Finite Group Databasebeta

← #34184364

Subgroups — #34184364

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773188C211yesmaximal, characteristic
2#23772367C211yesmaximal, characteristic
3#23761898C211yesmaximal, characteristic, complemented
4#23761854C211yesmaximal, characteristic, complemented
5#23761295C211yesmaximal, characteristic, complemented
6#23761241C211yesmaximal, characteristic, complemented
7C79×#1806C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098870—21no—
10#19098863—21no—
11#19098858C411yescharacteristic, complemented
12#19098853C411yescharacteristic, complemented
13#19098848C411yescharacteristic, complemented
14#19098843C411yescharacteristic, complemented
15#19098694—21no—
16#19098641—21no—
17#19098635—22no—
18#19098616—21no—
19#19098616C2211yescharacteristic
20#19098599—21no—
21C2×C4⋊(C79⋊C8)C2211yescharacteristic
22#19098559—21no—
23C4×#11929608—21no—
24C2×C4×C79⋊C8C2211yescharacteristic
25#19098299—22no—
26#19098293—21no—
27#19098293—21no—
28C4×#41×C79C2211yescharacteristic
29#423×C79C2211yescharacteristic
30C2×#104×C79—21no—
31C4×C4⋊C4×C79C2211yescharacteristic
32#339×C79—21no—
33C2×#11929608—22no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36#14579161C2×C411yescharacteristic
37#14579161C2×C411yescharacteristic
38#14579153—21no—
39#14579140—21no—
40#14579140—21no—
41C4×Dic158—21no—
42C4×Dic158—21no—
43#14579120—21no—
44#14579094—21no—
45#14579094—21no—
46#14579092—21no—
47#14579092—21no—
48C4⋊(C79⋊C8)—22no—
49C4⋊(C79⋊C8)—21no—
50C4⋊(C79⋊C8)—21no—
51C4⋊(C79⋊C8)C2×C411yescharacteristic
52C4⋊(C79⋊C8)C2×C411yescharacteristic
53#14579085—21no—
54#14579085—21no—
55#14579082—21no—
56#14579082—21no—
57C22⋊(C79⋊C8)—22no—
58C22⋊(C79⋊C8)D412yes—
59#14579072—22no—
60C4×C79⋊C8—21no—
61C4×C79⋊C8—21no—
62C4×C79⋊C8D411yescharacteristic
63C4×C79⋊C8D411yescharacteristic
64C2×#41×C79D411yescharacteristic
65C2×M16×C79—21no—
66C4×Q8×C79C2×C411yescharacteristic
67C4×Q8×C79D411yescharacteristic
68C4×D4×C79—21no—
69C4×D4×C79—21no—
70C4×D4×C79C2×C411yescharacteristic
71C4×D4×C79D411yescharacteristic
72#117×C79—21no—
73#117×C79D411yescharacteristic
74C2×C4⋊C4×C79—21no—
75C2×C42×C79—21no—
76C2×C42×C79—21no—
77C2×C42×C79C2311yescharacteristic
78C4⋊C8×C79—21no—
79C4⋊C8×C79—21no—
80#104×C79—42no—
81#95×C79—21no—
82C2×Dic158—21no—
83C2×Dic158—21no—
84#11929608—44no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—22no—
90C2×C79⋊C8—42no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C4⋊Dic79—21no—
94C4⋊Dic79—21no—
95C4⋊Dic79—21no—
96C4⋊Dic79—21no—
97Dic79⋊C4—41no—
98Dic79⋊C4—41no—
99C22⋊Dic79—41no—
100C4×Dic79—41no—
101#41×C79—21no—
102#41×C79—21no—
103#41×C79—21no—
104#41×C79—21no—
105C2×Q8×C79C22⋊C411yescharacteristic
106C2×D4×C79—21no—
107C2×D4×C79C22⋊C411yescharacteristic
108C22×C4×C79—21no—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—41no—
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79#4111yescharacteristic
115M16×C79—41no—
116M16×C79—41no—
117C2×C8×C79—41no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—22no—
121C4⋊C4×C79—41no—
122C4⋊C4×C79C22⋊C411yescharacteristic
123C4⋊C4×C79C22⋊C411yescharacteristic
124C22⋊C4×C79—21no—
125C22⋊C4×C79—21no—
