Finite Group Databasebeta

← #34260268

Subgroups — #34260268

Order 20224.

606 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773970C211yesmaximal, characteristic
2#23772556C212yesmaximal, complemented
3#23762296C211yesmaximal, characteristic, complemented
4#23762284C211yesmaximal, characteristic, complemented
5#23762277C212yesmaximal, complemented
6#23762256C212yesmaximal, complemented
7#23762206C211yesmaximal, characteristic, complemented
8#23762194C211yesmaximal, characteristic, complemented
9#23761811C212yesmaximal, complemented
10C8×#11929608C211yesmaximal, characteristic, complemented
11C2×#335×C79C211yesmaximal, characteristic, complemented
12#19099097C2212yes—
13#19098965C411yescharacteristic
14#19098947C411yescharacteristic
15#19098918C412yes—
16#19098804C411yescharacteristic
17#19098799C411yescharacteristic
18#19098792C412yes—
19#19098690C2212yes—
20#19098687C2212yes—
21C4×Dic316C2211yescharacteristic
22C4×#11929613C2212yes—
23C4×D632C2211yescharacteristic
24#19098640C2212yes—
25#19098634—22no—
26#19098628C2212yes—
27C8×C22⋊D79—22no—
28#19098623—22no—
29#19098616C2211yescharacteristic
30#19098588—22no—
31C4×#11929608C2211yescharacteristic
32#19098293C2212yes—
33#19098293C2212yes—
34#19098290C2212yes—
35#19098290C2212yes—
36#19098285C2211yescharacteristic
37#19098285C2211yescharacteristic
38C8×C79⋊C8C2211yescharacteristic
39C8×C79⋊C8C2211yescharacteristic
40C2×C4⋊C8×C79C2212yes—
41C2×C4×C8×C79C2211yescharacteristic
42#335×C79C2211yescharacteristic
43#335×C79C2212yes—
44#335×C79C2211yescharacteristic
45C2×Dic316C2×C411yescharacteristic
46C2×#11929613C2×C412yes—
47C2×D632C2×C411yescharacteristic
48C2×#11929608—21no—
49C2×#11929607—22no—
50C2×C8×D79—22no—
51#14579155—22no—
52#14579153C2×C412yes—
53#14579149C2×C412yes—
54#14579144—22no—
55#14579140C2×C412yes—
56#14579135C2×C412yes—
57C4×Dic158C2312yes—
58C4×D316C2312yes—
59C4×C22⋊D79—22no—
60#14579118—22no—
61#14579095C2×C411yescharacteristic
62#14579095C2×C411yescharacteristic
63#14579093C2×C412yes—
64C4⋊(C79⋊C8)D411yescharacteristic
65C4⋊(C79⋊C8)D411yescharacteristic
66Dic79⋊C8—22no—
67Dic79⋊C8—22no—
68#14579086C2×C412yes—
69#14579086C2×C412yes—
70#14579083C2×C412yes—
71#14579083C2×C412yes—
72C22⋊(C79⋊C8)D411yescharacteristic
73C22⋊(C79⋊C8)D411yescharacteristic
74(C2×D79)⋊C8—22no—
75(C2×D79)⋊C8—22no—
76#14579072—21no—
77#14579071—22no—
78C4×C79⋊C8C2311yescharacteristic
79C4×C79⋊C8C2311yescharacteristic
80C4×C79⋊C8C2311yescharacteristic
81C4×C79⋊C8C2311yescharacteristic
82C8×Dic79—22no—
83C22×C8×C79—22no—
84C22×C8×C79C2×C411yescharacteristic
85C22×C8×C79C2×C411yescharacteristic
86C2×C42×C79C2311yescharacteristic
87C4⋊C8×C79—22no—
88C4⋊C8×C79C2312yes—
89C4⋊C8×C79C2312yes—
90C4×C8×C79C2×C411yescharacteristic
91C4×C8×C79C2×C411yescharacteristic
92C4×C8×C79C2311yescharacteristic
93C4×C8×C79C2311yescharacteristic
94C2×Dic158C22×C412yes—
95C2×D316C22×C412yes—
96C2×C22⋊D79—22no—
97C2×C4×D79—22no—
98Dic316—22no—
99#11929613—42no—
100D632—22no—
101#11929608—24no—
102#11929607—82no—
103C2×C79⋊C8—21no—
104C2×C79⋊C8—21no—
105C2×C79⋊C8C2×D411yescharacteristic
106C2×C79⋊C8C2×D411yescharacteristic
107C2×C79⋊C8C2×D411yescharacteristic
108C2×C79⋊C8C2×D411yescharacteristic
109C8×D79—82no—
110C4⋊Dic79C22×C412yes—
111C4⋊Dic79C22×C412yes—
