Finite Group Databasebeta

← #34205185

Subgroups — #34205185

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772301C211yesmaximal, characteristic
2#23765028C211yesmaximal, characteristic, complemented
3C2×#19098346C211yesmaximal, characteristic, complemented
4#23765020C211yesmaximal, characteristic, complemented
5#23761488C211yesmaximal, characteristic, complemented
6#23761487C211yesmaximal, characteristic, complemented
7C2×#504×C79C211yesmaximal, characteristic, complemented
8#19099361C411yescharacteristic
9C2×#14579167C2211yescharacteristic
10C2×#14579101C2211yescharacteristic
11C2×#14579101C2211yescharacteristic
12#19098683—22no—
13#19098678—22no—
14#19098611C411yescharacteristic
15#19098580C411yescharacteristic
16#19098579C411yescharacteristic
17#19098518C2211yescharacteristic
18#19098347—22no—
19#19098346—21no—
20#19098346—21no—
21#19098344—22no—
22C2×#131×C79C2211yescharacteristic
23#504×C79—22no—
24#504×C79—22no—
25C2×M32×C79—21no—
26C2×M32×C79C2211yescharacteristic
27C22×C16×C79—21no—
28C22×C16×C79C2211yescharacteristic
29C2×#11929624C2×C411yescharacteristic
30#14579192—21no—
31#14579191C812yes—
32#14579187—21no—
33#14579186C812yes—
34#14579181—22no—
35#14579167—21no—
36#14579167—21no—
37#14579167—21no—
38#14579167—21no—
39C2×#11929608C2×C411yescharacteristic
40C2×#11929608C2×C411yescharacteristic
41C2×#11929607—21no—
42C2×C8×D79—21no—
43#14579120C2×C411yescharacteristic
44#14579101—21no—
45#14579101—21no—
46#14579101—21no—
47#14579101—21no—
48C2×C79⋊C16—21no—
49C2×C79⋊C16—21no—
50#14579096—22no—
51#14579088—22no—
52#14579087—22no—
53#14579071—21no—
54C8×Dic79—21no—
55C2×#41×C79C2×C411yescharacteristic
56#131×C79—21no—
57#131×C79—21no—
58#131×C79D412yes—
59#131×C79D412yes—
60C2×M16×C79—21no—
61C2×M16×C79C2×C411yescharacteristic
62C22×C8×C79—21no—
63C22×C8×C79C2311yescharacteristic
64M32×C79—24no—
65M32×C79—21no—
66M32×C79D412yes—
67C2×C16×C79—24no—
68C2×C16×C79—21no—
69C2×C16×C79—21no—
70C2×C16×C79—21no—
71C2×C16×C79D411yescharacteristic
72C2×C16×C79D411yescharacteristic
73#11929624—21no—
74#11929624—21no—
75#11929624—21no—
76#11929624—21no—
77C2×Dic158C2×C811yescharacteristic
78C2×D316C2×C811yescharacteristic
79C2×C22⋊D79—21no—
80C2×C4×D79—21no—
81#11929614—22no—
82#11929612—22no—
83#11929611—22no—
84#11929609—22no—
85#11929608—21no—
86#11929608—21no—
87#11929608—21no—
88#11929608C2×C812yes—
89#11929607—41no—
90#11929607—41no—
91C2×C79⋊C8—21no—
92C2×C79⋊C8—21no—
93C8×D79—41no—
94C8×D79—41no—
95C4⋊Dic79—21no—
96C22⋊Dic79—21no—
97C4×Dic79—21no—
98C79⋊C16—41no—
99C79⋊C16—41no—
100#41×C79—21no—
101#41×C79—21no—
102#41×C79C22⋊C412yes—
103#41×C79C2×C812yes—
104C2×Q8×C79M1611yescharacteristic
105C2×D4×C79—21no—
106C2×D4×C79M1611yescharacteristic
107C22×C4×C79—21no—
108C22×C4×C79C22×C411yescharacteristic
109M16×C79—22no—
110M16×C79—22no—
111M16×C79—21no—
112M16×C79C22⋊C412yes—
113C2×C8×C79—22no—
114C2×C8×C79—22no—
115C2×C8×C79—21no—
