Finite Group Databasebeta

← #34270779

Subgroups — #34270779

Order 20224.

770 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773396C212yesmaximal, complemented
2C2×#19098642C211yesmaximal, characteristic, complemented
3#23762886C212yesmaximal, complemented
4#23762876C212yesmaximal, complemented
5#23762862C214yesmaximal, complemented
6#23762148C211yesmaximal, characteristic, complemented
7#23762049C212yesmaximal, complemented
8C79×#3147C211yesmaximal, characteristic, complemented
9C22×#11929607—21no—
10C2×C4×D316C2211yescharacteristic
11#19098874—22no—
12#19098840C2214yes—
13#19098755C2214yes—
14#19098696—22no—
15C4×#11929609—24no—
16#19098642—21no—
17#19098642—21no—
18#19098642C2212yes—
19#19098642C2212yes—
20#19098616C2212yes—
21C2×C4⋊(C79⋊C8)C2211yescharacteristic
22C2×(C2×D79)⋊C8—21no—
23#19098316C2212yescomplemented
24#19098314C2212yescomplemented
25#19098309C2214yescomplemented
26#19098307C2214yescomplemented
27C2×#135×C79—21no—
28C2×#135×C79—21no—
29#532×C79C2212yes—
30#495×C79—21no—
31#495×C79—21no—
32#494×C79C2211yescharacteristic, complemented
33#493×C79C2211yescharacteristic, complemented
34#492×C79C2212yescomplemented
35C2×C4×C8×C79C2211yescharacteristic
36C22×D316D411yescharacteristic
37C22×C4×D79—21no—
38C2×#11929609—24no—
39C2×#11929607—22no—
40C2×#11929607—22no—
41C2×#11929607—44no—
42C22×C79⋊C8—21no—
43C4×D316—21no—
44C4×D316—21no—
45C4×D316D412yes—
46C4×D316C2312yes—
47C2×C4⋊Dic79D411yescharacteristic
48C2×(C2×D79)⋊C4—21no—
49#14579094—24no—
50C4⋊(C79⋊C8)—21no—
51C4⋊(C79⋊C8)C2312yes—
52C4⋊(C79⋊C8)C2314yes—
53#14579082—24no—
54#14579081—24no—
55C22⋊(C79⋊C8)—22no—
56(C2×D79)⋊C8—22no—
57(C2×D79)⋊C8—22no—
58C4×C79⋊C8—22no—
59C2×#41×C79—22no—
60#135×C79—44no—
61#135×C79—44no—
62C2×Q16×C79—21no—
63C2×Q16×C79—21no—
64C2×SD16×C79—22no—
65C2×SD16×C79—22no—
66C2×D8×C79—21no—
67C2×D8×C79—21no—
68C22×C8×C79D411yescharacteristic
69C22×C8×C79D411yescharacteristic
70C4⋊Q8×C79C2312yes—
71#127×C79C2312yes—
72#124×C79—22no—
73#121×C79—22no—
74#121×C79—22no—
75C4×D4×C79—22no—
76C4×D4×C79—22no—
77C2×C42×C79C2311yescharacteristic
78C4×C8×C79D411yescharacteristic
79C4×C8×C79D411yescharacteristic
80C4×C8×C79C2311yescharacteristic
81C4×C8×C79C2311yescharacteristic
82C23×D79—21no—
83C2×D316—21no—
84C2×D316—42no—
85C2×D316C2×D412yes—
86C22×Dic79—21no—
87C2×C4×D79—22no—
88C2×C4×D79—22no—
89C2×C4×D79—42no—
90C2×C4×D79—42no—
91#11929609—48no—
92#11929607—44no—
93#11929607—44no—
94C2×C79⋊C8—24no—
95C2×C79⋊C8—22no—
96C2×C79⋊C8—42no—
97C4⋊Dic79—21no—
98C4⋊Dic79C2×D412yes—
99(C2×D79)⋊C4—22no—
100(C2×D79)⋊C4—22no—
101#41×C79—44no—
102#41×C79—44no—
103C2×Q8×C79—22no—
104C2×D4×C79—22no—
105C2×D4×C79—22no—
106C2×D4×C79—22no—
107C2×D4×C79—42no—
108C22×C4×C79—42no—
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79#4111yescharacteristic
112Q16×C79—42no—
113Q16×C79—42no—
114SD16×C79—44no—
115SD16×C79—44no—
116D8×C79—42no—
117D8×C79—42no—
118C2×C8×C79—22no—
119C2×C8×C79D814yescomplemented
120C2×C8×C79C2×D411yescharacteristic
121C2×C8×C79C2×D411yescharacteristic
122C2×C8×C79C2×D411yescharacteristic
123C2×C8×C79C2×D411yescharacteristic
124C4⋊C4×C79—22no—
125C4⋊C4×C79—22no—
126C22⋊C4×C79—42no—
127C22⋊C4×C79—42no—
128C42×C79C2×D411yescharacteristic
