Finite Group Databasebeta

← #34181481

Subgroups — #34181481

Order 20224.

434 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772112C211yesmaximal, characteristic
2#23763562C211yesmaximal, characteristic, complemented
3#23762146C211yesmaximal, characteristic, complemented
4#23761960C211yesmaximal, characteristic
5#23761259C211yesmaximal, characteristic, complemented
6#23761258C211yesmaximal, characteristic, complemented
7C79×#1517C211yesmaximal, characteristic
8C2×#14579168—21no—
9#19098641—21no—
10#19098641C412yescomplemented
11D4×C79⋊C8—21no—
12D4×C79⋊C8C412yescomplemented
13#19098616C2211yescharacteristic
14C2×#14579094—21no—
15C2×#14579092—21no—
16#19098587C2211yescharacteristic
17#19098530—21no—
18C2×C4×C79⋊C8C2211yescharacteristic
19#19098429C2211yescharacteristic
20#19098315—21no—
21#19098313—21no—
22#19098297—22no—
23#532×C79C2211yescharacteristic
24#431×C79C2211yescharacteristic
25#382×C79C2211yescharacteristic
26#327×C79—22no—
27#14579168—42no—
28#14579168—42no—
29C2×#11929608—21no—
30C2×#11929608—22no—
31C22×C79⋊C8—22no—
32C22×C79⋊C8D411yescharacteristic
33C22×C79⋊C8D411yescharacteristic
34C2×C4⋊Dic79—21no—
35#14579094—42no—
36#14579092—42no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)C2×C412yes—
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)—22no—
42C22⋊(C79⋊C8)—21no—
43C4×C79⋊C8—21no—
44C4×C79⋊C8C2×C411yescharacteristic
45C4×C79⋊C8C2×C411yescharacteristic
46C4×C79⋊C8D411yescharacteristic
47C4×C79⋊C8D411yescharacteristic
48C79⋊#95—21no—
49C79⋊#95—21no—
50C2×#41×C79—21no—
51C4⋊Q8×C79Q811yescharacteristic
52#127×C79Q811yescharacteristic
53#124×C79—21no—
54#121×C79—21no—
55#121×C79D412yes—
56C4×D4×C79—21no—
57C4×D4×C79C2×C412yes—
58C2×C4⋊C4×C79—21no—
59C2×C42×C79C2311yescharacteristic
60C4⋊C8×C79—21no—
61C22⋊C8×C79—21no—
62C4×C8×C79—21no—
63#95×C79—21no—
64#95×C79—21no—
65C22×Dic79—41no—
66#11929608—44no—
67#11929608—41no—
68#11929608—41no—
69C2×C79⋊C8—22no—
70C2×C79⋊C8—21no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—44no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8C22⋊C411yescharacteristic
75C2×C79⋊C8C22⋊C411yescharacteristic
76C2×C79⋊C8C22⋊C411yescharacteristic
77C2×C79⋊C8C22⋊C411yescharacteristic
78C2×C79⋊C8D812yes—
79C2×C79⋊C8SD1612yes—
80C4⋊Dic79—41no—
81C4⋊Dic79—41no—
82#41×C79—41no—
83#41×C79—41no—
84#41×C79—41no—
85#41×C79—41no—
86C2×Q8×C79—21no—
87C2×D4×C79—22no—
88C2×D4×C79—21no—
89C2×D4×C79—21no—
90C2×D4×C79C4⋊C412yes—
91C22×C4×C79—22no—
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—41no—
99C4⋊C4×C79—41no—
100C4⋊C4×C79C4⋊C412yes—
101C22⋊C4×C79—22no—
102C22⋊C4×C79—22no—
103C42×C79C22×C411yescharacteristic
104C42×C79C2×D411yescharacteristic
105C42×C79C2×Q811yescharacteristic
106C42×C79#4111yescharacteristic
107C23×C79—22no—
108C23×C79#12111yescharacteristic
109Q8×C79—41no—
110Q8×C79—41no—
111D4×C79—42no—
112D4×C79—42no—
113D4×C79—42no—
114D4×C79—42no—
115D4×C79—41no—
116D4×C79—41no—
117C2×C4×C79—22no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—42no—
121C2×C4×C79—42no—
122C2×C4×C79—81no—
123C2×C4×C79—81no—
124C2×C4×C79—81no—
125C2×C4×C79C2×C22⋊C411yescharacteristic
126C2×C4×C79C2×C4⋊C411yescharacteristic
127C2×C4×C79#11711yescharacteristic
128C2×C4×C79#12111yescharacteristic
129C2×C4×C79C22⋊Q811yescharacteristic
130C2×C4×C79C22⋊Q811yescharacteristic
131C2×C4×C79C2×D811yescharacteristic
132C2×C4×C79C2×SD1611yescharacteristic
133C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C79#13711yescharacteristic
135C2×Dic79—81no—
136C2×Dic79—81no—
137C2×Dic79—81no—
138C8×C79—81no—
139C79⋊C8—21no—
140C79⋊C8—21no—
141C79⋊C8—42no—
142C79⋊C8—81no—
143C79⋊C8#10212yes—
144C79⋊C8#10212yes—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—42no—
148C22×C79—42no—
149C22×C79—42no—
150C22×C79#38011yescharacteristic
151C22×C79#45911yescharacteristic
152C22×C79#46111yescharacteristic
153C4×C79—21no—
154C4×C79—21no—
155C4×C79—21no—
156C4×C79—21no—
157C4×C79—42no—
158C4×C79—161no—
159C4×C79C2×#10211yescharacteristic
160C4×C79#41711yescharacteristic
161C4×C79#47411yescharacteristic
162C4×C79#47611yescharacteristic
