Finite Group Databasebeta

← #34205139

Subgroups — #34205139

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772301C211yesmaximal, characteristic
2#23765029C211yesmaximal, characteristic, complemented
3C2×#19098347C211yesmaximal, characteristic, complemented
4#23765015C211yesmaximal, characteristic, complemented
5#23761490C211yesmaximal, characteristic, complemented
6#23761485C211yesmaximal, characteristic, complemented
7C79×#3407C211yesmaximal, characteristic, complemented
8#19099361C411yescharacteristic
9C2×#14579167C2211yescharacteristic
10C2×#14579101C2211yescharacteristic
11C22×C79⋊C16C2211yescharacteristic
12#19098683—22no—
13#19098678—22no—
14#19098611C411yescharacteristic
15#19098580C411yescharacteristic
16#19098579C411yescharacteristic
17#19098518C2211yescharacteristic
18#19098348—22no—
19#19098347—22no—
20(C2×D79)⋊C16—22no—
21C2×#131×C79C2211yescharacteristic
22#504×C79—22no—
23#504×C79—22no—
24C2×M32×C79—22no—
25C2×M32×C79C2211yescharacteristic
26C2×M32×C79C2211yescharacteristic
27C2×#11929624C2×C411yescharacteristic
28#14579192—21no—
29#14579191C812yes—
30#14579187—21no—
31#14579186C812yes—
32#14579181—22no—
33#14579167—21no—
34#14579167—21no—
35#14579167—21no—
36#14579167—21no—
37C2×#11929608C2×C411yescharacteristic
38C2×#11929608C2×C411yescharacteristic
39C2×#11929607—21no—
40C2×C8×D79—21no—
41#14579120C2×C411yescharacteristic
42#14579101—21no—
43#14579101—21no—
44C2×C79⋊C16—21no—
45C2×C79⋊C16—21no—
46C2×C79⋊C16—21no—
47C2×C79⋊C16—21no—
48#14579096—22no—
49#14579088—22no—
50#14579087—22no—
51#14579071—21no—
52C8×Dic79—21no—
53C2×#41×C79C2×C411yescharacteristic
54#131×C79—21no—
55#131×C79—21no—
56#131×C79D412yes—
57#131×C79D412yes—
58C2×M16×C79—21no—
59C2×M16×C79C2×C411yescharacteristic
60C22×C8×C79—21no—
61C22×C8×C79C2311yescharacteristic
62M32×C79—24no—
63M32×C79—21no—
64M32×C79—21no—
65M32×C79—21no—
66M32×C79D412yes—
67C2×C16×C79—24no—
68C2×C16×C79—21no—
69C2×C16×C79D412yes—
70#11929624—21no—
71#11929624—21no—
72#11929624—21no—
73#11929624—21no—
74C2×Dic158C2×C811yescharacteristic
75C2×D316C2×C811yescharacteristic
76C2×C22⋊D79—21no—
77C2×C4×D79—21no—
78#11929614—22no—
79#11929612—22no—
80#11929611—22no—
81#11929609—22no—
82#11929608—21no—
83#11929608—21no—
84#11929608—21no—
85#11929608C2×C812yes—
86#11929607—41no—
87#11929607—41no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C8×D79—41no—
91C8×D79—41no—
92C4⋊Dic79—21no—
93C22⋊Dic79—21no—
94C4×Dic79—21no—
95C79⋊C16—41no—
96C79⋊C16—41no—
97#41×C79—21no—
98#41×C79—21no—
99#41×C79C22⋊C412yes—
100#41×C79C2×C812yes—
101C2×Q8×C79M1611yescharacteristic
102C2×D4×C79—21no—
103C2×D4×C79M1611yescharacteristic
104C22×C4×C79—21no—
105C22×C4×C79C22×C411yescharacteristic
106M16×C79—22no—
107M16×C79—22no—
108M16×C79—21no—
109M16×C79C22⋊C412yes—
110C2×C8×C79—22no—
111C2×C8×C79—22no—
112C2×C8×C79—21no—
113C2×C8×C79C2×D411yescharacteristic
114C2×C8×C79C2×D411yescharacteristic
