Finite Group Databasebeta

← #34306586

Subgroups — #34306586

Order 20224.

914 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098671C211yesmaximal, characteristic
2C2×#19098669C211yesmaximal, characteristic
3C2×#19098657C211yesmaximal, characteristic
4#23766314C212yesmaximal
5#23764139C211yesmaximal, characteristic
6#23763669C212yesmaximal
7#23763667C212yesmaximal
8#23763555C211yesmaximal, characteristic
9#23763404C212yesmaximal
10#23763268C211yesmaximal, characteristic
11C79×#1943C211yesmaximal, characteristic
12C22×Dic316C2211yescharacteristic
13C2×C4×Dic158C2212yes—
14C2×#14579126C2212yes—
15C2×#14579126C2212yes—
16#19098671—24no—
17#19098669—24no—
18#19098657C418yescomplemented
19C2×#14579093C2211yescharacteristic
20C2×#14579092C2212yes—
21C2×Dic79⋊C8C2211yescharacteristic
22C2×#14579086C2212yes—
23C2×#14579084C2212yes—
24C2×#14579084C2212yes—
25C2×#14579071C2211yescharacteristic
26C2×C8×Dic79C2211yescharacteristic
27#19098462C2212yes—
28#19098460—22no—
29#19098458—22no—
30#19098432—22no—
31#19098432C2212yes—
32C4×Dic79⋊C4C2211yescharacteristic
33#19098414—22no—
34#19098325C2212yes—
35#19098318C2211yescharacteristic
36#19098317C2211yescharacteristic
37C2×#107×C79C2211yescharacteristic
38C2×#106×C79C2211yescharacteristic
39C2×#97×C79C2211yescharacteristic
40#389×C79C2212yes—
41#340×C79C2212yes—
42C22×Dic158C2312yes—
43C2×Dic316C2×C414yes—
44C2×Dic316D414yes—
45C2×Dic316Q814yes—
46C22×C79⋊C8C2311yescharacteristic
47C22×C79⋊C8C2311yescharacteristic
48C4×Dic158—24no—
49C4×Dic158—42no—
50#14579126—24no—
51#14579126—24no—
52#14579126—42no—
53#14579126C2×C418yes—
54C2×C4⋊Dic79—22no—
55C2×C4⋊Dic79C2312yes—
56C2×Dic79⋊C4—22no—
57C2×Dic79⋊C4—22no—
58C2×Dic79⋊C4—22no—
59C2×Dic79⋊C4—22no—
60C2×Dic79⋊C4—22no—
61C2×Dic79⋊C4D412yes—
62C2×C4×Dic79—22no—
63C2×C4×Dic79—22no—
64C2×C4×Dic79—22no—
65C2×C4×Dic79C2311yescharacteristic
66C2×C4×Dic79C2311yescharacteristic
67#14579093—22no—
68#14579092—42no—
69Dic79⋊C8—22no—
70#14579086—24no—
71#14579084—24no—
72#14579084C2×C418yes—
73#14579071—22no—
74C8×Dic79C2×C414yes—
75C79⋊#95—22no—
76C79⋊#95D412yes—
77C22×C8×C79C2311yescharacteristic
78C22×C8×C79C2311yescharacteristic
79C2×C4⋊C4×C79—22no—
80C2×C4⋊C4×C79C2312yes—
81C2×C4⋊C4×C79C2312yes—
82C2×C42×C79C2311yescharacteristic
83#107×C79C2×C414yes—
84#106×C79—22no—
85#97×C79—22no—
86#95×C79—22no—
87C2×Dic158—22no—
88C2×Dic158—22no—
89C2×Dic158—22no—
90C2×Dic158C22×C414yes—
91C2×Dic158C2×D414yes—
92C2×Dic158C2×Q814yes—
93C22×Dic79—22no—
94C22×Dic79—22no—
95C22×Dic79C2×D412yes—
96C22×Dic79C2×D411yescharacteristic
97C22×Dic79C2×D411yescharacteristic
98Dic316C4⋊C4116yescomplemented
99C2×C79⋊C8—21no—
100C2×C79⋊C8—21no—
101C2×C79⋊C8—21no—
102C2×C79⋊C8—21no—
103C2×C79⋊C8—21no—
104C2×C79⋊C8C22×C412yes—
105C4⋊Dic79—22no—
106C4⋊Dic79—22no—
107C4⋊Dic79—82no—
108Dic79⋊C4—28no—
109Dic79⋊C4—24no—
110Dic79⋊C4—44no—
111Dic79⋊C4—44no—
112Dic79⋊C4—44no—
113Dic79⋊C4—82no—
114C4×Dic79—28no—
115C4×Dic79—21no—
116C4×Dic79—21no—
117C4×Dic79—44no—
118C4×Dic79—44no—
119C4×Dic79C22×C412yes—
120C4×Dic79#4112yes—
121C22×C4×C79—22no—
122C22×C4×C79—22no—
