Finite Group Databasebeta

← #34205709

Subgroups — #34205709

Order 20224.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098980C211yesmaximal, characteristic, complemented
2C2×#19098833C211yesmaximal, characteristic
3C2×#19098375C211yesmaximal, characteristic, complemented
4C2×#19098363C211yesmaximal, characteristic, complemented
5#23761508C211yesmaximal, characteristic, complemented
6#23761493C211yesmaximal, characteristic, complemented
7C79×#1816C211yesmaximal, characteristic, complemented
8C2×#14579186—21no—
9C22×D632C2211yescharacteristic
10#19098980—22no—
11#19098980—22no—
12C2×#14579101—21no—
13C2×#14579101C2211yescharacteristic
14C22×C79⋊C16—21no—
15C22×C79⋊C16C2211yescharacteristic
16#19098833—22no—
17#19098833C414yescomplemented
18C2×#14579097C2211yescharacteristic
19C2×#14579077—21no—
20#19098375—22no—
21#19098363—21no—
22#19098363—21no—
23C2×#131×C79C2211yescharacteristic
24C2×#108×C79C2211yescharacteristic
25C2×#107×C79C2211yescharacteristic
26#417×C79—21no—
27#343×C79—21no—
28C22×D316—21no—
29#14579186—24no—
30#14579186—42no—
31C2×#11929611—41no—
32C2×D632—21no—
33C2×D632—21no—
34C2×D632—21no—
35C2×D632—21no—
36C2×D632—21no—
37C2×D632C2×C412yes—
38C2×#11929609—41no—
39C2×#11929608—21no—
40#14579101—24no—
41#14579101—21no—
42#14579101D412yes—
43C2×C79⋊C16—24no—
44C2×C79⋊C16—21no—
45C2×C79⋊C16—21no—
46C2×C79⋊C16—21no—
47C2×C79⋊C16D411yescharacteristic
48C2×C79⋊C16D411yescharacteristic
49#14579097—21no—
50#14579097C2×C412yes—
51#14579077—24no—
52C2×#41×C79D411yescharacteristic
53#131×C79—21no—
54#131×C79—21no—
55#131×C79C2×C412yes—
56#131×C79D412yes—
57C2×M16×C79—21no—
58C2×M16×C79—21no—
59C2×M16×C79D411yescharacteristic
60C22×C8×C79—21no—
61C22×C8×C79C2311yescharacteristic
62C2×C4⋊C4×C79—21no—
63#108×C79—21no—
64#108×C79—21no—
65#107×C79—21no—
66#107×C79—21no—
67#103×C79—41no—
68#102×C79—41no—
69C23×D79—41no—
70C2×D316—22no—
71C2×D316—41no—
72C2×D316—41no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—41no—
76#11929611—44no—
77D632—22no—
78D632—22no—
79D632—41no—
80D632—41no—
81#11929609—44no—
82#11929608—22no—
83#11929608—41no—
84C2×C79⋊C8—41no—
85C79⋊C16—21no—
86C79⋊C16—22no—
87C79⋊C16—21no—
88#41×C79—21no—
89#41×C79—21no—
90#41×C79C22⋊C412yes—
91#41×C79SD1612yes—
92C2×Q8×C79D811yescharacteristic
93C2×D4×C79—21no—
94C2×D4×C79D811yescharacteristic
95C22×C4×C79—21no—
96C22×C4×C79—41no—
97C22×C4×C79C2×D411yescharacteristic
98M16×C79—22no—
99M16×C79—22no—
100M16×C79—21no—
101M16×C79—41no—
102M16×C79—41no—
103M16×C79C22⋊C412yes—
104C2×C8×C79—22no—
105C2×C8×C79—22no—
106C2×C8×C79—21no—
107C2×C8×C79—41no—
108C2×C8×C79C22×C411yescharacteristic
109C2×C8×C79C2×D411yescharacteristic
110C2×C8×C79C2×D411yescharacteristic
111C2×C8×C79C2×D411yescharacteristic
112C2×C8×C79#4111yescharacteristic
113C2×C8×C79#4111yescharacteristic
114C4⋊C4×C79—41no—
115C4⋊C4×C79—41no—
116C23×C79—21no—
117C23×C79#13511yescharacteristic
