Finite Group Databasebeta

← #34208686

Subgroups — #34208686

Order 20224.

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

normal subgroups only

This group has 316 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#23774461C211yesmaximal, characteristic
2#23765089C211yesmaximal, characteristic, complemented
3#23761572C212yesmaximal, complemented
4#23761566C212yesmaximal, complemented
5C79×#2135C211yesmaximal, characteristic, complemented
6#19099168C411yescharacteristic
7#19098953C411yescharacteristic
8#19098937C2211yescharacteristic
9#19098931C2211yescharacteristic
10#19098917C411yescharacteristic
11#19098775—21no—
12#19098765—21no—
13#19098708—21no—
14#19098702—21no—
15#19098624C411yescharacteristic
16C2×#14579080—21no—
17#19098362C2212yes—
18#19098360—21no—
19#536×C79C2211yescharacteristic
20#413×C79—21no—
21#356×C79—21no—
22#356×C79C2212yes—
23C2×#11929611—21no—
24C2×#11929611—21no—
25C2×#11929611—21no—
26C2×#11929609—21no—
27C2×#11929608—41no—
28#14579158C2×C411yescharacteristic
29#14579153C2×C411yescharacteristic
30#14579146—41no—
31#14579135—21no—
32#14579134—41no—
33C4×D316—21no—
34C4⋊(C79⋊C8)—21no—
35#14579085—21no—
36#14579085—21no—
37#14579085—21no—
38#14579082—21no—
39#14579080—22no—
40#14579080—41no—
41C22⋊(C79⋊C8)—21no—
42#14579072C2×C411yescharacteristic
43C4×C79⋊C8C2×C411yescharacteristic
44C2×#41×C79D411yescharacteristic
45C22×Q8×C79D411yescharacteristic
46C2×M16×C79—41no—
47C4⋊Q8×C79D411yescharacteristic
48C4⋊Q8×C79C2311yescharacteristic
49#124×C79C2×C411yescharacteristic
50#124×C79D411yescharacteristic
51#123×C79—41no—
52C22⋊Q8×C79—41no—
53#117×C79C2×C411yescharacteristic
54#101×C79—22no—
55#100×C79—41no—
56C2×D316—21no—
57C2×D316—41no—
58C2×C22⋊D79—41no—
59C2×C4×D79—41no—
60#11929611—81no—
61#11929611—81no—
62#11929611—81no—
63#11929609—81no—
64#11929608—42no—
65#11929608—81no—
66C2×C79⋊C8—21no—
67C2×C79⋊C8—21no—
68C2×C79⋊C8—81no—
69C4⋊Dic79—21no—
70Dic79⋊C4—41no—
71(C2×D79)⋊C4—41no—
72(C2×D79)⋊C4—41no—
73#41×C79—81no—
74C2×Q8×C79—21no—
75C2×Q8×C79—81no—
76C2×Q8×C79C22⋊C411yescharacteristic
77C2×Q8×C79C2×D411yescharacteristic
78C2×Q8×C79C2×D411yescharacteristic
79C2×D4×C79—21no—
80C2×D4×C79C22⋊C411yescharacteristic
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79C22⋊C411yescharacteristic
84M16×C79—42no—
85M16×C79—81no—
86C2×C8×C79—81no—
87C4⋊C4×C79—21no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—41no—
90C22⋊C4×C79—21no—
91C22⋊C4×C79—41no—
92C22⋊C4×C79—41no—
93C42×C79C22⋊C411yescharacteristic
94C42×C79C22×C411yescharacteristic
95C23×C79—21no—
96C23×C79#9911yescharacteristic
97C22×D79—41no—
98Q8×C79—41no—
99Q8×C79—41no—
100Q8×C79—41no—
101Q8×C79—81no—
102D4×C79—41no—
103D4×C79—81no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79#9911yescharacteristic
115C2×C4×C79C2×C22⋊C411yescharacteristic
116C22⋊D79—161no—
117C2×Dic79—41no—
118D316—41no—
119D316—41no—
120D316—161no—
121C4×D79—161no—
122C8×C79—82no—
123C79⋊C8—41no—
124C79⋊C8—41no—
125C79⋊C8—82no—
126C22×C79—21no—
127C22×C79—41no—
128C22×C79—41no—
129C22×C79—81no—
130C22×C79C2×#9911yescharacteristic
131C2×D79—81no—
132C2×D79—81no—
133C2×D79—81no—
134C4×C79—21no—
135C4×C79—21no—
136C4×C79—41no—
137C4×C79—41no—
138C4×C79—41no—
139C4×C79—41no—
140C4×C79—41no—
141Dic79—81no—
142#13611—791nomaximal, Sylow
143C2×C79—21no—
