Finite Group Databasebeta

← #34444614

Subgroups — #34444614

Order 20224.

490 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#23765564C212yesmaximal, complemented
3#23765549C211yesmaximal, characteristic, complemented
4#23765544C211yesmaximal, characteristic, complemented
5#23765515C212yesmaximal
6#23765506C212yesmaximal, complemented
7#23765462C212yesmaximal, complemented
8#23765447C211yesmaximal, characteristic, complemented
9#23765442C211yesmaximal, characteristic, complemented
10#23765156C211yesmaximal, characteristic, complemented
11C79×#3170C211yesmaximal, characteristic, complemented
12#19099097—21no—
13C2×#14579107C2212yes—
14C2×#14579104C2212yes—
15C2×#14579101C2212yes—
16#19098965C2211yescharacteristic
17#19098947C2211yescharacteristic
18#19098918—21no—
19#19098804C2211yescharacteristic
20#19098799C2211yescharacteristic
21#19098792—21no—
22C4×Dic316C2211yescharacteristic
23C4×#11929613—21no—
24C4×D632C2211yescharacteristic
25#19098390C2212yes—
26#19098388C2212yes—
27C4⋊(C79⋊C16)C2211yescharacteristic
28C4⋊(C79⋊C16)C2211yescharacteristic
29C4⋊(C79⋊C16)C2212yes—
30#19098367C2212yes—
31#19098367C2212yes—
32#19098366C2212yes—
33#19098363C2212yes—
34#533×C79—21no—
35#500×C79C2211yescharacteristic
36#494×C79C2211yescharacteristic
37#487×C79—21no—
38#467×C79—21no—
39#467×C79C2212yes—
40C4×Q16×C79—21no—
41C4×Q16×C79C2212yes—
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—42no—
57#14579104—42no—
58#14579101—42no—
59C2×C79⋊C16C2312yes—
60C2×C79⋊C16C2312yes—
61#14579095C2311yescharacteristic
62#14579095C2311yescharacteristic
63#14579093—21no—
64#14579086—21no—
65#14579086—21no—
66#14579083—21no—
67#14579083—21no—
68C2×Q16×C79—21no—
69C2×Q16×C79C2312yes—
70C22×C8×C79D411yescharacteristic
71C22×C8×C79C2311yescharacteristic
72C4⋊Q8×C79—21no—
73C4⋊Q8×C79—21no—
74#125×C79—41no—
75C22⋊Q8×C79—22no—
76C22⋊Q8×C79—41no—
77C4×Q8×C79—22no—
78C4×Q8×C79—41no—
79#117×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—21no—
94Dic316—21no—
95#11929613—42no—
96D632—21no—
97D632—21no—
98C4⋊Dic79—21no—
99C4⋊Dic79—21no—
100Dic79⋊C4—41no—
101Dic79⋊C4—41no—
102Dic79⋊C4—41no—
103Dic79⋊C4—41no—
104C22⋊Dic79—41no—
105(C2×D79)⋊C4—41no—
106(C2×D79)⋊C4—41no—
107(C2×D79)⋊C4—41no—
108(C2×D79)⋊C4—41no—
109C4×Dic79—41no—
110C79⋊C16—22no—
111C79⋊C16—22no—
112C2×Q8×C79—22no—
113C2×Q8×C79—41no—
114C22×C4×C79D811yescharacteristic
115C22×C4×C79D811yescharacteristic
116C22×C4×C79C2×D411yescharacteristic
117Q16×C79—22no—
118Q16×C79—22no—
119Q16×C79—42no—
120C2×C8×C79—21no—
121C2×C8×C79—21no—
122C2×C8×C79—41no—
123C2×C8×C79C2×D411yescharacteristic
124C2×C8×C79C2×D411yescharacteristic
125C2×C8×C79C2×D411yescharacteristic
126C2×C8×C79C2411yescharacteristic
127C4⋊C4×C79—22no—
128C4⋊C4×C79—41no—
129C4⋊C4×C79—41no—
130C4⋊C4×C79—42no—
131C4⋊C4×C79—42no—
132C4⋊C4×C79—41no—
133C22⋊C4×C79—42no—
134C42×C79—42no—
135C42×C79D811yescharacteristic
136C42×C79D811yescharacteristic
137C42×C79C2×D411yescharacteristic
138C42×C79C2×D411yescharacteristic
139C23×C79C2×D811yescharacteristic
140C22×D79—41no—
141Q8×C79—42no—
142Q8×C79—42no—
143Q8×C79—82no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—22no—
147C2×C4×C79—41no—
148C2×C4×C79—42no—
149C2×C4×C79—42no—
150C2×C4×C79C2×D811yescharacteristic
151C2×C4×C79C2×D811yescharacteristic
152C2×C4×C79C2×D811yescharacteristic
153C2×C4×C79C2×D811yescharacteristic
154C2×C4×C79C2×D811yescharacteristic
155C2×C4×C79C22×D411yescharacteristic
156C22⋊D79—82no—
157C2×Dic79—41no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160D316—41no—
161D316—41no—
162C4×D79—82no—
163Dic158—41no—
164Dic158—41no—
165C8×C79—21no—
166C8×C79—21no—
167C8×C79—21no—
168C8×C79H(5,𝔽2)11yescharacteristic
169C8×C792−1+411yescharacteristic
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—21no—
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—82no—
182C4×C79—82no—
183C4×C79#51012yes—
184C4×C79#53711yescharacteristic
185C4×C79#57511yescharacteristic
186Dic79—81no—
187Dic79—81no—
188Dic79—81no—
189#33818—791nomaximal, Sylow
190C2×C79—41no—
191C2×C79#324511yescharacteristic
192C2×C79C2×#51011yescharacteristic
193C2×C79#341411yescharacteristic
194D79—161no—
195#3170D7911yescharacteristic, complemented