129C42×C79C2×D411yescharacteristic
130C42×C79#4111yescharacteristic
131C42×C79C2411yescharacteristic
132C23×C79—42no—
133C23×C79#12111yescharacteristic
134C22×D79—22no—
135C22×D79—42no—
136C22×D79—42no—
137C22×D79—42no—
138Q8×C79—42no—
139Q8×C79—42no—
140D4×C79—42no—
141D4×C79—42no—
142D4×C79—44no—
143D4×C79—82no—
144D4×C79—84no—
145C2×C4×C79—22no—
146C2×C4×C79—41no—
147C2×C4×C79—42no—
148C2×C4×C79—84no—
149C2×C4×C79#12111yescharacteristic
150C2×C4×C79#12111yescharacteristic
151C2×C4×C79#12111yescharacteristic
152C2×C4×C79C2×D812yes—
153C2×C4×C79AGL(1,ℤ/8)12yes—
154C2×C4×C79C22×D411yescharacteristic
155C2×C4×C79C22×D411yescharacteristic
156C2×C4×C79C2×#4111yescharacteristic
157C2×Dic79—22no—
158C2×Dic79—42no—
159D316—24no—
160D316—42no—
161C4×D79—42no—
162C4×D79—42no—
163C4×D79—44no—
164C8×C79—21no—
165C8×C79—21no—
166C8×C79C2×D812yes—
167C8×C79C2×D812yes—
168C79⋊C8—42no—
169C79⋊C8—44no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—82no—
173C22×C79—82no—
174C22×C79—82no—
175C22×C79#45912yes—
176C22×C79C2×#12111yescharacteristic
177C2×D79—42no—
178C2×D79—42no—
179C2×D79—42no—
180C2×D79—84no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—21no—
184C4×C79—82no—
185C4×C79#45912yes—
186C4×C79C22×D811yescharacteristic
187C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
188C4×C79#57512yes—
189Dic79—42no—
190#19133—791nomaximal, Sylow
191C2×C79—22no—
192C2×C79—162no—
193C2×C79C2×#45911yescharacteristic
194C2×C79#305411yescharacteristic
195C2×C79#306711yescharacteristic
196D79—82no—
197#3147D7911yescharacteristic, complemented
198#3100—792no—
199C2×#436—791no—
200#1689—792no—
201#1687—792no—
202#1685—794no—
203#1564—791no—
204#1540—792no—
205C79#1913311yescharacteristic, complemented, Sylow
206C2×#135—21no—
207C2×#135—21no—
208C22×M16—1581no—
209#532C2×D7912yes—
210C2×C4×D4—791no—
211#495—21no—
212#495—21no—
213#494C2×D7911yescharacteristic, complemented
214#493C2×D7911yescharacteristic, complemented
215#492C2×D7912yescomplemented
216#480—1582no—
217#474—794no—
218#459—794no—
219#447—1582no—
220C4×D8—1584no—
221#436—792no—
222#436—792no—
223#436—1581no—
224#436—1581no—
225#431—792no—
226C2×C4⋊C8—791no—
227C2×C22⋊C8—1581no—
228C2×C4×C8C2×D7911yescharacteristic
229#335—792no—
230#334—792no—
231#332—794no—
232#331—794no—
233C2×#41—22no—
234C22×D4—791no—
235C23×C4—1581no—
236#135—44no—
237#135—44no—
238C2×Q16—21no—
239C2×Q16—21no—
240C2×SD16—22no—
241C2×SD16—22no—
242C2×D8—21no—
243C2×D8—21no—
244C2×D8—1584no—
245C2×M16—3164no—
246C2×M16—1582no—
247C2×M16—1582no—
248C22×C8—1581no—
249C22×C8C22⋊D7911yescharacteristic
250C22×C8C22⋊D7911yescharacteristic
251C4⋊Q8C22×D7912yes—
252#127C22×D7912yes—
253#124—22no—
254#121—22no—
255#121—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—792no—
259C4×D4—792no—
260C4×D4—1581no—
261C4×D4—1581no—
262C2×C4⋊C4—791no—
263C2×C22⋊C4—1581no—
264C2×C42C22×D7911yescharacteristic
265#107—1584no—
266C4⋊C8—792no—
267C4⋊C8—794no—
268C4⋊C8—1581no—
269#102—1584no—
270#102—1584no—
271C22⋊C8—1582no—
272C22⋊C8—1582no—
273C22⋊C8—1582no—
274C4×C8—1582no—
275C4×C8C22⋊D7911yescharacteristic
276C4×C8C22⋊D7911yescharacteristic
277C4×C8C22×D7911yescharacteristic
278C4×C8C22×D7911yescharacteristic
279C24—1581no—
280#41—44no—
281#41—44no—
