Finite Group Databasebeta

← #34187569

Subgroups — #34187569

Order 20224.

642 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098939C211yesmaximal, characteristic
2#23764327C211yesmaximal, characteristic, complemented
3#23763269C211yesmaximal, characteristic, complemented
4C2×#19098313C211yesmaximal, characteristic, complemented
5C2×#19098307C211yesmaximal, characteristic, complemented
6C2×#19098306C211yesmaximal, characteristic, complemented
7C79×#1982C211yesmaximal, characteristic, complemented
8C22×#11929609—21no—
9C22×#11929609—21no—
10C2×#14579158C2211yescharacteristic
11#19098939—22no—
12#19098939C414yescomplemented
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×C4⋊(C79⋊C8)C2211yescharacteristic
15C2×#14579081—22no—
16C2×#14579072—21no—
17C2×C4×C79⋊C8C2211yescharacteristic
18#19098318—21no—
19#19098313—22no—
20#19098307—22no—
21#19098306—22no—
22C2×#127×C79C2211yescharacteristic
23C2×C4⋊C8×C79C2211yescharacteristic
24C2×#102×C79—22no—
25C4×C4⋊C4×C79C2211yescharacteristic
26C22×D316—21no—
27C22×D316—21no—
28C22×D316—21no—
29C2×#11929609—24no—
30C2×#11929609—24no—
31C2×#11929609—42no—
32C2×#11929609—42no—
33C2×#11929609—42no—
34C2×#11929609—42no—
35C22×C79⋊C8—21no—
36C22×C79⋊C8—21no—
37C22×C79⋊C8D411yescharacteristic
38C22×C79⋊C8D411yescharacteristic
39#14579158—21no—
40#14579158—21no—
41#14579158—21no—
42#14579158C2×C412yes—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)D412yes—
47#14579081—44no—
48#14579072—42no—
49C4×C79⋊C8C2×C412yes—
50C4×C79⋊C8D412yes—
51C22×D4×C79—21no—
52C22×D4×C79—21no—
53C22×D4×C79—21no—
54C22×C8×C79—21no—
55#127×C79—21no—
56#127×C79—21no—
57#127×C79—21no—
58#127×C79C2×C412yes—
59C2×C4⋊C4×C79D412yes—
60C2×C42×C79—21no—
61C2×C42×C79C2311yescharacteristic
62C4⋊C8×C79—21no—
63C4⋊C8×C79—21no—
64#102×C79—44no—
65#95×C79—21no—
66C23×D79—41no—
67C2×D316—22no—
68C2×D316—22no—
69C2×D316—22no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—42no—
76C2×D316—41no—
77C2×D316—41no—
78C2×D316—42no—
79C2×D316—82no—
80#11929609—48no—
81#11929609—48no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—21no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—42no—
89C2×C79⋊C8C22⋊C412yes—
90C2×C79⋊C8C22⋊C412yes—
91C2×C79⋊C8D812yes—
92C2×C79⋊C8SD1612yes—
93C24×C79—41no—
94C2×D4×C79—22no—
95C2×D4×C79—22no—
96C2×D4×C79—22no—
97C2×D4×C79—41no—
98C2×D4×C79—41no—
99C2×D4×C79—41no—
100C2×D4×C79—41no—
101C2×D4×C79—41no—
102C2×D4×C79—42no—
103C2×D4×C79—42no—
104C2×D4×C79—41no—
105C2×D4×C79—41no—
106C2×D4×C79—82no—
107C22×C4×C79—22no—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79#4111yescharacteristic
111C2×C8×C79—41no—
112C2×C8×C79—42no—
113C4⋊C4×C79—22no—
114C4⋊C4×C79—22no—
115C42×C79—41no—
116C42×C79—41no—
117C42×C79C22×C411yescharacteristic
118C42×C79C2×D411yescharacteristic
119C42×C79C2×D411yescharacteristic
120C42×C79#4111yescharacteristic
121C23×C79—42no—
122C23×C79—81no—
123C23×C79—81no—
124C23×C79—82no—
125C23×C79—82no—
126C23×C79#12111yescharacteristic
127C22×D79—42no—
128C22×D79—81no—
129C22×D79—81no—
130C22×D79—82no—
131C22×D79—82no—
132D4×C79—42no—
133D4×C79—42no—