112Dic79⋊C4—22no—
113Dic79⋊C4—22no—
114Dic79⋊C4—22no—
115Dic79⋊C4—22no—
116C22⋊Dic79—22no—
117(C2×D79)⋊C4—22no—
118(C2×D79)⋊C4—22no—
119(C2×D79)⋊C4—22no—
120(C2×D79)⋊C4—22no—
121C4×Dic79—22no—
122C22×C4×C79C22×C411yescharacteristic
123C22×C4×C79C2×D411yescharacteristic
124C22×C4×C79#4111yescharacteristic
125C2×C8×C79—22no—
126C2×C8×C79—22no—
127C2×C8×C79—22no—
128C2×C8×C79—41no—
129C2×C8×C79—82no—
130C2×C8×C79C22×C411yescharacteristic
131C2×C8×C79C22×C411yescharacteristic
132C2×C8×C79C22×C411yescharacteristic
133C2×C8×C79C22×C411yescharacteristic
134C42×C79C22×C411yescharacteristic
135C42×C79C2×D411yescharacteristic
136C42×C79#4111yescharacteristic
137C42×C79C2411yescharacteristic
138C23×C79C4×D411yescharacteristic
139C22×D79—22no—
140C2×C4×C79—22no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79C4×D411yescharacteristic
146C2×C4×C79C4×D411yescharacteristic
147C2×C4×C79C4×D411yescharacteristic
148C2×C4×C79C23×C411yescharacteristic
149C2×C4×C79C22×D411yescharacteristic
150C2×C4×C79C2×#4111yescharacteristic
151C22⋊D79—44no—
152C2×Dic79—22no—
153C2×Dic79—22no—
154C2×Dic79—22no—
155D316—22no—
156D316—22no—
157C4×D79—42no—
158C4×D79—42no—
159Dic158—22no—
160Dic158—22no—
161C8×C79—21no—
162C8×C79—21no—
163C8×C79—21no—
164C8×C79—42no—
165C8×C79—42no—
166C8×C79#13112yes—
167C79⋊C8—22no—
168C79⋊C8—22no—
169C79⋊C8—21no—
170C79⋊C8—21no—
171C79⋊C8—21no—
172C79⋊C8—21no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79#44312yes—
176C22×C79C2×C4×D411yescharacteristic
177C2×D79—42no—
178C2×D79—42no—
179C2×D79—42no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—21no—
184C4×C79C2×#13111yescharacteristic
185C4×C79#56811yescharacteristic
186C4×C79#57611yescharacteristic
187C4×C79#57811yescharacteristic
188Dic79—42no—
189Dic79—42no—
190Dic79—42no—
191#18087—791nomaximal, Sylow
192C2×C79—22no—
193C2×C79#293511yescharacteristic
194C2×C79#295111yescharacteristic
195C2×C79C2×#44311yescharacteristic
196D79—82no—
197#3168—791no—
198#2994—792no—
199#1591—791no—
200#1589—791no—
201#1588—792no—
202#1585—792no—
203C8×Q16—791no—
204C8×D8—791no—
205C2×#335D7911yescharacteristic, complemented
206#1483—792no—
207C8×M16—791no—
208C79#1808711yescharacteristic, complemented, Sylow
209#529—792no—
210#499—791no—
211#495—791no—
212#488—792no—
213#467—791no—
214#466—791no—
215#465—792no—
216#446—792no—
217C8×Q8—792no—
218C4×Q16—791no—
219C4×SD16—792no—
220C4×D8—791no—
221#436—792no—
222#435—1582no—
223C8×D4—792no—
224C8×D4—1582no—
225#433—1582no—
226#431—791no—
227#424—1582no—
228C2×C4⋊C8C2×D7912yes—
229C4×M16—791no—
230C2×C4×C8C2×D7911yescharacteristic
231#335C2×D7911yescharacteristic
232#335C2×D7912yes—
233#335C2×D7911yescharacteristic
234Q8⋊C8—792no—
235Q8⋊C8—792no—
236D4⋊C8—792no—
237D4⋊C8—792no—
238#322—791no—
239#322—791no—
240C82—791no—
241C82—791no—
242C2×Q16—791no—
243C2×SD16—792no—
244C2×D8—791no—
245C2×M16—1582no—
246C2×M16—1581no—
247C22×C8—22no—
248C22×C8—1582no—
249C22×C8C4×D7911yescharacteristic
250C22×C8C4×D7911yescharacteristic
251#126—1582no—
252#125—792no—
253#124—792no—
254#123—1582no—
255C22⋊Q8—792no—
256#121—792no—
257C4×Q8—792no—
258C4×D4—792no—
259C4×D4—1582no—
260#117—1582no—
261C2×C42C22×D7911yescharacteristic
262#107—791no—
263#107—791no—
264#106—792no—
265C4⋊C8—22no—
266C4⋊C8—791no—
