Finite Group Databasebeta

← #34187427

Subgroups — #34187427

Order 20224.

506 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773975C211yesmaximal, characteristic
2#23764320C211yesmaximal, characteristic, complemented
3#23764318C211yesmaximal, characteristic, complemented
4C4×#14579094C211yesmaximal, characteristic, complemented
5#23761896C211yesmaximal, characteristic, complemented
6#23761681C211yesmaximal, characteristic, complemented
7C79×#1502C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098966C411yescharacteristic, complemented
10#19098939C411yescharacteristic, complemented
11#19098911C411yescharacteristic, complemented
12#19098909C411yescharacteristic, complemented
13#19098797—21no—
14#19098797—21no—
15C4×#11929609—21no—
16C4×#11929609—21no—
17C2×#14579094—21no—
18C2×#14579089—22no—
19C2×#14579088—22no—
20C4×#11929608C2211yescharacteristic
21C2×C4×C79⋊C8C2211yescharacteristic
22C4×C4⋊Dic79C2211yescharacteristic
23#19098325—21no—
24#19098298—22no—
25#19098285—21no—
26C8×C79⋊C8—21no—
27#532×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29C4×C4⋊C4×C79C2211yescharacteristic
30#327×C79—22no—
31C2×#11929624—21no—
32C2×#11929609—21no—
33C2×#11929609—21no—
34C2×#11929608D412yes—
35C22×C79⋊C8D411yescharacteristic
36C22×C79⋊C8D411yescharacteristic
37#14579161C2×C411yescharacteristic
38#14579158C2×C411yescharacteristic
39#14579149—21no—
40#14579135—21no—
41#14579135—21no—
42C4×D316—21no—
43C4×D316—21no—
44C2×C4⋊Dic79—21no—
45C2×C4×Dic79—21no—
46#14579094—24no—
47#14579094—21no—
48#14579094—21no—
49#14579089—44no—
50#14579088—44no—
51#14579082—21no—
52#14579082—21no—
53#14579081—21no—
54#14579081—21no—
55#14579072—21no—
56C4×C79⋊C8—21no—
57C4×C79⋊C8C2×C411yescharacteristic
58C4×C79⋊C8C2×C411yescharacteristic
59C4×C79⋊C8D411yescharacteristic
60C4×C79⋊C8D411yescharacteristic
61C79⋊#95—21no—
62C2×#41×C79—21no—
63C4⋊Q8×C79C2×C411yescharacteristic
64#127×C79C2×C411yescharacteristic
65#124×C79—21no—
66#121×C79—21no—
67#121×C79—21no—
68C4×D4×C79—21no—
69C4×D4×C79—21no—
70C2×C4⋊C4×C79—21no—
71C2×C42×C79—21no—
72C2×C42×C79C2311yescharacteristic
73C4⋊C8×C79—21no—
74C22⋊C8×C79D412yes—
75C4×C8×C79—21no—
76#95×C79—21no—
77#11929624—42no—
78#11929624—42no—
79C2×Dic158—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×D316—21no—
83C2×C22⋊D79—41no—
84C22×Dic79—41no—
85C2×C4×D79—41no—
86#11929609—42no—
87#11929609—42no—
88#11929608—22no—
89#11929608D814yescomplemented
90C2×C79⋊C8—22no—
91C2×C79⋊C8—22no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8C22⋊C411yescharacteristic
94C2×C79⋊C8C22⋊C411yescharacteristic
95C2×C79⋊C8C22⋊C411yescharacteristic
96C2×C79⋊C8C22⋊C411yescharacteristic
97C4⋊Dic79—22no—
98C4⋊Dic79—21no—
99C4⋊Dic79—21no—
100C4⋊Dic79—41no—
101(C2×D79)⋊C4—41no—
102(C2×D79)⋊C4—41no—
103C4×Dic79—42no—
104#41×C79—42no—
105#41×C79—42no—
106C2×Q8×C79—21no—
107C2×D4×C79—21no—
108C2×D4×C79—21no—
109C2×D4×C79—21no—
110C2×D4×C79—41no—
111C22×C4×C79—41no—
112C22×C4×C79—41no—
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C2×C8×C79—22no—
117C4⋊C4×C79—22no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—41no—
