Finite Group Databasebeta

← #34205438

Subgroups — #34205438

Order 20224.

476 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099003C211yesmaximal, characteristic
2C2×#19098378C211yesmaximal, characteristic, complemented
3#23765112C211yesmaximal, characteristic
4#23764261C211yesmaximal, characteristic, complemented
5#23761510C211yesmaximal, characteristic, complemented
6#23761482C211yesmaximal, characteristic, complemented
7C2×#443×C79C211yesmaximal, characteristic
8C22×Dic316C2211yescharacteristic
9#19099003—22no—
10#19099003C414yescomplemented
11C2×#14579107—21no—
12C2×#14579105—21no—
13C2×#14579101C2211yescharacteristic
14C2×#14579101C2211yescharacteristic
15C2×#14579095C2211yescharacteristic
16C2×#14579086—21no—
17#19098469—21no—
18#19098378—22no—
19#19098367—21no—
20#19098364—21no—
21C2×#135×C79C2211yescharacteristic
22C2×#131×C79—21no—
23#443×C79—22no—
24#443×C79—22no—
25C2×#108×C79C2211yescharacteristic
26C2×#104×C79—21no—
27C2×C4×C8×C79C2211yescharacteristic
28C22×Dic158—21no—
29C2×Dic316—21no—
30C2×Dic316—21no—
31C2×Dic316—21no—
32C2×Dic316—21no—
33C2×Dic316—21no—
34C2×Dic316C2×C412yes—
35C2×C4⋊Dic79—21no—
36#14579107—24no—
37#14579105—24no—
38#14579101—21no—
39#14579101—21no—
40#14579101—21no—
41#14579101C2×C412yes—
42C2×C79⋊C16—21no—
43C2×C79⋊C16—21no—
44#14579095—21no—
45#14579095—21no—
46#14579086—41no—
47#14579086—41no—
48C79⋊#95—42no—
49C2×#41×C79—21no—
50#135×C79—21no—
51#135×C79—21no—
52#135×C79C2×C412yes—
53#135×C79D412yes—
54C2×Q16×C79D411yescharacteristic
55C2×SD16×C79—21no—
56C2×D8×C79D411yescharacteristic
57#131×C79—24no—
58#131×C79—42no—
59C2×M16×C79—21no—
60C2×M16×C79—41no—
61C22×C8×C79—41no—
62C22×C8×C79D411yescharacteristic
63C22×C8×C79C2311yescharacteristic
64C2×C42×C79D411yescharacteristic
65#108×C79—21no—
66#108×C79D412yes—
67#104×C79—24no—
68C4×C8×C79—21no—
69C4×C8×C79—21no—
70C2×Dic158—22no—
71C2×Dic158—41no—
72C2×Dic158—41no—
73C2×Dic158—41no—
74C2×Dic158—41no—
75C2×Dic158—41no—
76C22×Dic79—41no—
77C22×Dic79—41no—
78Dic316—22no—
79Dic316—22no—
80Dic316—41no—
81Dic316—41no—
82C4⋊Dic79—41no—
83C4⋊Dic79—41no—
84C79⋊C16—22no—
85C79⋊C16—41no—
86#41×C79—22no—
87#41×C79—22no—
88#41×C79—41no—
89#41×C79—41no—
90C2×Q8×C79—21no—
91C2×D4×C79—21no—
92C2×D4×C79—41no—
93C22×C4×C79—21no—
94C22×C4×C79—41no—
95C22×C4×C79C2×D411yescharacteristic
96Q16×C79—21no—
97Q16×C79C22⋊C412yes—
98SD16×C79—22no—
99SD16×C79—22no—
100D8×C79—21no—
101D8×C79C22⋊C412yes—
102M16×C79—22no—
103M16×C79—44no—
104M16×C79—41no—
105C2×C8×C79—21no—
106C2×C8×C79—21no—
107C2×C8×C79—44no—
108C2×C8×C79—41no—
109C2×C8×C79D811yescharacteristic
110C2×C8×C79D811yescharacteristic
111C2×C8×C79C22×C411yescharacteristic
112C2×C8×C79C2×D411yescharacteristic
113C2×C8×C79C2×D411yescharacteristic
114C2×C8×C79C2×D411yescharacteristic
115C2×C8×C79#4111yescharacteristic
