Finite Group Databasebeta

← #34181586

Subgroups — #34181586

Order 20224.

384 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772376C211yesmaximal, characteristic
2C4×#14579094C211yesmaximal, characteristic, complemented
3#23762161C211yesmaximal, characteristic, complemented
4#23761925C211yesmaximal, characteristic
5#23761256C211yesmaximal, characteristic, complemented
6#23761255C211yesmaximal, characteristic, complemented
7C79×#1509C211yesmaximal, characteristic
8#19098694C411yescharacteristic, complemented
9Q8×C79⋊C8C411yescharacteristic, complemented
10#19098641C411yescharacteristic, complemented
11#19098635—21no—
12D4×C79⋊C8—21no—
13D4×C79⋊C8C411yescharacteristic, complemented
14#19098624—21no—
15#19098620C2211yescharacteristic
16#19098620C2211yescharacteristic
17C2×#14579094—21no—
18#19098519—21no—
19C2×C4×C79⋊C8C2211yescharacteristic
20C4×C4⋊Dic79C2211yescharacteristic
21#19098325—21no—
22#19098315—21no—
23#19098313—21no—
24#19098300—21no—
25#19098297—21no—
26#529×C79C2211yescharacteristic
27#432×C79C2211yescharacteristic
28C4×C4⋊C4×C79C2211yescharacteristic
29#328×C79—21no—
30#327×C79—21no—
31C2×#11929608—21no—
32C22×C79⋊C8—21no—
33C22×C79⋊C8D411yescharacteristic
34C22×C79⋊C8D411yescharacteristic
35C2×C4⋊Dic79—21no—
36C2×C4×Dic79—21no—
37#14579094—24no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)C2×C411yescharacteristic
42C4⋊(C79⋊C8)C2×C411yescharacteristic
43C22⋊(C79⋊C8)—21no—
44C22⋊(C79⋊C8)—21no—
45C22⋊(C79⋊C8)—21no—
46C22⋊(C79⋊C8)—21no—
47#14579072—21no—
48#14579072—21no—
49#14579072—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8C2×C411yescharacteristic
52C4×C79⋊C8C2×C411yescharacteristic
53C4×C79⋊C8D411yescharacteristic
54C4×C79⋊C8D411yescharacteristic
55C79⋊#95—21no—
56#126×C79—21no—
57#125×C79Q811yescharacteristic
58#124×C79Q811yescharacteristic
59#123×C79—21no—
60C22⋊Q8×C79D411yescharacteristic
61#121×C79D411yescharacteristic
62C4×Q8×C79C2×C411yescharacteristic
63C4×D4×C79—21no—
64C4×D4×C79C2×C411yescharacteristic
65#117×C79—21no—
66C2×C4⋊C4×C79—21no—
67C2×C42×C79—21no—
68C2×C42×C79C2311yescharacteristic
69C4⋊C8×C79—21no—
70C22⋊C8×C79—21no—
71#97×C79—21no—
72#95×C79—21no—
73C22×Dic79—41no—
74#11929608—42no—
75C2×C79⋊C8—22no—
76C2×C79⋊C8—21no—
77C2×C79⋊C8—21no—
78C2×C79⋊C8—41no—
79C2×C79⋊C8—42no—
80C2×C79⋊C8—41no—
81C2×C79⋊C8C22⋊C411yescharacteristic
82C2×C79⋊C8C22⋊C411yescharacteristic
83C2×C79⋊C8C22⋊C411yescharacteristic
84C2×C79⋊C8C22⋊C411yescharacteristic
85C4⋊Dic79—22no—
86C4⋊Dic79—41no—
87C4×Dic79—81no—
88C2×Q8×C79C4⋊C411yescharacteristic
89C2×D4×C79—21no—
90C2×D4×C79C4⋊C411yescharacteristic
91C22×C4×C79—21no—
92C22×C4×C79—41no—
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79#4111yescharacteristic
96C2×C8×C79—41no—
97C4⋊C4×C79—21no—
98C4⋊C4×C79—21no—
99C4⋊C4×C79—21no—
100C4⋊C4×C79—21no—
101C4⋊C4×C79—22no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79C4⋊C411yescharacteristic
104C4⋊C4×C79C4⋊C411yescharacteristic
105C22⋊C4×C79—21no—
106C22⋊C4×C79—21no—
107C22⋊C4×C79—21no—
108C22⋊C4×C79—21no—
109C22⋊C4×C79—21no—
110C42×C79—21no—
111C42×C79—81no—
112C42×C79C22×C411yescharacteristic
113C42×C79C2×D411yescharacteristic
114C42×C79C2×Q811yescharacteristic
115C42×C79#4111yescharacteristic
116C23×C79—21no—
117C23×C79#12111yescharacteristic
118Q8×C79—41no—
119D4×C79—41no—
120D4×C79—41no—
121D4×C79—41no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—42no—
133C2×C4×C79—81no—
134C2×C4×C79—81no—
135C2×C4×C79C2×C22⋊C411yescharacteristic
136C2×C4×C79C2×C4⋊C411yescharacteristic
137C2×C4×C79#11711yescharacteristic
138C2×C4×C79#12111yescharacteristic
139C2×C4×C79C22⋊Q811yescharacteristic
140C2×C4×C79C22⋊Q811yescharacteristic
141C2×Dic79—42no—
142C2×Dic79—81no—
143C2×Dic79—81no—
144C8×C79—81no—
145C79⋊C8—21no—
146C79⋊C8—21no—
147C79⋊C8—21no—
148C79⋊C8—41no—
149C79⋊C8—41no—
150C79⋊C8—81no—
151C79⋊C8#10412yes—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79#38011yescharacteristic
158C22×C79#47211yescharacteristic
159C22×C79#47311yescharacteristic
160C4×C79—21no—
161C4×C79—21no—
162C4×C79—21no—
163C4×C79—21no—
164C4×C79—21no—
165C4×C79—41no—
166C4×C79—41no—
