Finite Group Databasebeta

← #34183137

Subgroups — #34183137

Order 20224.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11929608C211yesmaximal, characteristic, complemented
2#23764320C211yesmaximal, characteristic
3C4×#14579085C211yesmaximal, characteristic, complemented
4#23761851C211yesmaximal, characteristic
5#23761302C211yesmaximal, characteristic, complemented
6#23761236C211yesmaximal, characteristic, complemented
7C79×#3103C211yesmaximal, characteristic
8#19098694—22no—
9#19098694—21no—
10Q8×C79⋊C8—21no—
11#19098587—21no—
12#19098587C2211yescharacteristic
13C2×#14579088—21no—
14C2×#14579088—21no—
15C2×#14579085—21no—
16C4×#11929608—21no—
17C4×#11929608C2211yescharacteristic
18C2×C4×C79⋊C8C2211yescharacteristic
19C4×C4⋊Dic79C2211yescharacteristic
20#19098325—21no—
21D4⋊(C79⋊C8)—22no—
22#532×C79C2211yescharacteristic
23C2×C4×Q8×C79C2211yescharacteristic
24#481×C79—21no—
25#475×C79C411yescharacteristic, complemented
26#475×C79C411yescharacteristic, complemented
27#460×C79C411yescharacteristic, complemented
28#460×C79C411yescharacteristic, complemented
29#448×C79—21no—
30C4×SD16×C79—21no—
31C4×SD16×C79—21no—
32#431×C79C2211yescharacteristic
33C2×#11929608—21no—
34C2×#11929608D411yescharacteristic
35C2×#11929608D411yescharacteristic
36C22×C79⋊C8—21no—
37C2×C4⋊Dic79—21no—
38C2×C4×Dic79—21no—
39C4⋊(C79⋊C8)—22no—
40C4⋊(C79⋊C8)—22no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)D412yes—
43#14579088—42no—
44#14579088—42no—
45#14579085—24no—
46#14579072—22no—
47#14579072—21no—
48C4×C79⋊C8—22no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C79⋊#95—21no—
53C2×#41×C79—21no—
54C22×Q8×C79D411yescharacteristic
55C2×SD16×C79—21no—
56C2×SD16×C79—21no—
57C4⋊Q8×C79C2×C411yescharacteristic
58#127×C79C2×C411yescharacteristic
59#124×C79—21no—
60#121×C79—21no—
61#121×C79—21no—
62C4×Q8×C79—21no—
63C4×Q8×C79—21no—
64C4×Q8×C79C2×C411yescharacteristic
65C4×Q8×C79C2×C411yescharacteristic
66C4×Q8×C79D411yescharacteristic
67C4×Q8×C79D411yescharacteristic
68C4×D4×C79—21no—
69C4×D4×C79—21no—
70C2×C4⋊C4×C79—21no—
71C2×C4⋊C4×C79D411yescharacteristic
72C2×C42×C79—21no—
73C2×C42×C79C2311yescharacteristic
74#106×C79—21no—
75#106×C79—21no—
76C4⋊C8×C79C2×C411yescharacteristic
77C4⋊C8×C79C2×C411yescharacteristic
78#103×C79—21no—
79#103×C79—21no—
80#102×C79—21no—
81#102×C79—21no—
82C22⋊C8×C79—21no—
83C4×C8×C79—21no—
84C22×Dic79—41no—
85#11929608—21no—
86#11929608—21no—
87#11929608—21no—
88#11929608—41no—
89#11929608—41no—
90#11929608SD1612yescomplemented
91C2×C79⋊C8—22no—
92C2×C79⋊C8—21no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—22no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C4⋊Dic79—22no—
98C4⋊Dic79—41no—
99C4×Dic79—41no—
100C4×Dic79—41no—
101#41×C79—41no—
102#41×C79—41no—
103#41×C79—41no—
104#41×C79—41no—
105C2×Q8×C79—21no—
106C2×Q8×C79—22no—
107C2×Q8×C79—21no—
108C2×Q8×C79—41no—
109C2×Q8×C79C22⋊C411yescharacteristic
110C2×Q8×C79C22⋊C411yescharacteristic
111C2×D4×C79—21no—
112C2×D4×C79—21no—
113C2×D4×C79—21no—
114C2×D4×C79—41no—
115C22×C4×C79—21no—
116C22×C4×C79—21no—
117C22×C4×C79—41no—
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79C2×D411yescharacteristic
120C22×C4×C79#4111yescharacteristic
121SD16×C79—42no—
122SD16×C79—42no—
123C2×C8×C79—21no—
124C2×C8×C79—21no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79—21no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79—21no—
