Finite Group Databasebeta

← #34197703

Subgroups — #34197703

Order 20224.

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

normal subgroups only

This group has 284 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#23772036C211yesmaximal, characteristic
2#23765212C211yesmaximal, characteristic, complemented
3#23761350C212yesmaximal, complemented
4#23761335C212yesmaximal
5C79×#2165C211yesmaximal, characteristic, complemented
6C2×#14579118C411yescharacteristic
7#19098625—21no—
8#19098625—21no—
9#19098622—21no—
10#19098622—21no—
11#19098586C411yescharacteristic
12#19098586C2211yescharacteristic
13#19098526C411yescharacteristic
14#19098526C2211yescharacteristic
15C4×#11929608C411yescharacteristic
16#19098384—21no—
17#19098345C2212yes—
18#19098341—21no—
19C4×M16×C79C2211yescharacteristic
20#364×C79—21no—
21#349×C79C2212yes—
22#347×C79—21no—
23C22×C4×D79C2×C411yescharacteristic
24C2×#11929608C2×C411yescharacteristic
25C2×#11929608C2×C411yescharacteristic
26#14579118—21no—
27#14579118—21no—
28#14579118—22no—
29C2×Dic79⋊C4—21no—
30C2×(C2×D79)⋊C4—21no—
31C2×C4×Dic79C2×C411yescharacteristic
32#14579101—22no—
33Dic79⋊C8—21no—
34Dic79⋊C8—21no—
35Dic79⋊C8—21no—
36Dic79⋊C8—21no—
37(C2×D79)⋊C8—21no—
38(C2×D79)⋊C8—21no—
39(C2×D79)⋊C8—21no—
40(C2×D79)⋊C8—21no—
41#14579072—21no—
42C4×C79⋊C8—21no—
43C2×M16×C79C2×C411yescharacteristic
44C2×M16×C79C2311yescharacteristic
45C2×C42×C79C2×C411yescharacteristic
46M32×C79—22no—
47#97×C79D411yescharacteristic
48#97×C79D411yescharacteristic
49C4×C8×C79D411yescharacteristic
50C4×C8×C79D411yescharacteristic
51C23×D79M1611yescharacteristic
52C22×Dic79—21no—
53C22×Dic79M1611yescharacteristic
54C2×C4×D79—21no—
55C2×C4×D79—21no—
56C2×C4×D79—41no—
57C2×C4×D79—41no—
58#11929608—41no—
59#11929608—41no—
60C2×C79⋊C8—21no—
61C2×C79⋊C8—21no—
62Dic79⋊C4—22no—
63Dic79⋊C4—22no—
64(C2×D79)⋊C4—22no—
65(C2×D79)⋊C4—22no—
66C4×Dic79—21no—
67C4×Dic79M1612yes—
68C79⋊C16—42no—
69C22×C4×C79—21no—
70C22×C4×C79C22×C411yescharacteristic
71M16×C79—41no—
72M16×C79—41no—
73C2×C8×C79C22⋊C411yescharacteristic
74C2×C8×C79C22⋊C411yescharacteristic
75C2×C8×C79C2×D411yescharacteristic
76C2×C8×C79C2×D411yescharacteristic
77C42×C79—21no—
78C42×C79C22⋊C411yescharacteristic
79C42×C79C22⋊C411yescharacteristic
80C16×C79—42no—
81C23×C79C2×M1611yescharacteristic
82C22×D79—21no—
83C22×D79—41no—
84C22×D79—81no—
85C2×C4×C79—21no—
86C2×C4×C79—21no—
87C2×C4×C79—21no—
88C2×C4×C79—21no—
89C2×C4×C79—81no—
90C2×C4×C79C2×C22⋊C411yescharacteristic
91C2×C4×C79C2×M1611yescharacteristic
92C2×Dic79—22no—
93C2×Dic79—21no—
94C2×Dic79—41no—
95C2×Dic79—81no—
96C4×D79—42no—
97C4×D79—41no—
98C4×D79—41no—
99C8×C79—21no—
100C8×C79—21no—
101C8×C79—21no—
102C8×C79—21no—
103C79⋊C8—41no—
104C79⋊C8—41no—
105C22×C79—21no—
106C22×C79—41no—
107C22×C79#40711yescharacteristic
108C2×D79—42no—
109C2×D79—41no—
110C2×D79—82no—
111C4×C79—21no—
112C4×C79—41no—
113C4×C79—41no—
114C4×C79#41011yescharacteristic
115C4×C79#41311yescharacteristic
116Dic79—42no—
117Dic79—41no—
118#11522—791nomaximal, Sylow