163Dic79—161no—
164#8676—791nomaximal, Sylow
165C2×C79—21no—
166C2×C79—21no—
167C2×C79—82no—
168C2×C79#180211yescharacteristic
169C2×C79#181211yescharacteristic
170C2×C79#181411yescharacteristic
171#2931—791no—
172#1846—791no—
173#1564—791no—
174#1517—791no—
175#1517D7911yescharacteristic
176#1286—791no—
177#1286—791no—
178C79#867611yescharacteristic, complemented, Sylow
179C2×#131—1581no—
180#532C2×D7911yescharacteristic
181#436—792no—
182#436—1581no—
183C8×D4—792no—
184C8×D4—1581no—
185#431—791no—
186#431C2×D7911yescharacteristic
187C2×#107—1581no—
188C2×#106—1581no—
189#423—791no—
190#408—1581no—
191C2×C4×C8—791no—
192#382—791no—
193#382C2×D7911yescharacteristic
194#335—1581no—
195#334—1581no—
196#327—22no—
197#327—1582no—
198C2×#41—21no—
199#131—3162no—
200#131—3162no—
201C2×M16—1582no—
202C2×M16—1581no—
203C22×C8—791no—
204C22×C8—791no—
205C22×C8—1582no—
206C4⋊Q8Dic15811yescharacteristic
207#127Dic15811yescharacteristic
208#124—21no—
209#121—21no—
210#121C22⋊D7912yes—
211C4×D4—21no—
212C4×D4C4×D7912yes—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—1581no—
215C2×C42C22×D7911yescharacteristic
216#107—3162no—
217#106—3162no—
218C4⋊C8—21no—
219C4⋊C8—792no—
220C4⋊C8—1581no—
221C4⋊C8—1581no—
222C22⋊C8—21no—
223C22⋊C8—1581no—
224C22⋊C8—1582no—
225C22⋊C8—1582no—
226C4×C8—21no—
227C4×C8—791no—
228C4×C8—791no—
229C4×C8—791no—
230C4×C8—791no—
231C4×C8—1581no—
232#95—21no—
233#95—21no—
234#95—1581no—
235#95—1581no—
236#41—41no—
237#41—41no—
238#41—41no—
239#41—41no—
240C2×Q8—21no—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4Dic79⋊C412yes—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—3161no—
248C22×C4D4×D7911yescharacteristic
249C22×C4C2×C22⋊D7911yescharacteristic
250C22×C4#1192962311yescharacteristic
251M16—3161no—
252M16—3161no—
253M16—3164no—
254C2×C8—41no—
255C2×C8—3164no—
256C2×C8—3161no—
257C2×C8—791no—
258C2×C8—791no—
259C2×C8—792no—
260C2×C8—792no—
261C2×C8—791no—
262C2×C8—791no—
263C2×C8—1582no—
264C2×C8—1581no—
265C2×C8—1581no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—3161no—
270C4⋊C4—3161no—
271C4⋊C4Dic79⋊C412yes—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C42C2×C4×D7911yescharacteristic
275C42D4×D7911yescharacteristic
276C42C2×Dic15811yescharacteristic
277C42#1192962311yescharacteristic
278C23—22no—
279C23#1457913311yescharacteristic
280Q8—41no—
281Q8—41no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—41no—
287D4—41no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—6321no—
297C2×C4—6321no—
298C2×C4—6321no—
299C2×C4C22⋊C4×D7911yescharacteristic
300C2×C4C2×Dic79⋊C411yescharacteristic
301C2×C4#1457911711yescharacteristic
302C2×C4#1457913311yescharacteristic
303C2×C4#1457913811yescharacteristic
304C2×C4#1457913811yescharacteristic
305C2×C4D8×D7911yescharacteristic
306C2×C4SD16×D7911yescharacteristic
307C2×C4#1457918511yescharacteristic
308C2×C4#1457919011yescharacteristic
309C8—81no—
310C8—3162no—
311C8—792no—
312C8—792no—
313C8—6321no—
314C8—1581no—
315C8—1581no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22#1909842111yescharacteristic
322C22#1909875611yescharacteristic
323C22#1909876611yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—161no—
330C4—12641no—
331C4#102×D7911yescharacteristic
332C4#1909856111yescharacteristic
333C4#1909884111yescharacteristic
334C4#1909885111yescharacteristic
335C2—21no—
336C2—21no—
337C2—82no—
338C2#2376339811yescharacteristic
339C2#2376344411yescharacteristic
340C2#2376345211yescharacteristic
subgroupG/H
centerC22#19098421
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1517C2
socleC22×C79#380
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#1517D79
hypocenterC79#8676
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #23763562C2semidirect
3. #23762146C2semidirect
5. #23761259C2semidirect
6. #23761258C2semidirect
10. #19098641C4semidirect
12. D4×C79⋊C8C4semidirect
178. C79#8676semidirect

Maximal subgroups

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

subgroupindex
#237721122
#237635622
#237621462
#237619602
#237612592
#237612582
C79×#15172
#867679