Finite Group Databasebeta

← #34205445

Subgroups — #34205445

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×#19098998C211yesmaximal, characteristic
2C2×#19098375C211yesmaximal, characteristic, complemented
3#23765112C211yesmaximal, characteristic, complemented
4#23764266C211yesmaximal, characteristic, complemented
5#23761510C211yesmaximal, characteristic, complemented
6#23761482C211yesmaximal, characteristic, complemented
7C2×#443×C79C211yesmaximal, characteristic, complemented
8C22×D632C2211yescharacteristic
9#19098998—22no—
10#19098998C414yescomplemented
11C2×#14579104—21no—
12C2×#14579102—21no—
13C2×#14579101C2211yescharacteristic
14C2×#14579101C2211yescharacteristic
15C2×#14579095C2211yescharacteristic
16C2×#14579083—21no—
17#19098465—21no—
18#19098375—22no—
19#19098367—21no—
20#19098364—21no—
21C2×#135×C79C2211yescharacteristic
22C2×#131×C79—21no—
23#443×C79—22no—
24#443×C79—22no—
25C2×#108×C79C2211yescharacteristic
26C2×#104×C79—21no—
27C2×C4×C8×C79C2211yescharacteristic
28C22×D316—21no—
29C2×D632—21no—
30C2×D632—21no—
31C2×D632—21no—
32C2×D632—21no—
33C2×D632—21no—
34C2×D632C2×C412yes—
35C2×C4⋊Dic79—21no—
36C2×(C2×D79)⋊C4—42no—
37#14579104—24no—
38#14579102—24no—
39#14579101—21no—
40#14579101—21no—
41#14579101—21no—
42#14579101C2×C412yes—
43C2×C79⋊C16—21no—
44C2×C79⋊C16—21no—
45#14579095—21no—
46#14579095—21no—
47#14579083—41no—
48#14579083—41no—
49C2×#41×C79—21no—
50#135×C79—21no—
51#135×C79—21no—
52#135×C79C2×C412yes—
53#135×C79D412yes—
54C2×Q16×C79D411yescharacteristic
55C2×SD16×C79—21no—
56C2×D8×C79D411yescharacteristic
57#131×C79—24no—
58#131×C79—42no—
59C2×M16×C79—21no—
60C2×M16×C79—41no—
61C22×C8×C79—41no—
62C22×C8×C79D411yescharacteristic
63C22×C8×C79C2311yescharacteristic
64C2×C42×C79D411yescharacteristic
65#108×C79—21no—
66#108×C79D412yes—
67#104×C79—24no—
68C4×C8×C79—21no—
69C4×C8×C79—21no—
70C23×D79—41no—
71C2×D316—22no—
72C2×D316—41no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—41no—
76C2×D316—41no—
77C22×Dic79—41no—
78D632—22no—
79D632—22no—
80D632—41no—
81D632—41no—
82C4⋊Dic79—41no—
83C4⋊Dic79—41no—
84(C2×D79)⋊C4—84no—
85C79⋊C16—22no—
86C79⋊C16—41no—
87#41×C79—22no—
88#41×C79—22no—
89#41×C79—41no—
90#41×C79—41no—
91C2×Q8×C79—21no—
92C2×D4×C79—21no—
93C2×D4×C79—41no—
94C22×C4×C79—21no—
95C22×C4×C79—41no—
96C22×C4×C79C2×D411yescharacteristic
97Q16×C79—21no—
98Q16×C79C22⋊C412yescomplemented
99SD16×C79—22no—
100SD16×C79—22no—
101D8×C79—21no—
102D8×C79C22⋊C412yescomplemented
103M16×C79—22no—
104M16×C79—44no—
105M16×C79—41no—
106C2×C8×C79—21no—
107C2×C8×C79—21no—
108C2×C8×C79—44no—
109C2×C8×C79—41no—
110C2×C8×C79D811yescharacteristic
111C2×C8×C79D811yescharacteristic
112C2×C8×C79C22×C411yescharacteristic
113C2×C8×C79C2×D411yescharacteristic
114C2×C8×C79C2×D411yescharacteristic
115C2×C8×C79C2×D411yescharacteristic
116C2×C8×C79#4111yescharacteristic
117C2×C8×C79#4111yescharacteristic
118C42×C79—21no—
119C42×C79—21no—
120C23×C79—41no—