282C2×Q8—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—42no—
287C2×D4—3162no—
288C2×D4—792no—
289C2×D4—1581no—
290C22×C4—42no—
291C22×C4—3162no—
292C22×C4—3162no—
293C22×C4—1581no—
294C22×C4—1582no—
295C22×C4—1582no—
296C22×C4D4×D7911yescharacteristic
297C22×C4C2×C22⋊D7911yescharacteristic
298C22×C4#1192962311yescharacteristic
299Q16—42no—
300Q16—42no—
301SD16—44no—
302SD16—44no—
303D8—42no—
304D8—42no—
305D8—3168no—
306M16—3164no—
307M16—3164no—
308C2×C8—22no—
309C2×C8—3162no—
310C2×C8—1584no—
311C2×C8—1582no—
312C2×C8#1192960914yescomplemented
313C2×C8C2×C22⋊D7911yescharacteristic
314C2×C8C2×C22⋊D7911yescharacteristic
315C2×C8C2×C22⋊D7911yescharacteristic
316C2×C8C2×C22⋊D7911yescharacteristic
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—792no—
320C4⋊C4—1581no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—1582no—
324C22⋊C4—1582no—
325C42D4×D7911yescharacteristic
326C42C2×C22⋊D7911yescharacteristic
327C42#1192962311yescharacteristic
328C42C23×D7911yescharacteristic
329C23—42no—
330C23—3162no—
331C23—3162no—
332C23—3162no—
333C23—1582no—
334C23#1457913211yescharacteristic
335Q8—42no—
336Q8—42no—
337D4—42no—
338D4—42no—
339D4—44no—
340D4—82no—
341D4—84no—
342D4—3162no—
343D4—1584no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—84no—
348C2×C4—3162no—
349C2×C4—3164no—
350C2×C4—3162no—
351C2×C4—3162no—
352C2×C4—1582no—
353C2×C4#1457913211yescharacteristic
354C2×C4#1457913211yescharacteristic
355C2×C4#1457913211yescharacteristic
356C2×C4C2×#1192960912yes—
357C2×C4#1457918712yes—
358C2×C4C2×D4×D7911yescharacteristic
359C2×C4C22×C22⋊D7911yescharacteristic
360C2×C4C2×#1192962311yescharacteristic
361C8—21no—
362C8—21no—
363C8—3162no—
364C8—3164no—
365C8C2×#1192960912yes—
366C8C2×#1192960912yes—
367C22—21no—
368C22—21no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—3162no—
373C22—3162no—
374C22—3162no—
375C22—6324no—
376C22#1909875512yes—
377C22C2×#1457913211yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—3162no—
383C4#1909875512yes—
384C4C22×#1192960911yescharacteristic
385C4C2×#1457918711yescharacteristic
386C4#1909936012yes—
387C2—22no—
388C2—162no—
389C2—6322no—
390C2C2×#1909875511yescharacteristic
391C2#2377300511yescharacteristic
392C2#2377311611yescharacteristic
subgroupG/H
centerC2×C4#14579132
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#3147C2
socleC22×C79C2×#121
hypercenter#3147D79
hypocenterC79#19133
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #23773396C2semidirect
2. C2×#19098642C2semidirect
3. #23762886C2semidirect
4. #23762876C2semidirect
5. #23762862C2semidirect
6. #23762148C2semidirect
7. #23762049C2semidirect
8. C79×#3147C2semidirect
23. #19098316C22semidirect
24. #19098314C22semidirect
25. #19098309C22semidirect
26. #19098307C22semidirect
32. #494×C79C22semidirect
33. #493×C79C22semidirect
34. #492×C79C22semidirect
119. C2×C8×C79D8semidirect
197. #3147D79semidirect
205. C79#19133semidirect
213. #494C2×D79semidirect
214. #493C2×D79semidirect
215. #492C2×D79semidirect
312. C2×C8#11929609semidirect

Maximal subgroups

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

subgroupindex
#237733962
C2×#190986422
#237628862
#237628762
#237628622
#237621482
#237620492
C79×#31472
#1913379