Finite Group Databasebeta

← #34198639

Subgroups — #34198639

Order 20224.

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

normal subgroups only

This group has 434 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#23766683C211yesmaximal, characteristic
2C4×#14579080C211yesmaximal, characteristic, complemented
3#23761428C212yesmaximal, complemented
4#23761404C212yesmaximal
5C79×#1461C211yesmaximal, characteristic, complemented
6C2×C4×D316C411yescharacteristic
7C2×#14579080—21no—
8C4×#11929608—21no—
9#19098502C411yescharacteristic
10#19098502C2211yescharacteristic
11#19098489—21no—
12#19098480C411yescharacteristic
13#19098480C2211yescharacteristic
14#19098474—21no—
15C4×(C2×D79)⋊C4—21no—
16C4×(C2×D79)⋊C4—21no—
17#19098409C411yescharacteristic
18#19098332C2212yes—
19C2×C4×Q8×C79C2211yescharacteristic
20#431×C79—21no—
21#324×C79—21no—
22#324×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C4×D79—21no—
25C2×#11929608—22no—
26C4×D316—41no—
27C4×D316—41no—
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×(C2×D79)⋊C4—21no—
30C2×(C2×D79)⋊C4—21no—
31C2×(C2×D79)⋊C4—21no—
32C2×(C2×D79)⋊C4—21no—
33C2×(C2×D79)⋊C4—21no—
34C2×C4×Dic79—21no—
35#14579080—24no—
36#14579072—41no—
37C4×C79⋊C8—41no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95C2×C411yescharacteristic
44C79⋊#95C2×C411yescharacteristic
45C22×Q8×C79D411yescharacteristic
46C4×Q8×C79—41no—
47C4×Q8×C79—41no—
48C2×C4⋊C4×C79C2×C411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C42×C79C2×C411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C4⋊C8×C79—41no—
55C22⋊C8×C79—22no—
56C4×C8×C79—41no—
57C23×D79—21no—
58C2×D316—41no—
59C2×D316—42no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C2×C4×D79—42no—
64C2×C4×D79—42no—
65C2×C4×D79—41no—
66C2×C4×D79—41no—
67#11929608—24no—
68#11929608—42no—
69C2×C79⋊C8—42no—
70C4⋊Dic79—41no—
71(C2×D79)⋊C4—42no—
72(C2×D79)⋊C4—42no—
73(C2×D79)⋊C4—42no—
74(C2×D79)⋊C4—42no—
75(C2×D79)⋊C4—42no—
76C4×Dic79—42no—
77C2×Q8×C79—22no—
78C2×Q8×C79—22no—
79C2×Q8×C79—41no—
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C2×C8×C79—42no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—41no—
90C4⋊C4×C79—41no—
91C42×C79—21no—
92C42×C79—21no—
93C42×C79—21no—
94C42×C79—21no—
95C42×C79—21no—
96C42×C79—21no—
97C23×C79C2×C22⋊C411yescharacteristic
98C22×D79—22no—
99C22×D79—41no—
100C22×D79—41no—
101C22×D79—41no—
102C22×D79—41no—
103C22×D79—41no—
104C22×D79—41no—
105Q8×C79—44no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—22no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79#9911yescharacteristic
121C2×C4×C79#9911yescharacteristic
122C2×C4×C79#10412yes—
123C2×C4×C79#13512yes—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133D316—42no—
134D316—42no—
135C4×D79—82no—
136C4×D79—82no—
137C8×C79—82no—
138C79⋊C8—44no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79C2×#9911yescharacteristic
142C22×C79#41611yescharacteristic
143C22×C79C2×#10411yescharacteristic
144C2×D79—42no—
145C2×D79—42no—
146C2×D79—42no—
147C2×D79—81no—
148C2×D79—81no—
149C2×D79—81no—
150C2×D79—81no—
151C4×C79—21no—
152C4×C79—22no—
153C4×C79—22no—
154C4×C79—41no—
155C4×C79—41no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79#35512yes—
159Dic79—81no—
160Dic79—81no—
161Dic79—81no—
162#11734—791nomaximal, Sylow
163C2×C79—21no—
164C2×C79—21no—
165C2×C79#150111yescharacteristic
166C2×C79#212011yescharacteristic
167C2×C79C2×#35511yescharacteristic
168D79—82no—
169#2398—791no—
170C4×#101—791no—