126C22⋊C4×C79—21no—
127C42×C79—21no—
128C42×C79—21no—
129C42×C79—21no—
130C42×C79—41no—
131C42×C79—41no—
132C42×C79C22×C411yescharacteristic
133C42×C79C2×D411yescharacteristic
134C42×C79C2×D411yescharacteristic
135C42×C79#4111yescharacteristic
136C23×C79—21no—
137C23×C79#12111yescharacteristic
138Q8×C79—21no—
139Q8×C79—21no—
140D4×C79—21no—
141D4×C79—21no—
142D4×C79—41no—
143D4×C79—41no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—41no—
150C2×C4×C79—42no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—42no—
156C2×C4×C79—81no—
157C2×C4×C79—81no—
158C2×C4×C79C2×C22⋊C411yescharacteristic
159C2×C4×C79C4×D411yescharacteristic
160C2×C4×C79C4×D411yescharacteristic
161C2×C4×C79#12011yescharacteristic
162C2×C4×C79#12111yescharacteristic
163C2×C4×C79#12311yescharacteristic
164C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C79#13711yescharacteristic
166C2×C4×C79#13711yescharacteristic
167C2×C4×C79#13711yescharacteristic
168C2×Dic79—41no—
169C2×Dic79—41no—
170Dic158—41no—
171Dic158—41no—
172Dic158—81no—
173C8×C79—81no—
174C79⋊C8—22no—
175C79⋊C8—41no—
176C79⋊C8—42no—
177C79⋊C8—41no—
178C79⋊C8—41no—
179C22×C79—21no—
180C22×C79—21no—
181C22×C79—41no—
182C22×C79—41no—
183C22×C79—41no—
184C22×C79#38611yescharacteristic
185C22×C79#46811yescharacteristic
186C22×C79#47011yescharacteristic
187C4×C79—21no—
188C4×C79—21no—
189C4×C79—21no—
190C4×C79—41no—
191C4×C79—41no—
192C4×C79—41no—
193C4×C79—82no—
194C4×C79#41711yescharacteristic
195C4×C79#41911yescharacteristic
196C4×C79#44311yescharacteristic
197C4×C79#44411yescharacteristic
198C4×C79#46811yescharacteristic
199C4×C79#47011yescharacteristic
200Dic79—81no—
201Dic79—81no—
202#9209—791nomaximal, Sylow
203C2×C79—21no—
204C2×C79—21no—
205C2×C79—81no—
206C2×C79#187611yescharacteristic
207C2×C79#189411yescharacteristic
208C2×C79#190611yescharacteristic
209#3076—791no—
210#2965—791no—
211#1806D7911yescharacteristic
212#1503—791no—
213#1493—791no—
214#1301—791no—
215#1279—791no—
216C79#920911yescharacteristic, complemented, Sylow
217#533—791no—
218C4×#41C2×D7911yescharacteristic
219#479—1581no—
220#478—1581no—
221#477—791no—
222#476—791no—
223#475—791no—
224#474—791no—
225#446—1581no—
226#436—1581no—
227#435—1582no—
228#431—791no—
229#431—1581no—
230#427—1581no—
231#423C2×D7911yescharacteristic
232C2×C4⋊C8—791no—
233C2×#104—21no—
234#416—1581no—
235C4×M16—1581no—
236C2×C4×C8—791no—
237C4×C4⋊C4C2×D7911yescharacteristic
238#339—21no—
239#328—1582no—
240Q8⋊C8—1581no—
241Q8⋊C8—1581no—
242C2×#41C22⋊D7911yescharacteristic
243C2×M16—21no—
244C2×M16—1582no—
245C22×C8—1581no—
246C22×C8—1581no—
247C4⋊Q8—791no—
248C4⋊Q8—791no—
249#125—1581no—
250C22⋊Q8—1581no—
251C22⋊Q8—1581no—
252C4×Q8—1581no—
253C4×Q8—1581no—
254C4×Q8C4×D7911yescharacteristic
255C4×Q8D31611yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C4×D7911yescharacteristic
259C4×D4D31611yescharacteristic
260#117—21no—
261#117—1581no—
262#117C22⋊D7911yescharacteristic
263C2×C4⋊C4—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C22×D7911yescharacteristic
267#107—1581no—
268#107—1581no—
269#106—1581no—
270#106—1581no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—791no—
274C4⋊C8—791no—
