Finite Group Databasebeta

← #34201531

Subgroups — #34201531

Order 20224.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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#23774710C211yesmaximal, characteristic
2#23765580C212yesmaximal, complemented
3#23761408C212yesmaximal, complemented
4#23761340C211yesmaximal, characteristic, complemented
5C79×#2086C211yesmaximal, characteristic, complemented
6#19099113—21no—
7#19098844—22no—
8#19098799—21no—
9#19098799C2212yes—
10#19098759—22no—
11#19098388—22no—
12#19098363—22no—
13C22⋊(C79⋊C16)C2212yes—
14C2×#107×C79C2211yescharacteristic
15C2×C4⋊C8×C79C2211yescharacteristic
16#389×C79—21no—
17#340×C79—21no—
18#340×C79C2211yescharacteristic
19C2×#11929624—41no—
20C2×D632—21no—
21C2×D632D412yes—
22#14579161—21no—
23#14579158—21no—
24#14579149—41no—
25#14579135—22no—
26#14579135—41no—
27C4×D316—22no—
28C4×D316—41no—
29C2×C79⋊C16—22no—
30#14579095—21no—
31#14579095D412yes—
32#14579083—22no—
33#14579083—22no—
34C22×C8×C79D411yescharacteristic
35C22×C8×C79C2311yescharacteristic
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—41no—
39C2×C42×C79D411yescharacteristic
40#107×C79—22no—
41C4⋊C8×C79—22no—
42#95×C79—41no—
43#11929624—82no—
44#11929624—82no—
45C2×Dic158—41no—
46C2×D316—22no—
47C2×D316—41no—
48C2×D316—41no—
49C2×C22⋊D79—42no—
50C2×C4×D79—42no—
51D632—42no—
52D632—81no—
53C4⋊Dic79—22no—
54C4⋊Dic79—41no—
55(C2×D79)⋊C4—42no—
56(C2×D79)⋊C4—42no—
57C79⋊C16—42no—
58C22×C4×C79—41no—
59C22×C4×C79—41no—
60C22×C4×C79D811yescharacteristic
61C22×C4×C79D811yescharacteristic
62C22×C4×C79C2×D411yescharacteristic
63C2×C8×C79—21no—
64C2×C8×C79—41no—
65C2×C8×C79—41no—
66C2×C8×C79C2×D412yes—
67C4⋊C4×C79—22no—
68C4⋊C4×C79—41no—
69C4⋊C4×C79—41no—
70C4⋊C4×C79—41no—
71C4⋊C4×C79—82no—
72C42×C79—21no—
73C42×C79—21no—
74C23×C79C2×D811yescharacteristic
75C22×D79—42no—
76C2×C4×C79—21no—
77C2×C4×C79—21no—
78C2×C4×C79—21no—
79C2×C4×C79—21no—
80C2×C4×C79—21no—
81C2×C4×C79—41no—
82C2×C4×C79—42no—
83C2×C4×C79—81no—
84C2×C4×C79—81no—
85C2×C4×C79—81no—
86C2×C4×C79—81no—
87C2×C4×C79—81no—
88C2×C4×C79D1612yes—
89C2×C4×C79#12011yescharacteristic
90C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
91C22⋊D79—82no—
92C22⋊D79—82no—
93C2×Dic79—42no—
94D316—42no—
95D316—42no—
96D316—82no—
97D316—82no—
98C4×D79—84no—
99Dic158—82no—
100C8×C79—22no—
101C8×C79—41no—
102C22×C79—21no—
103C22×C79—21no—
104C22×C79#44711yescharacteristic
105C22×C79C2×D1611yescharacteristic
106C22×C79#50911yescharacteristic
107C2×D79—82no—
108C2×D79—82no—
109C2×D79—82no—
110C4×C79—21no—
111C4×C79—22no—
112C4×C79—41no—
113C4×C79—82no—
114C4×C79—161no—
115C4×C79#45611yescharacteristic
116C4×C79#45811yescharacteristic
117Dic79—82no—
118#12288—791nomaximal, Sylow
119C2×C79—21no—
120C2×C79—21no—
121C2×C79#161011yescharacteristic
122C2×C79#218411yescharacteristic
123C2×C79#219111yescharacteristic
124D79—162no—
125#3245—791no—
126#2232—792no—
127#2086D7911yescharacteristic, complemented
128#1347—792no—
129#1322—791no—
130C79#1228811yescharacteristic, complemented, Sylow
131#532—1581no—
132#474—1582no—
133#466—792no—
134#466—1581no—
135#459—1582no—
136C2×#107C2×D7911yescharacteristic
137C2×C4⋊C8C2×D7911yescharacteristic
138#389—21no—
139#366—1582no—
140#357—1582no—
141C22⋊C16—792no—
142#340—21no—
143#340C2×D7911yescharacteristic
144C2×#41—3161no—
145C2×D8—792no—
146C2×D8—1581no—
147C22×C8C22⋊D7911yescharacteristic
148C22×C8C22×D7911yescharacteristic
149C4⋊Q8—1581no—
150#127—1581no—
151#124—3161no—
152#121—3161no—
153#121—1582no—
154C4×D4—3161no—
155C4×D4—1582no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C42C22⋊D7911yescharacteristic
160C2×C16—1582no—
161#107—22no—
162#107—792no—
163#107—1581no—
164C4⋊C8—22no—
165#102—1582no—
166#102—1582no—
167#95—41no—
168#41—6322no—
169#41—6322no—
170C2×Q8—3161no—
171C2×D4—3161no—
172C2×D4—3161no—
173C2×D4—3162no—
174C2×D4—1582no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—3162no—
178C22×C4#1192960911yescharacteristic
179C22×C4#1192960911yescharacteristic
180C22×C4C2×C22⋊D7911yescharacteristic
181D8—3162no—
182D8—6321no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8D4×D7912yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—82no—
192C4⋊C4—3161no—
193C4⋊C4—1582no—
194C22⋊C4—3162no—
195C22⋊C4—3162no—
196C42—21no—
197C42—21no—
198C16—3162no—
199C23—3162no—
200C23C2×#1192960911yescharacteristic
201Q8—6322no—
202D4—3162no—
203D4—3162no—
204D4—6322no—
205D4—6322no—
206D4—6322no—
207D4—6322no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—3162no—
221C2×C4—6324no—
222C2×C4#1457910212yes—
223C2×C4#1457912911yescharacteristic
224C2×C4#1457918611yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—6322no—
230C22—6322no—
231C22—6322no—
232C22#1909869611yescharacteristic
233C22C2×#1457910211yescharacteristic
234C22#1909899811yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—82no—
239C4—161no—
240C4—6322no—
241C4#1909874511yescharacteristic
242C4#1909875211yescharacteristic
243C2—21no—
244C2—21no—
245C2—12642no—
246C2#2376240211yescharacteristic
247C2#2376531111yescharacteristic
248C2#2376535011yescharacteristic
subgroupG/H
centerC22#19098696
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC79×#2086C2
socleC22×C79#447
prime radical#23774710C2
hypercenter#2086D79
hypocenterC79#12288
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2086

Complements

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

NH ≅ G/Ntype
2. #23765580C2semidirect
3. #23761408C2semidirect
4. #23761340C2semidirect
5. C79×#2086C2semidirect
127. #2086D79semidirect
130. C79#12288semidirect

Maximal subgroups

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

subgroupindex
#237747102
#237655802
#237614082
#237613402
C79×#20862
#1228879