131C4⋊C4×C79—21no—
132C4⋊C4×C79C22⋊C411yescharacteristic
133C4⋊C4×C79C22⋊C411yescharacteristic
134C22⋊C4×C79—41no—
135C22⋊C4×C79—41no—
136C42×C79—21no—
137C42×C79—21no—
138C42×C79—21no—
139C42×C79—21no—
140C42×C79C22×C411yescharacteristic
141C42×C79C2×D411yescharacteristic
142C42×C79C2×D411yescharacteristic
143C42×C79#4111yescharacteristic
144C23×C79—41no—
145C23×C79#12111yescharacteristic
146Q8×C79—21no—
147Q8×C79—21no—
148Q8×C79—21no—
149Q8×C79—42no—
150Q8×C79—41no—
151Q8×C79—41no—
152Q8×C79#10412yescomplemented
153D4×C79—42no—
154D4×C79—41no—
155D4×C79—41no—
156D4×C79—41no—
157D4×C79—41no—
158D4×C79—81no—
159D4×C79—81no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—21no—
165C2×C4×C79—21no—
166C2×C4×C79—41no—
167C2×C4×C79—42no—
168C2×C4×C79—41no—
169C2×C4×C79—81no—
170C2×C4×C79—81no—
171C2×C4×C79—81no—
172C2×C4×C79C2×C22⋊C411yescharacteristic
173C2×C4×C79C4×D411yescharacteristic
174C2×C4×C79C4×D411yescharacteristic
175C2×C4×C79#12011yescharacteristic
176C2×C4×C79#12111yescharacteristic
177C2×C4×C79#12311yescharacteristic
178C2×C4×C79C2×SD1611yescharacteristic
179C2×C4×C79#13511yescharacteristic
180C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C79#13711yescharacteristic
182C2×Dic79—42no—
183C2×Dic79—81no—
184C2×Dic79—81no—
185C8×C79—41no—
186C8×C79—41no—
187C79⋊C8—22no—
188C79⋊C8—42no—
189C79⋊C8—41no—
190C79⋊C8—42no—
191C22×C79—21no—
192C22×C79—21no—
193C22×C79—81no—
194C22×C79—81no—
195C22×C79—81no—
196C22×C79#38611yescharacteristic
197C22×C79#46011yescharacteristic
198C22×C79#46411yescharacteristic
199C4×C79—21no—
200C4×C79—22no—
201C4×C79—21no—
202C4×C79—21no—
203C4×C79—41no—
204C4×C79—41no—
205C4×C79—41no—
206C4×C79—81no—
207C4×C79C2×#10411yescharacteristic
208C4×C79#42111yescharacteristic
209C4×C79C4×SD1611yescharacteristic
210C4×C79#44011yescharacteristic
211C4×C79#44811yescharacteristic
212C4×C79#44911yescharacteristic
213Dic79—82no—
214#8992—791nomaximal, Sylow
215C2×C79—21no—
216C2×C79—21no—
217C2×C79—161no—
218C2×C79#187111yescharacteristic
219C2×C79#188011yescharacteristic
220C2×C79#188111yescharacteristic
221#3103D7911yescharacteristic
222Q8×M16—791no—
223#1980—791no—
224C4×#103—791no—
225#1492—791no—
226#1303—791no—
227#1278—791no—
228C79#899211yescharacteristic, complemented, Sylow
229#532C2×D7911yescharacteristic
230C2×C4×Q8C2×D7911yescharacteristic
231#481—21no—
232#475Dic7911yescharacteristic, complemented
233#475Dic7911yescharacteristic, complemented
234#460Dic7911yescharacteristic, complemented
235#460Dic7911yescharacteristic, complemented
236#448—21no—
237#446—1582no—
238#446—1581no—
239C8×Q8—1581no—
240C4×SD16—21no—
241C4×SD16—21no—
242#431C2×D7911yescharacteristic
243#423—791no—
244#423—1581no—
245C2×#104—1581no—
246C2×#104—1581no—
247C2×#103—1581no—
248C4×M16—791no—
249C4×M16—1581no—
250C2×C4×C8—791no—
251C4×C4⋊C4—791no—
252#340—1581no—
253D4⋊C8—1582no—
254C2×#41—21no—
255C22×Q8C22⋊D7911yescharacteristic
256C2×SD16—21no—
257C2×SD16—21no—
258C2×M16—791no—
259C2×M16—791no—
260C2×M16—1581no—
261C22×C8—1581no—
262C4⋊Q8C2×Dic7911yescharacteristic
263#127C2×Dic7911yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C2×Dic7911yescharacteristic
270C4×Q8C2×Dic7911yescharacteristic
271C4×Q8C22⋊D7911yescharacteristic
272C4×Q8C22⋊D7911yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1581no—
277C2×C4⋊C4C22⋊D7911yescharacteristic
278C2×C42—21no—
279C2×C42—1581no—
