Finite Group Databasebeta

← #34443364

Subgroups — #34443364

Order 20224.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773970C211yesmaximal, characteristic
2#23765602C211yesmaximal, characteristic, complemented
3#23765585C211yesmaximal, characteristic
4#23765564C212yesmaximal, complemented
5#23765462C212yesmaximal, complemented
6#23765334C211yesmaximal, characteristic, complemented
7#23765317C211yesmaximal, characteristic
8#23765269C212yesmaximal, complemented
9#23765262C212yesmaximal, complemented
10#23765203C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C79C211yesmaximal, characteristic, complemented
12#19099097—21no—
13C2×#14579107C2212yes—
14C2×#14579104C2212yes—
15#19098965C2211yescharacteristic
16#19098947C2211yescharacteristic
17#19098918—21no—
18#19098804C2211yescharacteristic
19#19098799C2211yescharacteristic
20#19098792—21no—
21C4×Dic316C2211yescharacteristic
22C4×#11929613—21no—
23C4×D632C2211yescharacteristic
24#19098390C2212yes—
25#19098388C2212yes—
26C4⋊(C79⋊C16)C2212yes—
27#19098367C2212yes—
28#19098367C2212yes—
29#19098366C2212yes—
30#19098363C2212yes—
31C22⋊(C79⋊C16)C2211yescharacteristic
32C22⋊(C79⋊C16)C2211yescharacteristic
33#19098339C2212yes—
34C22×Q16×C79C2211yescharacteristic
35C2×C4×Q8×C79—21no—
36C4×Q16×C79—21no—
37C4×Q16×C79—21no—
38C4×Q16×C79C2212yes—
39C4×Q16×C79C2212yes—
40C2×#107×C79C2211yescharacteristic
41C2×#103×C79—21no—
42C2×C4×C8×C79C2211yescharacteristic
43C2×Dic316C2311yescharacteristic
44C2×#11929613—21no—
45C2×D632C2311yescharacteristic
46#14579155—41no—
47#14579153—21no—
48#14579149—21no—
49#14579144—41no—
50#14579140—21no—
51#14579135—21no—
52C4×Dic158—21no—
53C4×D316—21no—
54C4×C22⋊D79—41no—
55#14579118—41no—
56#14579107—24no—
57#14579104—24no—
58C2×C79⋊C16C2312yes—
59C2×C79⋊C16C2312yes—
60#14579095C2311yescharacteristic
61#14579095C2311yescharacteristic
62#14579093—21no—
63#14579086—21no—
64#14579086—21no—
65#14579083—21no—
66#14579083—21no—
67C22×Q8×C79—21no—
68C2×Q16×C79—24no—
69C2×Q16×C79—21no—
70C2×Q16×C79C2312yes—
71C22×C8×C79D411yescharacteristic
72C22×C8×C79C2311yescharacteristic
73C4×Q8×C79—22no—
74C4×Q8×C79—22no—
75C4×Q8×C79—41no—
76C4×Q8×C79—41no—
77C2×C4⋊C4×C79—21no—
78C2×C4⋊C4×C79—41no—
79C2×C42×C79—41no—
80C2×C42×C79D411yescharacteristic
81#107×C79—21no—
82#107×C79C2312yes—
83#103×C79—22no—
84#103×C79—22no—
85C4×C8×C79D411yescharacteristic
86C4×C8×C79D411yescharacteristic
87C4×C8×C79C2311yescharacteristic
88C4×C8×C79C2311yescharacteristic
89C2×Dic158—21no—
90C2×D316—21no—
91C2×C22⋊D79—41no—
92C2×C4×D79—41no—
93Dic316—22no—
94#11929613—42no—
95D632—22no—
96C4⋊Dic79—21no—
97C4⋊Dic79—21no—
98Dic79⋊C4—41no—
99Dic79⋊C4—41no—
100Dic79⋊C4—41no—
101Dic79⋊C4—41no—
102C22⋊Dic79—41no—
103(C2×D79)⋊C4—41no—
104(C2×D79)⋊C4—41no—
105(C2×D79)⋊C4—41no—
106(C2×D79)⋊C4—41no—
107C4×Dic79—41no—
108C79⋊C16—22no—
109C79⋊C16—22no—
110C2×Q8×C79—24no—
111C2×Q8×C79—22no—
112C2×Q8×C79—42no—
113C2×Q8×C79—41no—
114C22×C4×C79—41no—
115C22×C4×C79—41no—
116C22×C4×C79D811yescharacteristic
117C22×C4×C79D811yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119Q16×C79—24no—
