Finite Group Databasebeta

← #34442541

Subgroups — #34442541

Order 20224.

770 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098998C212yesmaximal, complemented
2C2×C4×D632C211yesmaximal, characteristic, complemented
3#23765452C214yesmaximal, complemented
4#23765442C212yesmaximal, complemented
5#23765432C212yesmaximal, complemented
6#23765156C211yesmaximal, characteristic, complemented
7#23765112C212yesmaximal, complemented
8C79×#3147C211yesmaximal, characteristic, complemented
9C22×D632C2211yescharacteristic
10C2×C4×D316—21no—
11#19098998—28no—
12C2×#14579104C2214yes—
13C2×#14579102C2214yes—
14C2×#14579101C2212yes—
15C4×D632—21no—
16C4×D632—21no—
17C4×D632C2212yes—
18C4×D632C2212yes—
19C2×#14579095C2211yescharacteristic
20C2×#14579083—21no—
21C4⋊(C79⋊C16)C2212yescomplemented
22C4⋊(C79⋊C16)C2212yescomplemented
23#19098367C2214yescomplemented
24#19098364C2214yescomplemented
25C2×#135×C79—21no—
26C2×#135×C79C2212yes—
27#532×C79—21no—
28#495×C79—21no—
29#495×C79C2212yescomplemented
30#494×C79C2211yescharacteristic, complemented
31#493×C79C2211yescharacteristic, complemented
32#492×C79—21no—
33C2×C4×C8×C79C2211yescharacteristic
34C22×D316—21no—
35C22×C4×D79—41no—
36C2×D632—21no—
37C2×D632—22no—
38C2×D632D414yescomplemented
39C2×D632C2312yes—
40C4×D316—22no—
41C4×D316—22no—
42C4×D316—41no—
43C4×D316—41no—
44C2×C4⋊Dic79—21no—
45C2×(C2×D79)⋊C4—41no—
46#14579104—28no—
47#14579102—28no—
48#14579101—24no—
49C2×C79⋊C16C2314yes—
50#14579095—21no—
51#14579095C2312yes—
52#14579083—22no—
53#14579083—22no—
54C2×#41×C79—22no—
55#135×C79—24no—
56#135×C79—24no—
57#135×C79—44no—
58C2×Q16×C79—21no—
59C2×Q16×C79C2312yes—
60C2×SD16×C79—22no—
61C2×SD16×C79—22no—
62C2×D8×C79—21no—
63C2×D8×C79C2312yes—
64C22×C8×C79D411yescharacteristic
65C22×C8×C79C2311yescharacteristic
66C4⋊Q8×C79—21no—
67#127×C79—21no—
68#124×C79—22no—
69#121×C79—41no—
70#121×C79—41no—
71C4×D4×C79—41no—
72C4×D4×C79—41no—
73C2×C42×C79D411yescharacteristic
74C4×C8×C79D411yescharacteristic
75C4×C8×C79D411yescharacteristic
76C4×C8×C79C2311yescharacteristic
77C4×C8×C79C2311yescharacteristic
78C23×D79—41no—
79C2×D316—24no—
80C2×D316—22no—
81C2×D316—41no—
82C2×D316—42no—
83C22×Dic79—41no—
84C2×C4×D79—42no—
85C2×C4×D79—42no—
86C2×C4×D79—84no—
87D632—24no—
88D632—22no—
89D632C2×D414yes—
90C4⋊Dic79—22no—
91C4⋊Dic79—41no—
92(C2×D79)⋊C4—42no—
93(C2×D79)⋊C4—42no—
94C79⋊C16—24no—
95#41×C79—44no—
96#41×C79—44no—
97C2×Q8×C79—22no—
98C2×D4×C79—22no—
99C2×D4×C79—42no—
100C2×D4×C79—41no—
101C2×D4×C79—41no—
102C22×C4×C79—42no—
103C22×C4×C79D811yescharacteristic
104C22×C4×C79D811yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106Q16×C79—22no—
107Q16×C79—22no—
108Q16×C79—42no—
109SD16×C79—44no—
110SD16×C79—44no—
111D8×C79—22no—
112D8×C79—22no—
113D8×C79—42no—
114C2×C8×C79—22no—
115C2×C8×C79C2×D412yes—
116C2×C8×C79C2×D411yescharacteristic
117C2×C8×C79C2×D411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79#4112yes—
120C2×C8×C79C2411yescharacteristic
121C4⋊C4×C79—41no—
122C4⋊C4×C79—41no—
123C22⋊C4×C79—42no—
124C22⋊C4×C79—42no—
125C42×C79D811yescharacteristic
126C42×C79D811yescharacteristic
127C42×C79C2×D411yescharacteristic
128C42×C79C2×D411yescharacteristic