116C2×C8×C79#4111yescharacteristic
117C42×C79—21no—
118C42×C79—21no—
119C23×C79—41no—
120C23×C79AGL(1,ℤ/8)11yescharacteristic
121Q8×C79—22no—
122Q8×C79—41no—
123D4×C79—22no—
124D4×C79—42no—
125D4×C79—42no—
126D4×C79—41no—
127C2×C4×C79—22no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—42no—
132C2×C4×C79—42no—
133C2×C4×C79C4×D411yescharacteristic
134C2×C4×C79#12011yescharacteristic
135C2×C4×C79#12111yescharacteristic
136C2×C4×C79C2×D811yescharacteristic
137C2×C4×C79#13511yescharacteristic
138C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
139C2×Dic79—42no—
140C2×Dic79—81no—
141C2×Dic79—81no—
142C2×Dic79—82no—
143C2×Dic79—81no—
144Dic158—42no—
145Dic158—42no—
146Dic158—81no—
147Dic158—81no—
148C8×C79—21no—
149C8×C79—21no—
150C8×C79—42no—
151C8×C79C2×C22⋊C411yescharacteristic
152C8×C79C4×D411yescharacteristic
153C8×C79#12111yescharacteristic
154C8×C79#12311yescharacteristic
155C22×C79—21no—
156C22×C79—21no—
157C22×C79—42no—
158C22×C79—81no—
159C22×C79—81no—
160C22×C79#44211yescharacteristic
161C22×C79#44911yescharacteristic
162C22×C79#45911yescharacteristic
163C4×C79—42no—
164C4×C79—42no—
165C4×C79#38611yescharacteristic
166C4×C79C4×D811yescharacteristic
167C4×C79#44711yescharacteristic
168C4×C79#46311yescharacteristic
169Dic79—82no—
170Dic79—161no—
171#13028—791nomaximal, Sylow
172C2×C79—21no—
173C2×C79—21no—
174C2×C79—82no—
175C2×C79#187911yescharacteristic
176C2×C79#221211yescharacteristic
177C2×C79#221411yescharacteristic
178C2×#510—791no—
179C2×#443D7911yescharacteristic
180C2×#362—791no—
181#2139—791no—
182#1969—791no—
183#1387—791no—
184#1376—791no—
185C79#1302811yescharacteristic, complemented, Sylow
186C2×#135C2×D7911yescharacteristic
187C22×Q16—791no—
188C2×#131—21no—
189#510—794no—
190#510—1582no—
191C2×Q32—1581no—
192C2×SD32—1581no—
193C2×M32—791no—
194C2×M32—791no—
195#443—22no—
196#443—22no—
197C2×#108C2×D7911yescharacteristic
198C2×#107—791no—
199C2×#104—21no—
200C2×#103—1581no—
201C2×C4×C8C2×D7911yescharacteristic
202#391—1581no—
203#362—1582no—
204#358—1581no—
205#357—1581no—
206C2×#41—21no—
207C22×Q8—1581no—
208#135—21no—
209#135—21no—
210#135C4×D7912yes—
211#135C22⋊D7912yes—
212C2×Q16—792no—
213C2×Q16—1581no—
214C2×Q16—1581no—
215C2×Q16—1581no—
216C2×Q16—1581no—
217C2×Q16—1581no—
218C2×Q16D31611yescharacteristic
219C2×SD16—21no—
220C2×D8D31611yescharacteristic
221#131—24no—
222#131—42no—
223C2×M16—21no—
224C2×M16—41no—
225C22×C8—41no—
226C22×C8C22⋊D7911yescharacteristic
227C22×C8C22×D7911yescharacteristic
228C2×C4⋊C4—1581no—
229C2×C42C22⋊D7911yescharacteristic
230Q32—1584no—
231SD32—1584no—
232M32—792no—
233M32—1581no—
234M32—1581no—
235M32—1581no—
236C2×C16—1581no—
237C2×C16—1581no—
238#108—21no—
239#108C22⋊D7912yes—
240#107—1581no—
241#107—1581no—
242#104—24no—
243#103—3161no—
244#103—3161no—
