Finite Group Databasebeta

← #34201489

Subgroups — #34201489

Order 20224.

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

normal subgroups only

This group has 396 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#23774151C211yesmaximal, characteristic
2#23765586C212yesmaximal, complemented
3#23764828C211yesmaximal, characteristic, complemented
4#23761410C212yesmaximal, complemented
5C79×#1314C211yesmaximal, characteristic, complemented
6C2×#14579135—21no—
7#19098877—22no—
8#19098799—21no—
9#19098799C2212yes—
10#19098699—22no—
11C2×#14579095C2211yescharacteristic, complemented
12#19098423—21no—
13#19098391—22no—
14#19098365—22no—
15#19098326—21no—
16#19098326C2211yescharacteristic, complemented
17C2×C22⋊C8×C79C2211yescharacteristic
18C22⋊C16×C79C2212yescomplemented
19C22×D316—21no—
20C22×C22⋊D79—41no—
21C2×D632—21no—
22C2×D632D412yes—
23#14579135—22no—
24#14579135—22no—
25#14579135—41no—
26#14579135—41no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×C22⋊Dic79—41no—
31C2×(C2×D79)⋊C4—41no—
32#14579095—21no—
33#14579095—22no—
34#14579095D412yes—
35#14579083—22no—
36#14579083—22no—
37C79⋊#95—41no—
38C23×C4×C79D411yescharacteristic
39C22×C8×C79D411yescharacteristic
40C22×C8×C79C2311yescharacteristic
41C2×C16×C79—22no—
42C22⋊C8×C79—22no—
43C23×D79—41no—
44C2×D316—22no—
45C2×D316—41no—
46C2×D316—81no—
47C2×D316—81no—
48C2×C22⋊D79—42no—
49C2×C22⋊D79—42no—
50C2×C22⋊D79—82no—
51C2×C22⋊D79—82no—
52C22×Dic79—41no—
53C22×Dic79—41no—
54C22×Dic79—41no—
55D632—42no—
56D632—81no—
57C4⋊Dic79—22no—
58C4⋊Dic79—41no—
59C4⋊Dic79—41no—
60C4⋊Dic79—41no—
61C4⋊Dic79—41no—
62C4⋊Dic79—41no—
63C22⋊Dic79—82no—
64(C2×D79)⋊C4—42no—
65(C2×D79)⋊C4—42no—
66C24×C79D811yescharacteristic
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—41no—
70C22×C4×C79—41no—
71C22×C4×C79D811yescharacteristic
72C22×C4×C79C2×D411yescharacteristic
73C2×C8×C79—21no—
74C2×C8×C79—41no—
75C2×C8×C79—41no—
76C2×C8×C79C2×D412yes—
77C16×C79—42no—
78C23×C79—21no—
79C23×C79—21no—
80C23×C79—41no—
81C23×C79—41no—
82C23×C79SD3212yes—
83C23×C79C2×D811yescharacteristic
84C22×D79—42no—
85C22×D79—82no—
86C22×D79—82no—
87C22×D79—82no—
88C2×C4×C79—21no—
89C2×C4×C79—21no—
90C2×C4×C79—21no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79—41no—
94C2×C4×C79—81no—
95C2×C4×C79#12011yescharacteristic
96C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
97C22⋊D79—84no—
98C22⋊D79—82no—
99C22⋊D79—82no—
100C2×Dic79—42no—
101C2×Dic79—82no—
102C2×Dic79—81no—
103C2×Dic79—81no—
104C2×Dic79—81no—
105C2×Dic79—81no—
106C2×Dic79—81no—
107C2×Dic79—81no—
108D316—42no—
109D316—42no—
110D316—82no—
111C8×C79—22no—
112C8×C79—41no—
113C22×C79—21no—
114C22×C79—21no—
115C22×C79—22no—
116C22×C79—42no—
117C22×C79—41no—
118C22×C79—41no—
119C22×C79—81no—
120C22×C79#44711yescharacteristic
121C22×C79C2×SD3211yescharacteristic
122C22×C79#50911yescharacteristic
123C2×D79—82no—
124C2×D79—82no—
125C2×D79—82no—
126C2×D79—164no—
127C4×C79—21no—
128C4×C79—41no—
129C4×C79#45511yescharacteristic
130C4×C79#45811yescharacteristic
131Dic79—82no—
132Dic79—161no—
133Dic79—161no—
134#12281—791nomaximal, Sylow
135C2×C79—21no—
136C2×C79—21no—
137C2×C79—42no—
138C2×C79#160711yescharacteristic
