Finite Group Databasebeta

← #34443338

Subgroups — #34443338

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
1C4×#14579183C211yesmaximal, characteristic, complemented
2#23765588C212yesmaximal, complemented
3#23765539C211yesmaximal, characteristic, complemented
4#23765534C211yesmaximal, characteristic, complemented
5#23765437C211yesmaximal, characteristic, complemented
6#23765432C211yesmaximal, characteristic, complemented
7#23765321C212yesmaximal, complemented
8#23765274C212yesmaximal, complemented
9#23765264C212yesmaximal, complemented
10#23765203C211yesmaximal, characteristic, complemented
11C79×#3169C211yesmaximal, characteristic, complemented
12C2×#14579183C2211yescharacteristic
13C4×#11929624—21no—
14C2×#14579105C2212yes—
15C2×#14579102C2212yes—
16C4×Dic316C2211yescharacteristic
17C4×Dic316C2211yescharacteristic
18C4×#11929613—21no—
19C4×#11929613—21no—
20C4×D632C2211yescharacteristic
21C4×D632C2211yescharacteristic
22#19098601C2211yescharacteristic
23#19098560—21no—
24#19098390C2212yes—
25#19098388C2212yes—
26C4⋊(C79⋊C16)C2211yescharacteristic
27C4⋊(C79⋊C16)C2211yescharacteristic
28#19098366C2212yes—
29#19098364C2212yes—
30#19098364C2212yes—
31#19098363C2212yes—
32C22⋊(C79⋊C16)C2212yes—
33#19098339C2212yes—
34#532×C79—21no—
35#500×C79C2211yescharacteristic
36#493×C79C2211yescharacteristic
37#486×C79—21no—
38#466×C79—21no—
39#466×C79C2212yes—
40C4×D8×C79—21no—
41C4×D8×C79C2212yes—
42C2×C4×C8×C79C2211yescharacteristic
43C2×#11929624—21no—
44#14579183—22no—
45#14579183—22no—
46C2×Dic316C2311yescharacteristic
47C2×#11929613—21no—
48C2×D632C2311yescharacteristic
49C4×Dic158—21no—
50C4×Dic158—21no—
51C4×D316—21no—
52C4×D316—21no—
53C4×C22⋊D79—41no—
54C4×C22⋊D79—41no—
55#14579120—21no—
56#14579118—41no—
57C42×D79—41no—
58#14579105—24no—
59#14579102—24no—
60C2×C79⋊C16C2312yes—
61C2×C79⋊C16C2312yes—
62#14579095C2311yescharacteristic
63#14579095C2311yescharacteristic
64#14579093—21no—
65#14579086—21no—
66#14579086—21no—
67#14579083—21no—
68#14579083—21no—
69C2×#41×C79—41no—
70C2×D8×C79—21no—
71C2×D8×C79C2312yes—
72C22×C8×C79D411yescharacteristic
73C22×C8×C79C2311yescharacteristic
74C4⋊Q8×C79—21no—
75#127×C79—21no—
76#124×C79—41no—
77#121×C79—22no—
78#121×C79—41no—
79C4×D4×C79—22no—
80C4×D4×C79—41no—
81C2×C42×C79D411yescharacteristic
82#107×C79—21no—
83#107×C79C2312yes—
84#102×C79—22no—
85#102×C79—22no—
86C4×C8×C79D411yescharacteristic
87C4×C8×C79D411yescharacteristic
88C4×C8×C79C2311yescharacteristic
89C4×C8×C79C2311yescharacteristic
90#11929624—24no—
91#11929624—42no—
92C2×Dic158—21no—
93C2×D316—21no—
94C2×C22⋊D79—41no—
95C2×C4×D79—41no—
96C2×C4×D79—41no—
97C2×C4×D79—41no—
98Dic316—21no—
99Dic316#4112yescomplemented
100#11929613—24no—
101D632—21no—
102D632#4112yescomplemented
103C4⋊Dic79—21no—
104C4⋊Dic79—21no—
105Dic79⋊C4—41no—
106Dic79⋊C4—41no—
107C22⋊Dic79—41no—
108(C2×D79)⋊C4—41no—
109(C2×D79)⋊C4—41no—
110C4×Dic79—41no—
111C4×Dic79—41no—
112C4×Dic79—41no—
113C79⋊C16—22no—
114C79⋊C16—22no—
115#41×C79—84no—
