Finite Group Databasebeta

← #34443344

Subgroups — #34443344

Order 20224.

592 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773979C211yesmaximal, characteristic, complemented
2#23765591C212yesmaximal, complemented
3#23765542C211yesmaximal, characteristic, complemented
4#23765537C211yesmaximal, characteristic, complemented
5#23765440C211yesmaximal, characteristic, complemented
6#23765435C211yesmaximal, characteristic, complemented
7#23765323C212yesmaximal, complemented
8#23765274C212yesmaximal, complemented
9#23765262C212yesmaximal, complemented
10#23765203C211yesmaximal, characteristic, complemented
11C4×#135×C79C211yesmaximal, characteristic, complemented
12#19099113—21no—
13C2×#14579104C2212yes—
14C2×#14579102C2212yes—
15#19098968C2211yescharacteristic
16#19098940C2211yescharacteristic
17#19098910—21no—
18#19098799—21no—
19#19098799C2212yes—
20C4×D632—21no—
21C4×D632C2212yes—
22#19098390C2212yes—
23#19098388C2212yes—
24C4⋊(C79⋊C16)C2211yescharacteristic
25C4⋊(C79⋊C16)C2211yescharacteristic
26#19098367C2212yes—
27#19098364C2212yes—
28#19098363C2212yes—
29#19098363C2212yes—
30C22⋊(C79⋊C16)C2212yes—
31#19098339C2212yes—
32C2×#135×C79C2211yescharacteristic
33C4×#41×C79—21no—
34C4×Q16×C79C2211yescharacteristic
35C4×Q16×C79C2211yescharacteristic
36C4×SD16×C79—21no—
37C4×SD16×C79—21no—
38C4×D8×C79C2211yescharacteristic
39C4×D8×C79C2211yescharacteristic
40#427×C79C2211yescharacteristic
41#416×C79—21no—
42C2×C4×C8×C79C2211yescharacteristic
43C2×#11929624—41no—
44C2×D632—21no—
45C2×D632C2312yes—
46#14579161—21no—
47#14579158—21no—
48#14579149—41no—
49#14579135—22no—
50#14579135—41no—
51C4×D316—22no—
52C4×D316—41no—
53#14579104—24no—
54#14579102—24no—
55C2×C79⋊C16C2312yes—
56C2×C79⋊C16C2312yes—
57#14579095—21no—
58#14579095C2312yes—
59#14579083—22no—
60#14579083—22no—
61C2×#41×C79—21no—
62#135×C79—22no—
63#135×C79—22no—
64C2×Q16×C79C2311yescharacteristic
65C2×SD16×C79—21no—
66C2×D8×C79C2311yescharacteristic
67C22×C8×C79D411yescharacteristic
68C22×C8×C79C2311yescharacteristic
69C4×Q8×C79—21no—
70C4×Q8×C79—21no—
71C4×D4×C79—21no—
72C4×D4×C79—21no—
73C4×D4×C79—41no—
74C4×D4×C79—41no—
75#117×C79—21no—
76#117×C79—41no—
77C2×C42×C79—41no—
78C2×C42×C79D411yescharacteristic
79#107×C79C2311yescharacteristic
80#107×C79C2311yescharacteristic
81#106×C79—21no—
82#103×C79—21no—
83#103×C79—21no—
84#102×C79—21no—
85#102×C79—21no—
86C4×C8×C79D411yescharacteristic
87C4×C8×C79D411yescharacteristic
88C4×C8×C79C2311yescharacteristic
89C4×C8×C79C2311yescharacteristic
90#11929624—84no—
91C2×Dic158—41no—
92C2×D316—22no—
93C2×D316—41no—
94C2×D316—41no—
95C2×C22⋊D79—42no—
96C2×C4×D79—42no—
97D632—22no—
98D632—22no—
99D632—42no—
100C4⋊Dic79—22no—
101C4⋊Dic79—41no—
102(C2×D79)⋊C4—42no—
103(C2×D79)⋊C4—42no—
104C79⋊C16—22no—
105C79⋊C16—22no—
106#41×C79—24no—
107#41×C79—42no—
108C2×Q8×C79—21no—
109C2×D4×C79—21no—
110C2×D4×C79—41no—
111C22×C4×C79—41no—
112C22×C4×C79—41no—
113C22×C4×C79—41no—
114C22×C4×C79D811yescharacteristic
115C22×C4×C79D811yescharacteristic
116C22×C4×C79C2×D411yescharacteristic
117Q16×C79—21no—
118Q16×C79#4112yescomplemented
119SD16×C79—24no—
120D8×C79—21no—
121D8×C79#4112yescomplemented
122C2×C8×C79—21no—
123C2×C8×C79—21no—
124C2×C8×C79—41no—
125C2×C8×C79C2×D411yescharacteristic
126C2×C8×C79C2×D411yescharacteristic
