Finite Group Databasebeta

← #34182592

Subgroups — #34182592

Order 20224.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772066C211yesmaximal, characteristic
2#23763420C211yesmaximal, characteristic
3#23762175C211yesmaximal, characteristic, complemented
4#23762156C211yesmaximal, characteristic, complemented
5#23761302C211yesmaximal, characteristic, complemented
6#23761240C211yesmaximal, characteristic, complemented
7C79×#1476C211yesmaximal, characteristic
8#19098624—22no—
9#19098620—21no—
10#19098616—21no—
11#19098587C2211yescharacteristic
12#19098587C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579088—21no—
15#19098530—21no—
16#19098519—21no—
17C4×C4⋊Dic79C2211yescharacteristic
18#19098322—21no—
19#19098316—21no—
20#19098316—21no—
21#19098314—22no—
22C4×#41×C79C2211yescharacteristic
23#431×C79C2211yescharacteristic
24C2×C4×C8×C79C2211yescharacteristic
25Q8⋊C8×C79C411yescharacteristic, complemented
26Q8⋊C8×C79C411yescharacteristic, complemented
27D4⋊C8×C79C411yescharacteristic, complemented
28D4⋊C8×C79C411yescharacteristic, complemented
29C2×#11929608—21no—
30C2×#11929608—21no—
31C22×C79⋊C8—21no—
32C2×C4⋊Dic79—21no—
33C2×C4×Dic79—21no—
34C4⋊(C79⋊C8)—21no—
35C4⋊(C79⋊C8)—21no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)D412yes—
39C4⋊(C79⋊C8)D411yescharacteristic
40C4⋊(C79⋊C8)D411yescharacteristic
41#14579088—42no—
42C22⋊(C79⋊C8)—22no—
43C22⋊(C79⋊C8)—22no—
44#14579072—22no—
45C4×C79⋊C8—22no—
46C79⋊#95—21no—
47C2×#41×C79D411yescharacteristic
48C22×C8×C79—21no—
49C4×Q8×C79C2×C411yescharacteristic
50C4×Q8×C79Q811yescharacteristic
51C4×D4×C79—21no—
52C4×D4×C79—21no—
53C4×D4×C79C2×C411yescharacteristic
54C4×D4×C79Q811yescharacteristic
55#117×C79—21no—
56#117×C79D411yescharacteristic
57C2×C42×C79—21no—
58C2×C42×C79C2311yescharacteristic
59C4⋊C8×C79C2×C411yescharacteristic
60C4⋊C8×C79C2×C411yescharacteristic
61C22⋊C8×C79—21no—
62C4×C8×C79—21no—
63C4×C8×C79—21no—
64C4×C8×C79C2×C411yescharacteristic
65C4×C8×C79C2×C411yescharacteristic
66C22×Dic79—41no—
67#11929608—41no—
68#11929608—41no—
69#11929608—41no—
70#11929608—41no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—22no—
73C2×C79⋊C8—21no—
74C2×C79⋊C8—41no—
75C2×C79⋊C8—41no—
76C2×C79⋊C8—41no—
77C4⋊Dic79—22no—
78C4⋊Dic79—41no—
79C4×Dic79—41no—
80C4×Dic79—41no—
81#41×C79—21no—
82#41×C79—21no—
83#41×C79—21no—
84#41×C79—21no—
85C2×Q8×C79C4⋊C411yescharacteristic
86C2×D4×C79—21no—
87C2×D4×C79C4⋊C411yescharacteristic
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79#4111yescharacteristic
93C22×C4×C79#4111yescharacteristic
94C2×C8×C79—21no—
95C2×C8×C79—21no—
96C2×C8×C79—21no—
97C2×C8×C79—21no—
98C2×C8×C79—42no—
99C4⋊C4×C79—21no—
100C4⋊C4×C79—21no—
101C4⋊C4×C79C4⋊C411yescharacteristic
102C4⋊C4×C79C4⋊C411yescharacteristic
103C22⋊C4×C79—21no—
104C22⋊C4×C79—21no—
105C22⋊C4×C79—21no—
106C42×C79—21no—
107C42×C79—21no—
108C42×C79—21no—
109C42×C79C22×C411yescharacteristic
110C42×C79C2×D411yescharacteristic
111C42×C79C2×D411yescharacteristic
112C42×C79C2×Q811yescharacteristic
113C23×C79—21no—
114C23×C79#12311yescharacteristic
115Q8×C79—21no—
116Q8×C79—21no—
117D4×C79—21no—
118D4×C79—21no—
119D4×C79—41no—
120D4×C79—41no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—81no—
132C2×C4×C79C2×C4⋊C411yescharacteristic
133C2×C4×C79C4×D411yescharacteristic
134C2×C4×C79C4×D411yescharacteristic
135C2×C4×C79#12011yescharacteristic
136C2×C4×C79C22⋊Q811yescharacteristic
137C2×C4×C79C22⋊Q811yescharacteristic
138C2×C4×C79#13511yescharacteristic
139C2×C4×C79#13511yescharacteristic
140C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C79#13711yescharacteristic
142C2×Dic79—42no—
143C2×Dic79—81no—
144C2×Dic79—81no—
145C8×C79—22no—
146C8×C79—41no—
147C8×C79—41no—
148C8×C79—41no—
149C79⋊C8—41no—
150C79⋊C8—41no—
151C79⋊C8—42no—
152C79⋊C8—81no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—41no—
158C22×C79#38511yescharacteristic
159C22×C79#48211yescharacteristic
160C22×C79#48511yescharacteristic
161C4×C79—21no—
162C4×C79—21no—
163C4×C79—21no—
164C4×C79—41no—
165C4×C79—41no—
166C4×C79—41no—
167C4×C79#42711yescharacteristic
168C4×C79#42811yescharacteristic
169C4×C79#44311yescharacteristic
170C4×C79#44411yescharacteristic
171C4×C79#44911yescharacteristic
172C4×C79#45211yescharacteristic
173Dic79—82no—
174#8905—791nomaximal, Sylow
175C2×C79—21no—
176C2×C79—21no—
177C2×C79—81no—
