Subgroups — #34219469

Order 20224.

1422 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098429	C₂	1	2	yes	maximal
2	#23761630	C₂	1	4	yes	maximal
3	#23761627	C₂	1	8	yes	maximal
4	C₂×C₄³×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
6	#19098429	—	2	8	no	—
7	C₂×C₇₉⋊#95	C₂²	1	8	yes	—
8	C₂×C₇₉⋊#95	C₂²	1	16	yes	—
9	C₂²×C₄²×C₇₉	C₂²	1	4	yes	—
10	C₂²×C₄²×C₇₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
12	C₄³×C₇₉	C₄	1	8	yes	complemented
13	C₂³×Dic₇₉	C₂³	1	8	yes	—
14	C₂×C₄⋊Dic₇₉	—	2	8	no	—
15	C₂×C₄⋊Dic₇₉	—	2	16	no	—
16	C₇₉⋊#95	—	2	16	no	—
17	C₇₉⋊#95	—	2	32	no	—
18	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
19	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
20	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇₉	—	2	16	no	—
22	C₂×C₄²×C₇₉	—	2	8	no	—
23	C₂×C₄²×C₇₉	C₂×C₄	1	16	yes	—
24	C₂×C₄²×C₇₉	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇₉	D₄	1	4	yes	—
27	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
28	C₂²×Dic₇₉	—	2	8	no	—
29	C₂²×Dic₇₉	—	2	32	no	—
30	C₂²×Dic₇₉	—	2	16	no	—
31	C₄⋊Dic₇₉	—	4	16	no	—
32	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	—	2	8	no	—
34	C₂²×C₄×C₇₉	—	2	8	no	—
35	C₂²×C₄×C₇₉	—	2	8	no	—
36	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
37	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
38	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
40	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
43	C₂²×C₄×C₇₉	#41	1	16	yes	—
44	C₂²×C₄×C₇₉	#41	1	8	yes	—
45	C₄²×C₇₉	—	2	32	no	—
46	C₄²×C₇₉	—	2	16	no	—
47	C₄²×C₇₉	C₄⋊C₄	1	16	yes	—
48	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
49	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
50	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
51	C₂³×C₇₉	C₂×#41	1	4	yes	—
52	C₂³×C₇₉	C₂×#41	1	8	yes	—
53	C₂×C₄×C₇₉	—	2	8	no	—
54	C₂×C₄×C₇₉	—	2	8	no	—
55	C₂×C₄×C₇₉	—	2	8	no	—
56	C₂×C₄×C₇₉	—	2	32	no	—
57	C₂×C₄×C₇₉	—	2	8	no	—
58	C₂×C₄×C₇₉	—	2	8	no	—
59	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
60	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
61	C₂×C₄×C₇₉	#117	1	16	yes	—
62	C₂×C₄×C₇₉	#117	1	8	yes	—
63	C₂×C₄×C₇₉	#124	1	8	yes	—
64	C₂×C₄×C₇₉	#125	1	8	yes	—
65	C₂×Dic₇₉	—	4	8	no	—
66	C₂×Dic₇₉	—	4	32	no	—
67	C₂×Dic₇₉	—	4	16	no	—
68	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₇₉	C₂×#117	1	2	yes	—
70	C₂²×C₇₉	C₂×#117	1	4	yes	—
71	C₂²×C₇₉	C₂×#124	1	2	yes	—
72	C₂²×C₇₉	C₂×#125	1	2	yes	—
73	C₂²×C₇₉	#529	1	16	yes	—
74	C₂²×C₇₉	#532	1	4	yes	—
75	C₂²×C₇₉	#533	1	4	yes	—
76	C₄×C₇₉	—	2	16	no	—
77	C₄×C₇₉	—	2	8	no	—
78	C₄×C₇₉	#382	1	8	yes	—
79	Dic₇₉	—	8	8	no	—
80	#14689	—	79	1	no	maximal, Sylow
81	C₂×C₇₉	C₂×#382	1	1	yes	characteristic
82	C₂×C₇₉	#2283	1	4	yes	—
83	C₂×C₇₉	#2285	1	2	yes	—
84	C₂×C₇₉	#2861	1	4	yes	—
85	C₂×C₇₉	#2864	1	4	yes	—
86	C₂×#382	—	79	2	no	—
87	C₂×C₄³	D₇₉	1	1	yes	characteristic
88	C₄⋊#95	—	79	4	no	—
89	#1434	—	79	8	no	—
90	C₇₉	#14689	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₄⋊C₄	—	79	4	no	—
