Finite Group Databasebeta

← #34181541

Subgroups — #34181541

Order 20224.

414 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772112C211yesmaximal, characteristic
2#23763537C211yesmaximal, characteristic, complemented
3C2×#19098315C211yesmaximal, characteristic, complemented
4#23762077C211yesmaximal, characteristic
5#23761275C211yesmaximal, characteristic, complemented
6#23761274C211yesmaximal, characteristic, complemented
7C79×#1547C211yesmaximal, characteristic
8C2×#14579168—21no—
9#19098641—21no—
10#19098641C412yes—
11D4×C79⋊C8—21no—
12D4×C79⋊C8C412yes—
13#19098587C2211yescharacteristic
14#19098587C2211yescharacteristic
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16#19098530—21no—
17C2×C4×C79⋊C8C2211yescharacteristic
18#19098327—21no—
19#19098327—21no—
20#19098315—21no—
21#19098315—21no—
22#19098309—21no—
23#19098307—21no—
24#532×C79C2211yescharacteristic
25#423×C79C2211yescharacteristic
26C2×C4⋊C8×C79C2211yescharacteristic
27#332×C79—21no—
28#331×C79—21no—
29#14579168—42no—
30#14579168—42no—
31C2×#11929608—21no—
32C2×#11929608—22no—
33C2×#11929608—21no—
34C22×C79⋊C8—22no—
35C22×C79⋊C8—21no—
36C22×C79⋊C8D411yescharacteristic
37C22×C79⋊C8D411yescharacteristic
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)C2×C412yes—
44C22⋊(C79⋊C8)—22no—
45C22⋊(C79⋊C8)—22no—
46C4×C79⋊C8C2×C411yescharacteristic
47C4×C79⋊C8C2×C411yescharacteristic
48C4×C79⋊C8D411yescharacteristic
49C4×C79⋊C8D411yescharacteristic
50C2×#41×C79—21no—
51C2×M16×C79—21no—
52C22×C8×C79—21no—
53C4⋊Q8×C79D411yescharacteristic
54#127×C79D411yescharacteristic
55#124×C79—21no—
56#121×C79—21no—
57#121×C79Q812yes—
58C4×D4×C79—21no—
59C4×D4×C79C2×C412yes—
60C2×C42×C79C2311yescharacteristic
61C4⋊C8×C79—21no—
62C4⋊C8×C79—21no—
63C4⋊C8×C79—21no—
64C4⋊C8×C79—21no—
65#11929608—44no—
66#11929608—41no—
67#11929608—41no—
68#11929608—41no—
69#11929608—41no—
70C2×C79⋊C8—22no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—21no—
73C2×C79⋊C8—21no—
74C2×C79⋊C8—21no—
75C2×C79⋊C8—44no—
76C2×C79⋊C8—41no—
77C2×C79⋊C8—41no—
78C2×C79⋊C8—81no—
79C2×C79⋊C8C22⋊C411yescharacteristic
80C2×C79⋊C8C22⋊C411yescharacteristic
81C2×C79⋊C8C22⋊C411yescharacteristic
82C2×C79⋊C8C22⋊C411yescharacteristic
83#41×C79—41no—
84#41×C79—41no—
85#41×C79—41no—
86#41×C79—41no—
87C2×Q8×C79—21no—
88C2×D4×C79—22no—
89C2×D4×C79—21no—
90C2×D4×C79—21no—
91C2×D4×C79C4⋊C412yes—
92C22×C4×C79—22no—
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×Q811yescharacteristic
95C22×C4×C79#4111yescharacteristic
96M16×C79—41no—
97M16×C79—41no—
98C2×C8×C79—41no—
99C2×C8×C79—41no—
100C2×C8×C79—81no—
101C4⋊C4×C79—21no—
102C4⋊C4×C79C4⋊C412yes—
103C22⋊C4×C79—22no—
104C22⋊C4×C79—22no—
105C42×C79C22×C411yescharacteristic
106C42×C79C2×D411yescharacteristic
107C42×C79C2×D411yescharacteristic
108C42×C79#4111yescharacteristic
109C23×C79—22no—
110C23×C79C22⋊Q811yescharacteristic
111Q8×C79—41no—
112Q8×C79—41no—
113D4×C79—42no—
114D4×C79—42no—
115D4×C79—42no—
116D4×C79—41no—
117D4×C79—41no—
118D4×C79—41no—
119D4×C79—41no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—22no—
123C2×C4×C79—41no—
124C2×C4×C79—42no—
125C2×C4×C79—42no—
126C2×C4×C79C2×C22⋊C411yescharacteristic
127C2×C4×C79C2×C4⋊C411yescharacteristic
128C2×C4×C79#11711yescharacteristic
129C2×C4×C79#12111yescharacteristic
130C2×C4×C79#12111yescharacteristic
131C2×C4×C79C22⋊Q811yescharacteristic
132C2×C4×C79#13511yescharacteristic
133C2×C4×C79#13511yescharacteristic
134C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
136C8×C79—81no—
137C8×C79—81no—
138C79⋊C8—21no—
139C79⋊C8—21no—
140C79⋊C8—21no—
141C79⋊C8—21no—
142C79⋊C8—42no—
143C79⋊C8—81no—
144C79⋊C8—81no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—42no—
148C22×C79—42no—
149C22×C79—42no—
150C22×C79#38011yescharacteristic
151C22×C79#47811yescharacteristic
152C22×C79#47811yescharacteristic
153C4×C79—21no—
154C4×C79—21no—
155C4×C79—21no—
156C4×C79—42no—
157C4×C79#41611yescharacteristic
158C4×C79#41811yescharacteristic
159C4×C79#46311yescharacteristic
160C4×C79#46311yescharacteristic
161C4×C79#47211yescharacteristic
