Finite Group Databasebeta

← #34205447

Subgroups — #34205447

Order 20224.

512 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098680C211yesmaximal, characteristic, complemented
2C2×#19098376C211yesmaximal, characteristic, complemented
3#23765112C211yesmaximal, characteristic, complemented
4#23764267C211yesmaximal, characteristic, complemented
5#23761509C211yesmaximal, characteristic, complemented
6#23761481C211yesmaximal, characteristic, complemented
7C2×#509×C79C211yesmaximal, characteristic
8C2×#14579168—21no—
9C2×#14579101C2211yescharacteristic
10#19098680—22no—
11#19098680—22no—
12C2×#14579097C2211yescharacteristic
13C2×#14579095C2211yescharacteristic
14C2×#14579088—21no—
15C2×#14579082—21no—
16C2×C8×Dic79C2211yescharacteristic
17#19098466—21no—
18#19098376—22no—
19#19098367—21no—
20#19098364—21no—
21C2×#135×C79C2211yescharacteristic
22C22×D8×C79C2211yescharacteristic
23#509×C79—22no—
24#509×C79C414yescomplemented
25C2×SD32×C79—21no—
26C2×D16×C79—21no—
27C2×M32×C79C2211yescharacteristic
28#14579168—24no—
29#14579168—42no—
30C2×#11929608—21no—
31C2×#11929608—41no—
32C22×C79⋊C8—41no—
33C22×C79⋊C8D411yescharacteristic
34C2×C4⋊Dic79—21no—
35C2×C22⋊Dic79—42no—
36C2×C4×Dic79D411yescharacteristic
37#14579101—21no—
38#14579101—21no—
39C2×C79⋊C16—21no—
40#14579097—21no—
41#14579097D412yes—
42#14579095—21no—
43#14579095—21no—
44#14579088—24no—
45#14579082—41no—
46#14579082—41no—
47C8×Dic79—21no—
48C8×Dic79—21no—
49C2×#41×C79—21no—
50C22×D4×C79—21no—
51#135×C79—21no—
52#135×C79—21no—
53#135×C79C2×C412yes—
54#135×C79D412yes—
55C2×Q16×C79D411yescharacteristic
56C2×SD16×C79—21no—
57C2×D8×C79—21no—
58C2×D8×C79—21no—
59C2×D8×C79—21no—
60C2×D8×C79—21no—
61C2×D8×C79—21no—
62C2×D8×C79C2×C412yes—
63C2×D8×C79D411yescharacteristic
64C22×C8×C79C2311yescharacteristic
65SD32×C79—24no—
66D16×C79—24no—
67M32×C79—21no—
68M32×C79C2×C412yes—
69C2×C16×C79—21no—
70C22×Dic79—21no—
71C22×Dic79—41no—
72#11929608—22no—
73#11929608—44no—
74#11929608—41no—
75C2×C79⋊C8—21no—
76C2×C79⋊C8—21no—
77C2×C79⋊C8—44no—
78C2×C79⋊C8—41no—
79C2×C79⋊C8D811yescharacteristic
80C2×C79⋊C8D811yescharacteristic
81C4⋊Dic79—41no—
82C4⋊Dic79—41no—
83C22⋊Dic79—84no—
84C4×Dic79—21no—
85C4×Dic79—21no—
86C79⋊C16—41no—
87C24×C79—41no—
88#41×C79—22no—
89#41×C79—22no—
90#41×C79—41no—
91#41×C79—41no—
92C2×Q8×C79—21no—
93C2×D4×C79—22no—
94C2×D4×C79—21no—
95C2×D4×C79—41no—
96C2×D4×C79—41no—
97C2×D4×C79—41no—
98C2×D4×C79—41no—
99C2×D4×C79—41no—
100C2×D4×C79—41no—
101C22×C4×C79—41no—
102C22×C4×C79C2×D411yescharacteristic
103Q16×C79—21no—
104Q16×C79C22⋊C412yescomplemented
105SD16×C79—22no—
106SD16×C79—22no—
107D8×C79—21no—
108D8×C79—22no—
109D8×C79—22no—
110D8×C79—41no—
111D8×C79—41no—
112D8×C79C22⋊C412yescomplemented
113C2×C8×C79C22×C411yescharacteristic
114C2×C8×C79C2×D411yescharacteristic
115C2×C8×C79C2×D411yescharacteristic
116C2×C8×C79C2×D411yescharacteristic
117C2×C8×C79#4111yescharacteristic