115C2×C8×C79C2×D411yescharacteristic
116C2×C8×C79C2×D411yescharacteristic
117C2×C8×C79#4111yescharacteristic
118C2×C8×C79#4111yescharacteristic
119C16×C79—22no—
120C16×C79—22no—
121C23×C79—21no—
122C23×C79#13111yescharacteristic
123C22×D79—21no—
124Q8×C79—21no—
125Q8×C79C22⋊C812yes—
126D4×C79—22no—
127D4×C79—22no—
128D4×C79—21no—
129D4×C79C22⋊C812yes—
130C2×C4×C79—21no—
131C2×C4×C79—22no—
132C2×C4×C79—22no—
133C2×C4×C79C2×C22⋊C411yescharacteristic
134C2×C4×C79C4×D411yescharacteristic
135C2×C4×C79C4×D411yescharacteristic
136C2×C4×C79C22×C811yescharacteristic
137C2×C4×C79C2×M1611yescharacteristic
138C2×C4×C79#13111yescharacteristic
139C22⋊D79—41no—
140C22⋊D79—41no—
141C2×Dic79—21no—
142C2×Dic79—41no—
143D316—21no—
144D316—21no—
145C4×D79—41no—
146C4×D79—41no—
147Dic158—21no—
148Dic158—21no—
149C8×C79—22no—
150C8×C79#12011yescharacteristic
151C8×C79#12111yescharacteristic
152C8×C79#12111yescharacteristic
153C8×C79#12311yescharacteristic
154C79⋊C8—22no—
155C79⋊C8—41no—
156C22×C79—22no—
157C22×C79—41no—
158C22×C79—81no—
159C22×C79#40811yescharacteristic
160C22×C79C8×D411yescharacteristic
161C22×C79#43511yescharacteristic
162C2×D79—42no—
163C2×D79—41no—
164C4×C79—22no—
165C4×C79#38611yescharacteristic
166C4×C79C2×C22⋊C811yescharacteristic
167C4×C79C8×D411yescharacteristic
168C4×C79#43511yescharacteristic
169Dic79—41no—
170Dic79—81no—
171#12979—791nomaximal, Sylow
172C2×C79—21no—
173C2×C79—21no—
174C2×C79—21no—
175C2×C79—42no—
176C2×C79#186911yescharacteristic
177D79—81no—
178#3407D7911yescharacteristic, complemented
179#2955—791no—
180#2117—791no—
181C2×#350—791no—
182#2113—791no—
183#1379—791no—
184#1378—791no—
185C79#1297911yescharacteristic, complemented, Sylow
186#575—791no—
187C2×#131—791no—
188C2×#131C2×D7911yescharacteristic
189#504—22no—
190#504—22no—
191C2×M32—22no—
192C2×M32—791no—
193C2×M32C2×D7911yescharacteristic
194C2×M32C2×D7911yescharacteristic
195C22×C16—791no—
196#444—1582no—
197#443—1582no—
198#430—791no—
199#421—791no—
200#421—791no—
201#405—791no—
202#350—1582no—
203#350—1582no—
204C22⋊C16—1582no—
205C2×#41—791no—
206C2×#41C4×D7911yescharacteristic
207#137—792no—
208#137—1581no—
209AGL(1,ℤ/8)—792no—
210AGL(1,ℤ/8)—1581no—
211#135—1582no—
212#131—21no—
213#131—21no—
214#131—1581no—
215#131—1581no—
216#131—1581no—
217#131—1581no—
218#131D31612yes—
219#131C22⋊D7912yes—
220C2×M16—21no—
221C2×M16—791no—
222C2×M16—791no—
223C2×M16—1581no—
224C2×M16C4×D7911yescharacteristic
225C22×C8—21no—
226C22×C8—1581no—
227C22×C8C22×D7911yescharacteristic
228#117—791no—
229M32—24no—
230M32—21no—
231M32—21no—
232M32—21no—
233M32—1581no—
234M32—1581no—
235M32C22⋊D7912yes—
236C2×C16—24no—
237C2×C16—21no—
238C2×C16—1581no—
239C2×C16—1581no—
240C2×C16—1581no—
241C2×C16—1581no—
242C2×C16C22⋊D7912yes—
243#108—1582no—
244#104—1582no—
245#104—1582no—
246#97—1581no—
247C4×C8—1581no—
248#41—21no—
249#41—21no—
250#41—1581no—
251#41—1581no—
252#41—1581no—