92	C₂²×C₄²	C₂×D₇₉	1	4	yes	—
93	C₂²×C₄²	C₂×D₇₉	1	2	yes	—
94	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
95	#382	—	158	8	no	—
96	C₂×#95	—	79	16	no	—
97	C₂×#95	—	79	8	no	—
98	C₄³	Dic₇₉	1	8	yes	complemented
99	C₂³×C₄	—	79	8	no	—
100	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
101	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
102	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	158	8	no	—
104	C₂×C₄⋊C₄	—	158	16	no	—
105	C₂×C₄²	—	2	16	no	—
106	C₂×C₄²	—	2	8	no	—
107	C₂×C₄²	Dic₁₅₈	1	4	yes	—
108	C₂×C₄²	D₃₁₆	1	4	yes	—
109	C₂×C₄²	C₂×Dic₇₉	1	16	yes	—
110	C₂×C₄²	C₂×Dic₇₉	1	8	yes	—
111	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
112	#95	—	158	16	no	—
113	#95	—	158	32	no	—
114	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	2	8	no	—
118	C₂²×C₄	—	158	32	no	—
119	C₂²×C₄	—	158	16	no	—
120	C₂²×C₄	—	158	8	no	—
121	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
122	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
123	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
124	C₂²×C₄	C₂×D₃₁₆	1	4	yes	—
125	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
126	C₂²×C₄	C₂×Dic₁₅₈	1	4	yes	—
127	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
128	C₂²×C₄	#11929624	1	16	yes	—
129	C₂²×C₄	#11929624	1	8	yes	—
130	C₄⋊C₄	—	316	16	no	—
131	C₄²	—	2	32	no	—
132	C₄²	—	2	16	no	—
133	C₄²	C₄⋊Dic₇₉	1	16	yes	—
134	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
135	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
136	C₂³	C₂²×Dic₁₅₈	1	1	yes	characteristic
137	C₂³	C₂×#11929624	1	4	yes	—
138	C₂³	C₂×#11929624	1	8	yes	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	32	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	316	8	no	—
146	C₂×C₄	—	316	32	no	—
147	C₂×C₄	—	316	16	no	—
148	C₂×C₄	C₂×C₄⋊Dic₇₉	1	8	yes	—
149	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
150	C₂×C₄	#14579120	1	16	yes	—
151	C₂×C₄	#14579120	1	8	yes	—
152	C₂×C₄	#14579149	1	8	yes	—
153	C₂×C₄	#14579153	1	8	yes	—
154	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
155	C₂²	C₂×#14579120	1	2	yes	—
156	C₂²	C₂×#14579120	1	4	yes	—
157	C₂²	C₂×#14579149	1	2	yes	—
158	C₂²	C₂×#14579153	1	2	yes	—
159	C₂²	#19099097	1	16	yes	—
160	C₂²	#19099113	1	4	yes	—
161	C₂²	#19099120	1	4	yes	—
162	C₄	—	2	16	no	—
163	C₄	—	2	8	no	—
164	C₄	—	632	8	no	—
165	C₄	#19098429	1	8	yes	—
166	C₂	C₂×#19098429	1	1	yes	characteristic
167	C₂	#23765798	1	4	yes	—
168	C₂	#23765807	1	2	yes	—
169	C₂	#23771772	1	4	yes	—
170	C₂	#23771787	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₇₉
commutator	C₂³×C₇₉	C₂³×C₄
Fitting	C₂×C₄³×C₇₉	C₂
socle, prime radical, cosocle	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂×C₄³	D₇₉
hypocenter	C₇₉	#14689
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
12. C₄³×C₇₉	C₄	semidirect
90. C₇₉	#14689	semidirect
98. C₄³	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098429	2
#23761630	2
#23761627	2
C₂×C₄³×C₇₉	2
#14689	79