Finite Group Databasebeta

Subgroups — #34208773

Order 20224.

194 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098396	C₂	1	1	yes	maximal, characteristic
2	#23765181	C₂	1	1	yes	maximal, characteristic
3	#366×D₇₉	C₂	1	1	yes	maximal, characteristic
4	#23761602	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761601	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761599	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
9	#19098600	—	2	1	no	—
10	#107×D₇₉	—	2	1	no	—
11	#19098396	C₄	1	4	yes	complemented
12	C₂×C₇₉⋊C₃₂	C₂²	1	1	yes	characteristic
13	#19098390	—	2	1	no	—
14	#19098389	C₂²	1	1	yes	characteristic
15	#19098389	C₂²	1	1	yes	characteristic
16	#19098388	—	2	1	no	—
17	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
18	#366×C₇₉	C₂²	1	1	yes	characteristic
19	#366×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
21	#14579119	—	4	1	no	—
22	C₄⋊C₄×D₇₉	—	4	1	no	—
23	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₇₉	C₂×C₄	1	2	yes	—
25	C₁₆×D₇₉	Q₈	1	2	yes	—
26	#14579095	—	2	1	no	—
27	#14579095	—	2	1	no	—
28	#14579094	—	4	1	no	—
29	#14579092	—	4	1	no	—
30	C₇₉⋊C₃₂	C₂×C₄	1	2	yes	complemented
31	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
32	#107×C₇₉	—	2	1	no	—
33	#107×C₇₉	—	2	1	no	—
34	C₃₂×C₇₉	C₂×C₄	1	2	yes	complemented
35	C₂×C₄×D₇₉	—	8	1	no	—
36	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
37	C₂×C₇₉⋊C₈	D₈	1	1	yes	characteristic
38	C₈×D₇₉	C₄⋊C₄	1	2	yes	—
39	C₈×D₇₉	Q₁₆	1	2	yes	—
40	C₄⋊Dic₇₉	—	4	1	no	—
41	C₄⋊Dic₇₉	—	4	1	no	—
42	Dic₇₉⋊C₄	—	8	1	no	—
43	(C₂×D₇₉)⋊C₄	—	8	1	no	—
44	C₄×Dic₇₉	—	8	1	no	—
45	C₇₉⋊C₁₆	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₇₉	—	4	1	no	—
48	C₄⋊C₄×C₇₉	—	4	1	no	—
49	C₁₆×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	C₂²×D₇₉	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₉	—	8	1	no	—
53	C₂×C₄×C₇₉	—	8	1	no	—
54	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₇₉	—	8	1	no	—
56	C₂×Dic₇₉	—	8	1	no	—
57	C₂×Dic₇₉	D₁₆	1	1	yes	characteristic
58	C₄×D₇₉	—	16	2	no	—
59	C₄×D₇₉	#107	1	2	yes	—
60	C₄×D₇₉	Q₃₂	1	2	yes	—
61	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₇₉⋊C₈	#107	1	2	yes	—
64	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
65	C₂×D₇₉	—	2	1	no	—
66	C₂×D₇₉	—	2	1	no	—
67	C₂×D₇₉	#366	1	2	yes	—
68	C₄×C₇₉	—	16	1	no	—
69	C₄×C₇₉	—	16	1	no	—
70	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
71	C₄×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
72	Dic₇₉	—	16	1	no	—
73	Dic₇₉	—	16	1	no	—
74	Dic₇₉	#366	1	2	yes	—
75	#13634	—	79	1	no	maximal, Sylow
76	C₂×C₇₉	C₂×#366	1	1	yes	characteristic
77	C₂×C₇₉	#2263	1	1	yes	characteristic
78	C₂×C₇₉	#2264	1	1	yes	characteristic
79	D₇₉	—	2	2	no	—
80	C₂×M₆₄	—	79	1	no	—
81	#2158	—	79	1	no	—
82	C₂×#366	—	79	1	no	—
83	#1424	—	79	1	no	—
84	#1424	—	79	1	no	—
