Finite Group Databasebeta

← #34198902

Subgroups — #34198902

Order 20224.

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

normal subgroups only

This group has 572 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#23766765C211yesmaximal, characteristic
2#23761738C211yesmaximal, characteristic
3#23761429C214yesmaximal, complemented
4C4×#100×C79C211yesmaximal, characteristic
5C23×Dic158C411yescharacteristic
6C2×#14579140—21no—
7#19098527—21no—
8#19098483C2212yes—
9#19098475C412yes—
10#19098469—21no—
11#19098443—21no—
12#19098435—21no—
13#19098423C411yescharacteristic
14#19098289C2212yes—
15#19098287—21no—
16C2×C4×D4×C79C2211yescharacteristic
17C2×#100×C79—21no—
18C4×M16×C79—21no—
19#343×C79C2212yes—
20C22×Dic158—21no—
21C22×Dic158—44no—
22C22×Dic158—41no—
23C22×Dic158—41no—
24C22×Dic158C2×C412yes—
25C23×Dic79—21no—
26C2×#11929608—41no—
27#14579140—44no—
28C2×C4⋊Dic79—21no—
29C2×Dic79⋊C4—22no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—22no—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)—41no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95C2×C412yes—
37C22×D4×C79D411yescharacteristic
38C23×C4×C79C2×C411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C2×M16×C79—22no—
41C4×D4×C79—41no—
42C4×D4×C79—41no—
43C2×C4⋊C4×C79C2311yescharacteristic
44C2×C22⋊C4×C79C2×C411yescharacteristic
45C2×C22⋊C4×C79D411yescharacteristic
46C2×C42×C79D411yescharacteristic
47#100×C79—24no—
48#97×C79—41no—
49C4×C8×C79—41no—
50C2×Dic158—24no—
51C2×Dic158—42no—
52C2×Dic158—42no—
53C2×Dic158—48no—
54C2×Dic158—44no—
55C2×Dic158—44no—
56C2×Dic158—44no—
57C2×Dic158—41no—
58C2×Dic158—41no—
59C22×Dic79—22no—
60C22×Dic79—22no—
61C22×Dic79—42no—
62C22×Dic79—44no—
63#11929608—82no—
64C2×C79⋊C8—42no—
65C4⋊Dic79—42no—
66Dic79⋊C4—44no—
67C22⋊Dic79—44no—
68C22⋊Dic79—42no—
69C24×C79C22⋊C411yescharacteristic
70C24×C79C22⋊C411yescharacteristic
71C2×D4×C79—22no—
72C2×D4×C79—22no—
73C2×D4×C79—41no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—42no—
79C22×C4×C79—42no—
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85M16×C79—24no—
86M16×C79—42no—
87C2×C8×C79—42no—
88C4⋊C4×C79—41no—
89C22⋊C4×C79—41no—
90C22⋊C4×C79—41no—
91C42×C79—21no—
92C42×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79—41no—
102C23×C79#10412yes—
103C23×C79C2×C22⋊C411yescharacteristic
104D4×C79—44no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—22no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—81no—
118C2×C4×C79#9911yescharacteristic
119C2×C4×C79#9911yescharacteristic
120C2×C4×C79#13712yes—
121C2×Dic79—24no—
122C2×Dic79—42no—
123C2×Dic79—48no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128Dic158—44no—
129Dic158—44no—
130Dic158—44no—
131Dic158—44no—
132C8×C79—44no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—22no—
137C22×C79—42no—
138C22×C79—42no—
139C22×C79—42no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—41no—
143C22×C79—41no—
144C22×C79—41no—
145C22×C79—81no—
146C22×C79C2×#9911yescharacteristic
147C22×C79#41911yescharacteristic
148C22×C79C2×#10411yescharacteristic
149C4×C79—21no—
150C4×C79—22no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79—41no—
154C4×C79AΣL(1,𝔽16)12yes—
155Dic79—44no—
156Dic79—82no—
157#11791—791nomaximal, Sylow