120Q16×C79—22no—
121Q16×C79#4114yes—
122C2×C8×C79—22no—
123C2×C8×C79—41no—
124C2×C8×C79C2×D411yescharacteristic
125C2×C8×C79C2×D411yescharacteristic
126C2×C8×C79C2×D411yescharacteristic
127C2×C8×C79C2411yescharacteristic
128C4⋊C4×C79—22no—
129C4⋊C4×C79—41no—
130C4⋊C4×C79—42no—
131C4⋊C4×C79—42no—
132C42×C79—42no—
133C42×C79—42no—
134C42×C79D811yescharacteristic
135C42×C79D811yescharacteristic
136C42×C79C2×D411yescharacteristic
137C42×C79C2×D411yescharacteristic
138C23×C79C2×D811yescharacteristic
139C22×D79—41no—
140Q8×C79—24no—
141Q8×C79—44no—
142Q8×C79—42no—
143C2×C4×C79—22no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—41no—
147C2×C4×C79—44no—
148C2×C4×C79—42no—
149C2×C4×C79—42no—
150C2×C4×C79—82no—
151C2×C4×C79C2×D811yescharacteristic
152C2×C4×C79C2×D811yescharacteristic
153C2×C4×C79C2×D811yescharacteristic
154C2×C4×C79C2×D811yescharacteristic
155C2×C4×C79C2×D811yescharacteristic
156C2×C4×C79C22×D411yescharacteristic
157C22⋊D79—82no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—41no—
161D316—41no—
162D316—41no—
163C4×D79—82no—
164Dic158—41no—
165Dic158—41no—
166C8×C79—21no—
167C8×C79—21no—
168C8×C79—21no—
169C8×C79C2×#4112yes—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79#51012yes—
173C22×C79C22×D811yescharacteristic
174C2×D79—81no—
175C2×D79—81no—
176C2×D79—81no—
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—44no—
183C4×C79—82no—
184C4×C79#52511yescharacteristic
185C4×C79#57511yescharacteristic
186Dic79—81no—
187Dic79—81no—
188Dic79—81no—
189#33677—791nomaximal, Sylow
190C2×C79—22no—
191C2×C79#310011yescharacteristic
192C2×C79C2×#51011yescharacteristic
193C2×C79#341411yescharacteristic
194D79—161no—
195#3168—791no—
196C2×C4×Q16D7911yescharacteristic, complemented
197#2236—791no—
198#2233—791no—
199#2229—792no—
200#2211—792no—
201#2188—791no—
202#2185—791no—
203#2176—792no—
204#2175—792no—
205#2163—791no—
206C79#3367711yescharacteristic, complemented, Sylow
207C22×Q16C2×D7911yescharacteristic
208#529—1581no—
209C2×C4×Q8—21no—
210C2×Q32—792no—
211C2×SD32—792no—
212#499—791no—
213#495—791no—
214#488—1581no—
215#467—791no—
216#466—791no—
217#465—1581no—
218C4×Q16—21no—
219C4×Q16—21no—
220C4×Q16—791no—
221C4×Q16C2×D7912yes—
222C4×Q16C2×D7912yes—
223C4×SD16—1581no—
224C4×D8—791no—
225C2×#107C2×D7911yescharacteristic
226C2×#103—21no—
227C2×C4×C8C2×D7911yescharacteristic
228#367—792no—
229#366—792no—
230C4⋊C16—792no—
231#358—792no—
232#358—792no—
233#358—792no—
234#357—792no—
235C22⋊C16—791no—
236C22⋊C16—791no—
237#346—792no—
238C22×Q8—21no—
239C2×Q16—24no—
240C2×Q16—21no—
241C2×Q16—791no—
242C2×Q16C22×D7912yes—
243C2×SD16—1581no—
244C2×D8—791no—
245C22×C8C22⋊D7911yescharacteristic
246C22×C8C22×D7911yescharacteristic
247#126—3161no—
248#125—1581no—
249#124—1581no—
250#123—3161no—
251C22⋊Q8—1581no—
252#121—1581no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×Q8—1581no—
258C4×D4—3161no—
259C4×D4—1581no—
260#117—3161no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—41no—
263C2×C42—41no—
264C2×C42C22⋊D7911yescharacteristic