171#1461D7911yescharacteristic, complemented
172#1353—792no—
173#1345—792no—
174C79#1173411yescharacteristic, complemented, Sylow
175C2×C4×Q8C2×D7911yescharacteristic
176C2×C4×D4—791no—
177#431—21no—
178C2×#101—1581no—
179C4×M16—1581no—
180#400—791no—
181#400—791no—
182#397—1581no—
183#394—791no—
184#394—791no—
185#393—1581no—
186C4×C22⋊C4—1581no—
187C4×C22⋊C4—1581no—
188#376—791no—
189#343—792no—
190#324—21no—
191#324C2×D7912yes—
192C22×Q8D31611yescharacteristic
193C22×D4—791no—
194C23×C4—1581no—
195C2×M16—1582no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×D4—3161no—
199C4×D4—3161no—
200C2×C4⋊C4—791no—
201C2×C4⋊C4C4×D7911yescharacteristic
202C2×C4⋊C4C22⋊D7911yescharacteristic
203C2×C4⋊C4C22×D7911yescharacteristic
204C2×C22⋊C4—1581no—
205C2×C22⋊C4—1581no—
206C2×C22⋊C4—1581no—
207C2×C22⋊C4—1581no—
208C2×C22⋊C4—1581no—
209C2×C42—1581no—
210C2×C42C4×D7911yescharacteristic
211C2×C42D31611yescharacteristic
212C2×C42C22⋊D7911yescharacteristic
213C4⋊C8—41no—
214#101—1584no—
215C22⋊C8—22no—
216#97—3161no—
217C4×C8—41no—
218C4×C8—3161no—
219#95—791no—
220#95—791no—
221#95—1581no—
222#95—1581no—
223#95—1581no—
224#95—1581no—
225#95—1581no—
226C24—1581no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×D4—3161no—
231C2×D4—3162no—
232C22×C4—3162no—
233C22×C4—3162no—
234C22×C4—3161no—
235C22×C4—3161no—
236C22×C4—1581no—
237C22×C4—1581no—
238C22×C4—1581no—
239C22×C4(C2×D79)⋊C411yescharacteristic
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4(C2×D79)⋊C411yescharacteristic
243C22×C4C2×C4×D7911yescharacteristic
244C22×C4C2×C22⋊D7911yescharacteristic
245C22×C4C2×D31611yescharacteristic
246M16—3162no—
247M16—1584no—
248C2×C8—42no—
249C2×C8—3162no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—3161no—
254C22⋊C4—3162no—
255C22⋊C4—3162no—
256C22⋊C4—3162no—
257C22⋊C4—3162no—
258C22⋊C4—3162no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—3162no—
266C23—3161no—
267C23—3161no—
268C23—3161no—
269C23—3161no—
270C23—3161no—
271C23—3161no—
272C23—1582no—
273C23C2×(C2×D79)⋊C411yescharacteristic
274Q8—44no—
275D4—3162no—
276D4—3162no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4—3161no—
297C2×C4—3161no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—6322no—
301C2×C4—6322no—
302C2×C4#1457907511yescharacteristic
303C2×C4#1457907511yescharacteristic
304C2×C4#1457908712yes—
305C2×C4#1457918312yes—
306C8—82no—
307C8—3164no—
308C22—21no—
309C22—21no—
310C22—3162no—
311C22—3162no—
312C22—3162no—
313C22—6321no—
314C22—6321no—
315C22—6321no—
316C22—6321no—
317C22C2×#1457907511yescharacteristic
318C22#1909856011yescharacteristic
319C22C2×#1457908711yescharacteristic
320C4—21no—
321C4—22no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—6321no—
328C4—6321no—
329C4—6321no—
330C4#1909835912yes—
331C2—21no—
332C2—21no—
333C2—6322no—
334C2#2376189211yescharacteristic
335C2#2376503911yescharacteristic
336C2C2×#1909835911yescharacteristic
subgroupG/H
centerC2×C4#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1461C2
socleC22×C79C2×#99
prime radicalC22×D316C2×C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1461D79
hypocenterC79#11734
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. C4×#14579080C2semidirect
3. #23761428C2semidirect
5. C79×#1461C2semidirect
171. #1461D79semidirect
174. C79#11734semidirect

Maximal subgroups

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

subgroupindex
#237666832
C4×#145790802
#237614282
#237614042
C79×#14612
#1173479