Finite Group Databasebeta

← #34197921

Subgroups — #34197921

Order 20224.

480 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 87 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#23768353C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761345C214yesmaximal, complemented
4C79×#1465C211yesmaximal, characteristic
5C2×#14579161C411yescharacteristic
6C2×#14579140—21no—
7C2×#14579140—21no—
8C2×C22⋊(C79⋊C8)—21no—
9#19098499—21no—
10#19098499—21no—
11#19098466C412yes—
12#19098432C2212yes—
13C4×C4⋊Dic79C411yescharacteristic
14#19098289C2212yes—
15#19098287—21no—
16C2×C4×D4×C79C2211yescharacteristic
17#407×C79—21no—
18#324×C79C2212yes—
19#323×C79—21no—
20C22×Dic158—21no—
21C22×C79⋊C8—41no—
22#14579161—41no—
23#14579161—41no—
24#14579140—44no—
25#14579140—44no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79C2×C412yes—
29C2×Dic79⋊C4—22no—
30C2×Dic79⋊C4—22no—
31C2×C22⋊Dic79—22no—
32C2×C22⋊Dic79—22no—
33C2×C4×Dic79C2×C412yes—
34C22⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—41no—
36C79⋊#95—21no—
37C22×D4×C79C2×C411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C2×M16×C79—41no—
41C4×D4×C79—41no—
42C4×D4×C79—41no—
43C2×C4⋊C4×C79C2311yescharacteristic
44C2×C22⋊C4×C79D411yescharacteristic
45C2×C22⋊C4×C79D411yescharacteristic
46C2×C42×C79C2×C411yescharacteristic
47C22⋊C8×C79—22no—
48C22⋊C8×C79—41no—
49C2×Dic158—42no—
50C2×Dic158—42no—
51C2×Dic158—41no—
52C2×Dic158—41no—
53C22×Dic79—22no—
54C22×Dic79—22no—
55C2×C79⋊C8—42no—
56C2×C79⋊C8—82no—
57C4⋊Dic79—24no—
58C4⋊Dic79—42no—
59C4⋊Dic79—42no—
60Dic79⋊C4—44no—
61Dic79⋊C4—44no—
62C22⋊Dic79—44no—
63C22⋊Dic79—44no—
64C4×Dic79—24no—
65C24×C79C22⋊C411yescharacteristic
66C24×C79C22⋊C411yescharacteristic
67C2×D4×C79—22no—
68C2×D4×C79—22no—
69C2×D4×C79—41no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—42no—
75C22×C4×C79—42no—
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81M16×C79—82no—
82C2×C8×C79—42no—
83C4⋊C4×C79—41no—
84C22⋊C4×C79—41no—
85C22⋊C4×C79—41no—
86C42×C79—21no—
87C42×C79—21no—
88C23×C79—21no—
89C23×C79—21no—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—41no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79#10112yes—
98C23×C79C2×C22⋊C411yescharacteristic
99D4×C79—44no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—22no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
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—81no—
113C2×C4×C79#9911yescharacteristic
114C2×C4×C79#9911yescharacteristic
115C2×C4×C79#13112yes—
116C2×Dic79—24no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122Dic158—82no—
123Dic158—82no—
124C8×C79—82no—
125C79⋊C8—82no—
126C22×C79—21no—
127C22×C79—21no—
128C22×C79—22no—
129C22×C79—42no—
130C22×C79—42no—
131C22×C79—41no—
132C22×C79—41no—
133C22×C79—41no—
134C22×C79—41no—
135C22×C79—41no—
136C22×C79—81no—
137C22×C79—81no—
138C22×C79#40811yescharacteristic
139C22×C79C2×#9911yescharacteristic
140C22×C79C2×#10111yescharacteristic
141C4×C79—21no—
142C4×C79—22no—
143C4×C79—41no—
144C4×C79—41no—
145C4×C79—41no—
146C4×C79#35312yes—
147Dic79—44no—
148Dic79—82no—
149#11570—791nomaximal, Sylow
150C2×C79—21no—
151C2×C79—21no—
