Finite Group Databasebeta

← #34182762

Subgroups — #34182762

Order 20224.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772143C211yesmaximal, characteristic
2#23764176C211yesmaximal, characteristic, complemented
3#23762170C211yesmaximal, characteristic, complemented
4#23762050C211yesmaximal, characteristic
5#23761852C211yesmaximal, characteristic
6#23761237C211yesmaximal, characteristic, complemented
7C79×#1277C211yesmaximal, characteristic, complemented
8C2×#14579167—21no—
9#19099113C2211yescharacteristic
10#19098640—21no—
11#19098633—22no—
12#19098628—21no—
13#19098616C2211yescharacteristic
14C2×#14579097—21no—
15#19098587C2211yescharacteristic
16#19098529—21no—
17C4×#11929608C2211yescharacteristic
18#19098328—21no—
19#19098313—21no—
20#19098313—21no—
21#19098310C411yescharacteristic
22#19098310C411yescharacteristic
23#19098306C411yescharacteristic
24#19098306C411yescharacteristic
25#19098292—22no—
26C2×C4⋊C8×C79C2211yescharacteristic
27C2×C4⋊C8×C79C2211yescharacteristic
28C2×C4×C8×C79C2211yescharacteristic
29C2×#11929624—21no—
30#14579167—42no—
31#14579167—42no—
32C2×#11929608—21no—
33C2×#11929608D411yescharacteristic
34C2×#11929608D411yescharacteristic
35C2×#11929607—22no—
36C2×C8×D79—22no—
37#14579161C2×C411yescharacteristic
38#14579158C2×C411yescharacteristic
39#14579149—21no—
40#14579135—21no—
41#14579135Q812yes—
42C4×D316—21no—
43C4×D316D412yes—
44#14579097—42no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)C2×C411yescharacteristic
48C4⋊(C79⋊C8)C2×C411yescharacteristic
49C22⋊(C79⋊C8)—21no—
50(C2×D79)⋊C8—22no—
51(C2×D79)⋊C8—22no—
52#14579072—21no—
53C4×C79⋊C8—21no—
54C4×C79⋊C8—21no—
55C22×C8×C79—21no—
56C22×C8×C79—21no—
57C22×C8×C79—21no—
58C2×C42×C79C2311yescharacteristic
59C4⋊C8×C79—21no—
60C4⋊C8×C79—21no—
61C4⋊C8×C79C2×C411yescharacteristic
62C4⋊C8×C79C2×C411yescharacteristic
63C4⋊C8×C79D411yescharacteristic
64C4⋊C8×C79D411yescharacteristic
65C4×C8×C79—21no—
66C4×C8×C79—21no—
67#11929624—41no—
68#11929624—41no—
69#11929624—41no—
70#11929624—41no—
71C2×Dic158—21no—
72C2×D316—21no—
73C2×D316—21no—
74C2×D316C4⋊C412yes—
75C2×C22⋊D79—22no—
76C2×C4×D79—22no—
77#11929608—21no—
78#11929608—21no—
79#11929608—21no—
80#11929608—41no—
81#11929608—41no—
82#11929608SD1612yes—
83#11929607—44no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—41no—
89C8×D79—44no—
90C4⋊Dic79—21no—
91C4⋊Dic79C4⋊C412yes—
92(C2×D79)⋊C4—22no—
93(C2×D79)⋊C4—22no—
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79C2×Q811yescharacteristic
96C22×C4×C79#4111yescharacteristic
97C2×C8×C79—22no—
98C2×C8×C79—21no—
99C2×C8×C79—21no—
100C2×C8×C79—21no—
101C2×C8×C79—21no—
102C2×C8×C79—41no—
103C2×C8×C79—41no—
104C2×C8×C79—41no—
105C2×C8×C79—41no—
106C2×C8×C79—41no—
107C2×C8×C79—41no—
108C42×C79C22×C411yescharacteristic
109C42×C79C2×D411yescharacteristic
110C42×C79C2×D411yescharacteristic
111C42×C79#4111yescharacteristic
112C23×C79C22⋊Q811yescharacteristic
113C22×D79—22no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—41no—
117C2×C4×C79C2×C4⋊C411yescharacteristic
118C2×C4×C79C4×D411yescharacteristic
119C2×C4×C79C4×D411yescharacteristic
120C2×C4×C79#12011yescharacteristic
121C2×C4×C79C22⋊Q811yescharacteristic
122C2×C4×C79#12311yescharacteristic
123C2×C4×C79C2×SD1611yescharacteristic
124C2×C4×C79#13511yescharacteristic
125C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
127C22⋊D79—42no—
128C22⋊D79—42no—
129C2×Dic79—22no—
130D316—42no—
131D316—41no—
132D316—41no—
133D316—41no—
134D316—41no—
135C4×D79—42no—
136C4×D79—42no—
137Dic158—41no—
138Dic158—41no—
139C8×C79—42no—
140C8×C79—41no—
141C8×C79—41no—
142C8×C79—41no—
143C8×C79—41no—
144C79⋊C8—22no—
145C79⋊C8—41no—
146C79⋊C8—41no—
147C79⋊C8—41no—
148C79⋊C8—81no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79#38511yescharacteristic
152C22×C79#47611yescharacteristic
153C22×C79#47811yescharacteristic
154C2×D79—42no—
155C2×D79—42no—
156C2×D79—42no—
157C4×C79—21no—
158C4×C79—21no—
159C4×C79—21no—
160C4×C79C4×SD1611yescharacteristic
161C4×C79#44211yescharacteristic
162C4×C79#44311yescharacteristic
163C4×C79#44411yescharacteristic
164C4×C79#44911yescharacteristic
165C4×C79#45011yescharacteristic
