Finite Group Databasebeta

← #34205433

Subgroups — #34205433

Order 20224.

484 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×Dic79C211yesmaximal, characteristic, complemented
2#23765136C211yesmaximal, characteristic, complemented
3#23765128C211yesmaximal, characteristic, complemented
4#23764289C211yesmaximal, characteristic, complemented
5#23761515C211yesmaximal, characteristic, complemented
6#23761481C211yesmaximal, characteristic, complemented
7C79×#3414C211yesmaximal, characteristic
8#41×Dic79—21no—
9C2×#14579101C2211yescharacteristic
10Q16×Dic79—21no—
11SD16×Dic79—22no—
12D8×Dic79—21no—
13C2×#14579097C2211yescharacteristic
14#19098601C2211yescharacteristic
15C2×#14579088—21no—
16#19098559—21no—
17#19098530—21no—
18C2×C8×Dic79C2211yescharacteristic
19#19098373—22no—
20#19098370—22no—
21C2×#135×C79C2211yescharacteristic
22C2×#135×C79C2211yescharacteristic
23#510×C79—21no—
24#510×C79C412yescomplemented
25#509×C79—21no—
26#509×C79C412yescomplemented
27#508×C79—22no—
28C2×M32×C79C2211yescharacteristic
29C2×#11929608—21no—
30C22×C79⋊C8D411yescharacteristic
31Q8×Dic79—22no—
32D4×Dic79—22no—
33D4×Dic79—42no—
34#14579120—21no—
35#14579120—41no—
36C2×C4×Dic79—41no—
37C2×C4×Dic79D411yescharacteristic
38#14579101—21no—
39#14579101—21no—
40C2×C79⋊C16—21no—
41#14579097—21no—
42#14579097—21no—
43#14579095—21no—
44#14579093D412yes—
45#14579088—41no—
46#14579088—41no—
47#14579085—22no—
48#14579082—22no—
49C22⋊(C79⋊C8)—42no—
50C8×Dic79—21no—
51C8×Dic79—21no—
52C2×#41×C79—21no—
53C2×#41×C79—21no—
54#135×C79—21no—
55#135×C79—21no—
56#135×C79—21no—
57#135×C79—21no—
58#135×C79—21no—
59#135×C79—21no—
60#135×C79C2×C412yes—
61#135×C79D412yes—
62C2×Q16×C79C2×C411yescharacteristic, complemented
63C2×Q16×C79D411yescharacteristic
64C2×SD16×C79—21no—
65C2×SD16×C79—21no—
66C2×D8×C79C2×C411yescharacteristic, complemented
67C2×D8×C79D411yescharacteristic
68C22×C8×C79C2311yescharacteristic
69Q32×C79—22no—
70SD32×C79—22no—
71SD32×C79—22no—
72D16×C79—22no—
73M32×C79—21no—
74M32×C79C2×C412yescomplemented
75C2×C16×C79—21no—
76C22×Dic79—21no—
77C22×Dic79—42no—
78#11929608—41no—
79#11929608—41no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—41no—
84C4⋊Dic79—22no—
85C4⋊Dic79—42no—
86C22⋊Dic79—42no—
87C22⋊Dic79—41no—
88C4×Dic79—21no—
89C4×Dic79—42no—
90C4×Dic79—41no—
91C4×Dic79D811yescharacteristic
92C4×Dic79D811yescharacteristic
93C79⋊C16—41no—
94#41×C79—22no—
95#41×C79—22no—
96#41×C79—41no—
97#41×C79—41no—
98#41×C79—41no—
99#41×C79—41no—
100#41×C79—41no—
101#41×C79—41no—
102C2×Q8×C79—21no—
103C2×Q8×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79—41no—
107C2×D4×C79—41no—
108C22×C4×C79—41no—
109C22×C4×C79—41no—
110C22×C4×C79C2×D411yescharacteristic
111Q16×C79—21no—
112Q16×C79—21no—
113Q16×C79—21no—
114Q16×C79C22⋊C412yes—
115SD16×C79—22no—
116SD16×C79—22no—
117SD16×C79—41no—
118SD16×C79—41no—
119D8×C79—21no—
120D8×C79—21no—
121D8×C79—21no—
122D8×C79C22⋊C412yes—
123C2×C8×C79C22×C411yescharacteristic
124C2×C8×C79C2×D411yescharacteristic
