Finite Group Databasebeta

← #34199413

Subgroups — #34199413

Order 20224.

440 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 99 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#23767717C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761426C214yesmaximal, complemented
4C79×#1800C211yesmaximal, characteristic
5C22×C4⋊Dic79C411yescharacteristic
6C2×C22⋊(C79⋊C8)—21no—
7#19098502C2212yes—
8#19098499C412yes—
9#19098461—21no—
10#19098443—21no—
11#19098435—21no—
12#19098423—21no—
13#19098423C411yescharacteristic
14#19098289C2212yes—
15#19098287—21no—
16C2×C4×D4×C79C2211yescharacteristic
17#423×C79—21no—
18C2×#100×C79—21no—
19#343×C79C2212yes—
20C23×Dic79—21no—
21C22×C79⋊C8—41no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—41no—
25C2×C4⋊Dic79—41no—
26C2×C4⋊Dic79C2×C412yes—
27C2×Dic79⋊C4—22no—
28C2×C22⋊Dic79—21no—
29C2×C22⋊Dic79—22no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—41no—
32C79⋊#95—22no—
33C79⋊#95—22no—
34C79⋊#95C2×C412yes—
35C22×D4×C79D411yescharacteristic
36C23×C4×C79C2×C411yescharacteristic
37C23×C4×C79D411yescharacteristic
38C2×M16×C79—22no—
39C4×D4×C79—41no—
40C4×D4×C79—41no—
41C2×C4⋊C4×C79C2311yescharacteristic
42C2×C22⋊C4×C79C2×C411yescharacteristic
43C2×C22⋊C4×C79D411yescharacteristic
44C2×C42×C79D411yescharacteristic
45C4⋊C8×C79—41no—
46C4⋊C8×C79—41no—
47#100×C79—42no—
48C22×Dic79—22no—
49C22×Dic79—22no—
50C22×Dic79—42no—
51C22×Dic79—82no—
52C2×C79⋊C8—42no—
53C2×C79⋊C8—82no—
54C4⋊Dic79—42no—
55C4⋊Dic79—42no—
56C4⋊Dic79—41no—
57C4⋊Dic79—41no—
58Dic79⋊C4—82no—
59C22⋊Dic79—42no—
60C22⋊Dic79—82no—
61C24×C79C22⋊C411yescharacteristic
62C24×C79C22⋊C411yescharacteristic
63C2×D4×C79—41no—
64C2×D4×C79—41no—
65C2×D4×C79SD1612yes—
66C2×D4×C79Q1612yes—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—42no—
72C22×C4×C79—42no—
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22×C411yescharacteristic
76C22×C4×C79C2×D411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78M16×C79—42no—
79M16×C79—42no—
80C2×C8×C79—42no—
81C4⋊C4×C79—41no—
82C22⋊C4×C79—41no—
83C22⋊C4×C79—41no—
84C42×C79—21no—
85C42×C79—21no—
86C23×C79—21no—
87C23×C79—21no—
88C23×C79—21no—
89C23×C79—21no—
90C23×C79—41no—
91C23×C79—41no—
92C23×C79—41no—
93C23×C79—41no—
94C23×C79#10312yes—
95C23×C79#10312yes—
96C23×C79C2×C22⋊C411yescharacteristic
97D4×C79—42no—
98D4×C79—42no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—81no—
112C2×C4×C79#9911yescharacteristic
113C2×C4×C79#9911yescharacteristic
114C2×C4×C79C2×SD1611yescharacteristic
115C2×C4×C79C2×Q1611yescharacteristic
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—84no—
123C8×C79—82no—
124C79⋊C8—82no—
125C22×C79—21no—
126C22×C79—21no—
127C22×C79—22no—
128C22×C79—22no—
129C22×C79—42no—
130C22×C79—42no—
131C22×C79—42no—
132C22×C79—41no—
133C22×C79—41no—
134C22×C79—41no—
135C22×C79—41no—
136C22×C79—81no—
137C22×C79C2×#9911yescharacteristic
138C22×C79C2×#10311yescharacteristic
139C22×C79#42111yescharacteristic
140C4×C79—21no—
141C4×C79—21no—
142C4×C79—21no—
143C4×C79—41no—
144C4×C79—41no—
145C4×C79—41no—
146C4×C79#35212yes—
147Dic79—82no—
148Dic79—82no—
149#11903—791nomaximal, Sylow
150C2×C79—21no—
151C2×C79—21no—
152C2×C79—42no—
153C2×C79—42no—
