₂×M₁₆	1	8	yes	—
75	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
77	C₇₉⋊C₈	—	2	16	no	—
78	C₂²×C₇₉	—	2	4	no	—
79	C₂²×C₇₉	—	2	8	no	—
80	C₂²×C₇₉	#376	1	4	yes	—
81	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
82	C₂²×C₇₉	C₂×#117	1	2	yes	—
83	C₂²×C₇₉	C₂²×M₁₆	1	4	yes	—
84	C₄×C₇₉	—	2	4	no	—
85	C₄×C₇₉	—	2	16	no	—
86	C₄×C₇₉	#376	1	4	yes	—
87	C₄×C₇₉	C₄×M₁₆	1	8	yes	—
88	C₄×C₇₉	C₂×#117	1	4	yes	—
89	#20364	—	79	1	no	maximal, Sylow
90	C₂×C₇₉	—	2	4	no	—
91	C₂×C₇₉	C₂×#376	1	1	yes	characteristic
92	C₂×C₇₉	C₂×C₄×M₁₆	1	2	yes	—
93	C₂×C₇₉	#2920	1	4	yes	—
94	C₂×C₄×M₁₆	—	79	2	no	—
95	C₂×C₄³	D₇₉	1	1	yes	characteristic
96	#1745	—	79	4	no	—
97	#1744	—	79	4	no	—
98	#1728	—	79	4	no	—
99	C₇₉	#20364	1	1	yes	characteristic, complemented, Sylow
100	C₂²×M₁₆	—	79	4	no	—
101	C₂²×C₄²	Dic₇₉	1	4	yes	—
102	C₂²×C₄²	C₂×D₇₉	1	2	yes	—
103	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
104	C₄×M₁₆	—	79	16	no	—
105	C₂×#97	—	79	4	no	—
106	C₂×C₄×C₈	—	79	4	no	—
107	C₄³	Dic₇₉	1	4	yes	—
108	C₄³	C₂×D₇₉	1	2	yes	—
109	C₄³	C₂×D₇₉	1	2	yes	—
110	#336	—	79	16	no	—
111	C₂³×C₄	C₂×Dic₇₉	1	4	yes	—
112	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
113	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
114	C₂×M₁₆	—	79	16	no	—
115	C₂×M₁₆	—	158	16	no	—
116	C₂²×C₈	—	79	8	no	—
117	C₂×C₄²	—	2	8	no	—
118	C₂×C₄²	—	2	16	no	—
119	C₂×C₄²	C₄×D₇₉	1	8	yes	—
120	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
121	C₂×C₄²	C₂×Dic₇₉	1	8	yes	—
122	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
123	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
124	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
125	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
126	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
128	#97	—	79	16	no	—
129	C₄×C₈	—	79	16	no	—
130	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	16	no	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	C₄×Dic₇₉	1	8	yes	—
135	C₂²×C₄	C₂×C₄×D₇₉	1	4	yes	—
136	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
138	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
139	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
140	C₂²×C₄	#11929624	1	8	yes	—
141	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
142	M₁₆	—	158	32	no	—
143	C₂×C₈	—	79	16	no	—
144	C₂×C₈	—	158	16	no	—
145	C₄²	—	2	8	no	—
146	C₄²	—	2	8	no	—
147	C₄²	—	2	16	no	—
148	C₄²	—	2	8	no	—
149	C₄²	C₄×Dic₇₉	1	8	yes	—
150	C₄²	#11929608	1	16	yes	complemented
151	C₄²	C₂×C₄×D₇₉	1	4	yes	—
152	C₄²	C₂×C₄×D₇₉	1	4	yes	—
153	C₂³	—	2	4	no	—
154	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
155	C₂³	#14579120	1	4	yes	—
156	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
157	C₂×C₄	—	2	32	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	16	no	—
162	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
163	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
164	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
165	C₂×C₄	#14579118	1	16	yes	—
166	C₂×C₄	#14579120	1	4	yes	—
167	C₂×C₄	C₂×#11929608	1	16	yes	—
168	C₂×C₄	C₂×#11929608	1	8	yes	—
169	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
170	C₂×C₄	C₂×#11929624	1	4	yes	—
171	C₈	—	158	16	no	—
172	C₂²	—	2	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	#19098409	1	4	yes	—
175	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
176	C₂²	C₂×#14579120	1	2	yes	—
177	C₂²	C₂²×#11929608	1	4	yes	—
178	C₄	—	2	16	no	—
179	C₄	—	2	4	no	—
180	C₄	#19098409	1	4	yes	—
181	C₄	C₄×#11929608	1	8	yes	—
182	C₄	C₂×#14579118	1	4	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₂×#19098409	1	1	yes	characteristic
185	C₂	C₂×C₄×#11929608	1	2	yes	—
186	C₂	#23772033	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₄³×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄³	D₇₉
hypocenter	C₇₉	#20364
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #23763149	C₂	semidirect
3. #23763146	C₂	semidirect
7. C₄×#11929608	C₄	semidirect
20. #14579072	C₂×C₄	semidirect
21. C₄×C₇₉⋊C₈	C₂×C₄	semidirect
56. C₄²×C₇₉	M₁₆	semidirect
99. C₇₉	#20364	semidirect
150. C₄²	#11929608	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
#23763149	2
#23763146	2
#23763094	2
C₂×C₄³×C₇₉	2
#20364	79