125C2×C8×C79C2×D411yescharacteristic
126C2×C8×C79C2×D411yescharacteristic
127C2×C8×C79#4111yescharacteristic
128C2×C8×C79#4111yescharacteristic
129C16×C79—22no—
130C23×C79—41no—
131C23×C79—41no—
132C23×C79AGL(1,ℤ/8)11yescharacteristic
133Q8×C79—22no—
134Q8×C79—41no—
135Q8×C79—41no—
136Q8×C79—41no—
137D4×C79—22no—
138D4×C79—42no—
139D4×C79—42no—
140D4×C79—41no—
141D4×C79—41no—
142D4×C79—41no—
143D4×C79—81no—
144D4×C79—81no—
145C2×C4×C79—41no—
146C2×C4×C79—42no—
147C2×C4×C79—42no—
148C2×C4×C79—41no—
149C2×C4×C79—81no—
150C2×C4×C79—81no—
151C2×C4×C79C4×D411yescharacteristic
152C2×C4×C79#12011yescharacteristic
153C2×C4×C79#12111yescharacteristic
154C2×C4×C79C2×D811yescharacteristic
155C2×C4×C79#13511yescharacteristic
156C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
157C2×Dic79—21no—
158C2×Dic79—21no—
159C2×Dic79—42no—
160C2×Dic79—42no—
161C2×Dic79—84no—
162C8×C79C2×C22⋊C411yescharacteristic
163C8×C79C4×D411yescharacteristic
164C8×C79#12111yescharacteristic
165C8×C79#12311yescharacteristic
166C79⋊C8—41no—
167C79⋊C8—81no—
168C22×C79—41no—
169C22×C79—42no—
170C22×C79—82no—
171C22×C79—82no—
172C22×C79—81no—
173C22×C79#44211yescharacteristic
174C22×C79#44711yescharacteristic
175C22×C79#46311yescharacteristic
176C4×C79—42no—
177C4×C79—81no—
178C4×C79#38611yescharacteristic
179C4×C79C4×D811yescharacteristic
180C4×C79#44911yescharacteristic
181C4×C79#45911yescharacteristic
182Dic79—22no—
183Dic79—41no—
184Dic79—82no—
185#13027—791nomaximal, Sylow
186C2×C79—21no—
187C2×C79—21no—
188C2×C79—21no—
189C2×C79—82no—
190C2×C79—161no—
191C2×C79#187911yescharacteristic
192#3414D7911yescharacteristic
193C4×#135—791no—
194#2145—791no—
195#2143—791no—
196#1974—791no—
197#1389—791no—
198#1376—791no—
199C79#1302711yescharacteristic, complemented, Sylow
200C2×#135C2×D7911yescharacteristic
201C2×#135C2×D7911yescharacteristic
202C4×#41—1581no—
203#510—21no—
204#510Dic7912yescomplemented
205#509—21no—
206#509Dic7912yescomplemented
207#508—22no—
208C2×M32—791no—
209C2×M32C2×D7911yescharacteristic
210C4×Q16—1581no—
211C4×SD16—1582no—
212C4×D8—1581no—
213C2×#108—791no—
214#427—791no—
215C2×#104—1581no—
216#416—1581no—
217#408—1581no—
218C2×C4×C8—791no—
219#360—1582no—
220#359—1582no—
221C2×#41—21no—
222C2×#41—21no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135—21no—
228#135—21no—
229#135C2×Dic7912yes—
230#135C22⋊D7912yes—
231C2×Q16C2×Dic7911yescharacteristic, complemented
232C2×Q16C22⋊D7911yescharacteristic
233C2×SD16—21no—
234C2×SD16—21no—
235C2×D8C2×Dic7911yescharacteristic, complemented
236C2×D8C22⋊D7911yescharacteristic
237C2×M16—1581no—
238C22×C8—791no—
239C22×C8C22×D7911yescharacteristic
240C4×Q8—1582no—
241C4×D4—3162no—
242C4×D4—1582no—
243#117—3161no—
244#117—1581no—
245C2×C42—3161no—
246C2×C42—791no—
247Q32—22no—
248SD32—22no—
249SD32—22no—
250D16—22no—
251M32—21no—
252M32—1581no—
253M32—1581no—
254M32C2×Dic7912yescomplemented
255C2×C16—21no—
256C2×C16—1581no—
257#108—1581no—
258#108—1581no—
259#107—1581no—
260#106—792no—
261#104—3161no—