123C22×C4×C79C2×D412yes—
124C22×C4×C79C2411yescharacteristic
125C2×C8×C79—21no—
126C2×C8×C79—21no—
127C2×C8×C79—21no—
128C2×C8×C79C22×C412yes—
129C2×C8×C79C2×D412yes—
130C2×C8×C79C2×Q812yes—
131C4⋊C4×C79—22no—
132C4⋊C4×C79—22no—
133C4⋊C4×C79—22no—
134C4⋊C4×C79—82no—
135C4⋊C4×C79C22×C414yes—
136C42×C79—21no—
137C42×C79—21no—
138C23×C79C22×D411yescharacteristic
139C2×C4×C79—24no—
140C2×C4×C79—22no—
141C2×C4×C79—22no—
142C2×C4×C79—22no—
143C2×C4×C79—42no—
144C2×C4×C79—42no—
145C2×C4×C79—42no—
146C2×C4×C79—42no—
147C2×C4×C79—42no—
148C2×C4×C79C23×C411yescharacteristic
149C2×C4×C79C22×D411yescharacteristic
150C2×C4×C79C22×Q811yescharacteristic
151C2×C4×C79C2×#4111yescharacteristic
152C2×C4×C79H(5,𝔽2)11yescharacteristic
153C2×C4×C792−1+411yescharacteristic
154C2×Dic79—24no—
155C2×Dic79—24no—
156C2×Dic79—24no—
157C2×Dic79—22no—
158C2×Dic79—22no—
159C2×Dic79—22no—
160C2×Dic79—22no—
161C2×Dic79—42no—
162C2×Dic79—42no—
163C2×Dic79—42no—
164C2×Dic79C4×D412yes—
165C2×Dic79C4×D412yes—
166C2×Dic79#13714yes—
167Dic158—24no—
168Dic158—24no—
169Dic158—24no—
170Dic158C2×C4⋊C418yes—
171C8×C79—41no—
172C8×C79C2×C4⋊C414yes—
173C79⋊C8—22no—
174C79⋊C8—41no—
175C22×C79C2×C4×D411yescharacteristic
176C22×C79D4211yescharacteristic
177C22×C79D4×Q811yescharacteristic
178C22×C79C2×#13712yes—
179C22×C79#57712yes—
180C4×C79—44no—
181C4×C79—42no—
182C4×C79—82no—
183C4×C79C22×C4⋊C411yescharacteristic
184C4×C79#52011yescharacteristic
185C4×C79#54711yescharacteristic
186C4×C79#55411yescharacteristic
187Dic79—28no—
188Dic79—22no—
189Dic79—42no—
190Dic79—82no—
191Dic79#44114yescomplemented
192#22674—791nomaximal, Sylow
193C2×C79D4×C4⋊C411yescharacteristic
194C2×C79C2×#44111yescharacteristic
195C2×C79#295211yescharacteristic
196C2×C79#327812yes—
197C2×C79#338712yes—
198C2×#441—791no—
199C2×#441—791no—
200C2×C4×Q16—791no—
201Q8×C4⋊C4—792no—
202#1943—791no—
203#1943D7911yescharacteristic
204#1865—792no—
205#1865—792no—
206#1844—791no—
207#1803—792no—
208#1770—791no—
209C79#2267411yescharacteristic, complemented, Sylow
210C22×Q16—791no—
211C2×C4×Q8—792no—
212C2×C4×Q8—792no—
213C2×C4×Q8—792no—
214#441—1584no—
215#441—1584no—
216C4×Q16—798no—
217C2×#107C2×D7911yescharacteristic
218C2×#106—792no—
219C2×#106—791no—
220C2×#106C2×D7911yescharacteristic
221C2×C4⋊C8—791no—
222C2×#103—792no—
223C2×#103—792no—
224C2×#103—792no—
225C2×#97—791no—
226C2×#97C2×D7911yescharacteristic
227C2×C4×C8—791no—
228#389—792no—
229#389—1582no—
230#389—1582no—
231#389C2×D7912yes—
232#384—792no—
233#384—1582no—
234C4×C4⋊C4—791no—
235C4×C4⋊C4—1582no—
236#340—792no—
237#340C2×D7912yes—
238#336—791no—
239#336—791no—
240C22×Q8—792no—
241C2×Q16—794no—
242C2×Q16—794no—
243C2×Q16—794no—
244C22×C8—791no—
245C22×C8—791no—
246C22×C8C22×D7911yescharacteristic
247C22×C8C22×D7911yescharacteristic
248C4×Q8—3162no—
249C4×Q8—3162no—
250C4×Q8—798no—
251C4×Q8—1584no—
252C4×Q8—1584no—
253C4×Q8—1584no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—792no—
256C2×C4⋊C4—792no—
257C2×C4⋊C4—1582no—
258C2×C4⋊C4—1582no—
259C2×C4⋊C4—1582no—
260C2×C4⋊C4—1582no—
261C2×C4⋊C4—1582no—
262C2×C4⋊C4—1582no—
