Finite Group Databasebeta

← #34441614

Subgroups — #34441614

Order 20224.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098998C211yesmaximal, characteristic
2#23772276C211yesmaximal, characteristic, complemented
3#23772251C211yesmaximal, characteristic, complemented
4#23765273C212yesmaximal, complemented
5#23765267C212yesmaximal, complemented
6#23765249C212yesmaximal, complemented
7#23765244C212yesmaximal, complemented
8#23765189C211yesmaximal, characteristic, complemented
9C2×#19098365C211yesmaximal, characteristic, complemented
10#23764996C211yesmaximal, characteristic, complemented
11C79×#2138C211yesmaximal, characteristic, complemented
12C22×D632C2211yescharacteristic
13#19099046—21no—
14#19099037—21no—
15#19098998C418yescomplemented
16C2×#14579104C2212yes—
17C2×#14579102C2212yes—
18C2×#14579101C2211yescharacteristic
19#19098674—22no—
20#19098664—22no—
21Q16×Dic79C2212yescomplemented
22D8×Dic79C2212yescomplemented
23#19098645C2212yes—
24#19098645C2212yes—
25C2×#14579095C2211yescharacteristic
26#19098567—21no—
27#19098564—21no—
28#19098518C2211yescharacteristic
29#19098390C2212yescomplemented
30#19098388C2212yescomplemented
31#19098365C2212yes—
32#19098365C2212yes—
33#19098338C2212yescomplemented
34C16×Dic79C2212yescomplemented
35C2×#135×C79C2211yescharacteristic
36C22×C16×C79C2211yescharacteristic
37#427×C79C2211yescharacteristic
38#358×C79C2211yescharacteristic, complemented
39#358×C79C2211yescharacteristic, complemented
40#357×C79C2211yescharacteristic, complemented
41#357×C79C2211yescharacteristic, complemented
42C22×D316—21no—
43C2×D632—21no—
44C2×D632—22no—
45C2×D632C2×C414yescomplemented
46C2×D632C2312yes—
47C2×#11929608D411yescharacteristic
48Q8×Dic79—22no—
49C4×D316—41no—
50C4×D316—41no—
51D4×Dic79—22no—
52D4×Dic79—42no—
53#14579123—22no—
54#14579123—22no—
55#14579120—41no—
56#14579120D411yescharacteristic
57C2×C4⋊Dic79—21no—
58C2×C4⋊Dic79—41no—
59C22⋊C4×D79—42no—
60#14579104C2×C418yescomplemented
61#14579102C2×C418yescomplemented
62#14579101C2×C414yescomplemented
63C2×C79⋊C16C2312yes—
64#14579095—21no—
65#14579095C2312yes—
66#14579085—22no—
67#14579082—22no—
68#14579081—22no—
69#14579081—22no—
70#14579071D412yes—
71C8×Dic79C2312yes—
72C2×#41×C79—21no—
73#135×C79—22no—
74#135×C79C2×C414yescomplemented
75C2×Q16×C79C2311yescharacteristic
76C2×SD16×C79—21no—
77C2×D8×C79C2311yescharacteristic
78C22×C8×C79C2311yescharacteristic
79#117×C79—21no—
80C2×C16×C79—22no—
81C2×C16×C79C2311yescharacteristic
82C2×C16×C79C2311yescharacteristic
83#107×C79C2311yescharacteristic
84#107×C79C2311yescharacteristic
85#106×C79—21no—
86C23×D79—41no—
87C2×D316—24no—
88C2×D316—22no—
89C2×D316—41no—
90C2×D316—42no—
91C22×Dic79—42no—
92C22×Dic79—41no—
93C2×C4×D79—42no—
94C2×C4×D79—42no—
95D632—24no—
96D632—22no—
97D632C22×C414yes—
98#11929608—22no—
99C2×C79⋊C8C2×D412yes—
100C4⋊Dic79—22no—
101C4⋊Dic79—42no—
102C4⋊Dic79—41no—
103C4⋊Dic79D811yescharacteristic
104C4⋊Dic79D811yescharacteristic
105C22⋊Dic79—42no—
106C22⋊Dic79D812yes—
107(C2×D79)⋊C4—42no—
108(C2×D79)⋊C4—42no—
109C4×Dic79—42no—
110C4×Dic79C2×D412yes—
111C79⋊C16C22×C414yes—
112#41×C79—24no—
113#41×C79—42no—
114C2×Q8×C79—21no—
115C2×D4×C79—21no—
116C2×D4×C79—41no—
117C22×C4×C79—41no—
118C22×C4×C79C2×D411yescharacteristic
119Q16×C79—21no—
120Q16×C79C22×C412yes—
121SD16×C79—24no—
