Finite Group Databasebeta

← #34442866

Subgroups — #34442866

Order 20224.

636 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099003C211yesmaximal, characteristic, complemented
2#23776503C211yesmaximal, characteristic, complemented
3C2×#19098838C211yesmaximal, characteristic, complemented
4#23765528C214yesmaximal, complemented
5#23765523C214yesmaximal, complemented
6#23765194C211yesmaximal, characteristic, complemented
7#23765149C211yesmaximal, characteristic, complemented
8C2×#19098378C211yesmaximal, characteristic, complemented
9C79×#3067C211yesmaximal, characteristic
10C2×#14579191—21no—
11C22×Dic316C2211yescharacteristic
12#19099003—22no—
13#19099003—22no—
14C2×#14579107C2212yescomplemented
15C2×#14579105C2212yescomplemented
16#19098980—22no—
17#19098980C2214yes—
18C2×#14579101—22no—
19C2×#14579101C2211yescharacteristic
20C2×#14579101C2211yescharacteristic
21#19098838—22no—
22#19098838C2214yes—
23C2×#14579097C2211yescharacteristic
24C2×#14579079—21no—
25#19098392C2214yescomplemented
26#19098379C2212yescomplemented
27#19098378C2214yescomplemented
28#19098376C2212yescomplemented
29#575×C79—21no—
30C2×#135×C79C2211yescharacteristic
31C2×#131×C79C2211yescharacteristic
32#473×C79C2212yes—
33#472×C79C2212yes—
34#471×C79—22no—
35C2×#108×C79C2211yescharacteristic
36#413×C79—21no—
37C22×Dic158—21no—
38#14579191—24no—
39#14579191—42no—
40C2×Dic316—21no—
41C2×Dic316—21no—
42C2×Dic316—21no—
43C2×Dic316—21no—
44C2×Dic316—21no—
45C2×Dic316C2312yes—
46C2×#11929614—41no—
47C2×#11929612—41no—
48C2×#11929608—21no—
49#14579107—24no—
50#14579105—24no—
51#14579101—24no—
52#14579101—21no—
53#14579101D412yes—
54#14579101D412yes—
55#14579101C2312yes—
56C2×C79⋊C16—24no—
57C2×C79⋊C16—21no—
58C2×C79⋊C16C2312yes—
59#14579097—21no—
60#14579097C2312yes—
61#14579079—24no—
62C2×#41×C79—21no—
63C2×#41×C79D411yescharacteristic
64#137×C79—22no—
65#137×C79—41no—
66AGL(1,ℤ/8)×C79—22no—
67AGL(1,ℤ/8)×C79—41no—
68#135×C79—21no—
69#135×C79—21no—
70#135×C79—21no—
71#135×C79—21no—
72#135×C79—42no—
73C2×Q16×C79C2311yescharacteristic
74C2×SD16×C79—21no—
75C2×D8×C79C2311yescharacteristic
76#131×C79—21no—
77#131×C79—21no—
78#131×C79D412yes—
79#131×C79C2312yes—
80C2×M16×C79—21no—
81C2×M16×C79—21no—
82C2×M16×C79D411yescharacteristic
83C22×C8×C79—21no—
84C22×C8×C79C2311yescharacteristic
85#108×C79—21no—
86#108×C79C2312yes—
87#101×C79—22no—
88#100×C79—22no—
89C2×Dic158—22no—
90C2×Dic158—41no—
91C2×Dic158—41no—
92C2×Dic158—41no—
93C2×Dic158—41no—
94C2×Dic158—41no—
95C22×Dic79—41no—
96Dic316—22no—
97Dic316—22no—
98Dic316—41no—
99Dic316—41no—
100#11929614—44no—
101#11929612—44no—
102#11929608—22no—
103#11929608—41no—
104C2×C79⋊C8—41no—
105C79⋊C16—22no—
106C79⋊C16C2×D414yes—
107#41×C79—21no—
108#41×C79—21no—
109#41×C79—41no—
110#41×C79—41no—
111#41×C79—41no—
112#41×C79—41no—
113#41×C79D812yes—
114#41×C79C2×D412yes—
115C2×Q8×C79—21no—
116C2×Q8×C79D811yescharacteristic
117C2×D4×C79—21no—
118C2×D4×C79—21no—
119C2×D4×C79—41no—
120C2×D4×C79D811yescharacteristic
121C22×C4×C79—21no—
122C22×C4×C79—41no—
123C22×C4×C79C2×D411yescharacteristic
124Q16×C79—21no—
125Q16×C79—21no—
126Q16×C79—42no—
127SD16×C79—41no—
128SD16×C79—41no—
129SD16×C79—42no—
130SD16×C79—42no—
131D8×C79—21no—
132D8×C79—21no—
133D8×C79—42no—
134M16×C79—22no—
135M16×C79—22no—
136M16×C79—21no—
137M16×C79—22no—