196#3168—791no—
197#2229—792no—
198#2227—791no—
199#2226—791no—
200#2220—792no—
201#2219—792no—
202#2211—792no—
203#2209—791no—
204#2208—791no—
205#2150—791no—
206C79#3381811yescharacteristic, complemented, Sylow
207#533—21no—
208#529—1581no—
209C2×Q32—792no—
210C2×SD32—792no—
211C2×M32—792no—
212#500C2×D7911yescharacteristic
213#499—791no—
214#495—791no—
215#494C2×D7911yescharacteristic
216#488—1581no—
217#487—21no—
218#467—21no—
219#467—791no—
220#467C2×D7912yes—
221#466—791no—
222#465—1581no—
223C4×Q16—21no—
224C4×Q16—791no—
225C4×Q16C2×D7912yes—
226C4×SD16—1581no—
227C4×D8—791no—
228C2×C4×C8C2×D7911yescharacteristic
229#367—792no—
230#366—792no—
231C4⋊C16—791no—
232C4⋊C16—791no—
233C4⋊C16—792no—
234#358—792no—
235#358—792no—
236#358—792no—
237#357—792no—
238C2×Q16—21no—
239C2×Q16—791no—
240C2×Q16C22×D7912yes—
241C2×SD16—1581no—
242C2×D8—791no—
243C22×C8C22⋊D7911yescharacteristic
244C22×C8C22×D7911yescharacteristic
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247#126—3161no—
248#125—41no—
249#125—1581no—
250#124—1581no—
251#123—3161no—
252C22⋊Q8—22no—
253C22⋊Q8—41no—
254C22⋊Q8—1581no—
255#121—1581no—
256C4×Q8—22no—
257C4×Q8—41no—
258C4×Q8—1581no—
259C4×D4—3161no—
260C4×D4—1581no—
261#117—41no—
262#117—3161no—
263C2×C42C22⋊D7911yescharacteristic
264Q32—3162no—
265SD32—3162no—
266M32—3162no—
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—22no—
285C2×Q8—41no—
286C2×Q8—1581no—
287C2×D4—3161no—
288C2×D4—1581no—
289C22×C4—3161no—
290C22×C4#1192960911yescharacteristic
291C22×C4#1192960911yescharacteristic
292C22×C4C2×C22⋊D7911yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—42no—
296Q16—1581no—
297Q16—1581no—
298SD16—3162no—
299D8—1581no—
300D8—1581no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8C2×C22⋊D7911yescharacteristic
305C2×C8C2×C22⋊D7911yescharacteristic
306C2×C8C2×C22⋊D7911yescharacteristic
307C2×C8C23×D7911yescharacteristic
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—3161no—
315C4⋊C4—3161no—
316C4⋊C4—3161no—
317C4⋊C4—3161no—
318C4⋊C4—1581no—
319C4⋊C4—1581no—
320C22⋊C4—42no—
321C22⋊C4—3161no—
322C22⋊C4—3161no—
323C22⋊C4—3161no—
324C22⋊C4—3161no—
325C22⋊C4—3161no—
326C42—42no—
327C42—3161no—
328C42#1192960911yescharacteristic
329C42#1192960911yescharacteristic
330C42C2×C22⋊D7911yescharacteristic
331C42C2×C22⋊D7911yescharacteristic
332C16—1582no—
333C16—1582no—
334C23—3161no—
335C23C2×#1192960911yescharacteristic
336Q8—42no—
337Q8—42no—
338Q8—82no—
339Q8—3161no—
340Q8—3161no—
341D4—3161no—
342D4—3161no—
343D4—6322no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—3161no—
351C2×C4—3161no—
352C2×C4—3161no—
353C2×C4—6322no—
354C2×C4C2×#1192960911yescharacteristic
355C2×C4C2×#1192960911yescharacteristic
356C2×C4C2×#1192960911yescharacteristic
357C2×C4C2×#1192960911yescharacteristic
358C2×C4C2×#1192960911yescharacteristic
359C2×C4C22×C22⋊D7911yescharacteristic
360C8—21no—
361C8—21no—
362C8—21no—
363C8#1457920711yescharacteristic
364C8#1457920811yescharacteristic
365C22—21no—
366C22—21no—
367C22—21no—
368C22—6321no—
369C22—6321no—
370C22—6321no—
371C22C22×#1192960911yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—82no—
377C4—82no—
378C4—6321no—
379C4—6321no—
380C4—6321no—
381C4#1909900412yes—
382C4#1909914111yescharacteristic
383C4#1909936111yescharacteristic
384C2—41no—
385C2—12641no—
386C2#2377470711yescharacteristic
387C2C2×#1909900411yescharacteristic
388C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3170C2
socleC22×C79C22×D8
prime radical#19098799C22
hypercenter#3170D79
hypocenterC79#33818
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3170

Complements

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

NH ≅ G/Ntype
2. #23765564C2semidirect
3. #23765549C2semidirect
4. #23765544C2semidirect
6. #23765506C2semidirect
7. #23765462C2semidirect
8. #23765447C2semidirect
9. #23765442C2semidirect
10. #23765156C2semidirect
11. C79×#3170C2semidirect
195. #3170D79semidirect
206. C79#33818semidirect

Maximal subgroups

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

subgroupindex
#237739702
#237655642
#237655492
#237655442
#237655152
#237655062
#237654622
#237654472
#237654422
#237651562
C79×#31702
#3381879