122D8×C79—21no—
123D8×C79C22×C412yes—
124C2×C8×C79C22×C412yes—
125C2×C8×C79C2×D411yescharacteristic
126C2×C8×C79#4112yes—
127C2×C8×C79C2411yescharacteristic
128C4⋊C4×C79—21no—
129C4⋊C4×C79—21no—
130C22⋊C4×C79—41no—
131C42×C79—41no—
132C16×C79—21no—
133C16×C79—21no—
134C23×C79—41no—
135C23×C79C2×D811yescharacteristic
136C22×D79—42no—
137C22×D79—44no—
138C22×D79—161no—
139C22×D79—161no—
140Q8×C79—22no—
141Q8×C79—41no—
142D4×C79—22no—
143D4×C79—44no—
144D4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—44no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79C4×D412yes—
150C2×C4×C79C2×D811yescharacteristic
151C2×C4×C79#13512yes—
152C2×C4×C79C22×D411yescharacteristic
153C2×Dic79—42no—
154C2×Dic79—162no—
155C2×Dic79—161no—
156C2×Dic79C2×D812yes—
157C2×Dic79C2×D812yes—
158D316—24no—
159D316—42no—
160D316—44no—
161C4×D79—162no—
162C4×D79—162no—
163C8×C79C4×D412yes—
164C8×C79C23×C411yescharacteristic
165C8×C79C2×#4111yescharacteristic
166C79⋊C8—22no—
167C22×C79—21no—
168C22×C79—21no—
169C22×C79—42no—
170C22×C79—81no—
171C22×C79—81no—
172C22×C79C4×D812yes—
173C22×C79C22×D811yescharacteristic
174C2×D79—44no—
175C2×D79—82no—
176C2×D79—82no—
177C2×D79—164no—
178C4×C79—42no—
179C4×C79—81no—
180C4×C79—81no—
181C4×C79C4×D812yes—
182C4×C79C2×C4×D411yescharacteristic
183C4×C79C2×#13511yescharacteristic
184Dic79—22no—
185Dic79—22no—
186Dic79—82no—
187#33454—791nomaximal, Sylow
188C2×C79—22no—
189C2×C79—82no—
190C2×C79C2×C4×D811yescharacteristic
191C2×C79#341511yescharacteristic
192C2×C79#341611yescharacteristic
193D79—84no—
194C2×#509—791no—
195#2951—791no—
196#2948—791no—
197#2177—792no—
198#2175—792no—
199C4×SD32—792no—
200C4×D16—792no—
201#2159—791no—
202#2138D7911yescharacteristic, complemented
203C2×#357—791no—
204#2108—791no—
205C79#3345411yescharacteristic, complemented, Sylow
206C2×#135C2×D7911yescharacteristic
207C22×D8—791no—
208#520—1581no—
209#518—1581no—
210#509—798no—
211C2×SD32—792no—
212C2×D16—792no—
213C2×M32—791no—
214C22×C16C2×D7911yescharacteristic
215#442—1582no—
216#440—1582no—
217C4×Q16—792no—
218C4×D8—792no—
219C4×D8—792no—
220C4×D8—792no—
221#427C2×D7911yescharacteristic
222C2×#107—791no—
223#418—1581no—
224#417—1581no—
225#405—791no—
226#367—792no—
227#366—792no—
228#358C2×D7911yescharacteristic, complemented
229#358C2×D7911yescharacteristic, complemented
230#357—792no—
231#357—792no—
232#357C2×D7911yescharacteristic, complemented
233#357C2×D7911yescharacteristic, complemented
234#346—792no—
235C4×C16—792no—
236C2×#41—21no—
237C22×D4—1581no—
238#135—22no—
239#135C4×D7914yescomplemented
240C2×Q16C22×D7911yescharacteristic
241C2×SD16—21no—
242C2×D8—794no—
243C2×D8—792no—
244C2×D8—1581no—
245C2×D8—1582no—
246C2×D8C22×D7911yescharacteristic
247C2×M16—791no—
248C22×C8C22×D7911yescharacteristic
249C4×Q8—1582no—
250C4×D4—3161no—
251C4×D4—3161no—
252C4×D4—3162no—
253C4×D4—1582no—
254C4×D4—1582no—
255C4×D4—1582no—
256#117—21no—
257#117—3161no—
258#117—791no—
259C2×C4⋊C4—3161no—
260C2×C4⋊C4—1581no—
261C2×C22⋊C4—3162no—
262SD32—798no—
263D16—798no—
264M32—794no—
265C2×C16—22no—
266C2×C16—792no—
267C2×C16C22×D7911yescharacteristic
268C2×C16C22×D7911yescharacteristic
269#107—792no—
270#107—1581no—
271#107C22×D7911yescharacteristic
272#107C22×D7911yescharacteristic
273#106—21no—
274#103—1582no—
275#102—1582no—
276#102—1582no—