134D4×C79—44no—
135D4×C79—82no—
136D4×C79—82no—
137D4×C79—81no—
138D4×C79—81no—
139D4×C79—81no—
140D4×C79—81no—
141C2×C4×C79—22no—
142C2×C4×C79—42no—
143C2×C4×C79—42no—
144C2×C4×C79—42no—
145C2×C4×C79C2×C22⋊C411yescharacteristic
146C2×C4×C79C4×D411yescharacteristic
147C2×C4×C79C4×D411yescharacteristic
148C2×C4×C79#12111yescharacteristic
149C2×C4×C79#12411yescharacteristic
150C2×C4×C79#12711yescharacteristic
151C2×C4×C79C2×D811yescharacteristic
152C2×C4×C79C2×SD1611yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C79AGL(1,ℤ/8)12yes—
155C2×C4×C79AGL(1,ℤ/8)12yes—
156C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
157D316—42no—
158D316—42no—
159D316—44no—
160D316—82no—
161D316—82no—
162D316—81no—
163D316—81no—
164D316—81no—
165D316—81no—
166C8×C79—81no—
167C79⋊C8—22no—
168C79⋊C8—42no—
169C79⋊C8—81no—
170C79⋊C8#10214yescomplemented
171C22×C79—82no—
172C22×C79—82no—
173C22×C79—82no—
174C22×C79—162no—
175C22×C79—162no—
176C22×C79#39011yescharacteristic
177C22×C79#45911yescharacteristic
178C22×C79#46011yescharacteristic
179C22×C79#46912yes—
180C22×C79#47212yes—
181C2×D79—82no—
182C2×D79—82no—
183C2×D79—82no—
184C2×D79—162no—
185C2×D79—162no—
186C4×C79—21no—
187C4×C79—21no—
188C4×C79—82no—
189C4×C79C2×#10211yescharacteristic
190C4×C79#41811yescharacteristic
191C4×C79#44212yes—
192C4×C79#45911yescharacteristic
193C4×C79#46011yescharacteristic
194C4×C79#49611yescharacteristic
195C4×C79#49611yescharacteristic
196#9743—791nomaximal, Sylow
197C2×C79—162no—
198C2×C79#168712yes—
199C2×C79#168812yes—
200C2×C79#197211yescharacteristic
201C2×C79#197811yescharacteristic
202C2×C79#198211yescharacteristic
203D79—162no—
204C2×#493—791no—
205#1982—791no—
206#1982D7911yescharacteristic, complemented
207#1770—791no—
208C2×#334—791no—
209C2×#331—791no—
210C2×#331—791no—
211C79#974311yescharacteristic, complemented, Sylow
212C22×D8—1581no—
213C22×D8—1581no—
214C2×#127—791no—
215C2×#127C2×D7911yescharacteristic
216#493—794no—
217#493—1582no—
218C2×C4⋊C8—791no—
219C2×C4⋊C8—791no—
220C2×C4⋊C8C2×D7911yescharacteristic
221C2×#102—22no—
222C2×#102—1582no—
223C2×#97—1581no—
224C2×C4×C8—791no—
225C4×C4⋊C4C2×D7911yescharacteristic
226#336—1581no—
227#334—1582no—
228#331—1582no—
229#331—1582no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—1581no—
234C22×D4—1581no—
235C22×D4—1581no—
236C2×D8—3162no—
237C2×D8—3162no—
238C2×D8—3162no—
239C2×D8—3162no—
240C2×D8—1584no—
241C2×D8—1584no—
242C22×C8—21no—
243C22×C8—791no—
244C22×C8—791no—
245C22×C8—1581no—
246C22×C8—1581no—
247#127—21no—
248#127—21no—
249#127—21no—
250#127—792no—
251#127—1581no—
252#127—1581no—
253#127—1581no—
254#127C4×D7912yes—
255C2×C4⋊C4C22⋊D7912yes—
256C2×C42—21no—
257C2×C42C22×D7911yescharacteristic
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—792no—
261C4⋊C8—1581no—
262C4⋊C8—1581no—
263C4⋊C8—1581no—
264#102—44no—
265#102—3164no—
266#97—3162no—
267C4×C8—792no—
268C4×C8—792no—
269#95—21no—
270C24—41no—
271C24—3161no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—82no—
285C2×D4—3161no—
286C2×D4—3161no—
287C2×D4—3161no—
288C2×D4—3161no—
289C2×D4—3161no—