121C22⋊C4×C79—41no—
122C22⋊C4×C79—41no—
123C42×C79—42no—
124C42×C79C22×C411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79#4111yescharacteristic
127C42×C79#4111yescharacteristic
128C23×C79—41no—
129C23×C79#12711yescharacteristic
130C22×D79—41no—
131Q8×C79—42no—
132D4×C79—41no—
133D4×C79—41no—
134D4×C79—42no—
135D4×C79—81no—
136D4×C79—81no—
137D4×C79—81no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—41no—
141C2×C4×C79—42no—
142C2×C4×C79—41no—
143C2×C4×C79—81no—
144C2×C4×C79—81no—
145C2×C4×C79—82no—
146C2×C4×C79C2×C22⋊C411yescharacteristic
147C2×C4×C79C4×D411yescharacteristic
148C2×C4×C79C4×D411yescharacteristic
149C2×C4×C79#12111yescharacteristic
150C2×C4×C79#12111yescharacteristic
151C2×C4×C79#12411yescharacteristic
152C2×C4×C79C2×D812yes—
153C2×C4×C79#13512yes—
154C22⋊D79—81no—
155C22⋊D79—81no—
156C2×Dic79—42no—
157C2×Dic79—41no—
158C2×Dic79—81no—
159C2×Dic79—81no—
160D316—41no—
161D316—41no—
162D316—42no—
163D316—81no—
164C4×D79—82no—
165Dic158—42no—
166C8×C79—42no—
167C79⋊C8—21no—
168C79⋊C8—21no—
169C79⋊C8—21no—
170C79⋊C8—24no—
171C79⋊C8#10412yescomplemented
172C22×C79—21no—
173C22×C79—21no—
174C22×C79—81no—
175C22×C79—81no—
176C22×C79—81no—
177C22×C79#39011yescharacteristic
178C22×C79#49311yescharacteristic
179C22×C79#49511yescharacteristic
180C2×D79—81no—
181C2×D79—81no—
182C2×D79—81no—
183C4×C79—21no—
184C4×C79—21no—
185C4×C79—21no—
186C4×C79—82no—
187C4×C79—81no—
188C4×C79C2×#10411yescharacteristic
189C4×C79#42111yescharacteristic
190C4×C79C4×D812yes—
191C4×C79#46612yes—
192Dic79—82no—
193Dic79—81no—
194#9719—791nomaximal, Sylow
195C2×C79—21no—
196C2×C79—21no—
197C2×C79—161no—
198C2×C79#197011yescharacteristic
199C2×C79#197611yescharacteristic
200C2×C79#198011yescharacteristic
201D79—161no—
202#3169—791no—
203#1980—791no—
204#1980—791no—
205C4×#107—791no—
206#1502—791no—
207#1502D7911yescharacteristic, complemented
208C8×M16—791no—
209C79#971911yescharacteristic, complemented, Sylow
210#532—791no—
211#532C2×D7911yescharacteristic
212#500—791no—
213#493—791no—
214#486—791no—
215#486—791no—
216#466—1581no—
217#466—1581no—
218C4×D8—1581no—
219C4×D8—1581no—
220#431C2×D7911yescharacteristic
221C2×#107—1581no—
222C2×#104—1582no—
223C2×#104—1582no—
224C4×M16—791no—
225C2×C4×C8—791no—
226C4×C4⋊C4—791no—
227C4×C4⋊C4C2×D7911yescharacteristic
228#340—1581no—
229#327—22no—
230#327—1582no—
231#322—1581no—
232C82—1581no—
233C2×#41—21no—
234C2×#41—1581no—
235C2×D8—1581no—
236C2×D8—1581no—
237C2×M16—792no—
238C22×C8—791no—
239C22×C8—791no—
240C4⋊Q8—791no—
241C4⋊Q8C4×D7911yescharacteristic
242#127—791no—
243#127C4×D7911yescharacteristic
244#124—21no—
245#124—1581no—
246#121—21no—
247#121—21no—
248#121—1581no—
249#121—1581no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—1581no—
253C4×D4—1581no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1581no—
256C2×C42—21no—
257C2×C42—1581no—
258C2×C42C22×D7911yescharacteristic
259#107—1581no—
260#107—1581no—
261#107—1584no—
262C4⋊C8—21no—
263#104—3164no—
264#104—3164no—
265#102—1581no—
266#102—1581no—
267#102—1581no—
268#102—1581no—
269C22⋊C8C22⋊D7912yes—
