Subgroups — #34285694

Order 20224.

1978 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×#19099330	C₂	1	2	yes	maximal
2	#23763127	C₂	1	6	yes	maximal, complemented
3	C₂×#19098328	C₂	1	6	yes	maximal, complemented
4	C₂²×#131×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#19099330	—	2	8	no	—
6	#19099330	C₄	1	16	yes	—
7	C₂×#14579168	—	2	8	no	—
8	C₂²×#11929608	C₂²	1	12	yes	—
9	#19098328	C₂²	1	4	yes	—
10	#19098328	C₂²	1	12	yes	—
11	C₂²×#41×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×#131×C₇₉	—	2	8	no	—
13	C₂×#131×C₇₉	C₄	1	8	yes	—
14	C₂²×M₁₆×C₇₉	C₂²	1	3	yes	—
15	C₂³×C₈×C₇₉	C₂²	1	3	yes	—
16	#14579168	—	2	64	no	—
17	C₂×#11929608	—	2	12	no	—
18	C₂×#11929608	—	2	24	no	—
19	C₂×#11929608	C₂×C₄	1	48	yes	—
20	C₂×#11929608	C₂³	1	8	yes	—
21	C₂²×C₇₉⋊C₈	—	2	12	no	—
22	C₂×#41×C₇₉	C₂×C₄	1	8	yes	—
23	C₂×#41×C₇₉	C₂×C₄	1	4	yes	—
24	C₂×#41×C₇₉	D₄	1	8	yes	—
25	C₂×#41×C₇₉	D₄	1	4	yes	—
26	C₂²×Q₈×C₇₉	Q₈	1	1	yes	characteristic
27	C₂²×D₄×C₇₉	Q₈	1	3	yes	—
28	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
29	#131×C₇₉	—	2	32	no	—
30	C₂×M₁₆×C₇₉	—	2	12	no	—
31	C₂×M₁₆×C₇₉	C₂×C₄	1	12	yes	—
32	C₂²×C₈×C₇₉	—	2	12	no	—
33	C₂²×C₈×C₇₉	C₂×C₄	1	12	yes	—
34	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	C₂³	1	3	yes	—
36	#11929608	—	2	48	no	—
37	#11929608	—	2	8	no	—
38	#11929608	C₂²×C₄	1	16	yes	—
39	C₂×C₇₉⋊C₈	—	2	48	no	—
40	C₂×C₇₉⋊C₈	—	2	8	no	—
41	C₂⁴×C₇₉	C₂×Q₈	1	3	yes	—
42	#41×C₇₉	C₄²	1	16	yes	—
43	#41×C₇₉	C₂²⋊C₄	1	32	yes	—
44	#41×C₇₉	C₂²⋊C₄	1	16	yes	—
45	C₂×Q₈×C₇₉	C₄⋊C₄	1	8	yes	—
46	C₂×Q₈×C₇₉	C₄⋊C₄	1	4	yes	—
47	C₂×D₄×C₇₉	C₄⋊C₄	1	24	yes	—
48	C₂×D₄×C₇₉	C₄⋊C₄	1	12	yes	—
49	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
50	C₂²×C₄×C₇₉	C₂²×C₄	1	6	yes	—
51	C₂²×C₄×C₇₉	C₂×D₄	1	12	yes	—
52	C₂²×C₄×C₇₉	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₇₉	C₂×Q₈	1	3	yes	—
54	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
55	M₁₆×C₇₉	—	2	24	no	—
56	C₂×C₈×C₇₉	—	2	24	no	—
57	C₂×C₈×C₇₉	—	2	2	no	—
58	C₂×C₈×C₇₉	—	2	6	no	—
59	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₇₉	C₂²×C₄	1	6	yes	—
61	C₂³×C₇₉	—	4	6	no	—
62	C₂³×C₇₉	C₂×C₄⋊C₄	1	12	yes	—
63	C₂³×C₇₉	C₂×C₄⋊C₄	1	6	yes	—
64	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
65	Q₈×C₇₉	#95	1	16	yes	—
66	D₄×C₇₉	#95	1	48	yes	—
67	C₂×C₄×C₇₉	C₂×C₄²	1	12	yes	—
68	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	24	yes	—
69	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	12	yes	—
70	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	12	yes	—
71	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	6	yes	—
72	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
73	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
75	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
76	C₈×C₇₉	—	2	2	no	—
77	C₈×C₇₉	—	2	6	no	—
78	C₇₉⋊C₈	—	2	16	no	—
79	C₂²×C₇₉	—	2	2	no	—
80	C₂²×C₇₉	—	2	12	no	—
81	C₂²×C₇₉	—	4	12	no	—
82	C₂²×C₇₉	C₂×#95	1	12	yes	—
83	C₂²×C₇₉	C₂²×C₄⋊C₄	1	2	yes	—
84	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₇₉	C₂×#95	1	12	yes	—
86	C₄×C₇₉	C₂²×C₄²	1	1	yes	characteristic
87	C₄×C₇₉	C₂²×C₂²⋊C₄	1	2	yes	—
88	C₄×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
89	#20351	—	79	1	no	maximal, Sylow
90	C₂×C₇₉	—	2	2	no	—
91	C₂×C₇₉	—	2	12	no	—
92	C₂×C₇₉	C₂²×#95	1	1	yes	characteristic
93	C₂×C₇₉	#2916	1	2	yes	—
94	C₂×#568	—	79	2	no	—
95	C₂²×#131	D₇₉	1	1	yes	characteristic