154C2×C79#152011yescharacteristic
155C2×C79C2×#35211yescharacteristic
156C2×C79#213411yescharacteristic
157#2487—791no—
158#1800D7911yescharacteristic
159#1459—791no—
160#1353—794no—
161C79#1190311yescharacteristic, complemented, Sylow
162C2×C4×D4C2×D7911yescharacteristic
163C22×C4⋊C4—791no—
164#423—21no—
165C2×#100—21no—
166C2×C22⋊C8—1581no—
167#400—792no—
168#399—792no—
169#389—1581no—
170#386—1581no—
171#385—1581no—
172#380—791no—
173#380—1581no—
174#343C2×D7912yes—
175#324—792no—
176#323—1581no—
177C22×D4C22⋊D7911yescharacteristic
178C23×C4—1581no—
179C23×C4C4×D7911yescharacteristic
180C23×C4D31611yescharacteristic
181C2×M16—22no—
182C22×C8—3161no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—3161no—
186C2×C4⋊C4—3161no—
187C2×C4⋊C4—792no—
188C2×C4⋊C4—1581no—
189C2×C4⋊C4—1581no—
190C2×C4⋊C4—1582no—
191C2×C4⋊C4C22×D7911yescharacteristic
192C2×C22⋊C4—1581no—
193C2×C22⋊C4—1582no—
194C2×C22⋊C4C4×D7911yescharacteristic
195C2×C22⋊C4D31611yescharacteristic
196C2×C42C22⋊D7911yescharacteristic
197C4⋊C8—41no—
198C4⋊C8—41no—
199#100—42no—
200C22⋊C8—3161no—
201C22⋊C8—1582no—
202#95—792no—
203#95—1582no—
204#95—1582no—
205C24(C2×D79)⋊C411yescharacteristic
206C24(C2×D79)⋊C411yescharacteristic
207C2×D4—41no—
208C2×D4—41no—
209C2×D4#1192961212yes—
210C2×D4#1192961412yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—3162no—
218C22×C4—6322no—
219C22×C4—1582no—
220C22×C4—1582no—
221C22×C4(C2×D79)⋊C411yescharacteristic
222C22×C4(C2×D79)⋊C411yescharacteristic
223C22×C4C2×C4×D7911yescharacteristic
224C22×C4C2×C22⋊D7911yescharacteristic
225C22×C4C2×D31611yescharacteristic
226M16—42no—
227M16—42no—
228C2×C8—42no—
229C2×C8—3162no—
230C2×C8—6322no—
231C4⋊C4—41no—
232C4⋊C4—3162no—
233C4⋊C4—3162no—
234C4⋊C4—3161no—
235C4⋊C4—3161no—
236C4⋊C4—6322no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—3162no—
240C22⋊C4—6322no—
241C42—21no—
242C42—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23#1457908412yes—
252C23#1457908412yes—
253C23C2×(C2×D79)⋊C411yescharacteristic
254D4—42no—
255D4—42no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—81no—
269C2×C4—3162no—
270C2×C4—3162no—
271C2×C4—3162no—
272C2×C4—3162no—
273C2×C4—3162no—
274C2×C4—3162no—
275C2×C4—6324no—
276C2×C4#1457907511yescharacteristic
277C2×C4#1457907511yescharacteristic
278C2×C4C2×#1192961211yescharacteristic
279C2×C4C2×#1192961411yescharacteristic
280C8—82no—
281C8—6322no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22C2×#1457907511yescharacteristic
295C22C2×#1457908411yescharacteristic
296C22#1909858111yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—6322no—
304C4—6322no—
305C4#1909835412yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#2376197111yescharacteristic
311C2C2×#1909835411yescharacteristic
312C2#2376509111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1800C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C4⋊C4×C79C23
hypercenter#1800D79
hypocenterC79#11903
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #23761426C2semidirect
161. C79#11903semidirect

Maximal subgroups

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

subgroupindex
#237677172
#237617182
#237614262
C79×#18002
#1190379