263C2×C4⋊C4C22×D7912yes—
264C2×C4⋊C4C22×D7912yes—
265C2×C42—791no—
266C2×C42—791no—
267C2×C42—1582no—
268C2×C42—1582no—
269C2×C42—1582no—
270C2×C42C22×D7911yescharacteristic
271#107C4×D7914yes—
272#106—22no—
273#106—3162no—
274#106—1582no—
275C4⋊C8—1582no—
276#103—798no—
277#103—1584no—
278#103—1584no—
279#97—22no—
280#97—1582no—
281C4×C8—794no—
282#95—22no—
283#95—792no—
284#95—1582no—
285C2×Q8—794no—
286C2×Q8—794no—
287C2×Q8—794no—
288C2×Q8—1582no—
289C2×Q8—1582no—
290C2×Q8—1582no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—791no—
294C22×C4—791no—
295C22×C4—792no—
296C22×C4—1582no—
297C22×C4—1582no—
298C22×C4D4×D7912yes—
299C22×C4C23×D7911yescharacteristic
300Q16—7916no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—792no—
305C2×C8—1581no—
306C2×C8—1581no—
307C2×C8—1581no—
308C2×C8—1581no—
309C2×C8—1581no—
310C2×C8C2×C4×D7912yes—
311C2×C8D4×D7912yes—
312C2×C8Q8×D7912yes—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—82no—
317C4⋊C4—3164no—
318C4⋊C4—3164no—
319C4⋊C4—3164no—
320C4⋊C4—6322no—
321C4⋊C4—6322no—
322C4⋊C4—1584no—
323C4⋊C4—1588no—
324C4⋊C4—1582no—
325C4⋊C4—1582no—
326C4⋊C4C2×C4×D7914yes—
327C42—21no—
328C42—21no—
329C42—3164no—
330C42—3164no—
331C42—792no—
332C42—792no—
333C42—1588no—
334C42—1581no—
335C42—1581no—
336C23C2×D4×D7911yescharacteristic
337Q8—798no—
338Q8—1584no—
339Q8—1584no—
340Q8—1584no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—3162no—
351C2×C4—3162no—
352C2×C4—3162no—
353C2×C4—792no—
354C2×C4—794no—
355C2×C4—792no—
356C2×C4—1582no—
357C2×C4—1582no—
358C2×C4—1582no—
359C2×C4—1584no—
360C2×C4—1584no—
361C2×C4—1584no—
362C2×C4—1582no—
363C2×C4C22×C4×D7911yescharacteristic
364C2×C4C2×D4×D7911yescharacteristic
365C2×C4C2×Q8×D7911yescharacteristic
366C2×C4#41×D7911yescharacteristic
367C2×C4#1457920711yescharacteristic
368C2×C4#1457920911yescharacteristic
369C8—41no—
370C8—3161no—
371C8—1582no—
372C8C4⋊C4×D7914yes—
373C22C4×D4×D7911yescharacteristic
374C22#1909919711yescharacteristic
375C22#1909922811yescharacteristic
376C22#137×D7912yes—
377C22#1909937312yes—
378C4—42no—
379C4—44no—
380C4—82no—
381C4—3162no—
382C4—794no—
383C4—6322no—
384C4—1588no—
385C4—1582no—
386C4C2×C4⋊C4×D7911yescharacteristic
387C4#1909904511yescharacteristic
388C4#1909921511yescharacteristic
389C4#1909926911yescharacteristic
390C2#2376630111yescharacteristic
391C2#441×D7911yescharacteristic
392C2#2377228211yescharacteristic
393C2#2377506312yes—
394C2#2377632912yes—
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1943C2
socle, prime radicalC23×C79C22×D4
cosocleC22×C4×C79C24
hypercenter#1943D79
hypocenterC79#22674
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1943

Complements

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

NH ≅ G/Ntype
18. #19098657C4semidirect
98. Dic316C4⋊C4semidirect
191. Dic79#441semidirect
209. C79#22674semidirect

Maximal subgroups

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

subgroupindex
C2×#190986712
C2×#190986692
C2×#190986572
#237663142
#237641392
#237636692
#237636672
#237635552
#237634042
#237632682
C79×#19432
#2267479