158C2×C79—21no—
159C2×C79—21no—
160C2×C79—42no—
161C2×C79—81no—
162C2×C79#150811yescharacteristic
163C2×C79C2×AΣL(1,𝔽16)11yescharacteristic
164C2×C79#213511yescharacteristic
165#2409—791no—
166C4×#100D7911yescharacteristic
167#1465—791no—
168#1354—794no—
169C79#1179111yescharacteristic, complemented, Sylow
170C23×Q8—791no—
171C2×C22⋊Q8—1581no—
172C2×C4×D4C2×D7911yescharacteristic
173C2×#100—21no—
174#407—1581no—
175C4×M16—21no—
176#395—792no—
177#393—792no—
178#391—1581no—
179#386—1581no—
180#385—1581no—
181#380—791no—
182#343C2×D7912yes—
183#324—792no—
184#323—1581no—
185C22×Q8—3161no—
186C22×Q8—3161no—
187C22×Q8—3164no—
188C22×Q8—792no—
189C22×Q8—1581no—
190C22×D4C22⋊D7911yescharacteristic
191C23×C4—1581no—
192C23×C4C4×D7911yescharacteristic
193C23×C4D31611yescharacteristic
194C2×M16—22no—
195C2×M16—3161no—
196C22⋊Q8—3164no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—1581no—
200C2×C4⋊C4—1582no—
201C2×C4⋊C4C22×D7911yescharacteristic
202C2×C22⋊C4—1582no—
203C2×C22⋊C4—1581no—
204C2×C22⋊C4C4×D7911yescharacteristic
205C2×C22⋊C4D31611yescharacteristic
206C2×C42C22⋊D7911yescharacteristic
207#100—24no—
208C22⋊C8—3161no—
209C22⋊C8—1582no—
210#97—41no—
211C4×C8—41no—
212#95—792no—
213#95—1582no—
214#95—1582no—
215C24(C2×D79)⋊C411yescharacteristic
216C24(C2×D79)⋊C411yescharacteristic
217C2×Q8—3162no—
218C2×Q8—3162no—
219C2×Q8—3164no—
220C2×Q8—3168no—
221C2×Q8—3164no—
222C2×Q8—3164no—
223C2×Q8—3161no—
224C2×Q8—3161no—
225C2×Q8—1584no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—3162no—
236C22×C4—3164no—
237C22×C4—1582no—
238C22×C4—1582no—
239C22×C4(C2×D79)⋊C411yescharacteristic
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4C2×C4×D7911yescharacteristic
242C22×C4C2×C22⋊D7911yescharacteristic
243C22×C4C2×D31611yescharacteristic
244M16—24no—
245M16—42no—
246M16—6322no—
247C2×C8—42no—
248C2×C8—3162no—
249C4⋊C4—41no—
250C4⋊C4—3162no—
251C4⋊C4—3164no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—3162no—
255C22⋊C4—3164no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#1457908912yes—
268C23C2×(C2×D79)⋊C411yescharacteristic
269Q8—3164no—
270Q8—3164no—
271Q8—3164no—
272Q8—3164no—
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—3162no—
288C2×C4—3168no—
289C2×C4—3162no—
290C2×C4—3162no—
291C2×C4—3162no—
292C2×C4—3162no—
293C2×C4—1584no—
294C2×C4#1457907511yescharacteristic
295C2×C4#1457907511yescharacteristic
296C2×C4#1457919112yes—
297C8—44no—
298C8—6322no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1457907511yescharacteristic
312C22#1909857111yescharacteristic
313C22C2×#1457908911yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—3164no—
320C4—6322no—
321C4#1909835112yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2#2376191911yescharacteristic
327C2C2×#1909835111yescharacteristic
328C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC4×#100×C79C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C4⋊C4×C79C23
hypercenterC4×#100D79
hypocenterC79#11791
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. #23761429C2semidirect
169. C79#11791semidirect

Maximal subgroups

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

subgroupindex
#237667652
#237617382
#237614292
C4×#100×C792
#1179179