Finite Group Databasebeta

← #34204301

Subgroups — #34204301

Order 20224.

420 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 89 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#23772108C211yesmaximal, characteristic
2#23765083C212yesmaximal
3#23763448C212yesmaximal, complemented
4#23761523C211yesmaximal, characteristic, complemented
5C79×#1392C211yesmaximal, characteristic, complemented
6C2×#14579169C411yescharacteristic
7C2×#14579168C2211yescharacteristic
8#19099109C2211yescharacteristic
9#19098638—22no—
10(C22⋊D79)⋊C8—22no—
11C2×#14579096—22no—
12#19098586C411yescharacteristic
13#19098547C2212yes—
14#19098529C411yescharacteristic
15C2×C8×Dic79C411yescharacteristic
16#19098361—22no—
17#19098359—22no—
18C79⋊#338—22no—
19#413×C79C2212yes—
20#408×C79C2211yescharacteristic
21C2×#11929624C2×C411yescharacteristic
22#14579169—42no—
23#14579168—42no—
24C2×#11929608—22no—
25C2×#11929608C2×C411yescharacteristic
26C2×#11929608D412yes—
27C2×#11929607—21no—
28C22×C79⋊C8C2×C411yescharacteristic
29C22×C79⋊C8D412yes—
30C2×C8×D79—21no—
31#14579159D411yescharacteristic
32#14579157Q811yescharacteristic
33#14579150D411yescharacteristic
34#14579148Q811yescharacteristic
35#14579133—22no—
36C4×C22⋊D79—22no—
37C2×C4×Dic79C2×C411yescharacteristic
38#14579096—44no—
39Dic79⋊C8—22no—
40#14579080—22no—
41#14579078—22no—
42(C2×D79)⋊C8—22no—
43C8×Dic79—21no—
44C8×Dic79—21no—
45C2×#41×C79C2311yescharacteristic
46C2×M16×C79—22no—
47C2×M16×C79C2×C411yescharacteristic
48C22×C8×C79C2×C411yescharacteristic
49#101×C79—22no—
50#100×C79—22no—
51C22⋊C8×C79—22no—
52#11929624—41no—
53#11929624—41no—
54C2×Dic158C4⋊C411yescharacteristic
55C2×D316C4⋊C411yescharacteristic
56C2×C22⋊D79—21no—
57C2×C22⋊D79—22no—
58C22×Dic79—21no—
59C2×C4×D79—21no—
60#11929608—41no—
61#11929608—42no—
62#11929608—42no—
63#11929608—42no—
64#11929607—42no—
65C2×C79⋊C8—24no—
66C2×C79⋊C8—22no—
67C2×C79⋊C8—42no—
68C2×C79⋊C8C22⋊C411yescharacteristic
69C2×C79⋊C8C22⋊C412yes—
70C2×C79⋊C8C22⋊C411yescharacteristic
71C8×D79—42no—
72Dic79⋊C4—22no—
73C22⋊Dic79—22no—
74(C2×D79)⋊C4—22no—
75C4×Dic79—21no—
76C4×Dic79C4⋊C411yescharacteristic
77C4×Dic79C4⋊C411yescharacteristic
78#41×C79—41no—
79#41×C79—41no—
80C2×Q8×C79C2×D411yescharacteristic
81C2×D4×C79C2×Q811yescharacteristic
82C2×D4×C79#4112yes—
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D412yes—
85M16×C79—41no—
86M16×C79—42no—
87M16×C79—42no—
88C2×C8×C79—21no—
89C2×C8×C79—21no—
90C2×C8×C79—41no—
91C2×C8×C79—42no—
92C23×C79#11711yescharacteristic
93C23×C79C22⋊Q812yes—
94C22×D79—21no—
95Q8×C79—41no—
96D4×C79—41no—
97D4×C79—42no—
98C2×C4×C79—22no—
99C2×C4×C79—21no—
100C2×C4×C79—22no—
101C2×C4×C79—21no—
102C2×C4×C79C2×C22⋊C411yescharacteristic
103C2×C4×C79C2×C4⋊C411yescharacteristic
104C2×C4×C79#12112yes—
105C22⋊D79—44no—
106C22⋊D79—42no—
107C2×Dic79—21no—
108C2×Dic79—21no—
109C2×Dic79—21no—
110C2×Dic79—42no—
111D316—41no—
112C4×D79—41no—
113C4×D79—41no—
114Dic158—41no—
115C8×C79—41no—
116C8×C79—41no—
117C8×C79—82no—
118C79⋊C8—21no—
119C79⋊C8—22no—
120C79⋊C8—82no—
121C79⋊C8#9912yes—
122C22×C79—21no—
123C22×C79—22no—
124C22×C79—42no—
125C22×C79—41no—
126C22×C79#38011yescharacteristic
127C2×D79—42no—
128C2×D79—41no—
129C4×C79—21no—
130C4×C79—22no—
131C4×C79C2×#9911yescharacteristic
132C4×C79#41011yescharacteristic
133Dic79—22no—
134Dic79—41no—
