Finite Group Databasebeta

← #34197702

Subgroups — #34197702

Order 20224.

284 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 69 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#23772035C211yesmaximal, characteristic
2#23765209C212yesmaximal, complemented
3#23761350C212yesmaximal, complemented
4#23761336C211yesmaximal, characteristic
5C79×#1320C211yesmaximal, characteristic
6C2×#14579120C411yescharacteristic
7#19098621—22no—
8#19098621—22no—
9#19098586C411yescharacteristic
10#19098586C2211yescharacteristic
11#19098527C411yescharacteristic
12M16×Dic79C411yescharacteristic
13M16×Dic79C2211yescharacteristic
14#19098383—22no—
15#19098345C2212yes—
16#19098340—22no—
17#407×C79C2211yescharacteristic
18#349×C79C2212yes—
19C2×#11929608C2×C411yescharacteristic
20C2×#11929608C2×C411yescharacteristic
21#14579120—21no—
22#14579120—21no—
23#14579120—22no—
24C2×C4⋊Dic79—21no—
25C2×C22⋊Dic79—21no—
26C2×C4×Dic79C2×C411yescharacteristic
27C2×C4×Dic79C2×C411yescharacteristic
28#14579101—22no—
29Dic79⋊C8—22no—
30Dic79⋊C8—22no—
31C22⋊(C79⋊C8)—22no—
32#14579071—21no—
33#14579071D412yes—
34C8×Dic79—21no—
35C8×Dic79D412yes—
36C23×C4×C79C2×C411yescharacteristic
37C2×M16×C79C2×C411yescharacteristic
38C2×M16×C79C2311yescharacteristic
39M32×C79—22no—
40C22⋊C8×C79—22no—
41C22×Dic79—21no—
42C22×Dic79—21no—
43#11929608—41no—
44#11929608—41no—
45C2×C79⋊C8—21no—
46C2×C79⋊C8C22⋊C412yes—
47C4⋊Dic79—22no—
48C4⋊Dic79—22no—
49C22⋊Dic79—22no—
50C22⋊Dic79—22no—
51C4×Dic79—21no—
52C4×Dic79—21no—
53C4×Dic79C22⋊C411yescharacteristic
54C4×Dic79C22⋊C411yescharacteristic
55C4×Dic79M1612yes—
56C79⋊C16—42no—
57C24×C79M1611yescharacteristic
58C22×C4×C79—21no—
59C22×C4×C79—21no—
60C22×C4×C79—41no—
61C22×C4×C79—41no—
62C22×C4×C79M1611yescharacteristic
63C22×C4×C79C22×C411yescharacteristic
64M16×C79—41no—
65M16×C79—41no—
66C2×C8×C79—21no—
67C2×C8×C79C2×D412yes—
68C16×C79—42no—
69C23×C79—21no—
70C23×C79—41no—
71C23×C79—81no—
72C23×C79C2×M1611yescharacteristic
73C2×C4×C79—21no—
74C2×C4×C79—21no—
75C2×C4×C79—21no—
76C2×C4×C79—42no—
77C2×C4×C79—41no—
78C2×C4×C79—41no—
79C2×C4×C79—41no—
80C2×C4×C79C2×C22⋊C411yescharacteristic
81C2×C4×C79C2×M1611yescharacteristic
82C2×Dic79—21no—
83C2×Dic79—21no—
84C2×Dic79—22no—
85C2×Dic79—81no—
86C2×Dic79—81no—
87C8×C79—22no—
88C8×C79—41no—
89C79⋊C8—22no—
90C79⋊C8—41no—
91C22×C79—21no—
92C22×C79—41no—
93C22×C79—42no—
94C22×C79—41no—
95C22×C79—82no—
96C22×C79#40711yescharacteristic
97C4×C79—21no—
98C4×C79—41no—
99C4×C79#41011yescharacteristic
100C4×C79#41311yescharacteristic
101Dic79—41no—
102Dic79—41no—
103Dic79—42no—
104#11522—791nomaximal, Sylow
105C2×C79—21no—
106C2×C79—41no—
107C2×C79—81no—
108C2×C79#146411yescharacteristic
109#2921—791no—
110#2165—792no—
111#1327—792no—
112#1320—791no—
113#1320D7911yescharacteristic
114C79#1152211yescharacteristic, complemented, Sylow
115C2×#117—791no—
116#433—1582no—
117#433—1582no—
118#423—791no—
119#423—791no—
120#407—791no—
121#407C2×D7911yescharacteristic
122C4×M16—791no—
123C4×M16—791no—
124#364—1582no—
125#349—792no—
126#349C2×D7912yes—
127#347—1582no—
128C23×C4C4×D7911yescharacteristic
129C2×M16—791no—
130C2×M16—791no—
131C2×M16C4×D7911yescharacteristic
132C2×M16C22×D7911yescharacteristic
133#117—1582no—
134#117—1581no—
135#117—1581no—
136C2×C4⋊C4—1581no—
137C2×C22⋊C4—1581no—
138C2×C42—791no—
139C2×C42—791no—
140M32—22no—
141M32—1582no—
142C4⋊C8—1582no—
143C4⋊C8—1582no—
144C22⋊C8—22no—
145C22⋊C8—1582no—
146#97—792no—
147#97—1581no—
148C4×C8—792no—
149C4×C8—1581no—
150C24#1192960711yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—1581no—
156C22×C4—1581no—
157C22×C4#1192960711yescharacteristic
158C22×C4C2×C4×D7911yescharacteristic
159M16—41no—
160M16—41no—
161M16—3161no—
162M16—3161no—
163C2×C8—21no—
164C2×C8—792no—
165C2×C8—1581no—
166C2×C8D4×D7912yes—
167C4⋊C4—1582no—
168C4⋊C4—1582no—
169C22⋊C4—1582no—
170C22⋊C4—1582no—
171C42—791no—
172C42—791no—
173C42—792no—
174C42—1581no—
175C42—1581no—
176C16—42no—
177C16—3162no—
178C23—21no—
179C23—41no—
180C23—81no—
181C23C2×#1192960711yescharacteristic
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—6321no—
190C2×C4—6321no—
191C2×C4—1581no—
192C2×C4—1581no—
193C2×C4—1582no—
194C2×C4C22⋊C4×D7911yescharacteristic
195C2×C4M16×D7911yescharacteristic
196C8—22no—
197C8—41no—
198C8—3161no—
199C8—1582no—
200C22—21no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22—82no—
205C22#1909852511yescharacteristic
206C4—21no—
207C4—41no—
208C4—3161no—
209C4—3161no—
210C4—3162no—
211C4#1909853511yescharacteristic
212C4#1909854411yescharacteristic
213C2—21no—
214C2—41no—
215C2—81no—
216C2#2376173111yescharacteristic
subgroupG/H
centerC4#19098535
commutatorC22×C4×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#1320C2
socleC2×C79#1464
prime radicalC24×C79M16
cosocleC2×M16×C79C23
hypercenter#1320D79
hypocenterC79#11522
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #23765209C2semidirect
3. #23761350C2semidirect
114. C79#11522semidirect

Maximal subgroups

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

subgroupindex
#237720352
#237652092
#237613502
#237613362
C79×#13202
#1152279