116C2×Q8×C79—41no—
117C2×D4×C79—22no—
118C2×D4×C79—42no—
119C2×D4×C79—41no—
120C2×D4×C79—41no—
121C22×C4×C79—42no—
122C22×C4×C79D811yescharacteristic
123C22×C4×C79D811yescharacteristic
124C22×C4×C79C2×D411yescharacteristic
125D8×C79—22no—
126D8×C79—22no—
127D8×C79—42no—
128C2×C8×C79—21no—
129C2×C8×C79—21no—
130C2×C8×C79—41no—
131C2×C8×C79C2×D411yescharacteristic
132C2×C8×C79C2×D411yescharacteristic
133C2×C8×C79C2×D411yescharacteristic
134C2×C8×C79C2411yescharacteristic
135C4⋊C4×C79—22no—
136C4⋊C4×C79—41no—
137C22⋊C4×C79—42no—
138C22⋊C4×C79—42no—
139C42×C79D811yescharacteristic
140C42×C79D811yescharacteristic
141C42×C79C2×D411yescharacteristic
142C42×C79C2×D411yescharacteristic
143C23×C79—42no—
144C23×C79C2×D811yescharacteristic
145C22×D79—41no—
146Q8×C79—82no—
147D4×C79—42no—
148D4×C79—42no—
149D4×C79—82no—
150D4×C79—84no—
151D4×C79—82no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—22no—
155C2×C4×C79—41no—
156C2×C4×C79—42no—
157C2×C4×C79—84no—
158C2×C4×C79C2×D811yescharacteristic
159C2×C4×C79C2×D811yescharacteristic
160C2×C4×C79C2×D811yescharacteristic
161C2×C4×C79C2×D811yescharacteristic
162C2×C4×C79C2×D811yescharacteristic
163C2×C4×C79C22×D411yescharacteristic
164C22⋊D79—44no—
165C2×Dic79—41no—
166C2×Dic79—41no—
167C2×Dic79—41no—
168D316—22no—
169D316—41no—
170C4×D79—44no—
171C4×D79—82no—
172C4×D79—82no—
173Dic158—22no—
174Dic158—41no—
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—82no—
184C22×C79—82no—
185C22×C79—82no—
186C22×C79C22×D811yescharacteristic
187C2×D79—42no—
188C2×D79—81no—
189C2×D79—81no—
190C4×C79—21no—
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79—82no—
195C4×C79#50912yes—
196C4×C79#52511yescharacteristic
197C4×C79#57511yescharacteristic
198Dic79—42no—
199Dic79—81no—
200Dic79—81no—
201#33675—791nomaximal, Sylow
202C2×C79—41no—
203C2×C79—162no—
204C2×C79#310011yescharacteristic
205C2×C79C2×#50911yescharacteristic
206C2×C79#341411yescharacteristic
207D79—82no—
208#3169D7911yescharacteristic, complemented
209C4×#135—791no—
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×#135—791no—
221#532—21no—
222C4×#41—1581no—
223C2×SD32—792no—
224C2×D16—792no—
225#500C2×D7911yescharacteristic
226#493C2×D7911yescharacteristic
227#486—21no—
228#466—21no—
229#466C2×D7912yes—
230C4×Q16—791no—
231C4×Q16—791no—
232C4×SD16—1581no—
233C4×SD16—1581no—
234C4×D8—21no—
235C4×D8—791no—
236C4×D8—791no—
237C4×D8C2×D7912yes—
238#427—791no—
239#416—1581no—
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—41no—
252C2×#41—1581no—
253#135—1582no—
254#135—1582no—
255C2×Q16—791no—
256C2×SD16—1581no—
257C2×D8—21no—
258C2×D8—791no—
259C2×D8C22×D7912yes—
260C22×C8C22⋊D7911yescharacteristic
261C22×C8C22×D7911yescharacteristic
262C4⋊Q8—21no—
263#127—21no—
264#124—41no—
265#121—22no—
266#121—41no—
267C4×Q8—1581no—
268C4×Q8—1581no—
269C4×D4—22no—
270C4×D4—41no—
271C4×D4—3161no—
272C4×D4—3161no—
273C4×D4—1581no—
274C4×D4—1581no—
275#117—3161no—
276#117—1581no—
277C2×C42—3161no—
278C2×C42C22⋊D7911yescharacteristic