270#97—1581no—
271C4×C8—21no—
272C4×C8—791no—
273C4×C8—791no—
274C4×C8—791no—
275C4×C8—791no—
276C4×C8—1581no—
277#95—21no—
278#95—1581no—
279#41—42no—
280#41—42no—
281#41—3162no—
282#41—3162no—
283C2×Q8—21no—
284C2×Q8—1581no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—3161no—
290C2×D4—1581no—
291C2×D4—1581no—
292C2×D4—1581no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—3161no—
296C22×C4—3161no—
297C22×C4D4×D7911yescharacteristic
298C22×C4D4×D7911yescharacteristic
299C22×C4C2×C22⋊D7911yescharacteristic
300D8—3162no—
301D8—3162no—
302M16—794no—
303M16—1582no—
304C2×C8—22no—
305C2×C8—3161no—
306C2×C8—791no—
307C2×C8—791no—
308C2×C8—791no—
309C2×C8—791no—
310C2×C8—1582no—
311C2×C8—1582no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—3161no—
317C4⋊C4—1582no—
318C4⋊C4—1581no—
319C4⋊C4—1581no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—3161no—
323C22⋊C4—3161no—
324C42—42no—
325C42—3162no—
326C42C2×C4×D7911yescharacteristic
327C42D4×D7911yescharacteristic
328C42#1192962311yescharacteristic
329C42#1192962411yescharacteristic
330C23—41no—
331C23—3161no—
332C23#1457915711yescharacteristic
333Q8—42no—
334Q8—3162no—
335D4—42no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—3162no—
342D4—3161no—
343D4—3161no—
344D4—6321no—
345D4—6321no—
346D4—6321no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—3162no—
356C2×C4—3161no—
357C2×C4—6321no—
358C2×C4—6321no—
359C2×C4—6322no—
360C2×C4C22⋊C4×D7911yescharacteristic
361C2×C4(C22⋊D79)⋊C411yescharacteristic
362C2×C4C4×C22⋊D7911yescharacteristic
363C2×C4#1457913311yescharacteristic
364C2×C4#1457913411yescharacteristic
365C2×C4#1457914711yescharacteristic
366C2×C4D8×D7912yes—
367C2×C4#1457918212yes—
368C8—42no—
369C8—792no—
370C8—1584no—
371C8—1581no—
372C8—1581no—
373C8—1581no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—6321no—
380C22—6321no—
381C22—6321no—
382C22#1909846311yescharacteristic
383C22#1909893811yescharacteristic
384C22#1909894411yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—81no—
390C4—6322no—
391C4—6321no—
392C4#104×D7911yescharacteristic
393C4#1909857811yescharacteristic
394C4#1909864312yes—
395C4#1909879812yes—
396C2—21no—
397C2—21no—
398C2—161no—
399C2—12641no—
400C2#2376426311yescharacteristic
401C2#2376429511yescharacteristic
402C2#2376431711yescharacteristic
subgroupG/H
centerC2×C4#14579134
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1502C2
socleC22×C79#390
prime radical#23773975C2
cosocleC2×C42×C79C23
hypercenter#1502D79
hypocenterC79#9719
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #23764320C2semidirect
3. #23764318C2semidirect
4. C4×#14579094C2semidirect
5. #23761896C2semidirect
6. #23761681C2semidirect
7. C79×#1502C2semidirect
9. #19098966C4semidirect
10. #19098939C4semidirect
11. #19098911C4semidirect
12. #19098909C4semidirect
89. #11929608D8semidirect
171. C79⋊C8#104semidirect
207. #1502D79semidirect
209. C79#9719semidirect

Maximal subgroups

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

subgroupindex
#237739752
#237643202
#237643182
C4×#145790942
#237618962
#237616812
C79×#15022
#971979