138M16×C79—41no—
139M16×C79C2×D412yes—
140C2×C8×C79—22no—
141C2×C8×C79—22no—
142C2×C8×C79—21no—
143C2×C8×C79—41no—
144C2×C8×C79C2×D411yescharacteristic
145C2×C8×C79C2×D411yescharacteristic
146C2×C8×C79C2×D411yescharacteristic
147C2×C8×C79#4111yescharacteristic
148C2×C8×C79#4111yescharacteristic
149C2×C8×C79C2411yescharacteristic
150C23×C79—21no—
151C23×C79—41no—
152C23×C79#13511yescharacteristic
153Q8×C79—21no—
154Q8×C79—41no—
155Q8×C79—41no—
156Q8×C79C2×D812yes—
157D4×C79—22no—
158D4×C79—22no—
159D4×C79—21no—
160D4×C79—41no—
161D4×C79—41no—
162D4×C79—81no—
163D4×C79—81no—
164D4×C79C2×D812yes—
165C2×C4×C79—21no—
166C2×C4×C79—22no—
167C2×C4×C79—22no—
168C2×C4×C79—41no—
169C2×C4×C79—81no—
170C2×C4×C79—81no—
171C2×C4×C79#12111yescharacteristic
172C2×C4×C79#12111yescharacteristic
173C2×C4×C79C2×D811yescharacteristic
174C2×C4×C79C2×D811yescharacteristic
175C2×C4×C79#13511yescharacteristic
176C2×C4×C79C22×D411yescharacteristic
177C2×Dic79—42no—
178C2×Dic79—81no—
179C2×Dic79—81no—
180Dic158—42no—
181Dic158—42no—
182Dic158—81no—
183Dic158—81no—
184C8×C79—22no—
185C8×C79—42no—
186C8×C79#12111yescharacteristic
187C8×C79#12111yescharacteristic
188C8×C79C22×D411yescharacteristic
189C8×C79C2×#4111yescharacteristic
190C79⋊C8—42no—
191C22×C79—22no—
192C22×C79—41no—
193C22×C79—81no—
194C22×C79—81no—
195C22×C79—161no—
196C22×C79#46611yescharacteristic
197C22×C79#46611yescharacteristic
198C22×C79C2×#13511yescharacteristic
199C4×C79—22no—
200C4×C79—81no—
201C4×C79#46611yescharacteristic
202C4×C79#46611yescharacteristic
203C4×C79C2×#12111yescharacteristic
204C4×C79C22×D811yescharacteristic
205Dic79—82no—
206#33619—791nomaximal, Sylow
207C2×C79—21no—
208C2×C79—21no—
209C2×C79—21no—
210C2×C79—42no—
211C2×C79—161no—
212C2×C79C2×#46611yescharacteristic
213C2×#510—791no—
214#3407—791no—
215#3067D7911yescharacteristic
216C2×#473—791no—
217#2223—794no—
218#2222—794no—
219#2161—791no—
220#2148—791no—
221C2×#362—791no—
222C79#3361911yescharacteristic, complemented, Sylow
223#575—21no—
224C2×#137—1581no—
225C2×#135C2×D7911yescharacteristic
226C22×Q16—791no—
227C2×#131C2×D7911yescharacteristic
228#510—1582no—
229#510—1582no—
230C2×Q32—792no—
231C2×SD32—792no—
232#504—794no—
233#504—1582no—
234C2×M32—791no—
235C2×M32—791no—
236C2×M32—1582no—
237#473—794no—
238#473—1582no—
239#473C2×D7912yes—
240#472C2×D7912yes—
241#471—22no—
242C2×#108—791no—
243C2×#108C2×D7911yescharacteristic
244#413—21no—
245C2×#101—1581no—
246#368—794no—
247#362—792no—
248#362—794no—
249#361—792no—
250C2×#41—21no—
251C2×#41C22⋊D7911yescharacteristic
252C22×Q8—1581no—
253#137—22no—
254#137—41no—
255#137—3162no—
256#137—1584no—
257AGL(1,ℤ/8)—22no—
258AGL(1,ℤ/8)—41no—
259#135—21no—
260#135—21no—
261#135—21no—
262#135—21no—
263#135—42no—
264C2×Q16—3161no—
265C2×Q16—792no—
266C2×Q16—1581no—
267C2×Q16—1581no—
268C2×Q16—1581no—
269C2×Q16—1581no—
270C2×Q16—1581no—
271C2×Q16C22×D7911yescharacteristic
272C2×SD16—21no—
273C2×SD16—3161no—
274C2×D8C22×D7911yescharacteristic
275#131—21no—
276#131—21no—
277#131C22⋊D7912yes—
278#131C22×D7912yes—
279C2×M16—21no—
280C2×M16—21no—
281C2×M16—1581no—
282C2×M16C22⋊D7911yescharacteristic
283C22×C8—21no—
284C22×C8C22×D7911yescharacteristic
285Q32—1584no—
286SD32—1584no—
287M32—792no—
288M32—792no—
289M32—792no—
290M32—1584no—
291M32—1581no—
292C2×C16—792no—
293C2×C16—1584no—