253#41—1581no—
254#41(C2×D79)⋊C412yes—
255#41C8×D7912yes—
256C2×Q8—791no—
257C2×Q8#1192960711yescharacteristic
258C2×D4—21no—
259C2×D4—791no—
260C2×D4—1581no—
261C2×D4#1192960711yescharacteristic
262C22×C4—21no—
263C22×C4—1581no—
264C22×C4C2×C4×D7911yescharacteristic
265Q16—1582no—
266SD16—1582no—
267SD16—1582no—
268D8—1582no—
269M16—21no—
270M16—22no—
271M16—22no—
272M16—3161no—
273M16—3161no—
274M16—792no—
275M16—1581no—
276M16—1581no—
277M16—1581no—
278M16(C2×D79)⋊C412yes—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—3161no—
283C2×C8—3161no—
284C2×C8—1581no—
285C2×C8—1581no—
286C2×C8C2×C22⋊D7911yescharacteristic
287C2×C8C2×C22⋊D7911yescharacteristic
288C2×C8C2×C22⋊D7911yescharacteristic
289C2×C8C2×D31611yescharacteristic
290C2×C8#1192962411yescharacteristic
291C2×C8#1192962411yescharacteristic
292C4⋊C4—1581no—
293C22⋊C4—1581no—
294C42—1581no—
295C16—22no—
296C16—22no—
297C16—3161no—
298C16—3161no—
299C23—21no—
300C23—1581no—
301C23#1457916711yescharacteristic
302Q8—21no—
303Q8—1581no—
304Q8—1581no—
305Q8(C2×D79)⋊C812yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4—3161no—
310D4—3161no—
311D4—1581no—
312D4—1581no—
313D4(C2×D79)⋊C812yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—3161no—
318C2×C4—3161no—
319C2×C4—3161no—
320C2×C4—1581no—
321C2×C4C2×(C2×D79)⋊C411yescharacteristic
322C2×C4C4×C22⋊D7911yescharacteristic
323C2×C4C4×C22⋊D7911yescharacteristic
324C2×C4C2×C8×D7911yescharacteristic
325C2×C4C2×#1192960711yescharacteristic
326C2×C4#1457916711yescharacteristic
327C8—22no—
328C8—3161no—
329C8—1582no—
330C8#1457913011yescharacteristic
331C8#1457913511yescharacteristic
332C8#1457913511yescharacteristic
333C8#1457914411yescharacteristic
334C22—22no—
335C22—41no—
336C22—81no—
337C22—3162no—
338C22—3161no—
339C22#1909852911yescharacteristic
340C22C8×C22⋊D7911yescharacteristic
341C22#1909863411yescharacteristic
342C4—22no—
343C4—3161no—
344C4—6321no—
345C4#1909844211yescharacteristic
346C4C2×(C2×D79)⋊C811yescharacteristic
347C4C8×C22⋊D7911yescharacteristic
348C4#1909863411yescharacteristic
349C2—21no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—6321no—
354C2#2376369211yescharacteristic
subgroupG/H
centerC8#14579130
commutatorC2×C4×C79C22×C8
FrattiniC22×C8C22×D79
FittingC79×#3407C2
socleC2×C79#1869
prime radical#19099361C4
cosocleC22×C8×C79C23
hypercenter#3407D79
hypocenterC79#12979
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C8 ◁ C22×C8 ◁ #3407

Complements

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

NH ≅ G/Ntype
2. #23765029C2semidirect
3. C2×#19098347C2semidirect
4. #23765015C2semidirect
5. #23761490C2semidirect
6. #23761485C2semidirect
7. C79×#3407C2semidirect
178. #3407D79semidirect
185. C79#12979semidirect

Maximal subgroups

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

subgroupindex
#237723012
#237650292
C2×#190983472
#237650152
#237614902
#237614852
C79×#34072
#1297979