Subgroups — #34197279

Order 20224.

282 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579101	C₂	1	1	yes	maximal, characteristic
2	#23761683	C₂	1	1	yes	maximal, characteristic
3	#23761322	C₂	1	4	yes	maximal, complemented
4	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	C₄	1	2	yes	—
6	#19098616	C₄	1	2	yes	—
7	#19098339	C₂²	1	2	yes	—
8	C₄×C₇₉⋊C₁₆	C₄	1	2	yes	—
9	#19098284	C₂²	1	2	yes	—
10	C₈×C₇₉⋊C₈	C₄	1	2	yes	—
11	C₂²×C₁₆×C₇₉	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	#346×C₇₉	C₄	1	2	yes	—
14	#346×C₇₉	C₂²	1	2	yes	—
15	#14579101	C₈	1	8	yes	complemented
16	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
19	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₉	—	2	4	no	—
24	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₂×C₇₉⋊C₈	—	2	4	no	—
30	C₇₉⋊C₁₆	C₂×C₈	1	8	yes	—
31	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
33	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₇₉	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
37	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₇₉	C₄²	1	1	yes	characteristic
40	C₄²×C₇₉	M₁₆	1	2	yes	—
41	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₇₉	—	2	4	no	—
43	C₂³×C₇₉	#97	1	1	yes	characteristic
44	C₂×C₄×C₇₉	—	4	2	no	—
45	C₂×C₄×C₇₉	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₇₉	#97	1	2	yes	—
47	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
48	C₂×C₄×C₇₉	#97	1	2	yes	—
49	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
51	C₈×C₇₉	C₄×C₈	1	2	yes	—
52	C₈×C₇₉	#97	1	2	yes	—
53	C₈×C₇₉	C₂²×C₈	1	2	yes	—
54	C₈×C₇₉	C₂×M₁₆	1	2	yes	—
55	C₇₉⋊C₈	—	4	4	no	—
56	C₂²×C₇₉	—	2	2	no	—
57	C₂²×C₇₉	#322	1	2	yes	—
58	C₂²×C₇₉	C₂×#97	1	1	yes	characteristic
59	C₄×C₇₉	—	2	2	no	—
60	C₄×C₇₉	—	2	2	no	—
61	C₄×C₇₉	#322	1	2	yes	—
62	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
64	#11406	—	79	1	no	maximal, Sylow
65	C₂×C₇₉	—	2	2	no	—
66	C₂×C₇₉	C₂×#322	1	1	yes	characteristic
67	C₂×C₇₉	#2110	1	2	yes	—
68	C₄×M₃₂	—	79	1	no	—
69	C₂×#346	D₇₉	1	1	yes	characteristic
70	#1450	—	79	1	no	—
71	#1313	—	79	4	no	—
72	C₇₉	#11406	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₃₂	—	79	2	no	—
74	C₂²×C₁₆	Dic₇₉	1	2	yes	—
75	#431	—	79	2	no	—
76	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
77	#346	—	79	2	no	—
78	#346	Dic₇₉	1	2	yes	—
79	#346	C₂×D₇₉	1	2	yes	—
80	C₄×C₁₆	—	79	2	no	—
81	#322	—	79	2	no	—
82	C₈²	—	79	2	no	—
83	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
85	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
86	M₃₂	—	79	8	no	—
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	—	79	4	no	—
89	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
90	C₄⋊C₈	—	158	2	no	—
91	C₂²⋊C₈	—	158	2	no	—
92	C₄×C₈	—	79	4	no	—
93	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
94	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
96	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
97	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
98	C₂²×C₄	#11929607	1	2	yes	—
99	C₂×C₈	—	158	4	no	—
100	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
101	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
102	C₂×C₈	C₈×D₇₉	1	4	yes	—
103	C₂×C₈	#11929608	1	4	yes	—
104	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
105	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
106	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
107	C₄²	#11929607	1	2	yes	—
108	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
109	C₁₆	—	2	4	no	—
110	C₁₆	—	79	8	no	—
111	C₂³	#14579071	1	1	yes	characteristic
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₈×Dic₇₉	1	2	yes	—
114	C₂×C₄	#14579071	1	2	yes	—
115	C₂×C₄	#14579071	1	1	yes	characteristic
116	C₂×C₄	#14579072	1	2	yes	—
117	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
118	C₂×C₄	C₂×#11929607	1	2	yes	—
119	C₈	—	316	4	no	—
120	C₈	C₈×Dic₇₉	1	2	yes	—
121	C₈	#14579072	1	2	yes	—
122	C₈	C₂×C₈×D₇₉	1	2	yes	—
123	C₈	C₂×#11929608	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	#19098285	1	2	yes	—
126	C₂²	C₂×#14579071	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	#19098285	1	2	yes	—
130	C₄	C₂×C₈×Dic₇₉	1	1	yes	characteristic
131	C₄	C₂×#14579072	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#19098285	1	1	yes	characteristic
134	C₂	#23765004	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#97
prime radical	C₂³×C₇₉	#97
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#11406
LCS	G ▷ C₄×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #23761322	C₂	semidirect
15. #14579101	C₈	semidirect
72. C₇₉	#11406	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579101	2
#23761683	2
#23761322	2
C₂×#346×C₇₉	2
#11406	79