144C2×C79—41no—
145C2×C79—81no—
146C2×C79#212711yescharacteristic
147D79—161no—
148#3221—791no—
149#2135D7911yescharacteristic, complemented
150#2133—791no—
151#1413—792no—
152#1410—792no—
153C79#1361111yescharacteristic, complemented, Sylow
154#540—791no—
155#536C2×D7911yescharacteristic
156#496—791no—
157#492—791no—
158#491—791no—
159#488—791no—
160#463—1581no—
161#461—1581no—
162#449—1581no—
163#448—1581no—
164#433—791no—
165#413—21no—
166C2×#101—1581no—
167#356—21no—
168#356—792no—
169#356C2×D7912yes—
170#355—1581no—
171C2×#41C22⋊D7911yescharacteristic
172C22×Q8D31611yescharacteristic
173C2×SD16—1581no—
174C2×SD16—1581no—
175C2×SD16—1581no—
176C2×D8—1581no—
177C2×M16—41no—
178C2×M16—3161no—
179C4⋊Q8C22⋊D7911yescharacteristic
180C4⋊Q8C22×D7911yescharacteristic
181#127—791no—
182#125—791no—
183#124C4×D7911yescharacteristic
184#124D31611yescharacteristic
185#123—41no—
186#123—3161no—
187C22⋊Q8—41no—
188#121—3161no—
189#121—1581no—
190C4×D4—1581no—
191#117C4×D7911yescharacteristic
192C4⋊C8—1581no—
193#103—1581no—
194#103—1581no—
195#103—1581no—
196#102—1581no—
197#101—22no—
198#101—3161no—
199#101—1582no—
200#100—41no—
201C22⋊C8—1581no—
202#97—791no—
203C4×C8—791no—
204#41—81no—
205C2×Q8—21no—
206C2×Q8—81no—
207C2×Q8(C2×D79)⋊C411yescharacteristic
208C2×Q8C2×C22⋊D7911yescharacteristic
209C2×Q8C2×D31611yescharacteristic
210C2×D4—21no—
211C2×D4—3161no—
212C2×D4—3161no—
213C2×D4—1581no—
214C2×D4(C2×D79)⋊C411yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—3161no—
218C22×C4(C2×D79)⋊C411yescharacteristic
219SD16—6321no—
220SD16—6321no—
221SD16—6321no—
222D8—6321no—
223M16—42no—
224M16—81no—
225M16—3162no—
226M16—6321no—
227C2×C8—81no—
228C2×C8—6321no—
229C2×C8—1581no—
230C2×C8—1581no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—3161no—
235C4⋊C4—1581no—
236C22⋊C4—21no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—3161no—
240C22⋊C4—3161no—
241C42(C2×D79)⋊C411yescharacteristic
242C42C2×C4×D7911yescharacteristic
243C23—21no—
244C23—3161no—
245C23#1457907511yescharacteristic
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—81no—
250D4—41no—
251D4—81no—
252D4—3161no—
253D4—3161no—
254D4—12641no—
255D4—12641no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—3161no—
267C2×C4—12641no—
268C2×C4#1457907511yescharacteristic
269C2×C4C2×(C2×D79)⋊C411yescharacteristic
270C8—82no—
271C8—3161no—
272C8—3161no—
273C8—6322no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—6321no—
279C22—6321no—
280C22—6321no—
281C22C2×#1457907511yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—6321no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—12641no—
294C2#2376506711yescharacteristic
subgroupG/H
centerC2#23765067
commutatorC42×C79C22×C4
FrattiniC4⋊Q8C22×D79
FittingC79×#2135C2
socleC2×C79#2127
prime radical#23774461C2
cosocleC4⋊Q8×C79C23
hypercenter#2135D79
hypocenterC79#13611
LCSG ▷ C42×C79 ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #2135

Complements

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

NH ≅ G/Ntype
2. #23765089C2semidirect
3. #23761572C2semidirect
4. #23761566C2semidirect
5. C79×#2135C2semidirect
149. #2135D79semidirect
153. C79#13611semidirect

Maximal subgroups

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

subgroupindex
#237744612
#237650892
#237615722
#237615662
C79×#21352
#1361179