118C2×C8×C79#4111yescharacteristic
119C16×C79—22no—
120C23×C79—41no—
121C23×C79—42no—
122C23×C79—82no—
123C23×C79—82no—
124C23×C79—81no—
125C23×C79—81no—
126C23×C79AGL(1,ℤ/8)11yescharacteristic
127Q8×C79—22no—
128Q8×C79—41no—
129D4×C79—22no—
130D4×C79—42no—
131D4×C79—42no—
132D4×C79—42no—
133D4×C79—42no—
134D4×C79—41no—
135D4×C79—81no—
136D4×C79—81no—
137C2×C4×C79—41no—
138C2×C4×C79—42no—
139C2×C4×C79—42no—
140C2×C4×C79C4×D411yescharacteristic
141C2×C4×C79#12011yescharacteristic
142C2×C4×C79#12111yescharacteristic
143C2×C4×C79C2×D811yescharacteristic
144C2×C4×C79#13511yescharacteristic
145C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
146C2×Dic79—22no—
147C2×Dic79—41no—
148C2×Dic79—41no—
149C2×Dic79—82no—
150C2×Dic79—81no—
151C8×C79C2×C22⋊C411yescharacteristic
152C8×C79C4×D411yescharacteristic
153C8×C79#12111yescharacteristic
154C8×C79#12311yescharacteristic
155C79⋊C8—21no—
156C79⋊C8—21no—
157C79⋊C8—42no—
158C22×C79—21no—
159C22×C79—21no—
160C22×C79—42no—
161C22×C79—84no—
162C22×C79—81no—
163C22×C79—81no—
164C22×C79—82no—
165C22×C79—162no—
166C22×C79—161no—
167C22×C79—161no—
168C22×C79#44211yescharacteristic
169C22×C79#44911yescharacteristic
170C22×C79#45911yescharacteristic
171C4×C79—42no—
172C4×C79#38611yescharacteristic
173C4×C79C4×D811yescharacteristic
174C4×C79#44711yescharacteristic
175C4×C79#46311yescharacteristic
176Dic79—42no—
177Dic79—161no—
178#13029—791nomaximal, Sylow
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—82no—
182C2×C79—162no—
183C2×C79#187911yescharacteristic
184C2×C79#221211yescharacteristic
185C2×C79#221311yescharacteristic
186C2×#509D7911yescharacteristic
187C2×#443—791no—
188C2×#361—791no—
189#2139—791no—
190#1970—791no—
191#1387—791no—
192#1376—791no—
193C79#1302911yescharacteristic, complemented, Sylow
194C2×#135C2×D7911yescharacteristic
195C22×D8C2×D7911yescharacteristic
196C2×#131—1581no—
197#509—22no—
198#509Dic7914yescomplemented
199C2×SD32—21no—
200C2×D16—21no—
201C2×M32—791no—
202C2×M32C2×D7911yescharacteristic
203#443—1582no—
204#443—1582no—
205C2×#108—791no—
206C2×#107—791no—
207C2×#104—1581no—
208C2×#102—1581no—
209C2×C4×C8—791no—
210#390—1581no—
211#361—1582no—
212#358—1581no—
213#357—1581no—
214C2×#41—21no—
215C22×D4—21no—
216#135—21no—
217#135—21no—
218#135C2×Dic7912yes—
219#135C22⋊D7912yes—
220C2×Q16C22⋊D7911yescharacteristic
221C2×SD16—21no—
222C2×D8—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8C2×Dic7912yes—
228C2×D8C22⋊D7911yescharacteristic
229#131—3162no—
230#131—1584no—
231C2×M16—3161no—
232C2×M16—1581no—
233C22×C8—3161no—
234C22×C8—791no—
235C22×C8C22×D7911yescharacteristic
236C2×C4⋊C4—1581no—
237C2×C22⋊C4—3162no—
238C2×C42—791no—
239SD32—24no—
240D16—24no—
241M32—21no—
242M32—1581no—
243M32—1581no—
244M32C2×Dic7912yes—
245C2×C16—21no—
246C2×C16—1581no—
247#108—792no—
248#108—1581no—
249#107—1581no—
250#107—1581no—
251#104—1584no—