118C22×D79—42no—
119C22×D79—81no—
120C22×D79—81no—
121C22×D79—161no—
122C22×D79—161no—
123Q8×C79—21no—
124Q8×C79#10212yes—
125D4×C79—22no—
126D4×C79—22no—
127D4×C79—21no—
128D4×C79#10212yes—
129C2×C4×C79—21no—
130C2×C4×C79—22no—
131C2×C4×C79—22no—
132C2×C4×C79—81no—
133C2×C4×C79—161no—
134C2×C4×C79C2×C22⋊C411yescharacteristic
135C2×C4×C79#12111yescharacteristic
136C2×C4×C79#12111yescharacteristic
137C2×C4×C79C2×D811yescharacteristic
138C2×C4×C79C2×SD1611yescharacteristic
139C2×C4×C79#13511yescharacteristic
140D316—42no—
141D316—42no—
142D316—81no—
143D316—81no—
144C8×C79—22no—
145C8×C79—81no—
146C8×C79C4×D411yescharacteristic
147C8×C79C4×D411yescharacteristic
148C8×C79#12011yescharacteristic
149C8×C79#12311yescharacteristic
150C79⋊C8—42no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—22no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79#41611yescharacteristic
157C22×C79#46511yescharacteristic
158C22×C79#46611yescharacteristic
159C2×D79—82no—
160C2×D79—162no—
161C2×D79—161no—
162C2×D79—161no—
163C2×D79—162no—
164C4×C79—22no—
165C4×C79—161no—
166C4×C79#38611yescharacteristic
167C4×C79C2×#10211yescharacteristic
168C4×C79#46511yescharacteristic
169C4×C79#46611yescharacteristic
170#13073—791nomaximal, Sylow
171C2×C79—21no—
172C2×C79—21no—
173C2×C79—42no—
174C2×C79#189311yescharacteristic
175C2×C79#222111yescharacteristic
176C2×C79#222311yescharacteristic
177D79—162no—
178C2×#504—791no—
179C2×#472—791no—
180C2×#361—791no—
181C2×#357—791no—
182#1816D7911yescharacteristic, complemented
183#1387—791no—
184#1381—791no—
185C79#1307311yescharacteristic, complemented, Sylow
186C2×AGL(1,ℤ/8)—1581no—
187C22×D8—791no—
188C2×#131C2×D7911yescharacteristic
189#504—1582no—
190#504—1582no—
191C2×M32—791no—
192C2×M32—1581no—
193C22×C16—791no—
194C22×C16—1581no—
195#472—794no—
196#472—1582no—
197C2×#108—791no—
198C2×#108C2×D7911yescharacteristic
199C2×#107C2×D7911yescharacteristic
200#417—21no—
201C2×#100—1581no—
202#361—1582no—
203#357—1581no—
204#357—1581no—
205#343—21no—
206C2×#41C22⋊D7911yescharacteristic
207C22×D4—1581no—
208AGL(1,ℤ/8)—3162no—
209AGL(1,ℤ/8)—1584no—
210C2×SD16—3161no—
211C2×D8—3161no—
212C2×D8—792no—
213C2×D8—1581no—
214C2×D8—1581no—
215C2×D8—1581no—
216C2×D8—1581no—
217C2×D8—1581no—
218#131—21no—
219#131—21no—
220#131C4×D7912yes—
221#131D31612yes—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—1581no—
225C2×M16C22⋊D7911yescharacteristic
226C22×C8—21no—
227C22×C8C22×D7911yescharacteristic
228C2×C4⋊C4—21no—
229M32—792no—
230M32—1584no—
231M32—1581no—
232C2×C16—791no—
233C2×C16—791no—
234C2×C16—1584no—
235C2×C16—1581no—
236C2×C16—1581no—
237C2×C16—1581no—
238#108—21no—
239#108—21no—
240#108—792no—
241#108—1581no—
242#107—21no—
243#107—21no—
244#103—41no—
245#102—41no—
246#100—1584no—
247C24—3161no—
248#41—21no—
249#41—21no—
250#41(C2×D79)⋊C412yes—
251#41#1192961112yes—