279SD32—1584no—
280D16—1584no—
281C2×C16—792no—
282C2×C16—792no—
283#107—21no—
284#107—791no—
285#107—791no—
286#107C22×D7912yes—
287#106—1581no—
288#103—1581no—
289#103—1581no—
290#102—22no—
291#102—22no—
292#102—1581no—
293#102—1581no—
294C4×C8C22⋊D7911yescharacteristic
295C4×C8C22⋊D7911yescharacteristic
296C4×C8C22×D7911yescharacteristic
297C4×C8C22×D7911yescharacteristic
298#41—84no—
299#41—3162no—
300#41—1584no—
301C2×Q8—41no—
302C2×Q8—1581no—
303C2×D4—22no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—3161no—
308C2×D4—1581no—
309C22×C4—42no—
310C22×C4—3161no—
311C22×C4—3161no—
312C22×C4—3161no—
313C22×C4#1192960911yescharacteristic
314C22×C4#1192960911yescharacteristic
315C22×C4C2×C22⋊D7911yescharacteristic
316Q16—792no—
317Q16—1581no—
318SD16—1584no—
319D8—22no—
320D8—22no—
321D8—42no—
322D8—792no—
323D8—1581no—
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—22no—
332C4⋊C4—41no—
333C4⋊C4—3161no—
334C4⋊C4—3161no—
335C4⋊C4—1581no—
336C4⋊C4—1581no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—3161no—
340C22⋊C4—3161no—
341C22⋊C4—3161no—
342C42—3161no—
343C42—3161no—
344C42—3161no—
345C42#1192960911yescharacteristic
346C42#1192960911yescharacteristic
347C42C2×C22⋊D7911yescharacteristic
348C42C2×C22⋊D7911yescharacteristic
349C16—1582no—
350C16—1582no—
351C23—42no—
352C23—3161no—
353C23C2×#1192960911yescharacteristic
354Q8—82no—
355Q8—3161no—
356Q8—1582no—
357D4—42no—
358D4—42no—
359D4—82no—
360D4—84no—
361D4—82no—
362D4—3164no—
363D4—3161no—
364D4—1582no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—84no—
371C2×C4—3164no—
372C2×C4—3161no—
373C2×C4—3161no—
374C2×C4—3161no—
375C2×C4—6322no—
376C2×C4—6322no—
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—
386C8#41×D7911yescharacteristic
387C8#41×D7911yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—3162no—
395C22—6321no—
396C22—6321no—
397C22C22×#1192960911yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—82no—
403C4—3162no—
404C4—6321no—
405C4—6321no—
406C4#1909899912yes—
407C4#1909907011yescharacteristic
408C4#1909936111yescharacteristic
409C2—41no—
410C2—162no—
411C2—6322no—
412C2#2377339611yescharacteristic
413C2C2×#1909899911yescharacteristic
414C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3169C2
socleC22×C79C22×D8
hypercenter#3169D79
hypocenterC79#33675
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. C4×#14579183C2semidirect
2. #23765588C2semidirect
3. #23765539C2semidirect
4. #23765534C2semidirect
5. #23765437C2semidirect
6. #23765432C2semidirect
7. #23765321C2semidirect
8. #23765274C2semidirect
9. #23765264C2semidirect
10. #23765203C2semidirect
11. C79×#3169C2semidirect
99. Dic316#41semidirect
102. D632#41semidirect
208. #3169D79semidirect
219. C79#33675semidirect

Maximal subgroups

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

subgroupindex
C4×#145791832
#237655882
#237655392
#237655342
#237654372
#237654322
#237653212
#237652742
#237652642
#237652032
C79×#31692
#3367579