129C23×C79—42no—
130C23×C79C2×D811yescharacteristic
131C22×D79—42no—
132C22×D79—44no—
133C22×D79—82no—
134C22×D79—82no—
135Q8×C79—42no—
136Q8×C79—42no—
137D4×C79—42no—
138D4×C79—42no—
139D4×C79—82no—
140D4×C79—84no—
141D4×C79—82no—
142C2×C4×C79—22no—
143C2×C4×C79—41no—
144C2×C4×C79—42no—
145C2×C4×C79—84no—
146C2×C4×C79#12112yes—
147C2×C4×C79C2×D811yescharacteristic
148C2×C4×C79C2×D811yescharacteristic
149C2×C4×C79C2×D811yescharacteristic
150C2×C4×C79C2×D811yescharacteristic
151C2×C4×C79C2×D811yescharacteristic
152C2×C4×C79AGL(1,ℤ/8)12yes—
153C2×C4×C79C22×D411yescharacteristic
154C2×Dic79—42no—
155C2×Dic79—82no—
156D316—24no—
157D316—42no—
158D316—44no—
159C4×D79—84no—
160C4×D79—84no—
161C8×C79—21no—
162C8×C79—21no—
163C8×C79#12112yes—
164C8×C79C22×D411yescharacteristic
165C8×C79C2×#4111yescharacteristic
166C22×C79—21no—
167C22×C79—21no—
168C22×C79—82no—
169C22×C79—82no—
170C22×C79—82no—
171C22×C79#45912yes—
172C22×C79C22×D811yescharacteristic
173C2×D79—44no—
174C2×D79—88no—
175C2×D79—82no—
176C2×D79—82no—
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—82no—
181C4×C79#45912yes—
182C4×C79#50912yescomplemented
183C4×C79C2×#12111yescharacteristic
184C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
185Dic79—82no—
186#33580—791nomaximal, Sylow
187C2×C79—22no—
188C2×C79—162no—
189C2×C79C2×#45911yescharacteristic
190C2×C79C2×#50911yescharacteristic
191C2×C79#341411yescharacteristic
192D79—84no—
193C2×#509—792no—
194#3147D7911yescharacteristic, complemented
195C2×C4×D8—791no—
196#2210—794no—
197#2208—792no—
198#2206—792no—
199#2150—791no—
200#2139—792no—
201C79#3358011yescharacteristic, complemented, Sylow
202C2×#135—21no—
203C2×#135C2×D7912yes—
204C22×D8—791no—
205#532—21no—
206C2×C4×D4—1581no—
207#509—1588no—
208C2×SD32—794no—
209C2×D16—794no—
210C2×M32—792no—
211#495—21no—
212#495C2×D7912yescomplemented
213#494C2×D7911yescharacteristic, complemented
214#493C2×D7911yescharacteristic, complemented
215#492—21no—
216C4×D8—792no—
217C4×D8—792no—
218C4×D8—1581no—
219C4×D8—1581no—
220C2×#107—791no—
221C2×#102—1581no—
222C2×C4×C8C2×D7911yescharacteristic
223C4⋊C16—792no—
224C4⋊C16—792no—
225#358—794no—
226#357—794no—
227C2×#41—22no—
228C22×D4—1581no—
229C23×C4—3161no—
230#135—24no—
231#135—24no—
232#135—44no—
233C2×Q16—21no—
234C2×Q16C22×D7912yes—
235C2×SD16—22no—
236C2×SD16—22no—
237C2×D8—21no—
238C2×D8—794no—
239C2×D8—792no—
240C2×D8—1581no—
241C2×D8—1582no—
242C2×D8C22×D7912yes—
243C22×C8C22⋊D7911yescharacteristic
244C22×C8C22×D7911yescharacteristic
245C4⋊Q8—21no—
246#127—21no—
247#124—22no—
248#121—41no—
249#121—41no—
250C4×D4—41no—
251C4×D4—41no—
252C4×D4—3161no—
253C4×D4—3161no—
254C4×D4—1582no—
255C4×D4—1582no—
256C2×C4⋊C4—1581no—
257C2×C22⋊C4—3161no—
258C2×C42C22⋊D7911yescharacteristic
259SD32—1588no—
260D16—1588no—
261M32—1584no—
262C2×C16—794no—
263#107—792no—
264#107—1581no—
265#102—1582no—
266#102—1582no—
267C4×C8C22⋊D7911yescharacteristic
268C4×C8C22⋊D7911yescharacteristic
269C4×C8C22×D7911yescharacteristic
270C4×C8C22×D7911yescharacteristic
271C24—3161no—
272#41—44no—
273#41—44no—
274C2×Q8—22no—
275C2×D4—22no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—3161no—