277#102—1582no—
278#97—792no—
279C4×C8—792no—
280C24—3161no—
281#41—24no—
282#41—42no—
283C2×Q8—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—3161no—
287C2×D4—3162no—
288C2×D4—1584no—
289C2×D4—1582no—
290C22×C4—41no—
291C22×C4—3162no—
292C22×C4—3162no—
293C22×C4—3161no—
294C22×C4—3162no—
295C22×C4D4×D7911yescharacteristic
296Q16—21no—
297Q16C2×C4×D7912yes—
298SD16—24no—
299D8—21no—
300D8—794no—
301D8—1582no—
302D8—1584no—
303D8C2×C4×D7912yes—
304M16—1582no—
305C2×C8—792no—
306C2×C8C2×C4×D7912yes—
307C2×C8D4×D7911yescharacteristic
308C2×C8#1192962312yes—
309C2×C8C23×D7911yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—3162no—
313C4⋊C4—3161no—
314C4⋊C4—791no—
315C4⋊C4—791no—
316C4⋊C4—1582no—
317C22⋊C4—41no—
318C22⋊C4—3162no—
319C22⋊C4—3162no—
320C22⋊C4—3162no—
321C22⋊C4—792no—
322C42—41no—
323C42—3162no—
324C42—792no—
325C16—21no—
326C16—21no—
327C16—794no—
328C23—41no—
329C23—3162no—
330C23—3164no—
331C23—12641no—
332C23—12641no—
333C23D8×D7911yescharacteristic
334Q8—22no—
335Q8—41no—
336D4—22no—
337D4—44no—
338D4—41no—
339D4—3164no—
340D4—3162no—
341D4—1584no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—3162no—
347C2×C4—792no—
348C2×C4—792no—
349C2×C4—12642no—
350C2×C4—12642no—
351C2×C4—12641no—
352C2×C4—12642no—
353C2×C4(C22⋊D79)⋊C412yes—
354C2×C4D8×D7911yescharacteristic
355C2×C4#1457918212yes—
356C2×C4C2×D4×D7911yescharacteristic
357C8—1582no—
358C8(C22⋊D79)⋊C412yes—
359C8C22×C4×D7911yescharacteristic
360C8C2×#1192962311yescharacteristic
361C22—21no—
362C22—21no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—3164no—
367C22—6322no—
368C22—6322no—
369C22—12644no—
370C22#1909864312yes—
371C22C2×D8×D7911yescharacteristic
372C4—42no—
373C4—81no—
374C4—81no—
375C4—6322no—
376C4—1582no—
377C4—1582no—
378C4#1909864312yes—
379C4C2×(C22⋊D79)⋊C411yescharacteristic
380C4C2×#1457918211yescharacteristic
381C2—22no—
382C2—82no—
383C2—6324no—
384C2C2×#1909864311yescharacteristic
385C2#2377653811yescharacteristic
386C2#2377654311yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutatorC8×C79C23×C4
FrattiniC2×C8C23×D79
FittingC79×#2138C2
socleC22×C79C22×D8
prime radicalC2×#19098998C2
cosocleC2×C8×C79C24
hypercenter#2138D79
hypocenterC79#33454
LCSG ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. #23772276C2semidirect
3. #23772251C2semidirect
4. #23765273C2semidirect
5. #23765267C2semidirect
6. #23765249C2semidirect
7. #23765244C2semidirect
8. #23765189C2semidirect
9. C2×#19098365C2semidirect
10. #23764996C2semidirect
11. C79×#2138C2semidirect
15. #19098998C4semidirect
21. Q16×Dic79C22semidirect
22. D8×Dic79C22semidirect
29. #19098390C22semidirect
30. #19098388C22semidirect
33. #19098338C22semidirect
34. C16×Dic79C22semidirect
38. #358×C79C22semidirect
39. #358×C79C22semidirect
40. #357×C79C22semidirect
41. #357×C79C22semidirect
45. C2×D632C2×C4semidirect
60. #14579104C2×C4semidirect
61. #14579102C2×C4semidirect
62. #14579101C2×C4semidirect
74. #135×C79C2×C4semidirect
202. #2138D79semidirect
205. C79#33454semidirect
228. #358C2×D79semidirect
229. #358C2×D79semidirect
232. #357C2×D79semidirect
233. #357C2×D79semidirect
239. #135C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989982
#237722762
#237722512
#237652732
#237652672
#237652492
#237652442
#237651892
C2×#190983652
#237649962
C79×#21382
#3345479