290C2×D4—3161no—
291C2×D4—3162no—
292C2×D4—3162no—
293C2×D4—3161no—
294C2×D4—6322no—
295C2×D4—1582no—
296C2×D4—1582no—
297C2×D4—1582no—
298C22×C4—22no—
299C22×C4D4×D7911yescharacteristic
300C22×C4C2×C22⋊D7911yescharacteristic
301C22×C4#1192962311yescharacteristic
302D8—3168no—
303D8—3168no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—3161no—
307C2×C8—3161no—
308C2×C8—3161no—
309C2×C8—3162no—
310C2×C8—792no—
311C2×C8—792no—
312C2×C8—792no—
313C2×C8—792no—
314C2×C8—1582no—
315C2×C8—1581no—
316C2×C8—1581no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C42—41no—
320C42—41no—
321C42C2×C4×D7911yescharacteristic
322C42D4×D7911yescharacteristic
323C42D4×D7911yescharacteristic
324C42#1192962411yescharacteristic
325C23—42no—
326C23—81no—
327C23—81no—
328C23—82no—
329C23—82no—
330C23—3162no—
331C23—6321no—
332C23—6321no—
333C23—6322no—
334C23—6322no—
335C23#1457913311yescharacteristic
336D4—44no—
337D4—42no—
338D4—42no—
339D4—82no—
340D4—82no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—3164no—
346D4—3162no—
347D4—3162no—
348D4—6322no—
349D4—6322no—
350D4—6321no—
351D4—6321no—
352D4—6321no—
353D4—6321no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4C22⋊C4×D7911yescharacteristic
359C2×C4(C22⋊D79)⋊C411yescharacteristic
360C2×C4C4×C22⋊D7911yescharacteristic
361C2×C4#1457913411yescharacteristic
362C2×C4#1457914711yescharacteristic
363C2×C4#1457915711yescharacteristic
364C2×C4D8×D7911yescharacteristic
365C2×C4SD16×D7911yescharacteristic
366C2×C4#1457918511yescharacteristic
367C2×C4#1457918612yes—
368C2×C4#1457918712yes—
369C2×C4#1457918811yescharacteristic
370C8—81no—
371C8—3162no—
372C8—794no—
373C8—6321no—
374C8—1582no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—162no—
379C22—162no—
380C22—6322no—
381C22—6322no—
382C22—6322no—
383C22—12642no—
384C22—12642no—
385C22#1909846311yescharacteristic
386C22#1909875611yescharacteristic
387C22#1909876111yescharacteristic
388C22#1909881412yes—
389C22#1909883112yes—
390C4—21no—
391C4—21no—
392C4—82no—
393C4#102×D7911yescharacteristic
394C4#1909856611yescharacteristic
395C4#1909867312yes—
396C4#1909875711yescharacteristic
397C4#1909876211yescharacteristic
398C4#1909895011yescharacteristic
399C4#1909895311yescharacteristic
400C2—162no—
401C2—12642no—
402C2#2376287712yes—
403C2#2376288212yes—
404C2#2376427511yescharacteristic
405C2#2376430911yescharacteristic
406C2#2376432511yescharacteristic
subgroupG/H
centerC23#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1982C2
socleC23×C79#121
prime radicalC2×#19098939C2
cosocleC2×C42×C79C23
hypercenter#1982D79
hypocenterC79#9743
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. #23764327C2semidirect
3. #23763269C2semidirect
4. C2×#19098313C2semidirect
5. C2×#19098307C2semidirect
6. C2×#19098306C2semidirect
7. C79×#1982C2semidirect
12. #19098939C4semidirect
170. C79⋊C8#102semidirect
206. #1982D79semidirect
211. C79#9743semidirect

Maximal subgroups

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

subgroupindex
C2×#190989392
#237643272
#237632692
C2×#190983132
C2×#190983072
C2×#190983062
C79×#19822
#974379