121C23×C79AGL(1,ℤ/8)11yescharacteristic
122C22×D79—42no—
123C22×D79—82no—
124C22×D79—82no—
125C22×D79—81no—
126C22×D79—81no—
127Q8×C79—22no—
128Q8×C79—41no—
129D4×C79—22no—
130D4×C79—42no—
131D4×C79—42no—
132D4×C79—41no—
133C2×C4×C79—22no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—42no—
138C2×C4×C79—42no—
139C2×C4×C79C4×D411yescharacteristic
140C2×C4×C79#12011yescharacteristic
141C2×C4×C79#12111yescharacteristic
142C2×C4×C79C2×D811yescharacteristic
143C2×C4×C79#13511yescharacteristic
144C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
145C2×Dic79—82no—
146C2×Dic79—81no—
147D316—42no—
148D316—42no—
149D316—81no—
150D316—81no—
151C8×C79—21no—
152C8×C79—21no—
153C8×C79—42no—
154C8×C79C2×C22⋊C411yescharacteristic
155C8×C79C4×D411yescharacteristic
156C8×C79#12111yescharacteristic
157C8×C79#12311yescharacteristic
158C22×C79—21no—
159C22×C79—21no—
160C22×C79—42no—
161C22×C79—81no—
162C22×C79—81no—
163C22×C79#44211yescharacteristic
164C22×C79#44911yescharacteristic
165C22×C79#45911yescharacteristic
166C2×D79—84no—
167C2×D79—82no—
168C2×D79—162no—
169C2×D79—161no—
170C2×D79—161no—
171C4×C79—42no—
172C4×C79—42no—
173C4×C79#38611yescharacteristic
174C4×C79C4×D811yescharacteristic
175C4×C79#44711yescharacteristic
176C4×C79#46311yescharacteristic
177Dic79—161no—
178#13029—791nomaximal, Sylow
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—82no—
182C2×C79#187911yescharacteristic
183C2×C79#221211yescharacteristic
184C2×C79#221311yescharacteristic
185D79—162no—
186C2×#509—791no—
187C2×#443D7911yescharacteristic, complemented
188C2×#361—791no—
189#2139—791no—
190#1970—791no—
191#1387—791no—
192#1376—791no—
193C79#1302911yescharacteristic, complemented, Sylow
194C2×#135C2×D7911yescharacteristic
195C22×D8—791no—
196C2×#131—21no—
197#509—794no—
198#509—1582no—
199C2×SD32—1581no—
200C2×D16—1581no—
201C2×M32—791no—
202C2×M32—791no—
203#443—22no—
204#443—22no—
205C2×#108C2×D7911yescharacteristic
206C2×#107—791no—
207C2×#104—21no—
208C2×#102—1581no—
209C2×C4×C8C2×D7911yescharacteristic
210#390—1581no—
211#361—1582no—
212#358—1581no—
213#357—1581no—
214C2×#41—21no—
215C22×D4—1581no—
216#135—21no—
217#135—21no—
218#135C4×D7912yes—
219#135C22⋊D7912yes—
220C2×Q16D31611yescharacteristic
221C2×SD16—21no—
222C2×D8—792no—
223C2×D8—1581no—
224C2×D8—1581no—
225C2×D8—1581no—
226C2×D8—1581no—
227C2×D8—1581no—
228C2×D8D31611yescharacteristic
229#131—24no—
230#131—42no—
231C2×M16—21no—
232C2×M16—41no—
233C22×C8—41no—
234C22×C8C22⋊D7911yescharacteristic
235C22×C8C22×D7911yescharacteristic
236C2×C4⋊C4—1581no—
237C2×C22⋊C4—3162no—
238C2×C42C22⋊D7911yescharacteristic
239SD32—1584no—
240D16—1584no—
241M32—792no—
242M32—1581no—
243M32—1581no—
244M32—1581no—
245C2×C16—1581no—
246C2×C16—1581no—
247#108—21no—
248#108C22⋊D7912yes—
249#107—1581no—
250#107—1581no—
251#104—24no—
252#102—3161no—
253#102—3161no—
254C4×C8—21no—
255C4×C8—21no—
256C24—3161no—
257#41—22no—
258#41—22no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—3161no—