116C2×C8×C79C2×D411yescharacteristic
117C2×C8×C79C2×D411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79C2×D411yescharacteristic
120C2×C8×C79#4111yescharacteristic
121C2×C8×C79#4111yescharacteristic
122C16×C79—21no—
123C16×C79—22no—
124C16×C79—21no—
125C23×C79—21no—
126C23×C79#13111yescharacteristic
127C22×D79—21no—
128Q8×C79—21no—
129Q8×C79C22⋊C812yes—
130D4×C79—22no—
131D4×C79—22no—
132D4×C79—21no—
133D4×C79C22⋊C812yes—
134C2×C4×C79—21no—
135C2×C4×C79—22no—
136C2×C4×C79—22no—
137C2×C4×C79C2×C22⋊C411yescharacteristic
138C2×C4×C79C4×D411yescharacteristic
139C2×C4×C79C4×D411yescharacteristic
140C2×C4×C79C22×C811yescharacteristic
141C2×C4×C79C2×M1611yescharacteristic
142C2×C4×C79#13111yescharacteristic
143C22⋊D79—41no—
144C22⋊D79—41no—
145C2×Dic79—21no—
146C2×Dic79—41no—
147D316—21no—
148D316—21no—
149C4×D79—41no—
150C4×D79—41no—
151Dic158—21no—
152Dic158—21no—
153C8×C79—22no—
154C8×C79#12011yescharacteristic
155C8×C79#12111yescharacteristic
156C8×C79#12111yescharacteristic
157C8×C79#12311yescharacteristic
158C79⋊C8—22no—
159C79⋊C8—41no—
160C22×C79—22no—
161C22×C79—41no—
162C22×C79—42no—
163C22×C79#40811yescharacteristic
164C22×C79C8×D411yescharacteristic
165C22×C79#43511yescharacteristic
166C2×D79—41no—
167C2×D79—81no—
168C4×C79—22no—
169C4×C79#38611yescharacteristic
170C4×C79C2×C22⋊C811yescharacteristic
171C4×C79C8×D411yescharacteristic
172C4×C79#43511yescharacteristic
173Dic79—41no—
174Dic79—81no—
175#12987—791nomaximal, Sylow
176C2×C79—21no—
177C2×C79—21no—
178C2×C79—21no—
179C2×C79—42no—
180C2×C79#186911yescharacteristic
181D79—81no—
182C2×#504D7911yescharacteristic, complemented
183#2955—791no—
184#2117—791no—
185C2×#350—791no—
186#2115—791no—
187#1379—791no—
188#1378—791no—
189C79#1298711yescharacteristic, complemented, Sylow
190#575—791no—
191C2×#131—791no—
192C2×#131C2×D7911yescharacteristic
193#504—22no—
194#504—22no—
195C2×M32—21no—
196C2×M32—791no—
197C2×M32—791no—
198C2×M32C2×D7911yescharacteristic
199C22×C16—21no—
200C22×C16C2×D7911yescharacteristic
201#444—1582no—
202#443—1582no—
203#430—791no—
204#421—791no—
205#421—791no—
206#405—791no—
207#350—1582no—
208#350—1581no—
209#350—1581no—
210#349—1582no—
211C2×#41—791no—
212C2×#41C4×D7911yescharacteristic
213#137—792no—
214#137—1581no—
215AGL(1,ℤ/8)—792no—
216AGL(1,ℤ/8)—1581no—
217#135—1582no—
218#131—21no—
219#131—21no—
220#131—1581no—
221#131—1581no—
222#131—1581no—
223#131—1581no—
224#131D31612yes—
225#131C22⋊D7912yes—
226C2×M16—21no—
227C2×M16—791no—
228C2×M16—791no—
229C2×M16—1581no—
230C2×M16C4×D7911yescharacteristic
231C22×C8—21no—
232C22×C8—1581no—
233C22×C8C22×D7911yescharacteristic
234#117—791no—
235M32—24no—
236M32—21no—
237M32—1581no—
238M32—1581no—
239M32—1581no—
240M32—1581no—
241M32C22⋊D7912yes—
242C2×C16—24no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—1581no—
247C2×C16—1581no—
248C2×C16C22⋊D7911yescharacteristic
249C2×C16C22⋊D7911yescharacteristic
250#108—1582no—