119C2×C79—21no—
120C2×C79—41no—
121C2×C79#146411yescharacteristic
122D79—81no—
123#2921—791no—
124#2165—791no—
125#2165D7911yescharacteristic, complemented
126#1327—792no—
127#1320—792no—
128C79#1152211yescharacteristic, complemented, Sylow
129C2×#117—791no—
130#433—1581no—
131#433—1581no—
132#433—1581no—
133#433—1581no—
134#423—791no—
135#423—791no—
136#407—791no—
137#407—791no—
138C4×M16—791no—
139C4×M16C2×D7911yescharacteristic
140#364—21no—
141#364—1581no—
142#349—792no—
143#349C2×D7912yes—
144#347—21no—
145#347—1581no—
146C23×C4—791no—
147C2×M16—791no—
148C2×M16—791no—
149C2×M16C4×D7911yescharacteristic
150C2×M16C22×D7911yescharacteristic
151#117—1582no—
152#117—1581no—
153#117—1581no—
154C2×C4⋊C4—1581no—
155C2×C22⋊C4—1581no—
156C2×C42—791no—
157C2×C42C4×D7911yescharacteristic
158M32—22no—
159M32—1582no—
160C4⋊C8—1581no—
161C4⋊C8—1581no—
162C4⋊C8—1581no—
163C4⋊C8—1581no—
164C22⋊C8—1581no—
165C22⋊C8—1581no—
166C22⋊C8—1581no—
167C22⋊C8—1581no—
168#97—1581no—
169#97D31611yescharacteristic
170#97C22⋊D7911yescharacteristic
171C4×C8—1581no—
172C4×C8D31611yescharacteristic
173C4×C8C22⋊D7911yescharacteristic
174C24—791no—
175C22×C4—21no—
176C22×C4—3161no—
177C22×C4—3161no—
178C22×C4—791no—
179C22×C4—1581no—
180C22×C4—1581no—
181C22×C4—1581no—
182C22×C4C2×C4×D7911yescharacteristic
183M16—41no—
184M16—41no—
185M16—3161no—
186M16—3161no—
187C2×C8—1581no—
188C2×C8—1581no—
189C2×C8(C2×D79)⋊C411yescharacteristic
190C2×C8(C2×D79)⋊C411yescharacteristic
191C2×C8C2×C22⋊D7911yescharacteristic
192C2×C8C2×D31611yescharacteristic
193C4⋊C4—1582no—
194C4⋊C4—1582no—
195C22⋊C4—1582no—
196C22⋊C4—1582no—
197C42—21no—
198C42—792no—
199C42—1581no—
200C42(C2×D79)⋊C411yescharacteristic
201C42(C2×D79)⋊C411yescharacteristic
202C16—42no—
203C16—3162no—
204C23—3161no—
205C23—6321no—
206C23—1581no—
207C23M16×D7911yescharacteristic
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—81no—
213C2×C4—3162no—
214C2×C4—3161no—
215C2×C4—3161no—
216C2×C4—3161no—
217C2×C4—6321no—
218C2×C4—1582no—
219C2×C4—1581no—
220C2×C4C2×(C2×D79)⋊C411yescharacteristic
221C2×C4M16×D7911yescharacteristic
222C8—21no—
223C8—21no—
224C8—21no—
225C8—21no—
226C8—3161no—
227C8—3161no—
228C22—21no—
229C22—41no—
230C22—3162no—
231C22—3161no—
232C22—6322no—
233C22#1909852611yescharacteristic
234C4—21no—
235C4—41no—
236C4—41no—
237C4—3162no—
238C4—3161no—
239C4#1909853611yescharacteristic
240C4#1909854511yescharacteristic
241C2—21no—
242C2—41no—
243C2—6321no—
244C2#2376173211yescharacteristic
subgroupG/H
centerC4#19098536
commutatorC22×C4×C79C22×C4
FrattiniC2×M16C22×D79
FittingC79×#2165C2
socleC2×C79#1464
prime radicalC23×D79M16
cosocleC2×M16×C79C23
hypercenter#2165D79
hypocenterC79#11522
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #23765212C2semidirect
3. #23761350C2semidirect
5. C79×#2165C2semidirect
125. #2165D79semidirect
128. C79#11522semidirect

Maximal subgroups

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

subgroupindex
#237720362
#237652122
#237613502
#237613352
C79×#21652
#1152279