152C2×C79—42no—
153C2×C79—81no—
154C2×C79#147211yescharacteristic
155C2×C79C2×#35311yescharacteristic
156C2×C79#213511yescharacteristic
157#2542—791no—
158#1465D7911yescharacteristic
159#1459—791no—
160#1325—794no—
161C79#1157011yescharacteristic, complemented, Sylow
162C2×C4⋊Q8—791no—
163C2×C22⋊Q8—1581no—
164C2×C22⋊Q8—1581no—
165C2×C4×D4C2×D7911yescharacteristic
166#407—21no—
167C2×C22⋊C8—1581no—
168#399—1581no—
169#399—1581no—
170#390—792no—
171#384—792no—
172C4×C4⋊C4—791no—
173#324—792no—
174#324C2×D7912yes—
175#323—21no—
176#323—1581no—
177C22×Q8—1581no—
178C22×D4C4×D7911yescharacteristic
179C23×C4D31611yescharacteristic
180C23×C4C22⋊D7911yescharacteristic
181C2×M16—41no—
182C22×C8—3161no—
183C4⋊Q8—3161no—
184C4⋊Q8—3161no—
185C22⋊Q8—3164no—
186C22⋊Q8—3164no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—792no—
190C2×C4⋊C4—1581no—
191C2×C4⋊C4—1582no—
192C2×C4⋊C4—1582no—
193C2×C4⋊C4—1581no—
194C2×C4⋊C4C22×D7911yescharacteristic
195C2×C22⋊C4—1582no—
196C2×C22⋊C4—1582no—
197C2×C22⋊C4D31611yescharacteristic
198C2×C22⋊C4C22⋊D7911yescharacteristic
199C2×C42—792no—
200C2×C42C4×D7911yescharacteristic
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—3161no—
204C22⋊C8—1582no—
205#95—1581no—
206C24(C2×D79)⋊C411yescharacteristic
207C24(C2×D79)⋊C411yescharacteristic
208C2×Q8—3162no—
209C2×Q8—3162no—
210C2×Q8—3161no—
211C2×Q8—3161no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—1582no—
222C22×C4—1582no—
223C22×C4(C2×D79)⋊C411yescharacteristic
224C22×C4(C2×D79)⋊C411yescharacteristic
225C22×C4C2×C4×D7911yescharacteristic
226C22×C4C2×C22⋊D7911yescharacteristic
227C22×C4C2×D31611yescharacteristic
228M16—82no—
229C2×C8—42no—
230C2×C8—3162no—
231C2×C8—6322no—
232C4⋊C4—41no—
233C4⋊C4—3162no—
234C4⋊C4—3162no—
235C4⋊C4—3164no—
236C4⋊C4—3164no—
237C4⋊C4—1584no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—3164no—
241C22⋊C4—3164no—
242C42—21no—
243C42—21no—
244C42—1584no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1457907912yes—
255C23C2×(C2×D79)⋊C411yescharacteristic
256Q8—6322no—
257Q8—6322no—
258D4—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—3162no—
273C2×C4—3162no—
274C2×C4—3162no—
275C2×C4—3162no—
276C2×C4—3162no—
277C2×C4—1584no—
278C2×C4#1457907511yescharacteristic
279C2×C4#1457907511yescharacteristic
280C2×C4#1457916912yes—
281C8—82no—
282C8—6322no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22#1909853111yescharacteristic
296C22C2×#1457907511yescharacteristic
297C22C2×#1457907911yescharacteristic
298C4—21no—
299C4—22no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—3164no—
304C4—6322no—
305C4#1909835512yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2376176511yescharacteristic
311C2C2×#1909835511yescharacteristic
312C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1465C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1465D79
hypocenterC79#11570
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #23761345C2semidirect
161. C79#11570semidirect

Maximal subgroups

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

subgroupindex
#237683532
#237617182
#237613452
C79×#14652
#1157079