265Q32—1584no—
266SD32—1584no—
267C2×C16—792no—
268C2×C16—792no—
269#107—21no—
270#107—791no—
271#107—791no—
272#107C22×D7912yes—
273#106—1581no—
274#103—22no—
275#103—22no—
276#103—1581no—
277#103—1581no—
278#102—1581no—
279#102—1581no—
280C4×C8C22⋊D7911yescharacteristic
281C4×C8C22⋊D7911yescharacteristic
282C4×C8C22×D7911yescharacteristic
283C4×C8C22×D7911yescharacteristic
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—1581no—
289C2×D4—3161no—
290C2×D4—1581no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—3161no—
294C22×C4#1192960911yescharacteristic
295C22×C4#1192960911yescharacteristic
296C22×C4C2×C22⋊D7911yescharacteristic
297Q16—24no—
298Q16—22no—
299Q16—1582no—
300Q16#1192962414yes—
301SD16—3162no—
302D8—1582no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8C2×C22⋊D7911yescharacteristic
306C2×C8C2×C22⋊D7911yescharacteristic
307C2×C8C2×C22⋊D7911yescharacteristic
308C2×C8C23×D7911yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—3161no—
314C4⋊C4—3161no—
315C4⋊C4—3161no—
316C4⋊C4—3161no—
317C4⋊C4—1581no—
318C4⋊C4—1581no—
319C22⋊C4—3161no—
320C22⋊C4—3161no—
321C22⋊C4—3161no—
322C22⋊C4—3161no—
323C22⋊C4—3161no—
324C42—42no—
325C42—42no—
326C42—3161no—
327C42#1192960911yescharacteristic
328C42#1192960911yescharacteristic
329C42C2×C22⋊D7911yescharacteristic
330C42C2×C22⋊D7911yescharacteristic
331C16—1582no—
332C16—1582no—
333C23—3161no—
334C23C2×#1192960911yescharacteristic
335Q8—24no—
336Q8—44no—
337Q8—42no—
338Q8—3161no—
339Q8—3161no—
340D4—3161no—
341D4—3161no—
342D4—6322no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—3161no—
352C2×C4—3161no—
353C2×C4—3161no—
354C2×C4—6322no—
355C2×C4C2×#1192960911yescharacteristic
356C2×C4C2×#1192960911yescharacteristic
357C2×C4C2×#1192960911yescharacteristic
358C2×C4C2×#1192960911yescharacteristic
359C2×C4C2×#1192960911yescharacteristic
360C2×C4C22×C22⋊D7911yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8C2×#1192962412yes—
365C22—21no—
366C22—21no—
367C22—6321no—
368C22—6321no—
369C22—6321no—
370C22#1909900412yes—
371C22C22×#1192960911yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—44no—
378C4—82no—
379C4—6321no—
380C4—6321no—
381C4—6321no—
382C4#1909907411yescharacteristic
383C4#1909936111yescharacteristic
384C2—22no—
385C2—12641no—
386C2#2377340011yescharacteristic
387C2C2×#1909900411yescharacteristic
388C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC2×C4×Q16×C79C2
socleC22×C79C22×D8
prime radical#19098799C22
hypercenterC2×C4×Q16D79
hypocenterC79#33677
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. #23765602C2semidirect
4. #23765564C2semidirect
5. #23765462C2semidirect
6. #23765334C2semidirect
8. #23765269C2semidirect
9. #23765262C2semidirect
10. #23765203C2semidirect
11. C2×C4×Q16×C79C2semidirect
196. C2×C4×Q16D79semidirect
206. C79#33677semidirect

Maximal subgroups

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

subgroupindex
#237739702
#237656022
#237655852
#237655642
#237654622
#237653342
#237653172
#237652692
#237652622
#237652032
C2×C4×Q16×C792
#3367779