178C2×C79#185411yescharacteristic
179C2×C79#186111yescharacteristic
180C2×C79#186611yescharacteristic
181#2925—791no—
182#1806—791no—
183#1571—791no—
184#1566—791no—
185#1476D7911yescharacteristic
186#1303—791no—
187#1279—791no—
188C79#890511yescharacteristic, complemented, Sylow
189C4×#41C2×D7911yescharacteristic
190#433—1582no—
191#432—1581no—
192#431—1581no—
193#431C2×D7911yescharacteristic
194#423—791no—
195#423—791no—
196C2×C4⋊C8—791no—
197C2×#104—1581no—
198#408—1581no—
199#405—1581no—
200C2×C4×C8C2×D7911yescharacteristic
201C4×C4⋊C4—791no—
202#339—1581no—
203#335—1581no—
204#335—1581no—
205#334—1582no—
206Q8⋊C8Dic7911yescharacteristic, complemented
207Q8⋊C8Dic7911yescharacteristic, complemented
208D4⋊C8Dic7911yescharacteristic, complemented
209D4⋊C8Dic7911yescharacteristic, complemented
210C2×#41D31611yescharacteristic
211C2×M16—1581no—
212C2×M16—1581no—
213C22×C8—21no—
214C22×C8—1581no—
215C4×Q8Dic15811yescharacteristic
216C4×Q8C2×Dic7911yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4Dic15811yescharacteristic
220C4×D4C2×Dic7911yescharacteristic
221#117—21no—
222#117D31611yescharacteristic
223C2×C4⋊C4—1581no—
224C2×C42—21no—
225C2×C42—1581no—
226C2×C42C22×D7911yescharacteristic
227C4⋊C8—792no—
228C4⋊C8—791no—
229C4⋊C8—791no—
230C4⋊C8—1581no—
231C4⋊C8—1581no—
232C4⋊C8—1581no—
233C4⋊C8—1581no—
234C4⋊C8C2×Dic7911yescharacteristic
235C4⋊C8C2×Dic7911yescharacteristic
236#104—3162no—
237C22⋊C8—21no—
238C22⋊C8—1582no—
239C22⋊C8—1582no—
240#97—1582no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—1582no—
244C4×C8C2×Dic7911yescharacteristic
245C4×C8C2×Dic7911yescharacteristic
246#95—1581no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8C4⋊Dic7911yescharacteristic
252C2×D4—21no—
253C2×D4C4⋊Dic7911yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—3161no—
258C22×C4C2×D31611yescharacteristic
259C22×C4#1192962311yescharacteristic
260C22×C4#1192962311yescharacteristic
261M16—3161no—
262M16—3161no—
263M16—3161no—
264M16—3161no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—3161no—
271C2×C8—3161no—
272C2×C8—3161no—
273C2×C8—1581no—
274C2×C8—1582no—
275C2×C8—1581no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3161no—
279C4⋊C4—1582no—
280C4⋊C4C4⋊Dic7911yescharacteristic
281C4⋊C4C4⋊Dic7911yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—3161no—
289C42—3161no—
290C42C22×Dic7911yescharacteristic
291C42D4×D7911yescharacteristic
292C42D4×D7911yescharacteristic
293C42C2×Dic15811yescharacteristic
294C23—21no—
295C23#1457914511yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—3162no—
314C2×C4—6321no—
315C2×C4—6321no—
316C2×C4C2×C4⋊Dic7911yescharacteristic
317C2×C4D4×Dic7911yescharacteristic
318C2×C4D4×Dic7911yescharacteristic
319C2×C4#1457913111yescharacteristic
320C2×C4#1457913811yescharacteristic
321C2×C4#1457913811yescharacteristic
322C2×C4#1457918311yescharacteristic
323C2×C4#1457918311yescharacteristic
324C2×C4#1457918911yescharacteristic
325C2×C4#1457919411yescharacteristic
326C8—22no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—3161no—
331C8—3161no—
332C8—3162no—
333C8—6321no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1909843811yescharacteristic
340C22#1909888811yescharacteristic
341C22#1909890511yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—6322no—
349C4#1909860111yescharacteristic
350C4#1909860511yescharacteristic
351C4#1909868011yescharacteristic
352C4#1909868511yescharacteristic
353C4#1909871011yescharacteristic
354C4#1909872711yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2#2376360111yescharacteristic
359C2#2376364211yescharacteristic
360C2#2376367711yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1476C2
socleC22×C79#385
prime radicalC2×#41×C79D4
cosocleC2×C42×C79C23
hypercenter#1476D79
hypocenterC79#8905
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
3. #23762175C2semidirect
4. #23762156C2semidirect
5. #23761302C2semidirect
6. #23761240C2semidirect
25. Q8⋊C8×C79C4semidirect
26. Q8⋊C8×C79C4semidirect
27. D4⋊C8×C79C4semidirect
28. D4⋊C8×C79C4semidirect
188. C79#8905semidirect
206. Q8⋊C8Dic79semidirect
207. Q8⋊C8Dic79semidirect
208. D4⋊C8Dic79semidirect
209. D4⋊C8Dic79semidirect

Maximal subgroups

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

subgroupindex
#237720662
#237634202
#237621752
#237621562
#237613022
#237612402
C79×#14762
#890579