267C4⋊C8—791no—
268C4⋊C8—1582no—
269C4⋊C8—1582no—
270C4⋊C8C22×D7912yes—
271C4⋊C8C22×D7912yes—
272#103—792no—
273#103—792no—
274#102—792no—
275#102—792no—
276C22⋊C8—791no—
277C22⋊C8—791no—
278C22⋊C8—1582no—
279C22⋊C8—1582no—
280#97—1582no—
281#97—1581no—
282C4×C8—791no—
283C4×C8—791no—
284C4×C8—791no—
285C4×C8—791no—
286C4×C8—1582no—
287C4×C8C4×D7911yescharacteristic
288C4×C8C4×D7911yescharacteristic
289C4×C8C22×D7911yescharacteristic
290C4×C8C22×D7911yescharacteristic
291C2×Q8—792no—
292C2×D4—792no—
293C2×D4—1582no—
294C22×C4—1582no—
295C22×C4C2×C4×D7911yescharacteristic
296C22×C4D4×D7911yescharacteristic
297C22×C4#1192962511yescharacteristic
298Q16—1582no—
299SD16—3162no—
300D8—1582no—
301M16—6322no—
302M16—1584no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—82no—
308C2×C8—791no—
309C2×C8—791no—
310C2×C8—791no—
311C2×C8—791no—
312C2×C8—6322no—
313C2×C8—1581no—
314C2×C8—1581no—
315C2×C8C2×C4×D7911yescharacteristic
316C2×C8C2×C4×D7911yescharacteristic
317C2×C8C2×C4×D7911yescharacteristic
318C2×C8C2×C4×D7911yescharacteristic
319C4⋊C4—792no—
320C4⋊C4—792no—
321C4⋊C4—1582no—
322C4⋊C4—1582no—
323C4⋊C4—1582no—
324C4⋊C4—1582no—
325C22⋊C4—1582no—
326C22⋊C4—1582no—
327C22⋊C4—1582no—
328C22⋊C4—1582no—
329C22⋊C4—1582no—
330C42—1582no—
331C42C2×C4×D7911yescharacteristic
332C42D4×D7911yescharacteristic
333C42#1192962511yescharacteristic
334C42C23×D7911yescharacteristic
335C23—1582no—
336C23#1457912311yescharacteristic
337Q8—1582no—
338Q8—1582no—
339D4—3164no—
340D4—1582no—
341D4—1582no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—3162no—
348C2×C4—3162no—
349C2×C4—1582no—
350C2×C4—1582no—
351C2×C4—1582no—
352C2×C4#1457912311yescharacteristic
353C2×C4#1457912311yescharacteristic
354C2×C4#1457912311yescharacteristic
355C2×C4C22×C4×D7911yescharacteristic
356C2×C4C2×D4×D7911yescharacteristic
357C2×C4C2×#1192962511yescharacteristic
358C8—21no—
359C8—21no—
360C8—21no—
361C8—42no—
362C8—42no—
363C8—1582no—
364C8—1582no—
365C8—1581no—
366C8—1581no—
367C8—1581no—
368C8—1581no—
369C8#1457916712yes—
370C22—21no—
371C22—21no—
372C22—3162no—
373C22—3162no—
374C22—3162no—
375C22#1909867912yes—
376C22C2×#1457912311yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—3162no—
382C4—3162no—
383C4—3162no—
384C4C2×#1457916711yescharacteristic
385C4#1909932911yescharacteristic
386C4#1909936411yescharacteristic
387C4#1909937611yescharacteristic
388C2—22no—
389C2—6322no—
390C2#2377214311yescharacteristic
391C2#2377227411yescharacteristic
392C2C2×#1909867911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC2×#335×C79C2
socleC22×C79C2×C4×D4
prime radical#19098799C4
cosocleC42×C79C24
hypercenterC2×#335D79
hypocenterC79#18087
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #23772556C2semidirect
3. #23762296C2semidirect
4. #23762284C2semidirect
5. #23762277C2semidirect
6. #23762256C2semidirect
7. #23762206C2semidirect
8. #23762194C2semidirect
9. #23761811C2semidirect
10. C8×#11929608C2semidirect
11. C2×#335×C79C2semidirect
205. C2×#335D79semidirect
208. C79#18087semidirect

Maximal subgroups

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

subgroupindex
#237739702
#237725562
#237622962
#237622842
#237622772
#237622562
#237622062
#237621942
#237618112
C8×#119296082
C2×#335×C792
#1808779