280C2×D4—3162no—
281C2×D4—1584no—
282C2×D4—1582no—
283C22×C4—42no—
284C22×C4—3161no—
285C22×C4—3162no—
286C22×C4—3162no—
287C22×C4—6324no—
288C22×C4#1192960911yescharacteristic
289C22×C4#1192960911yescharacteristic
290C22×C4C2×C22⋊D7911yescharacteristic
291Q16—22no—
292Q16—22no—
293Q16—42no—
294SD16—44no—
295SD16—44no—
296D8—22no—
297D8—22no—
298D8—42no—
299D8—794no—
300D8—1582no—
301D8—1584no—
302C2×C8—22no—
303C2×C8D4×D7912yes—
304C2×C8C2×C22⋊D7911yescharacteristic
305C2×C8C2×C22⋊D7911yescharacteristic
306C2×C8C2×C22⋊D7911yescharacteristic
307C2×C8#1192962312yes—
308C2×C8C23×D7911yescharacteristic
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—3161no—
312C4⋊C4—1582no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—3162no—
316C22⋊C4—3162no—
317C42#1192960911yescharacteristic
318C42#1192960911yescharacteristic
319C42C2×C22⋊D7911yescharacteristic
320C42C2×C22⋊D7911yescharacteristic
321C16—1584no—
322C23—42no—
323C23—3164no—
324C23—3162no—
325C23—6322no—
326C23—6322no—
327C23C2×#1192960911yescharacteristic
328Q8—42no—
329Q8—42no—
330D4—42no—
331D4—42no—
332D4—82no—
333D4—84no—
334D4—82no—
335D4—3164no—
336D4—3162no—
337D4—1584no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—84no—
342C2×C4—3162no—
343C2×C4—6324no—
344C2×C4—6322no—
345C2×C4—6324no—
346C2×C4#1457913212yes—
347C2×C4C2×#1192960911yescharacteristic
348C2×C4C2×#1192960911yescharacteristic
349C2×C4C2×#1192960911yescharacteristic
350C2×C4C2×#1192960911yescharacteristic
351C2×C4C2×#1192960911yescharacteristic
352C2×C4#1457918712yes—
353C2×C4C22×C22⋊D7911yescharacteristic
354C8—21no—
355C8—21no—
356C8#1457913212yes—
357C8C2×D4×D7911yescharacteristic
358C8C2×#1192962311yescharacteristic
359C22—21no—
360C22—21no—
361C22—82no—
362C22—82no—
363C22—82no—
364C22—3164no—
365C22—6328no—
366C22—6322no—
367C22—6322no—
368C22#1909875512yes—
369C22C22×#1192960911yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—6322no—
375C4#1909875512yes—
376C4#1909899812yescomplemented
377C4C2×#1457913211yescharacteristic
378C4C2×#1457918711yescharacteristic
379C2—22no—
380C2—162no—
381C2—6324no—
382C2C2×#1909875511yescharacteristic
383C2C2×#1909899811yescharacteristic
384C2#2377653411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3147C2
socleC22×C79C22×D8
hypercenter#3147D79
hypocenterC79#33580
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. C2×#19098998C2semidirect
2. C2×C4×D632C2semidirect
3. #23765452C2semidirect
4. #23765442C2semidirect
5. #23765432C2semidirect
6. #23765156C2semidirect
7. #23765112C2semidirect
8. C79×#3147C2semidirect
21. C4⋊(C79⋊C16)C22semidirect
22. C4⋊(C79⋊C16)C22semidirect
23. #19098367C22semidirect
24. #19098364C22semidirect
29. #495×C79C22semidirect
30. #494×C79C22semidirect
31. #493×C79C22semidirect
38. C2×D632D4semidirect
182. C4×C79#509semidirect
194. #3147D79semidirect
201. C79#33580semidirect
212. #495C2×D79semidirect
213. #494C2×D79semidirect
214. #493C2×D79semidirect
376. C4#19098998semidirect

Maximal subgroups

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

subgroupindex
C2×#190989982
C2×C4×D6322
#237654522
#237654422
#237654322
#237651562
#237651122
C79×#31472
#3358079