265C2×D4—3161no—
266C2×D4—3161no—
267C2×D4—3161no—
268C2×D4—3161no—
269C2×D4—1582no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—3161no—
273C22×C4C2×C22⋊D7911yescharacteristic
274Q16—21no—
275Q16(C2×D79)⋊C412yescomplemented
276SD16—22no—
277SD16—22no—
278D8—21no—
279D8—3161no—
280D8—3161no—
281D8—1582no—
282D8—1582no—
283D8(C2×D79)⋊C412yescomplemented
284M16—22no—
285M16—44no—
286M16—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—44no—
290C2×C8—41no—
291C2×C8#1192960911yescharacteristic
292C2×C8#1192960911yescharacteristic
293C2×C8C2×C4×D7911yescharacteristic
294C2×C8C2×C22⋊D7911yescharacteristic
295C2×C8C2×C22⋊D7911yescharacteristic
296C2×C8C2×D31611yescharacteristic
297C2×C8#1192962411yescharacteristic
298C2×C8#1192962411yescharacteristic
299C4⋊C4—3161no—
300C4⋊C4—3161no—
301C22⋊C4—6324no—
302C42—21no—
303C42—21no—
304C16—3161no—
305C16—1582no—
306C23—41no—
307C23—3162no—
308C23—6322no—
309C23—6322no—
310C23—6321no—
311C23—6321no—
312C23#1457918611yescharacteristic
313Q8—22no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—3162no—
320D4—3162no—
321D4—6321no—
322D4—6321no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—6322no—
330C2×C4—6321no—
331C2×C4C4×C22⋊D7911yescharacteristic
332C2×C4#1457913011yescharacteristic
333C2×C4#1457913511yescharacteristic
334C2×C4C2×#1192960911yescharacteristic
335C2×C4#1457918111yescharacteristic
336C2×C4#1457918711yescharacteristic
337C8—21no—
338C8—21no—
339C8—42no—
340C8C2×(C2×D79)⋊C411yescharacteristic
341C8C4×C22⋊D7911yescharacteristic
342C8#1457913511yescharacteristic
343C8#1457914411yescharacteristic
344C22—21no—
345C22—21no—
346C22—42no—
347C22—81no—
348C22—81no—
349C22—6324no—
350C22—6322no—
351C22—12642no—
352C22—12641no—
353C22—12641no—
354C22#1909867311yescharacteristic
355C22#1909870811yescharacteristic
356C22#1909875811yescharacteristic
357C4—42no—
358C4—42no—
359C4—12641no—
360C4#1909844211yescharacteristic
361C4C4×#1192960911yescharacteristic
362C4#1909869711yescharacteristic
363C4#1909877811yescharacteristic
364C2—21no—
365C2—21no—
366C2—82no—
367C2—12642no—
368C2#2376374811yescharacteristic
369C2#2376546911yescharacteristic
370C2#2376547611yescharacteristic
subgroupG/H
centerC22#19098758
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC2×#443×C79C2
socleC22×C79#459
prime radicalC2×#19098998C2
cosocleC22×C8×C79C23
hypercenterC2×#443D79
hypocenterC79#13029
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
2. C2×#19098375C2semidirect
3. #23765112C2semidirect
4. #23764266C2semidirect
5. #23761510C2semidirect
6. #23761482C2semidirect
7. C2×#443×C79C2semidirect
10. #19098998C4semidirect
98. Q16×C79C22⋊C4semidirect
102. D8×C79C22⋊C4semidirect
187. C2×#443D79semidirect
193. C79#13029semidirect
275. Q16(C2×D79)⋊C4semidirect
283. D8(C2×D79)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#190989982
C2×#190983752
#237651122
#237642662
#237615102
#237614822
C2×#443×C792
#1302979