Finite Group Databasebeta

← #34198815

Subgroups — #34198815

Order 20224.

292 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 75 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#23774059C211yesmaximal, characteristic
2#23765216C211yesmaximal, characteristic
3#23761393C212yesmaximal
4#23761387C212yesmaximal
5C79×#1747C211yesmaximal, characteristic
6#19099175—21no—
7#19098965C411yescharacteristic, complemented
8#19098943C411yescharacteristic, complemented
9#19098932—21no—
10#19098821C2212yes—
11#19098804—21no—
12#19098671C412yescomplemented
13C4×Dic316—21no—
14#19098384—21no—
15#19098345C2212yes—
16#19098341—21no—
17#405×C79—21no—
18#405×C79C2211yescharacteristic
19#338×C79C2211yescharacteristic
20H(3,ℤ/4)×C79—21no—
21H(3,ℤ/4)×C79C2211yescharacteristic
22C2×Dic316—21no—
23C2×Dic316C2×C412yes—
24#14579161—21no—
25#14579154—42no—
26#14579153—21no—
27#14579151—42no—
28#14579140—22no—
29C4×Dic158—22no—
30#14579101—22no—
31#14579095—21no—
32#14579093C2×C412yes—
33#14579086—22no—
34#14579086—22no—
35C2×M16×C79—21no—
36C2×M16×C79C2311yescharacteristic
37C22×C8×C79D411yescharacteristic
38#117×C79—21no—
39#117×C79D411yescharacteristic
40#97×C79—22no—
41#97×C79C2×C411yescharacteristic
42#97×C79D411yescharacteristic
43C4×C8×C79—22no—
44C4×C8×C79C2×C411yescharacteristic
45C4×C8×C79D411yescharacteristic
46C2×Dic158—22no—
47Dic316—42no—
48Dic316—41no—
49Dic316—41no—
50C4⋊Dic79—22no—
51C4⋊Dic79—42no—
52Dic79⋊C4—42no—
53Dic79⋊C4—42no—
54Dic79⋊C4—42no—
55C22⋊Dic79—42no—
56C4×Dic79—42no—
57C79⋊C16—42no—
58C22×C4×C79C2×D411yescharacteristic
59M16×C79—41no—
60M16×C79—41no—
61M16×C79—41no—
62C2×C8×C79—21no—
63C2×C8×C79—21no—
64C2×C8×C79—22no—
65C2×C8×C79C22⋊C411yescharacteristic
66C2×C8×C79C22⋊C411yescharacteristic
67C2×C8×C79C22×C411yescharacteristic
68C2×C8×C79C2×D411yescharacteristic
69C4⋊C4×C79—21no—
70C4⋊C4×C79—41no—
71C22⋊C4×C79—21no—
72C22⋊C4×C79—41no—
73C42×C79—22no—
74C42×C79C22⋊C411yescharacteristic
75C42×C79C22⋊C411yescharacteristic
76C23×C79AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C79—21no—
78C2×C4×C79—21no—
79C2×C4×C79—21no—
80C2×C4×C79—21no—
81C2×C4×C79—42no—
82C2×C4×C79C2×C22⋊C411yescharacteristic
83C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
84C2×Dic79—42no—
85C2×Dic79—42no—
86Dic158—42no—
87Dic158—42no—
88C8×C79—21no—
89C8×C79—21no—
90C8×C79—21no—
91C8×C79—42no—
92C8×C79#10411yescharacteristic
93C8×C79#10411yescharacteristic
94C22×C79—21no—
95C22×C79—41no—
96C22×C79#41811yescharacteristic
97C4×C79—21no—
98C4×C79—41no—
99C4×C79—41no—
100C4×C79—82no—
101C4×C79#41011yescharacteristic
102C4×C79C2×#10411yescharacteristic
103Dic79—82no—
104Dic79—82no—
105#11771—791nomaximal, Sylow
106C2×C79—21no—
107C2×C79—41no—
108C2×C79#150511yescharacteristic
109#3177—791no—
110#2166—791no—
111#1747D7911yescharacteristic
112#1342—792no—
113#1340—792no—
114C79#1177111yescharacteristic, complemented, Sylow
115#541—1581no—
116#499—791no—
117#494—791no—
118#491—1581no—
119#470—792no—
120#467—1581no—
121#441—792no—
122C4×Q16—1581no—
123#405—21no—
124#405C2×D7911yescharacteristic
125#364—1581no—
126#349—792no—
127#347—1581no—
128#338C2×D7911yescharacteristic
129H(3,ℤ/4)—21no—
130H(3,ℤ/4)C2×D7911yescharacteristic
131C2×Q16—792no—
132C2×Q16—1581no—
133C2×M16—21no—
134C2×M16C22×D7911yescharacteristic
135C22×C8C22⋊D7911yescharacteristic
136C4⋊Q8—1581no—
137#126—3162no—
138#125—3162no—
139#125—1581no—
140C22⋊Q8—1582no—
141C4×Q8—1582no—
142#117—21no—
143#117C22⋊D7911yescharacteristic
144M32—1582no—
145#107—1581no—
146#106—792no—
147#103—1582no—
148#103—1582no—
149#97—22no—
150#97C4×D7911yescharacteristic
151#97D31611yescharacteristic
152C4×C8—22no—
153C4×C8C4×D7911yescharacteristic
154C4×C8D31611yescharacteristic
155C2×Q8—1582no—
156C22×C4C2×C22⋊D7911yescharacteristic
157Q16—3162no—
158Q16—3161no—
159Q16—3161no—
160M16—41no—
161M16—41no—
162M16—41no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8(C2×D79)⋊C411yescharacteristic
167C2×C8(C2×D79)⋊C411yescharacteristic
168C2×C8C2×C4×D7911yescharacteristic
169C2×C8C2×D31611yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—3162no—
173C4⋊C4—3162no—
174C4⋊C4—3162no—
175C4⋊C4—3162no—
176C4⋊C4—1582no—
177C22⋊C4—21no—
178C22⋊C4—41no—
179C22⋊C4—3162no—
180C42—22no—
181C42—3162no—
182C42(C2×D79)⋊C411yescharacteristic
183C42(C2×D79)⋊C411yescharacteristic
184C16—3162no—
185C23#1457918611yescharacteristic
186Q8—3162no—
187Q8—3162no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—3162no—
194C2×C4—3162no—
195C2×C4C2×(C2×D79)⋊C411yescharacteristic
196C2×C4#1457918611yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—42no—
201C8#1457908911yescharacteristic
202C8#1457908911yescharacteristic
203C22—21no—
204C22—41no—
205C22#1909856711yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—6322no—
211C4—6322no—
212C4#1909853611yescharacteristic
213C4C2×#1457908911yescharacteristic
214C2—21no—
215C2—41no—
216C2#2376190711yescharacteristic
subgroupG/H
centerC2#23761907
commutatorC2×C8×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#1747C2
socleC2×C79#1505
prime radicalC23×C79AGL(1,ℤ/8)
cosocleC2×M16×C79C23
hypercenter#1747D79
hypocenterC79#11771
LCSG ▷ C2×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #1747

Complements

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

NH ≅ G/Ntype
7. #19098965C4semidirect
8. #19098943C4semidirect
12. #19098671C4semidirect
114. C79#11771semidirect

Maximal subgroups

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

subgroupindex
#237740592
#237652162
#237613932
#237613872
C79×#17472
#1177179