85	#1423	—	79	1	no	—
86	#1423	D₇₉	1	1	yes	characteristic, complemented
87	C₇₉	#13634	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	79	1	no	—
89	#427	—	158	1	no	—
90	C₂×#107	—	158	1	no	—
91	M₆₄	—	79	4	no	—
92	C₂×C₃₂	—	79	1	no	—
93	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
94	#367	—	158	1	no	—
95	#366	—	79	1	no	—
96	#366	—	79	1	no	—
97	#366	—	158	1	no	—
98	#366	C₂×D₇₉	1	1	yes	characteristic
99	#366	C₂×D₇₉	1	1	yes	characteristic
100	C₂²×C₈	—	79	1	no	—
101	#117	—	316	1	no	—
102	C₂×C₄⋊C₄	—	316	1	no	—
103	C₂×C₁₆	—	79	1	no	—
104	C₂×C₁₆	—	79	2	no	—
105	C₂×C₁₆	—	79	2	no	—
106	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
107	#107	—	2	1	no	—
108	#107	—	2	1	no	—
109	#107	—	316	1	no	—
110	#107	—	158	1	no	—
111	#107	—	158	1	no	—
112	#106	—	316	1	no	—
113	C₃₂	—	79	2	no	—
114	C₃₂	C₄×D₇₉	1	2	yes	complemented
115	C₂²×C₄	—	79	1	no	—
116	C₂²×C₄	—	632	1	no	—
117	C₂×C₈	—	79	1	no	—
118	C₂×C₈	—	79	2	no	—
119	C₂×C₈	—	79	2	no	—
120	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	316	1	no	—
124	C₄⋊C₄	—	316	1	no	—
125	C₄⋊C₄	—	632	1	no	—
126	C₂²⋊C₄	—	632	1	no	—
127	C₄²	—	632	1	no	—
128	C₁₆	—	79	2	no	—
129	C₁₆	C₂×C₄×D₇₉	1	1	yes	characteristic
130	C₁₆	Q₈×D₇₉	1	1	yes	characteristic
131	C₂³	—	79	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	79	1	no	—
135	C₂×C₄	—	79	2	no	—
136	C₂×C₄	—	79	2	no	—
137	C₂×C₄	—	632	1	no	—
138	C₂×C₄	—	632	1	no	—
139	C₂×C₄	—	1264	2	no	—
140	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
141	C₈	—	79	2	no	—
142	C₈	C₄⋊C₄×D₇₉	1	1	yes	characteristic
143	C₈	Q₁₆×D₇₉	1	1	yes	characteristic
144	C₂²	—	79	2	no	—
145	C₂²	—	158	1	no	—
146	C₂²	—	158	1	no	—
147	C₂²	D₁₆×D₇₉	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	16	1	no	—
150	C₄	—	79	2	no	—
151	C₄	—	1264	1	no	—
152	C₄	—	1264	1	no	—
153	C₄	#107×D₇₉	1	1	yes	characteristic
154	C₄	Q₃₂×D₇₉	1	1	yes	characteristic
155	C₂	—	158	2	no	—
156	C₂	#366×D₇₉	1	1	yes	characteristic
157	C₂	#23765711	1	1	yes	characteristic
158	C₂	#23765719	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₉
commutator	C₁₆×C₇₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₇₉×#1423	C₂
socle	C₂²×C₇₉	C₂×D₁₆
prime radical	C₂²×D₇₉	D₁₆
cosocle	C₂×C₁₆×C₇₉	C₂³
hypercenter	#1423	D₇₉
hypocenter	C₇₉	#13634
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

Complements

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

N	H ≅ G/N	type
4. #23761602	C₂	semidirect
5. #23761601	C₂	semidirect
6. #23761599	C₂	semidirect
7. C₇₉×#1423	C₂	semidirect
11. #19098396	C₄	semidirect
30. C₇₉⋊C₃₂	C₂×C₄	semidirect
34. C₃₂×C₇₉	C₂×C₄	semidirect
86. #1423	D₇₉	semidirect
87. C₇₉	#13634	semidirect
114. C₃₂	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098396	2
#23765181	2
#366×D₇₉	2
#23761602	2
#23761601	2
#23761599	2
C₇₉×#1423	2
#13634	79