280C2×C42C22×D7911yescharacteristic
281#106—21no—
282#106—21no—
283C4⋊C8—792no—
284C4⋊C8—1582no—
285C4⋊C8—1582no—
286C4⋊C8—1581no—
287C4⋊C8C2×Dic7911yescharacteristic
288C4⋊C8C2×Dic7911yescharacteristic
289#104—3162no—
290#104—3162no—
291#103—21no—
292#103—21no—
293#103—1584no—
294#102—21no—
295#102—21no—
296C22⋊C8—21no—
297#97—1582no—
298#97—1581no—
299C4×C8—21no—
300C4×C8—1582no—
301C4×C8—1581no—
302C4×C8—1581no—
303C4×C8—1581no—
304#95—1581no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8C22⋊Dic7911yescharacteristic
314C2×Q8C22⋊Dic7911yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—3161no—
323C22×C4D4×D7911yescharacteristic
324C22×C4C2×C22⋊D7911yescharacteristic
325C22×C4#1192962311yescharacteristic
326SD16—42no—
327SD16—42no—
328M16—3161no—
329M16—3161no—
330M16—792no—
331M16—1581no—
332M16—1581no—
333M16—1581no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—3161no—
337C2×C8—3161no—
338C2×C8—1582no—
339C2×C8—1582no—
340C2×C8—1581no—
341C2×C8—1581no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—3161no—
350C4⋊C4—1582no—
351C4⋊C4C22⋊Dic7911yescharacteristic
352C4⋊C4C22⋊Dic7911yescharacteristic
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—3161no—
360C42—3161no—
361C42C22×Dic7911yescharacteristic
362C42D4×D7911yescharacteristic
363C42C2×C22⋊D7911yescharacteristic
364C42#1192962311yescharacteristic
365C23—41no—
366C23#1457913311yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—41no—
372Q8—41no—
373Q8#1457908812yescomplemented
374D4—42no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—3162no—
394C2×C4—6321no—
395C2×C4—6321no—
396C2×C4C2×C22⋊Dic7911yescharacteristic
397C2×C4D4×Dic7911yescharacteristic
398C2×C4D4×Dic7911yescharacteristic
399C2×C4#1457912911yescharacteristic
400C2×C4#1457913311yescharacteristic
401C2×C4#1457914211yescharacteristic
402C2×C4SD16×D7911yescharacteristic
403C2×C4#1457918211yescharacteristic
404C2×C4#1457918811yescharacteristic
405C2×C4#1457919011yescharacteristic
406C8—41no—
407C8—41no—
408C8—3162no—
409C8—3162no—
410C8—3161no—
411C8—1582no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#1909844311yescharacteristic
418C22#1909876111yescharacteristic
419C22#1909878311yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—81no—
428C4—6322no—
429C4C2×#1457908811yescharacteristic
430C4#1909858011yescharacteristic
431C4SD16×Dic7911yescharacteristic
432C4#1909866411yescharacteristic
433C4#1909870011yescharacteristic
434C4#1909870711yescharacteristic
435C2—21no—
436C2—21no—
437C2—161no—
438C2#2376370311yescharacteristic
439C2#2376375711yescharacteristic
440C2#2376376411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3103C2
socleC22×C79#386
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#3103D79
hypocenterC79#8992
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. Q8×#11929608C2semidirect
3. C4×#14579085C2semidirect
5. #23761302C2semidirect
6. #23761236C2semidirect
25. #475×C79C4semidirect
26. #475×C79C4semidirect
27. #460×C79C4semidirect
28. #460×C79C4semidirect
90. #11929608SD16semidirect
152. Q8×C79#104semidirect
228. C79#8992semidirect
232. #475Dic79semidirect
233. #475Dic79semidirect
234. #460Dic79semidirect
235. #460Dic79semidirect
373. Q8#14579088semidirect

Maximal subgroups

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

subgroupindex
Q8×#119296082
#237643202
C4×#145790852
#237618512
#237613022
#237612362
C79×#31032
#899279