96	#1738	—	79	6	no	—
97	C₂×#341	—	79	6	no	—
98	C₇₉	#20351	1	1	yes	characteristic, complemented, Sylow
99	C₂²×#41	C₂×D₇₉	1	1	yes	characteristic
100	#568	—	79	16	no	—
101	#568	—	158	8	no	—
102	C₂×#131	—	2	8	no	—
103	C₂×#131	—	158	8	no	—
104	C₂×#131	Dic₇₉	1	8	yes	—
105	C₂²×M₁₆	—	79	12	no	—
106	C₂²×M₁₆	C₂×D₇₉	1	3	yes	—
107	C₂³×C₈	C₂×D₇₉	1	3	yes	—
108	#341	—	79	4	no	—
109	#341	—	79	12	no	—
110	C₂×#41	C₄×D₇₉	1	8	yes	—
111	C₂×#41	D₃₁₆	1	4	yes	—
112	C₂×#41	C₂×Dic₇₉	1	4	yes	—
113	C₂×#41	C₂²⋊D₇₉	1	8	yes	—
114	C₂²×Q₈	Dic₁₅₈	1	1	yes	characteristic
115	C₂²×D₄	Dic₁₅₈	1	3	yes	—
116	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
117	#131	—	2	32	no	—
118	#131	—	158	64	no	—
119	C₂×M₁₆	—	2	12	no	—
120	C₂×M₁₆	—	79	48	no	—
121	C₂×M₁₆	—	79	8	no	—
122	C₂×M₁₆	—	158	24	no	—
123	C₂×M₁₆	—	158	12	no	—
124	C₂×M₁₆	C₂×Dic₇₉	1	12	yes	—
125	C₂²×C₈	—	2	12	no	—
126	C₂²×C₈	—	158	12	no	—
127	C₂²×C₈	C₂×Dic₇₉	1	12	yes	—
128	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₇₉	1	3	yes	—
130	C₂⁴	C₂×Dic₁₅₈	1	3	yes	—
131	#41	C₄×Dic₇₉	1	16	yes	—
132	#41	(C₂×D₇₉)⋊C₄	1	32	yes	—
133	#41	C₂²⋊Dic₇₉	1	16	yes	—
134	C₂×Q₈	Dic₇₉⋊C₄	1	8	yes	—
135	C₂×Q₈	C₄⋊Dic₇₉	1	4	yes	—
136	C₂×D₄	Dic₇₉⋊C₄	1	24	yes	—
137	C₂×D₄	C₄⋊Dic₇₉	1	12	yes	—
138	C₂²×C₄	C₂×C₄×D₇₉	1	12	yes	—
139	C₂²×C₄	C₂²×Dic₇₉	1	6	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₇₉	1	12	yes	—
141	C₂²×C₄	C₂×D₃₁₆	1	6	yes	—
142	C₂²×C₄	C₂×Dic₁₅₈	1	3	yes	—
143	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
144	M₁₆	—	2	24	no	—
145	M₁₆	—	79	16	no	—
146	M₁₆	—	158	48	no	—
147	M₁₆	—	158	8	no	—
148	C₂×C₈	—	2	24	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	6	no	—
151	C₂×C₈	—	158	48	no	—
152	C₂×C₈	—	158	8	no	—
153	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
154	C₂×C₈	C₂²×Dic₇₉	1	6	yes	—
155	C₂³	—	4	6	no	—
156	C₂³	C₂×Dic₇₉⋊C₄	1	12	yes	—
157	C₂³	C₂×C₄⋊Dic₇₉	1	6	yes	—
158	C₂³	C₂²×Dic₁₅₈	1	1	yes	characteristic
159	Q₈	C₇₉⋊#95	1	16	yes	—
160	D₄	C₇₉⋊#95	1	48	yes	—
161	C₂×C₄	C₂×C₄×Dic₇₉	1	12	yes	—
162	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	24	yes	—
163	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	12	yes	—
164	C₂×C₄	C₂×Dic₇₉⋊C₄	1	12	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₇₉	1	6	yes	—
166	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
167	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₂²⋊D₇₉	1	2	yes	—
169	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	6	no	—
172	C₈	—	158	16	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	12	no	—
175	C₂²	—	4	12	no	—
176	C₂²	C₂×C₇₉⋊#95	1	12	yes	—
177	C₂²	C₂²×Dic₇₉⋊C₄	1	2	yes	—
178	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
179	C₄	C₂×C₇₉⋊#95	1	12	yes	—
180	C₄	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
181	C₄	C₂²×(C₂×D₇₉)⋊C₄	1	2	yes	—
182	C₄	C₂²×C₂²⋊Dic₇₉	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	2	12	no	—
185	C₂	C₂²×C₇₉⋊#95	1	1	yes	characteristic
186	C₂	#23772011	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₄×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂²×#131×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂²×#131	D₇₉
hypocenter	C₇₉	#20351
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #23763127	C₂	semidirect
3. C₂×#19098328	C₂	semidirect
98. C₇₉	#20351	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099330	2
#23763127	2
C₂×#19098328	2
C₂²×#131×C₇₉	2
#20351	79