252#102—3161no—
253#102—3161no—
254C4×C8—1581no—
255C4×C8—1581no—
256C24—41no—
257#41—22no—
258#41—22no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×D4—22no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4—3161no—
272C22×C4—1581no—
273C22×C4D4×D7911yescharacteristic
274Q16—21no—
275Q16C22⋊Dic7912yescomplemented
276SD16—22no—
277SD16—22no—
278D8—22no—
279D8—22no—
280D8—21no—
281D8—41no—
282D8—41no—
283D8C22⋊Dic7912yescomplemented
284M16—3161no—
285M16—3164no—
286M16—1582no—
287C2×C8—3161no—
288C2×C8—3164no—
289C2×C8—791no—
290C2×C8—791no—
291C2×C8—1581no—
292C2×C8—1581no—
293C2×C8C22×Dic7911yescharacteristic
294C2×C8D4×D7911yescharacteristic
295C2×C8C2×C22⋊D7911yescharacteristic
296C2×C8C2×C22⋊D7911yescharacteristic
297C2×C8#1192962311yescharacteristic
298C2×C8#1192962311yescharacteristic
299C4⋊C4—3161no—
300C4⋊C4—3161no—
301C22⋊C4—6324no—
302C42—1581no—
303C42—1581no—
304C16—22no—
305C16—3161no—
306C23—41no—
307C23—42no—
308C23—82no—
309C23—82no—
310C23—81no—
311C23—81no—
312C23#1457918511yescharacteristic
313Q8—22no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—41no—
321D4—81no—
322D4—81no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—6322no—
329C2×C4—6321no—
330C2×C4—1582no—
331C2×C4D4×Dic7911yescharacteristic
332C2×C4#1457912911yescharacteristic
333C2×C4#1457913311yescharacteristic
334C2×C4D8×D7911yescharacteristic
335C2×C4#1457918011yescharacteristic
336C2×C4#1457918511yescharacteristic
337C8—3162no—
338C8—1581no—
339C8—1581no—
340C8C2×C22⋊Dic7911yescharacteristic
341C8D4×Dic7911yescharacteristic
342C8#1457913311yescharacteristic
343C8#1457914211yescharacteristic
344C22—21no—
345C22—21no—
346C22—42no—
347C22—84no—
348C22—81no—
349C22—81no—
350C22—82no—
351C22—162no—
352C22—161no—
353C22—161no—
354C22#1909867511yescharacteristic
355C22#1909870511yescharacteristic
356C22#1909875611yescharacteristic
357C4—42no—
358C4—3162no—
359C4—12641no—
360C4#1909844311yescharacteristic
361C4D8×Dic7911yescharacteristic
362C4#1909869511yescharacteristic
363C4#1909877611yescharacteristic
364C2—21no—
365C2—21no—
366C2—82no—
367C2—162no—
368C2#2376375011yescharacteristic
369C2#2376546711yescharacteristic
370C2#2376547411yescharacteristic
subgroupG/H
centerC22#19098756
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC2×#509×C79C2
socleC22×C79#459
cosocleC22×C8×C79C23
hypercenterC2×#509D79
hypocenterC79#13029
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#19098680C2semidirect
2. C2×#19098376C2semidirect
3. #23765112C2semidirect
4. #23764267C2semidirect
5. #23761509C2semidirect
6. #23761481C2semidirect
24. #509×C79C4semidirect
104. Q16×C79C22⋊C4semidirect
112. D8×C79C22⋊C4semidirect
193. C79#13029semidirect
198. #509Dic79semidirect
275. Q16C22⋊Dic79semidirect
283. D8C22⋊Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190986802
C2×#190983762
#237651122
#237642672
#237615092
#237614812
C2×#509×C792
#1302979