127C2×C8×C79C2×D411yescharacteristic
128C2×C8×C79C2411yescharacteristic
129C4⋊C4×C79—21no—
130C4⋊C4×C79—21no—
131C4⋊C4×C79—41no—
132C4⋊C4×C79—41no—
133C22⋊C4×C79—41no—
134C22⋊C4×C79—41no—
135C22⋊C4×C79—41no—
136C42×C79—41no—
137C42×C79—41no—
138C42×C79—41no—
139C42×C79D811yescharacteristic
140C42×C79D811yescharacteristic
141C42×C79C2×D411yescharacteristic
142C42×C79C2×D411yescharacteristic
143C23×C79—41no—
144C23×C79C2×D811yescharacteristic
145C22×D79—42no—
146Q8×C79—22no—
147Q8×C79—41no—
148D4×C79—22no—
149D4×C79—44no—
150D4×C79—41no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—22no—
154C2×C4×C79—41no—
155C2×C4×C79—44no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—82no—
160C2×C4×C79—82no—
161C2×C4×C79C2×D811yescharacteristic
162C2×C4×C79C2×D811yescharacteristic
163C2×C4×C79C2×D811yescharacteristic
164C2×C4×C79C2×D811yescharacteristic
165C2×C4×C79C2×D811yescharacteristic
166C2×C4×C79C22×D411yescharacteristic
167C22⋊D79—84no—
168C2×Dic79—42no—
169D316—42no—
170D316—42no—
171D316—82no—
172D316—82no—
173C4×D79—84no—
174Dic158—82no—
175C8×C79—21no—
176C8×C79—21no—
177C8×C79—21no—
178C8×C79C2×#4111yescharacteristic
179C8×C79C2×#4111yescharacteristic
180C22×C79—21no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—42no—
184C22×C79—81no—
185C22×C79—81no—
186C22×C79C22×D811yescharacteristic
187C2×D79—82no—
188C2×D79—82no—
189C2×D79—82no—
190C4×C79—21no—
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79—42no—
195C4×C79—81no—
196C4×C79—81no—
197C4×C79#50912yes—
198C4×C79#52511yescharacteristic
199C4×C79#57511yescharacteristic
200Dic79—82no—
201#33675—791nomaximal, Sylow
202C2×C79—41no—
203C2×C79—82no—
204C2×C79#310011yescharacteristic
205C2×C79C2×#50911yescharacteristic
206C2×C79#341411yescharacteristic
207D79—162no—
208#3169—791no—
209C4×#135D7911yescharacteristic, complemented
210#2234—792no—
211#2225—791no—
212#2224—791no—
213#2207—791no—
214#2206—791no—
215#2186—792no—
216#2177—792no—
217#2175—792no—
218#2163—791no—
219C79#3367511yescharacteristic, complemented, Sylow
220C2×#135C2×D7911yescharacteristic
221#532—1581no—
222C4×#41—21no—
223C2×SD32—792no—
224C2×D16—792no—
225#500—791no—
226#493—791no—
227#486—1581no—
228#466—792no—
229#466—1581no—
230C4×Q16C2×D7911yescharacteristic
231C4×Q16C2×D7911yescharacteristic
232C4×SD16—21no—
233C4×SD16—21no—
234C4×D8—792no—
235C4×D8—1581no—
236C4×D8C2×D7911yescharacteristic
237C4×D8C2×D7911yescharacteristic
238#427C2×D7911yescharacteristic
239#416—21no—
240C2×C4×C8C2×D7911yescharacteristic
241#367—792no—
242#366—792no—
243C4⋊C16—791no—
244C4⋊C16—791no—
245#358—792no—
246#357—792no—
247#357—792no—
248#357—792no—
249C22⋊C16—792no—
250#346—792no—
251C2×#41—21no—
252C2×#41—3161no—
253#135—22no—
254#135—22no—
255C2×Q16C22×D7911yescharacteristic
256C2×SD16—21no—
257C2×D8—792no—
258C2×D8—1581no—
259C2×D8C22×D7911yescharacteristic
260C22×C8C22⋊D7911yescharacteristic
261C22×C8C22×D7911yescharacteristic
262C4⋊Q8—1581no—
263#127—1581no—
264#124—3161no—
265#121—3161no—
266#121—1582no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—3161no—
274C4×D4—1582no—
275#117—21no—
276#117—41no—
277C2×C42—41no—
278C2×C42C22⋊D7911yescharacteristic
279SD32—1584no—
280D16—1584no—
281C2×C16—792no—
282C2×C16—792no—
283#107—792no—
284#107—1581no—
