Finite Group Databasebeta

← #34201645

Subgroups — #34201645

Order 20224.

318 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 59 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#23765612C212yesmaximal, complemented
2#23764975C211yesmaximal, characteristic, complemented
3#23761385C212yesmaximal, complemented
4#23761351C211yesmaximal, characteristic, complemented
5C79×#3253C211yesmaximal, characteristic
6#19098612C2211yescharacteristic
7#19098385—22no—
8#19098376—22no—
9#19098345C2212yes—
10#19098335—21no—
11#19098323—21no—
12#19098320C2211yescharacteristic
13#532×C79—21no—
14#476×C79—22no—
15#469×C79—21no—
16#469×C79C2212yes—
17#459×C79—22no—
18#423×C79C2211yescharacteristic
19C2×#11929608—21no—
20#14579120—21no—
21#14579101—22no—
22#14579097—22no—
23C2×#41×C79—41no—
24C2×D8×C79—21no—
25C2×D8×C79D412yes—
26C2×M16×C79D411yescharacteristic
27C2×M16×C79C2311yescharacteristic
28C4⋊Q8×C79—21no—
29#127×C79—21no—
30#124×C79—41no—
31#121×C79—22no—
32#121×C79—41no—
33C4×D4×C79—22no—
34C4×D4×C79—41no—
35C2×C42×C79D411yescharacteristic
36#106×C79—21no—
37#106×C79D412yes—
38C4⋊C8×C79—22no—
39#102×C79—22no—
40#102×C79—22no—
41#11929608—41no—
42#11929608—41no—
43C2×C79⋊C8—41no—
44C4⋊Dic79—41no—
45C22⋊Dic79—41no—
46C4×Dic79—41no—
47C79⋊C16—42no—
48#41×C79—84no—
49C2×Q8×C79—41no—
50C2×D4×C79—22no—
51C2×D4×C79—41no—
52C2×D4×C79—41no—
53C2×D4×C79—42no—
54C22×C4×C79—21no—
55C22×C4×C79—42no—
56C22×C4×C79C2×D411yescharacteristic
57D8×C79—42no—
58D8×C79—81no—
59M16×C79—41no—
60M16×C79—41no—
61C2×C8×C79—21no—
62C2×C8×C79C2×D412yes—
63C4⋊C4×C79—22no—
64C4⋊C4×C79—41no—
65C22⋊C4×C79—42no—
66C22⋊C4×C79—42no—
67C42×C79—21no—
68C42×C79SD1611yescharacteristic
69C42×C79SD1611yescharacteristic
70C23×C79—42no—
71C23×C79AGL(1,ℤ/8)11yescharacteristic
72Q8×C79—82no—
73D4×C79—42no—
74D4×C79—42no—
75D4×C79—82no—
76D4×C79—82no—
77D4×C79—84no—
78C2×C4×C79—21no—
79C2×C4×C79—21no—
80C2×C4×C79—21no—
81C2×C4×C79—21no—
82C2×C4×C79—42no—
83C2×C4×C79—42no—
84C2×C4×C79—84no—
85C2×C4×C79#12011yescharacteristic
86C2×C4×C79C2×SD1611yescharacteristic
87C2×Dic79—81no—
88C8×C79—22no—
89C8×C79—41no—
90C79⋊C8—81no—
91C22×C79—21no—
92C22×C79—41no—
93C22×C79—82no—
94C22×C79—82no—
95C22×C79—82no—
96C22×C79#45011yescharacteristic
97C4×C79—22no—
98C4×C79—21no—
99C4×C79—41no—
100C4×C79—82no—
101C4×C79#45411yescharacteristic
102C4×C79#45811yescharacteristic
103Dic79—161no—
104#12307—791nomaximal, Sylow
105C2×C79—21no—
106C2×C79—41no—
107C2×C79—162no—
108C2×C79#161611yescharacteristic
109#3253D7911yescharacteristic
110#2238—792no—
111#2102—791no—
112#1339—792no—
113#1327—791no—
114C79#1230711yescharacteristic, complemented, Sylow
115#532—21no—
116#476—22no—
117#469—21no—
118#469C2×D7912yes—
119#459—22no—
120#430—791no—
121#423C2×D7911yescharacteristic
122#365—1582no—
123#361—1582no—
124#349—792no—
125#344—1581no—
126#339—1581no—
127H(3,ℤ/4)—791no—
128C2×#41—41no—
129C2×D8—21no—
130C2×D8C22⋊D7912yes—
131C2×M16—1581no—
132C2×M16C22⋊D7911yescharacteristic
133C2×M16C22×D7911yescharacteristic
134C4⋊Q8—21no—
135#127—21no—
136#124—41no—
137#121—22no—
138#121—41no—
139C4×D4—22no—
140C4×D4—41no—
141#117—1581no—
142C2×C42C22⋊D7911yescharacteristic
143M32—1582no—
144#108—1582no—
145#106—21no—
146#106C22⋊D7912yes—
147C4⋊C8—22no—
148#102—22no—
149#102—22no—
150#41—84no—
151C2×Q8—41no—
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—42no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4C2×C22⋊D7911yescharacteristic
159D8—42no—
160D8—81no—
161M16—41no—
162M16—41no—
163M16—3161no—
164M16—3161no—
165C2×C8—21no—
166C2×C8—3161no—
167C2×C8C2×C22⋊D7912yes—
168C4⋊C4—22no—
169C4⋊C4—41no—
170C4⋊C4—3161no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—3161no—
174C42—21no—
175C42—3161no—
176C42#1192961211yescharacteristic
177C42#1192961211yescharacteristic
178C16—3162no—
179C23—42no—
180C23#1457918711yescharacteristic
181Q8—82no—
182D4—42no—
183D4—42no—
184D4—82no—
185D4—82no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4—6321no—
195C2×C4#1457913011yescharacteristic
196C2×C4C2×#1192961211yescharacteristic
197C8—22no—
198C8—41no—
199C8—6321no—
200C22—21no—
201C22—41no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22#1909871411yescharacteristic
206C4—22no—
207C4—21no—
208C4—41no—
209C4—82no—
210C4—12641no—
211C4#1909873611yescharacteristic
212C4#1909875311yescharacteristic
213C2—21no—
214C2—41no—
215C2—162no—
216C2#2376243711yescharacteristic
subgroupG/H
centerC2#23762437
commutator, cosocleC2×M16×C79C23
FrattiniC2×M16C22×D79
Fitting, prime radicalC79×#3253C2
socleC2×C79#1616
hypercenter#3253D79
hypocenterC79#12307
LCSG ▷ C2×M16×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #23765612C2semidirect
2. #23764975C2semidirect
3. #23761385C2semidirect
4. #23761351C2semidirect
114. C79#12307semidirect

Maximal subgroups

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

subgroupindex
#237656122
#237649752
#237613852
#237613512
C79×#32532
#1230779