167C4×C79—41no—
168C4×C79—82no—
169C4×C79C2×#10411yescharacteristic
170C4×C79#42111yescharacteristic
171Dic79—82no—
172#8697—791nomaximal, Sylow
173C2×C79—21no—
174C2×C79—21no—
175C2×C79—81no—
176C2×C79#180611yescharacteristic
177C2×C79#181211yescharacteristic
178C2×C79#181611yescharacteristic
179#2966—791no—
180C4×#107—791no—
181#1568—791no—
182#1509—791no—
183#1509D7911yescharacteristic
184#1285—791no—
185#1285—791no—
186C79#869711yescharacteristic, complemented, Sylow
187#529C2×D7911yescharacteristic
188#446—791no—
189C8×Q8—791no—
190#436—791no—
191#435—1581no—
192C8×D4—791no—
193C8×D4—1581no—
194#433—1581no—
195#432—791no—
196#432—791no—
197#432C2×D7911yescharacteristic
198C2×#107—1581no—
199#405—1581no—
200C2×C4×C8—791no—
201C4×C4⋊C4—791no—
202C4×C4⋊C4C2×D7911yescharacteristic
203#340—1581no—
204#335—1581no—
205#334—1581no—
206#328—21no—
207#328—1581no—
208#327—21no—
209#327—1581no—
210C2×M16—1581no—
211C22×C8—791no—
212C22×C8—791no—
213C22×C8—1581no—
214#126—21no—
215#125Dic15811yescharacteristic
216#124Dic15811yescharacteristic
217#123—21no—
218C22⋊Q8C22⋊D7911yescharacteristic
219#121C22⋊D7911yescharacteristic
220C4×Q8C4×D7911yescharacteristic
221C4×D4—21no—
222C4×D4C4×D7911yescharacteristic
223#117—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1581no—
226C2×C42—21no—
227C2×C42—1581no—
228C2×C42C22×D7911yescharacteristic
229#107—1584no—
230C4⋊C8—21no—
231C4⋊C8—791no—
232C4⋊C8—791no—
233C4⋊C8—1581no—
234C4⋊C8—1581no—
235C4⋊C8—1581no—
236C22⋊C8—21no—
237C22⋊C8—1581no—
238C22⋊C8—1581no—
239C22⋊C8—1581no—
240C22⋊C8—1581no—
241#97—21no—
242#97—1581no—
243#97—1581no—
244#97—1581no—
245C4×C8—791no—
246C4×C8—791no—
247C4×C8—791no—
248C4×C8—791no—
249C4×C8—1581no—
250#95—21no—
251#95—1581no—
252C2×Q8Dic79⋊C411yescharacteristic
253C2×D4—21no—
254C2×D4Dic79⋊C411yescharacteristic
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—3161no—
258C22×C4D4×D7911yescharacteristic
259C22×C4C2×C22⋊D7911yescharacteristic
260C22×C4#1192962311yescharacteristic
261M16—3162no—
262C2×C8—41no—
263C2×C8—3162no—
264C2×C8—3161no—
265C2×C8—3161no—
266C2×C8—791no—
267C2×C8—791no—
268C2×C8—791no—
269C2×C8—791no—
270C2×C8—1581no—
271C2×C8—1581no—
272C2×C8—1582no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—3161no—
280C4⋊C4—1582no—
281C4⋊C4Dic79⋊C411yescharacteristic
282C4⋊C4Dic79⋊C411yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C42—21no—
289C42—81no—
290C42—6321no—
291C42C2×C4×D7911yescharacteristic
292C42D4×D7911yescharacteristic
293C42C2×Dic15811yescharacteristic
294C42#1192962311yescharacteristic
295C23—21no—
296C23#1457913311yescharacteristic
297Q8—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—3162no—
315C2×C4—6321no—
316C2×C4—6321no—
317C2×C4C22⋊C4×D7911yescharacteristic
318C2×C4C2×Dic79⋊C411yescharacteristic
319C2×C4#1457911711yescharacteristic
320C2×C4#1457913311yescharacteristic
321C2×C4#1457913811yescharacteristic
322C2×C4#1457913811yescharacteristic
323C8—81no—
324C8—3161no—
325C8—3161no—
326C8—792no—
327C8—6321no—
328C8—1581no—
329C8—1581no—
330C8—1581no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1909842111yescharacteristic
337C22#1909883111yescharacteristic
338C22#1909883611yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—82no—
348C4—6322no—
349C4#104×D7911yescharacteristic
350C4#1909857811yescharacteristic
351C2—21no—
352C2—21no—
353C2—81no—
354C2#2376341811yescharacteristic
355C2#2376344411yescharacteristic
356C2#2376346211yescharacteristic
subgroupG/H
centerC22#19098421
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1509C2
socleC22×C79#380
prime radical#121×C79D4
cosocleC2×C42×C79C23
hypercenter#1509D79
hypocenterC79#8697
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. C4×#14579094C2semidirect
3. #23762161C2semidirect
5. #23761256C2semidirect
6. #23761255C2semidirect
8. #19098694C4semidirect
9. Q8×C79⋊C8C4semidirect
10. #19098641C4semidirect
13. D4×C79⋊C8C4semidirect
186. C79#8697semidirect

Maximal subgroups

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

subgroupindex
#237723762
C4×#145790942
#237621612
#237619252
#237612562
#237612552
C79×#15092
#869779