252C2×Q8#1192960911yescharacteristic
253C2×D4—21no—
254C2×D4—3161no—
255C2×D4—3161no—
256C2×D4—3161no—
257C2×D4—3161no—
258C2×D4—3161no—
259C2×D4—1582no—
260C2×D4#1192960911yescharacteristic
261C22×C4—21no—
262C22×C4—41no—
263C22×C4C2×C22⋊D7911yescharacteristic
264SD16—3164no—
265D8—3164no—
266D8—3161no—
267D8—3161no—
268D8—1582no—
269D8—1582no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—41no—
274M16—41no—
275M16—3161no—
276M16—1582no—
277M16(C2×D79)⋊C412yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—3161no—
283C2×C8C2×C4×D7911yescharacteristic
284C2×C8D4×D7911yescharacteristic
285C2×C8D4×D7911yescharacteristic
286C2×C8C2×D31611yescharacteristic
287C2×C8#1192962311yescharacteristic
288C2×C8#1192962311yescharacteristic
289C4⋊C4—41no—
290C4⋊C4—41no—
291C16—1581no—
292C16—1581no—
293C16—1582no—
294C23—21no—
295C23—3162no—
296C23—6321no—
297C23—6321no—
298C23—12641no—
299C23—12641no—
300C23#1457918111yescharacteristic
301Q8—21no—
302Q8#1457908112yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—3162no—
307D4—3162no—
308D4—6321no—
309D4—6321no—
310D4#1457908112yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—81no—
315C2×C4—161no—
316C2×C4C2×(C2×D79)⋊C411yescharacteristic
317C2×C4#1457913311yescharacteristic
318C2×C4#1457913311yescharacteristic
319C2×C4C2×#1192960911yescharacteristic
320C2×C4C2×#1192961111yescharacteristic
321C2×C4#1457918111yescharacteristic
322C8—22no—
323C8—81no—
324C8—3162no—
325C8(C22⋊D79)⋊C411yescharacteristic
326C8(C22⋊D79)⋊C411yescharacteristic
327C8#1457913111yescharacteristic
328C8#1457914511yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—6322no—
335C22—12642no—
336C22—12642no—
337C22—12641no—
338C22—12641no—
339C22#1909855711yescharacteristic
340C22#1909879411yescharacteristic
341C22#1909879711yescharacteristic
342C4—22no—
343C4—161no—
344C4#1909844011yescharacteristic
345C4C2×#1457908111yescharacteristic
346C4#1909879411yescharacteristic
347C4#1909879711yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—12642no—
352C2#2376382611yescharacteristic
353C2#2376551811yescharacteristic
354C2#2376553211yescharacteristic
subgroupG/H
centerC22#19098797
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC79×#1816C2
socleC22×C79#466
prime radicalC2×#19098833C2
cosocleC22×C8×C79C23
hypercenter#1816D79
hypocenterC79#13073
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ #1816

Complements

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

NH ≅ G/Ntype
1. C2×#19098980C2semidirect
3. C2×#19098375C2semidirect
4. C2×#19098363C2semidirect
5. #23761508C2semidirect
6. #23761493C2semidirect
7. C79×#1816C2semidirect
17. #19098833C4semidirect
182. #1816D79semidirect
185. C79#13073semidirect

Maximal subgroups

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

subgroupindex
C2×#190989802
C2×#190988332
C2×#190983752
C2×#190983632
#237615082
#237614932
C79×#18162
#1307379