162C4×C79#47311yescharacteristic
163#8688—791nomaximal, Sylow
164C2×C79—21no—
165C2×C79—21no—
166C2×C79—82no—
167C2×C79#180411yescharacteristic
168C2×C79#181311yescharacteristic
169C2×C79#181611yescharacteristic
170#2931—791no—
171#1837—791no—
172C2×#335—791no—
173#1547—791no—
174#1547D7911yescharacteristic
175#1292—791no—
176#1292—791no—
177C79#868811yescharacteristic, complemented, Sylow
178C2×#131—1581no—
179#532C2×D7911yescharacteristic
180#436—792no—
181#436—1581no—
182C8×D4—792no—
183C8×D4—1581no—
184#423—791no—
185#423—791no—
186#423C2×D7911yescharacteristic
187C2×C4⋊C8—791no—
188C2×C4⋊C8C2×D7911yescharacteristic
189#408—1581no—
190C2×C4×C8—791no—
191#341—1581no—
192#341—1581no—
193#335—1581no—
194#335—1581no—
195#332—21no—
196#332—1581no—
197#331—21no—
198#331—1581no—
199C2×#41—21no—
200#131—3162no—
201#131—3162no—
202C2×M16—21no—
203C2×M16—1581no—
204C2×M16—1582no—
205C2×M16—1581no—
206C22×C8—21no—
207C22×C8—791no—
208C22×C8—791no—
209C22×C8—1582no—
210C22×C8—1581no—
211C4⋊Q8C22⋊D7911yescharacteristic
212#127C22⋊D7911yescharacteristic
213#124—21no—
214#121—21no—
215#121Dic15812yes—
216C4×D4—21no—
217C4×D4C4×D7912yes—
218C2×C42C22×D7911yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—792no—
224C4⋊C8—1581no—
225C4⋊C8—1581no—
226C4⋊C8—1581no—
227C4⋊C8—1581no—
228C4⋊C8—1581no—
229C22⋊C8—1582no—
230C22⋊C8—1582no—
231C4×C8—791no—
232C4×C8—791no—
233C4×C8—791no—
234C4×C8—791no—
235#41—41no—
236#41—41no—
237#41—41no—
238#41—41no—
239C2×Q8—21no—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4Dic79⋊C412yes—
244C22×C4—22no—
245C22×C4D4×D7911yescharacteristic
246C22×C4C2×Dic15811yescharacteristic
247C22×C4#1192962311yescharacteristic
248M16—41no—
249M16—41no—
250M16—3161no—
251M16—3161no—
252M16—3164no—
253M16—3161no—
254M16—3161no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—81no—
258C2×C8—3164no—
259C2×C8—3161no—
260C2×C8—3161no—
261C2×C8—791no—
262C2×C8—791no—
263C2×C8—791no—
264C2×C8—791no—
265C2×C8—6321no—
266C2×C8—1582no—
267C2×C8—1581no—
268C2×C8—1581no—
269C2×C8—1581no—
270C2×C8—1581no—
271C4⋊C4—21no—
272C4⋊C4Dic79⋊C412yes—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C42C2×C4×D7911yescharacteristic
276C42D4×D7911yescharacteristic
277C42C2×C22⋊D7911yescharacteristic
278C42#1192962311yescharacteristic
279C23—22no—
280C23#1457913811yescharacteristic
281Q8—41no—
282Q8—41no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—41no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4C22⋊C4×D7911yescharacteristic
297C2×C4C2×Dic79⋊C411yescharacteristic
298C2×C4#1457911711yescharacteristic
299C2×C4#1457913311yescharacteristic
300C2×C4#1457913311yescharacteristic
301C2×C4#1457913811yescharacteristic
302C2×C4#1457918011yescharacteristic
303C2×C4#1457918211yescharacteristic
304C2×C4#1457918511yescharacteristic
305C2×C4#1457918811yescharacteristic
306C8—81no—
307C8—81no—
308C8—3162no—
309C8—6321no—
310C8—6321no—
311C8—1581no—
312C8—1581no—
313C8—1581no—
314C8—1581no—
315C22—21no—
316C22—21no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22#1909842111yescharacteristic
321C22#1909886111yescharacteristic
322C22#1909886511yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4#1909855611yescharacteristic
328C4#1909856611yescharacteristic
329C4#1909877611yescharacteristic
330C4#1909878011yescharacteristic
331C4#1909883111yescharacteristic
332C4#1909883611yescharacteristic
333C2—21no—
334C2—21no—
335C2—82no—
336C2#2376340811yescharacteristic
337C2#2376344811yescharacteristic
338C2#2376346211yescharacteristic
subgroupG/H
centerC22#19098421
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1547C2
socleC22×C79#380
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#1547D79
hypocenterC79#8688
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #23763537C2semidirect
3. C2×#19098315C2semidirect
5. #23761275C2semidirect
6. #23761274C2semidirect
177. C79#8688semidirect

Maximal subgroups

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

subgroupindex
#237721122
#237635372
C2×#190983152
#237620772
#237612752
#237612742
C79×#15472
#868879