285#107C22×D7911yescharacteristic
286#107C22×D7911yescharacteristic
287#106—21no—
288#103—21no—
289#103—21no—
290#102—21no—
291#102—21no—
292#102—1582no—
293#102—1582no—
294C4×C8C22⋊D7911yescharacteristic
295C4×C8C22⋊D7911yescharacteristic
296C4×C8C22×D7911yescharacteristic
297C4×C8C22×D7911yescharacteristic
298#41—24no—
299#41—42no—
300#41—6324no—
301C2×Q8—21no—
302C2×Q8—3161no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—3162no—
306C2×D4—3161no—
307C2×D4—3161no—
308C2×D4—1582no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—3162no—
313C22×C4#1192960911yescharacteristic
314C22×C4#1192960911yescharacteristic
315C22×C4C2×C22⋊D7911yescharacteristic
316Q16—21no—
317Q16#1192962412yescomplemented
318SD16—24no—
319D8—21no—
320D8—3162no—
321D8—1582no—
322D8—1582no—
323D8#1192962412yescomplemented
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8C2×C22⋊D7911yescharacteristic
328C2×C8C2×C22⋊D7911yescharacteristic
329C2×C8C2×C22⋊D7911yescharacteristic
330C2×C8C23×D7911yescharacteristic
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—3161no—
336C4⋊C4—1582no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—3162no—
341C22⋊C4—3162no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42#1192960911yescharacteristic
346C42#1192960911yescharacteristic
347C42C2×C22⋊D7911yescharacteristic
348C42C2×C22⋊D7911yescharacteristic
349C16—1582no—
350C16—1582no—
351C23—41no—
352C23—3162no—
353C23C2×#1192960911yescharacteristic
354Q8—22no—
355Q8—41no—
356Q8—6322no—
357D4—22no—
358D4—44no—
359D4—41no—
360D4—3162no—
361D4—3162no—
362D4—6322no—
363D4—6324no—
364D4—6322no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—3162no—
376C2×C4—6324no—
377C2×C4C2×#1192960911yescharacteristic
378C2×C4C2×#1192960911yescharacteristic
379C2×C4C2×#1192960911yescharacteristic
380C2×C4C2×#1192960911yescharacteristic
381C2×C4C2×#1192960911yescharacteristic
382C2×C4C22×C22⋊D7911yescharacteristic
383C8—21no—
384C8—21no—
385C8—21no—
386C8C2×#1192962411yescharacteristic
387C8C2×#1192962411yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—81no—
393C22—81no—
394C22—6322no—
395C22—6322no—
396C22—6322no—
397C22C22×#1192960911yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4—6322no—
406C4#1909899812yes—
407C4#1909907411yescharacteristic
408C4#1909936111yescharacteristic
409C2—41no—
410C2—82no—
411C2—12642no—
412C2#2377340011yescharacteristic
413C2C2×#1909899811yescharacteristic
414C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC4×#135×C79C2
socleC22×C79C22×D8
hypercenterC4×#135D79
hypocenterC79#33675
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #23773979C2semidirect
2. #23765591C2semidirect
3. #23765542C2semidirect
4. #23765537C2semidirect
5. #23765440C2semidirect
6. #23765435C2semidirect
7. #23765323C2semidirect
8. #23765274C2semidirect
9. #23765262C2semidirect
10. #23765203C2semidirect
11. C4×#135×C79C2semidirect
118. Q16×C79#41semidirect
121. D8×C79#41semidirect
209. C4×#135D79semidirect
219. C79#33675semidirect
317. Q16#11929624semidirect
323. D8#11929624semidirect

Maximal subgroups

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

subgroupindex
#237739792
#237655912
#237655422
#237655372
#237654402
#237654352
#237653232
#237652742
#237652622
#237652032
C4×#135×C792
#3367579