251#104—1582no—
252#104—1582no—
253#97—1581no—
254C4×C8—1581no—
255#41—21no—
256#41—21no—
257#41—1581no—
258#41—1581no—
259#41—1581no—
260#41—1581no—
261#41(C2×D79)⋊C412yes—
262#41C8×D7912yes—
263C2×Q8—791no—
264C2×Q8#1192960711yescharacteristic
265C2×D4—21no—
266C2×D4—791no—
267C2×D4—1581no—
268C2×D4#1192960711yescharacteristic
269C22×C4—21no—
270C22×C4—1581no—
271C22×C4C2×C4×D7911yescharacteristic
272Q16—1582no—
273SD16—1582no—
274SD16—1582no—
275D8—1582no—
276M16—21no—
277M16—22no—
278M16—22no—
279M16—3161no—
280M16—3161no—
281M16—792no—
282M16—1581no—
283M16—1581no—
284M16—1581no—
285M16(C2×D79)⋊C412yes—
286C2×C8—21no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—3161no—
290C2×C8—3161no—
291C2×C8—1581no—
292C2×C8—1581no—
293C2×C8C2×C22⋊D7911yescharacteristic
294C2×C8C2×C22⋊D7911yescharacteristic
295C2×C8C2×C22⋊D7911yescharacteristic
296C2×C8C2×D31611yescharacteristic
297C2×C8#1192962411yescharacteristic
298C2×C8#1192962411yescharacteristic
299C4⋊C4—1581no—
300C22⋊C4—1581no—
301C42—1581no—
302C16—21no—
303C16—22no—
304C16—21no—
305C16—3161no—
306C16—3161no—
307C23—21no—
308C23—1581no—
309C23#1457916711yescharacteristic
310Q8—21no—
311Q8—1581no—
312Q8—1581no—
313Q8(C2×D79)⋊C812yes—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—3161no—
318D4—3161no—
319D4—1581no—
320D4—1581no—
321D4(C2×D79)⋊C812yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4—3161no—
328C2×C4—1581no—
329C2×C4C2×(C2×D79)⋊C411yescharacteristic
330C2×C4C4×C22⋊D7911yescharacteristic
331C2×C4C4×C22⋊D7911yescharacteristic
332C2×C4C2×C8×D7911yescharacteristic
333C2×C4C2×#1192960711yescharacteristic
334C2×C4#1457916711yescharacteristic
335C8—22no—
336C8—3161no—
337C8—1582no—
338C8#1457913011yescharacteristic
339C8#1457913511yescharacteristic
340C8#1457913511yescharacteristic
341C8#1457914411yescharacteristic
342C22—22no—
343C22—41no—
344C22—42no—
345C22—3161no—
346C22—6321no—
347C22#1909852911yescharacteristic
348C22C8×C22⋊D7911yescharacteristic
349C22#1909863411yescharacteristic
350C4—22no—
351C4—3161no—
352C4—6321no—
353C4#1909844211yescharacteristic
354C4C2×(C2×D79)⋊C811yescharacteristic
355C4C8×C22⋊D7911yescharacteristic
356C4#1909863411yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—6321no—
362C2#2376369211yescharacteristic
subgroupG/H
centerC8#14579135
commutatorC2×C4×C79C22×C8
FrattiniC22×C8C22×D79
FittingC2×#504×C79C2
socleC2×C79#1869
prime radical#19099361C4
cosocleC22×C8×C79C23
hypercenterC2×#504D79
hypocenterC79#12987
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
2. #23765028C2semidirect
3. C2×#19098346C2semidirect
4. #23765020C2semidirect
5. #23761488C2semidirect
6. #23761487C2semidirect
7. C2×#504×C79C2semidirect
182. C2×#504D79semidirect
189. C79#12987semidirect

Maximal subgroups

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

subgroupindex
#237723012
#237650282
C2×#190983462
#237650202
#237614882
#237614872
C2×#504×C792
#1298779