135Dic79—41no—
136#12809—791nomaximal, Sylow
137C2×C79—21no—
138C2×C79—41no—
139C2×C79—42no—
140C2×C79#179911yescharacteristic
141D79—81no—
142#2931—791no—
143#2132—792no—
144#1813—792no—
145#1392—791no—
146#1392D7911yescharacteristic, complemented
147C79#1280911yescharacteristic, complemented, Sylow
148C2×#131—791no—
149C2×#131—791no—
150#532—791no—
151#436—1582no—
152C8×D4—1582no—
153C2×#108—1582no—
154#423—791no—
155#413—792no—
156#413C2×D7912yes—
157#408—791no—
158#408C2×D7911yescharacteristic
159C2×C4×C8—791no—
160#356—1582no—
161#355—1582no—
162#338—1582no—
163C2×#41—791no—
164C2×#41C22×D7911yescharacteristic
165#131—3162no—
166#131—3162no—
167C2×M16—22no—
168C2×M16—791no—
169C2×M16—792no—
170C2×M16—1581no—
171C2×M16—1582no—
172C2×M16C4×D7911yescharacteristic
173C22×C8—791no—
174C22×C8—792no—
175C22×C8—1581no—
176C22×C8C4×D7911yescharacteristic
177C4⋊Q8—791no—
178#127—791no—
179#124—791no—
180#124—791no—
181#121—1582no—
182C4×D4—1582no—
183C2×C42—791no—
184#108—3164no—
185C4⋊C8—1582no—
186#101—22no—
187#101—1582no—
188#100—22no—
189#100—1582no—
190C22⋊C8—22no—
191C22⋊C8—1582no—
192C4×C8—1581no—
193C4×C8—1581no—
194#41—41no—
195#41—41no—
196#41—3161no—
197#41—3161no—
198C2×Q8—791no—
199C2×Q8D4×D7911yescharacteristic
200C2×D4—791no—
201C2×D4—1581no—
202C2×D4—1582no—
203C2×D4Q8×D7911yescharacteristic
204C2×D4#1192962412yes—
205C22×C4—1581no—
206C22×C4—1581no—
207C22×C4C2×C4×D7911yescharacteristic
208C22×C4D4×D7912yes—
209M16—41no—
210M16—42no—
211M16—42no—
212M16—3162no—
213M16—3162no—
214M16—3162no—
215M16—3162no—
216M16—3161no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—41no—
220C2×C8—42no—
221C2×C8—3162no—
222C2×C8—3162no—
223C2×C8—791no—
224C2×C8—791no—
225C2×C8—792no—
226C2×C8—1584no—
227C2×C8—1582no—
228C4⋊C4—1582no—
229C22⋊C4—1582no—
230C22⋊C4—1582no—
231C42—791no—
232C42—791no—
233C42—1581no—
234C23—1581no—
235C23#1457911811yescharacteristic
236C23#1457913912yes—
237Q8—41no—
238Q8—3161no—
239D4—41no—
240D4—42no—
241D4—3161no—
242D4—3162no—
243D4—3164no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—3162no—
249C2×C4—3161no—
250C2×C4—3161no—
251C2×C4—1581no—
252C2×C4—1581no—
253C2×C4—1581no—
254C2×C4C22⋊C4×D7911yescharacteristic
255C2×C4C4⋊C4×D7911yescharacteristic
256C2×C4#1457913412yes—
257C8—41no—
258C8—41no—
259C8—82no—
260C8—792no—
261C8—6322no—
262C8—1581no—
263C8—1582no—
264C22—21no—
265C22—22no—
266C22—41no—
267C22—42no—
268C22—3162no—
269C22—3161no—
270C22#1909842211yescharacteristic
271C4—22no—
272C4—21no—
273C4—3161no—
274C4—3161no—
275C4—1582no—
276C4#99×D7911yescharacteristic
277C4#1909853511yescharacteristic
278C2—21no—
279C2—41no—
280C2—42no—
281C2—6321no—
282C2#2376338711yescharacteristic
subgroupG/H
centerC4#19098535
commutatorC22×C4×C79C22×C4
FrattiniC2×#41C22×D79
FittingC79×#1392C2
socleC2×C79#1799
prime radical#19099109C22
cosocleC2×#41×C79C23
hypercenter#1392D79
hypocenterC79#12809
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1392

Complements

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

NH ≅ G/Ntype
3. #23763448C2semidirect
4. #23761523C2semidirect
5. C79×#1392C2semidirect
146. #1392D79semidirect
147. C79#12809semidirect

Maximal subgroups

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

subgroupindex
#237721082
#237650832
#237634482
#237615232
C79×#13922
#1280979