Finite Group Databasebeta

← #34209089

Subgroups — #34209089

Order 20224.

216 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23765715C211yesmaximal, characteristic, complemented
2#23765713C211yesmaximal, characteristic, complemented
3#23765648C211yesmaximal, characteristic, complemented
4#23765280C211yesmaximal, characteristic, complemented
5#23761597C211yesmaximal, characteristic, complemented
6#23761589C211yesmaximal, characteristic, complemented
7C79×#1419C211yesmaximal, characteristic, complemented
8#19098995C2211yescharacteristic
9C2×D1264C2211yescharacteristic
10C2×#14579102—21no—
11#19098938—21no—
12#19098679—21no—
13#19098402—21no—
14#19098401—21no—
15D2528—21no—
16#19098398—21no—
17#19098397C2211yescharacteristic
18#19098369—21no—
19C16×Dic79C2211yescharacteristic
20C2×D16×C79C2211yescharacteristic
21M64×C79C2211yescharacteristic
22#368×C79C2211yescharacteristic
23#14579183—21no—
24C2×D632—21no—
25C2×#11929609—41no—
26#14579169—41no—
27#14579157—41no—
28Dic632D411yescharacteristic, complemented
29#14579106—21no—
30D1264—21no—
31D1264—21no—
32D1264D411yescharacteristic, complemented
33#14579102—42no—
34C2×C79⋊C16D411yescharacteristic
35#14579087—41no—
36C8×Dic79D411yescharacteristic
37C79⋊C32—21no—
38C2×D8×C79—21no—
39D16×C79—21no—
40D16×C79—21no—
41C2×C16×C79C2311yescharacteristic
42#108×C79—21no—
43C32×C79—21no—
44#11929624—41no—
45C2×D316—41no—
46C2×C22⋊D79—82no—
47Dic316—21no—
48#11929613—41no—
49D632—21no—
50D632—41no—
51D632—41no—
52#11929609—81no—
53#11929609—81no—
54#11929607—81no—
55C2×C79⋊C8D811yescharacteristic
56C8×D79—81no—
57C4×Dic79D811yescharacteristic
58C79⋊C16—21no—
59C2×D4×C79—41no—
60D8×C79—41no—
61D8×C79—41no—
62M16×C79—41no—
63C2×C8×C79C2×D411yescharacteristic
64C16×C79C2×D411yescharacteristic
65C16×C79#4111yescharacteristic
66C23×C79—81no—
67C22×D79—81no—
68D4×C79—81no—
69D4×C79—81no—
70C2×C4×C79C2×D811yescharacteristic
71C22⋊D79—81no—
72C22⋊D79—164no—
73C2×Dic79—21no—
74D316—41no—
75D316—81no—
76D316—81no—
77C4×D79—81no—
78Dic158—41no—
79C8×C79—81no—
80C8×C79#12111yescharacteristic
81C8×C79AGL(1,ℤ/8)11yescharacteristic
82C79⋊C8D1611yescharacteristic
83C79⋊C8D1611yescharacteristic
84C22×C79—162no—
85C22×C79—161no—
86C22×C79#50911yescharacteristic
87C2×D79—81no—
88C2×D79—162no—
89C2×D79—161no—
90C4×C79#45911yescharacteristic
91C4×C79C2×D1611yescharacteristic
92Dic79—22no—
93Dic79—81no—
94#13692—791nomaximal, Sylow
95C2×C79—21no—
96C2×C79—321no—
97C2×C79#220611yescharacteristic
98D79—161no—
99D79—321no—
100#2263—791no—
101#2263—791no—
102#2246—791no—
103#2178—791no—
104#1422—791no—
105#1419—791no—
106#1419D7911yescharacteristic, complemented
107C79#1369211yescharacteristic, complemented, Sylow
108#508—791no—
109C2×D16—791no—
110C2×D16—1581no—
111C2×D16C2×D7911yescharacteristic
112#493—1581no—
113#443—1581no—
114SD64—1581no—
115SD64—1581no—
116D32—1581no—
117D32—1581no—
118M64—791no—
119M64C2×D7911yescharacteristic
120#368C2×D7911yescharacteristic
121#359—1581no—
122C4×C16—791no—
123#135—1581no—
124C2×D8—21no—
125C2×D8—3161no—
126C2×D8—1581no—
127#131—3161no—
128#127—3161no—
129Q32—791no—
130SD32—1581no—
131D16—21no—
132D16—21no—
133D16—3162no—
134D16—791no—
135D16—1581no—
136D16—1581no—
137C2×C16—791no—
138C2×C16C22×D7911yescharacteristic
139#108—21no—
140#104—3161no—
141C4×C8—791no—
142C32—21no—
143C32—1581no—
144#41—3161no—
145C2×D4—41no—
146C2×D4—3161no—
147C2×D4—6322no—
148Q16—1581no—
149SD16—3161no—
150D8—41no—
151D8—41no—
152D8—3161no—
153D8—3161no—
154D8—6321no—
155D8—6321no—
156D8—1581no—
157M16—41no—
158M16—6321no—
159C2×C8—791no—
160C2×C8—6321no—
161C2×C8D4×D7911yescharacteristic
162C42—791no—
163C16—1581no—
164C16D4×D7911yescharacteristic
165C16#1192962411yescharacteristic
166C23—81no—
167C23—6321no—
168Q8—3161no—
169D4—81no—
170D4—81no—
171D4—3161no—
172D4—6321no—
173D4—6321no—
174D4—6321no—
175D4—12644no—
176C2×C4—6321no—
177C2×C4—1581no—
178C2×C4D8×D7911yescharacteristic
179C8—81no—
180C8—791no—
181C8—791no—
182C8#1457913411yescharacteristic
183C8#1457918811yescharacteristic
184C22—162no—
185C22—161no—
186C22—6321no—
187C22—12642no—
188C22—12641no—
189C22#1909900011yescharacteristic
190C4—6321no—
191C4—1582no—
192C4#1909875711yescharacteristic
193C4D16×D7911yescharacteristic
194C2—21no—
195C2—321no—
196C2—25281no—
197C2—12641no—
198C2#2376543411yescharacteristic
subgroupG/H
centerC2#23765434
commutator, cosocleC2×C16×C79C23
FrattiniC2×C16C22×D79
FittingC79×#1419C2
socleC2×C79#2206
hypercenter#1419D79
hypocenterC79#13692
LCSG ▷ C2×C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1419

Complements

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

NH ≅ G/Ntype
1. #23765715C2semidirect
2. #23765713C2semidirect
3. #23765648C2semidirect
4. #23765280C2semidirect
5. #23761597C2semidirect
6. #23761589C2semidirect
7. C79×#1419C2semidirect
28. Dic632D4semidirect
32. D1264D4semidirect
106. #1419D79semidirect
107. C79#13692semidirect

Maximal subgroups

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

subgroupindex
#237657152
#237657132
#237656482
#237652802
#237615972
#237615892
C79×#14192
#1369279