166Dic79—42no—
167#8932—791nomaximal, Sylow
168C2×C79—21no—
169C2×C79—21no—
170C2×C79#185811yescharacteristic
171C2×C79#186411yescharacteristic
172C2×C79#186611yescharacteristic
173D79—82no—
174#2935—791no—
175#1950—791no—
176#1570—791no—
177#1540—791no—
178#1492—791no—
179#1278—791no—
180#1277D7911yescharacteristic, complemented
181C79#893211yescharacteristic, complemented, Sylow
182C2×#131—1581no—
183#532—791no—
184#436—1581no—
185#435—1582no—
186C8×D4—1581no—
187#431—791no—
188C2×#108—1581no—
189#423—791no—
190C2×C4⋊C8C2×D7911yescharacteristic
191C2×C4⋊C8C2×D7911yescharacteristic
192#408—1581no—
193C4×M16—791no—
194C2×C4×C8C2×D7911yescharacteristic
195#341—1581no—
196#334—1581no—
197#334—1581no—
198#332—791no—
199#332—791no—
200#331—791no—
201#331—791no—
202D4⋊C8—1582no—
203C2×#41—1581no—
204#131—3162no—
205#131—3162no—
206C2×M16—791no—
207C2×M16—791no—
208C2×M16—1582no—
209C2×M16—1581no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—1582no—
214C4⋊Q8—791no—
215#127—791no—
216#124—1581no—
217#121—792no—
218#121—1581no—
219C4×D4—792no—
220C4×D4—1581no—
221C2×C42C22×D7911yescharacteristic
222#108—3162no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—791no—
226C4⋊C8—791no—
227C4⋊C8—1581no—
228C4⋊C8—1581no—
229C4⋊C8C4×D7911yescharacteristic
230C4⋊C8C4×D7911yescharacteristic
231C4⋊C8D31611yescharacteristic
232C4⋊C8D31611yescharacteristic
233C22⋊C8—1582no—
234C22⋊C8—1582no—
235C22⋊C8—1581no—
236#97—1581no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—1581no—
240C4×C8—1581no—
241#41—3161no—
242#41—3161no—
243#41—3161no—
244#41—3161no—
245C2×Q8—1581no—
246C2×D4—792no—
247C2×D4—1582no—
248C2×D4—1581no—
249C2×D4—1581no—
250C22×C4—1582no—
251C22×C4D4×D7911yescharacteristic
252C22×C4Q8×D7911yescharacteristic
253C22×C4#1192962411yescharacteristic
254M16—3164no—
255M16—3161no—
256M16—3161no—
257M16—792no—
258M16—1581no—
259M16—1581no—
260M16—1581no—
261C2×C8—22no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—3164no—
273C2×C8—3161no—
274C2×C8—1581no—
275C2×C8—1581no—
276C2×C8—1581no—
277C2×C8—1581no—
278C4⋊C4—792no—
279C4⋊C4—1581no—
280C22⋊C4—1582no—
281C22⋊C4—1582no—
282C42C2×C4×D7911yescharacteristic
283C42D4×D7911yescharacteristic
284C42C2×D31611yescharacteristic
285C42#1192962311yescharacteristic
286C23—1582no—
287C23#1457913911yescharacteristic
288Q8—3161no—
289Q8—3161no—
290D4—3162no—
291D4—3162no—
292D4—3162no—
293D4—3161no—
294D4—3161no—
295D4—3161no—
296D4—3161no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—3162no—
301C2×C4—3162no—
302C2×C4—1582no—
303C2×C4C4⋊C4×D7911yescharacteristic
304C2×C4(C22⋊D79)⋊C411yescharacteristic
305C2×C4C4×D31611yescharacteristic
306C2×C4#1457913111yescharacteristic
307C2×C4#1457914111yescharacteristic
308C2×C4#1457914311yescharacteristic
309C2×C4SD16×D7911yescharacteristic
310C2×C4#1457918411yescharacteristic
311C2×C4#1457918511yescharacteristic
312C2×C4#1457918811yescharacteristic
313C8—42no—
314C8—41no—
315C8—41no—
316C8—41no—
317C8—41no—
318C8—3161no—
319C8—3161no—
320C8—3161no—
321C8—6321no—
322C8—1582no—
323C22—21no—
324C22—21no—
325C22—3162no—
326C22—3162no—
327C22—3162no—
328C22#1909843711yescharacteristic
329C22#1909885211yescharacteristic
330C22#1909886611yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—3162no—
335C4#1909864811yescharacteristic
336C4#1909867211yescharacteristic
337C4#1909868111yescharacteristic
338C4#1909868611yescharacteristic
339C4#1909871111yescharacteristic
340C4#1909871511yescharacteristic
341C2—21no—
342C2—21no—
343C2—6322no—
344C2#2376362111yescharacteristic
345C2#2376366211yescharacteristic
346C2#2376367811yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1277C2
socleC22×C79#385
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#1277D79
hypocenterC79#8932
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. #23764176C2semidirect
3. #23762170C2semidirect
6. #23761237C2semidirect
7. C79×#1277C2semidirect
180. #1277D79semidirect
181. C79#8932semidirect

Maximal subgroups

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

subgroupindex
#237721432
#237641762
#237621702
#237620502
#237618522
#237612372
C79×#12772
#893279