245C4×C8—21no—
246C4×C8—21no—
247#95—3162no—
248#41—22no—
249#41—22no—
250#41—41no—
251#41—41no—
252C2×Q8—21no—
253C2×Q8—3161no—
254C2×Q8—3161no—
255C2×Q8—3161no—
256C2×Q8—3161no—
257C2×Q8—3161no—
258C2×Q8—1582no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—3161no—
264C22×C4—3161no—
265C22×C4C2×C22⋊D7911yescharacteristic
266Q16—21no—
267Q16—3161no—
268Q16—3161no—
269Q16—1582no—
270Q16—1582no—
271Q16(C2×D79)⋊C412yes—
272SD16—22no—
273SD16—22no—
274D8—21no—
275D8(C2×D79)⋊C412yes—
276M16—22no—
277M16—44no—
278M16—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—44no—
282C2×C8—41no—
283C2×C8#1192960911yescharacteristic
284C2×C8#1192960911yescharacteristic
285C2×C8C2×C4×D7911yescharacteristic
286C2×C8C2×C22⋊D7911yescharacteristic
287C2×C8C2×C22⋊D7911yescharacteristic
288C2×C8C2×D31611yescharacteristic
289C2×C8#1192962411yescharacteristic
290C2×C8#1192962411yescharacteristic
291C4⋊C4—3161no—
292C4⋊C4—3161no—
293C42—21no—
294C42—21no—
295C16—3161no—
296C16—1582no—
297C23—41no—
298C23#1457918611yescharacteristic
299Q8—22no—
300Q8—41no—
301Q8—3162no—
302Q8—3162no—
303Q8—6321no—
304Q8—6321no—
305D4—22no—
306D4—42no—
307D4—42no—
308D4—41no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—3162no—
316C2×C4—6321no—
317C2×C4—6321no—
318C2×C4—6322no—
319C2×C4—6321no—
320C2×C4C4×C22⋊D7911yescharacteristic
321C2×C4#1457913011yescharacteristic
322C2×C4#1457913511yescharacteristic
323C2×C4C2×#1192960911yescharacteristic
324C2×C4#1457918111yescharacteristic
325C2×C4#1457918711yescharacteristic
326C8—21no—
327C8—21no—
328C8—42no—
329C8C2×(C2×D79)⋊C411yescharacteristic
330C8C4×C22⋊D7911yescharacteristic
331C8#1457913511yescharacteristic
332C8#1457914411yescharacteristic
333C22—21no—
334C22—21no—
335C22—42no—
336C22—81no—
337C22—81no—
338C22#1909867311yescharacteristic
339C22#1909870811yescharacteristic
340C22#1909875811yescharacteristic
341C4—42no—
342C4—42no—
343C4—6322no—
344C4—12641no—
345C4#1909844211yescharacteristic
346C4C4×#1192960911yescharacteristic
347C4#1909869711yescharacteristic
348C4#1909877811yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2#2376374811yescharacteristic
353C2#2376546911yescharacteristic
354C2#2376548111yescharacteristic
subgroupG/H
centerC22#19098758
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC2×#443×C79C2
socleC22×C79#459
prime radicalC2×#135×C79C22
cosocleC22×C8×C79C23
hypercenterC2×#443D79
hypocenterC79#13028
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
2. C2×#19098378C2semidirect
4. #23764261C2semidirect
5. #23761510C2semidirect
6. #23761482C2semidirect
10. #19099003C4semidirect
185. C79#13028semidirect

Maximal subgroups

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

subgroupindex
C2×#190990032
C2×#190983782
#237651122
#237642612
#237615102
#237614822
C2×#443×C792
#1302879