275C4⋊C8—1582no—
276C4⋊C8—1581no—
277C4⋊C8—1581no—
278#104—42no—
279#103—1581no—
280#103—1581no—
281#102—1581no—
282#102—1581no—
283C22⋊C8—792no—
284C22⋊C8—1582no—
285#97—1582no—
286C4×C8—791no—
287C4×C8—791no—
288C4×C8—1581no—
289C4×C8—1581no—
290#95—21no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295C2×Q8—1581no—
296C2×Q8—1581no—
297C2×Q8(C2×D79)⋊C411yescharacteristic
298C2×D4—21no—
299C2×D4(C2×D79)⋊C411yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4D4×D7911yescharacteristic
305C22×C4C2×C22⋊D7911yescharacteristic
306C22×C4#1192962311yescharacteristic
307M16—41no—
308M16—41no—
309M16—3164no—
310C2×C8—41no—
311C2×C8—3161no—
312C2×C8—3161no—
313C2×C8—3162no—
314C2×C8—1581no—
315C2×C8—1581no—
316C2×C8—1581no—
317C2×C8—1582no—
318C2×C8—1581no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—3161no—
324C4⋊C4—3161no—
325C4⋊C4—1581no—
326C4⋊C4—1581no—
327C4⋊C4—1581no—
328C4⋊C4—1581no—
329C4⋊C4(C2×D79)⋊C411yescharacteristic
330C4⋊C4(C2×D79)⋊C411yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—3161no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—41no—
339C42—41no—
340C42—3161no—
341C42C2×C4×D7911yescharacteristic
342C42D4×D7911yescharacteristic
343C42C2×D31611yescharacteristic
344C42#1192962311yescharacteristic
345C23—21no—
346C23#1457913311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—3161no—
350Q8—3161no—
351Q8—6321no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—41no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—3161no—
371C2×C4—3161no—
372C2×C4C2×(C2×D79)⋊C411yescharacteristic
373C2×C4(C22⋊D79)⋊C411yescharacteristic
374C2×C4(C22⋊D79)⋊C411yescharacteristic
375C2×C4#1457913111yescharacteristic
376C2×C4#1457913311yescharacteristic
377C2×C4#1457914511yescharacteristic
378C2×C4#1457918711yescharacteristic
379C2×C4#1457919111yescharacteristic
380C2×C4#1457919111yescharacteristic
381C2×C4#1457919211yescharacteristic
382C8—81no—
383C8—3161no—
384C8—3162no—
385C8—3161no—
386C8—3161no—
387C8—1582no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22#1909844011yescharacteristic
394C22#1909880711yescharacteristic
395C22#1909881911yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—82no—
403C4—6321no—
404C4—6321no—
405C4#1909856211yescharacteristic
406C4#1909857111yescharacteristic
407C4#1909867711yescharacteristic
408C4#1909868211yescharacteristic
409C4#1909880711yescharacteristic
410C4#1909881911yescharacteristic
411C2—21no—
412C2—21no—
413C2—81no—
414C2#2376372511yescharacteristic
415C2#2376383111yescharacteristic
416C2#2376389911yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1806C2
socleC22×C79#386
prime radicalC2×#41×C79D4
cosocleC2×C42×C79C23
hypercenter#1806D79
hypocenterC79#9209
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
3. #23761898C2semidirect
4. #23761854C2semidirect
5. #23761295C2semidirect
6. #23761241C2semidirect
11. #19098858C4semidirect
12. #19098853C4semidirect
13. #19098848C4semidirect
14. #19098843C4semidirect
216. C79#9209semidirect

Maximal subgroups

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

subgroupindex
#237731882
#237723672
#237618982
#237618542
#237612952
#237612412
C79×#18062
#920979