262#104—3161no—
263#103—1582no—
264#102—1582no—
265C22⋊C8—3162no—
266C4×C8—1581no—
267C4×C8—1581no—
268#41—22no—
269#41—22no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—41no—
281C2×D4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—3162no—
285C22×C4—1581no—
286C22×C4D4×D7911yescharacteristic
287Q16—21no—
288Q16—21no—
289Q16—21no—
290Q16C22⋊Dic7912yes—
291SD16—22no—
292SD16—22no—
293SD16—41no—
294SD16—41no—
295D8—21no—
296D8—21no—
297D8—21no—
298D8C22⋊Dic7912yes—
299M16—3161no—
300M16—3161no—
301C2×C8—3161no—
302C2×C8—1581no—
303C2×C8—1581no—
304C2×C8—1581no—
305C2×C8C22×Dic7911yescharacteristic
306C2×C8D4×D7911yescharacteristic
307C2×C8C2×C22⋊D7911yescharacteristic
308C2×C8C2×C22⋊D7911yescharacteristic
309C2×C8#1192962311yescharacteristic
310C2×C8#1192962311yescharacteristic
311C4⋊C4—3162no—
312C4⋊C4—1582no—
313C22⋊C4—3161no—
314C22⋊C4—3162no—
315C42—3162no—
316C42—3161no—
317C42—791no—
318C42—791no—
319C42—1581no—
320C16—22no—
321C16—3161no—
322C23—41no—
323C23—41no—
324C23#1457918511yescharacteristic
325Q8—22no—
326Q8—41no—
327Q8—41no—
328Q8—41no—
329D4—22no—
330D4—42no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—81no—
336D4—81no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—3162no—
344C2×C4—3162no—
345C2×C4—6324no—
346C2×C4—1581no—
347C2×C4—1581no—
348C2×C4D4×Dic7911yescharacteristic
349C2×C4#1457912911yescharacteristic
350C2×C4#1457913311yescharacteristic
351C2×C4D8×D7911yescharacteristic
352C2×C4#1457918011yescharacteristic
353C2×C4#1457918511yescharacteristic
354C8—3161no—
355C8—6321no—
356C8C2×C22⋊Dic7911yescharacteristic
357C8D4×Dic7911yescharacteristic
358C8#1457913311yescharacteristic
359C8#1457914211yescharacteristic
360C22—41no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—81no—
365C22#1909867511yescharacteristic
366C22#1909869511yescharacteristic
367C22#1909877611yescharacteristic
368C4—42no—
369C4—81no—
370C4—3161no—
371C4—6322no—
372C4—1582no—
373C4#1909844311yescharacteristic
374C4D8×Dic7911yescharacteristic
375C4#1909870511yescharacteristic
376C4#1909875611yescharacteristic
377C2—21no—
378C2—21no—
379C2—21no—
380C2—82no—
381C2—161no—
382C2#2376375011yescharacteristic
subgroupG/H
centerC4#19098756
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
Fitting, prime radicalC79×#3414C2
socleC2×C79#1879
cosocleC22×C8×C79C23
hypercenter#3414D79
hypocenterC79#13027
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #135×Dic79C2semidirect
2. #23765136C2semidirect
3. #23765128C2semidirect
4. #23764289C2semidirect
5. #23761515C2semidirect
6. #23761481C2semidirect
24. #510×C79C4semidirect
26. #509×C79C4semidirect
62. C2×Q16×C79C2×C4semidirect
66. C2×D8×C79C2×C4semidirect
74. M32×C79C2×C4semidirect
199. C79#13027semidirect
204. #510Dic79semidirect
206. #509Dic79semidirect
231. C2×Q16C2×Dic79semidirect
235. C2×D8C2×Dic79semidirect
254. M32C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#135×Dic792
#237651362
#237651282
#237642892
#237615152
#237614812
C79×#34142
#1302779