139C2×C79#218711yescharacteristic
140C2×C79#219011yescharacteristic
141D79—162no—
142#3186—791no—
143#2233—792no—
144#2076—791no—
145#1347—792no—
146#1314D7911yescharacteristic, complemented
147C79#1228111yescharacteristic, complemented, Sylow
148C2×#121—1581no—
149#480—1582no—
150#466—792no—
151#466—1581no—
152#447—1582no—
153C2×#107—791no—
154C2×C22⋊C8C2×D7911yescharacteristic
155#380—1581no—
156#367—1582no—
157#357—1582no—
158C22⋊C16C2×D7912yescomplemented
159#340—791no—
160#340—1581no—
161C22×D4—3161no—
162C22×D4—1581no—
163C23×C4D31611yescharacteristic
164C2×D8—792no—
165C2×D8—1581no—
166C22×C8D31611yescharacteristic
167C22×C8C22×D7911yescharacteristic
168#121—3161no—
169#121—3161no—
170#121—1582no—
171#121—1582no—
172C2×C4⋊C4—1581no—
173C2×C4⋊C4—1581no—
174C2×C4⋊C4—1581no—
175C2×C22⋊C4—3161no—
176C2×C22⋊C4—3161no—
177C2×C16—22no—
178#107—792no—
179#107—1582no—
180#107—1581no—
181#102—1582no—
182#102—1582no—
183C22⋊C8—22no—
184#95—3161no—
185C24—3161no—
186C24D63211yescharacteristic
187C2×D4—3161no—
188C2×D4—3162no—
189C2×D4—3162no—
190C2×D4—6321no—
191C2×D4—6321no—
192C2×D4—6322no—
193C2×D4—6322no—
194C2×D4—1582no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—3161no—
200C22×C4—3161no—
201C22×C4—3161no—
202C22×C4D63211yescharacteristic
203C22×C4C2×D31611yescharacteristic
204D8—3162no—
205D8—6321no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D7912yes—
210C4⋊C4—3161no—
211C4⋊C4—3161no—
212C4⋊C4—3161no—
213C4⋊C4—3161no—
214C4⋊C4—3161no—
215C4⋊C4—1582no—
216C22⋊C4—3162no—
217C22⋊C4—3162no—
218C22⋊C4—6322no—
219C16—42no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—3162no—
225C23—6322no—
226C23—6322no—
227C23—6322no—
228C23#1457910612yes—
229C23C2×D63211yescharacteristic
230D4—3162no—
231D4—3162no—
232D4—6322no—
233D4—6324no—
234D4—6322no—
235D4—6322no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—81no—
243C2×C4—3162no—
244C2×C4—6322no—
245C2×C4—6321no—
246C2×C4—6321no—
247C2×C4—6321no—
248C2×C4—6321no—
249C2×C4—6321no—
250C2×C4—6321no—
251C2×C4#1457913111yescharacteristic
252C2×C4#1457918911yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—6322no—
263C22—6322no—
264C22—6322no—
265C22—12644no—
266C22#1909869911yescharacteristic
267C22C2×#1457910611yescharacteristic
268C22#1909900111yescharacteristic
269C4—21no—
270C4—41no—
271C4—6322no—
272C4—12641no—
273C4—12641no—
274C4#1909874311yescharacteristic
275C4#1909875411yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—12642no—
280C2#2376238811yescharacteristic
281C2#2376533111yescharacteristic
282C2#2376534811yescharacteristic
subgroupG/H
centerC22#19098699
commutator, cosocleC22×C8×C79C23
FrattiniC22×C8C22×D79
FittingC79×#1314C2
socleC22×C79#447
prime radical#23774151C2
hypercenter#1314D79
hypocenterC79#12281
LCSG ▷ C22×C8×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #23765586C2semidirect
3. #23764828C2semidirect
4. #23761410C2semidirect
5. C79×#1314C2semidirect
11. C2×#14579095C22semidirect
16. #19098326C22semidirect
18. C22⋊C16×C79C22semidirect
146. #1314D79semidirect
147. C79#12281semidirect
158. C22⋊C16C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237741512
#237655862
#237648282
#237614102
C79×#13142
#1228179