294C2×C16—1581no—
295#108—21no—
296#108—792no—
297#108—1581no—
298#108C22×D7912yes—
299#101—22no—
300#101—1584no—
301#100—22no—
302#41—21no—
303#41—21no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41#1192960912yes—
309#41C2×C22⋊D7912yes—
310C2×Q8—21no—
311C2×Q8—3161no—
312C2×Q8—3161no—
313C2×Q8—3161no—
314C2×Q8—3161no—
315C2×Q8—3161no—
316C2×Q8—1582no—
317C2×Q8#1192960911yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4#1192960911yescharacteristic
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—3161no—
325C22×C4C2×C22⋊D7911yescharacteristic
326Q16—21no—
327Q16—21no—
328Q16—42no—
329Q16—3164no—
330Q16—3161no—
331Q16—3161no—
332Q16—1582no—
333Q16—1582no—
334SD16—42no—
335SD16—42no—
336SD16—41no—
337SD16—41no—
338SD16—3164no—
339D8—21no—
340D8—21no—
341D8—42no—
342M16—21no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—41no—
347M16—3161no—
348M16—1582no—
349M16C2×C22⋊D7912yes—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—3161no—
355C2×C8D4×D7911yescharacteristic
356C2×C8D4×D7911yescharacteristic
357C2×C8C2×C22⋊D7911yescharacteristic
358C2×C8#1192962311yescharacteristic
359C2×C8#1192962311yescharacteristic
360C2×C8C23×D7911yescharacteristic
361C16—794no—
362C16—1582no—
363C23—21no—
364C23—41no—
365C23#1457918111yescharacteristic
366Q8—21no—
367Q8—41no—
368Q8—41no—
369Q8—3162no—
370Q8—3162no—
371Q8—6321no—
372Q8—6321no—
373Q8C2×#1192960912yes—
374D4—22no—
375D4—22no—
376D4—21no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381D4C2×#1192960912yes—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—3162no—
389C2×C4—6321no—
390C2×C4—6321no—
391C2×C4#1457913311yescharacteristic
392C2×C4#1457913311yescharacteristic
393C2×C4C2×#1192960911yescharacteristic
394C2×C4C2×#1192960911yescharacteristic
395C2×C4#1457918111yescharacteristic
396C2×C4C22×C22⋊D7911yescharacteristic
397C8—22no—
398C8—42no—
399C8—3162no—
400C8#1457913311yescharacteristic
401C8#1457913311yescharacteristic
402C8C2×D4×D7911yescharacteristic
403C8C2×#1192962311yescharacteristic
404C22—22no—
405C22—41no—
406C22—81no—
407C22—81no—
408C22—161no—
409C22#1909879711yescharacteristic
410C22#1909879711yescharacteristic
411C22C2×#1457918111yescharacteristic
412C4—22no—
413C4—81no—
414C4—6322no—
415C4#1909879711yescharacteristic
416C4#1909879711yescharacteristic
417C4C2×#1457913311yescharacteristic
418C4C22×#1192960911yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—161no—
424C2C2×#1909879711yescharacteristic
subgroupG/H
centerC2C2×#19098797
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
Fitting, prime radicalC79×#3067C2
socleC2×C79C2×#466
hypercenter#3067D79
hypocenterC79#33619
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C2×#19099003C2semidirect
2. #23776503C2semidirect
3. C2×#19098838C2semidirect
4. #23765528C2semidirect
5. #23765523C2semidirect
6. #23765194C2semidirect
7. #23765149C2semidirect
8. C2×#19098378C2semidirect
14. C2×#14579107C22semidirect
15. C2×#14579105C22semidirect
25. #19098392C22semidirect
26. #19098379C22semidirect
27. #19098378C22semidirect
28. #19098376C22semidirect
222. C79#33619semidirect

Maximal subgroups

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

subgroupindex
C2×#190990032
#237765032
C2×#190988382
#237655282
#237655232
#237651942
#237651492
C2×#190983782
C79×#30672
#3361979