Finite Group Databasebeta

← #34181421

Subgroups — #34181421

Order 20224.

434 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098641C211yesmaximal, characteristic
2#23763537C211yesmaximal, characteristic, complemented
3#23762146C211yesmaximal, characteristic, complemented
4#23761825C211yesmaximal, characteristic
5#23761296C211yesmaximal, characteristic, complemented
6#23761295C211yesmaximal, characteristic, complemented
7C79×#1486C211yesmaximal, characteristic
8C22×#11929608—21no—
9#19098641—21no—
10#19098641—21no—
11#19098641C412yes—
12#19098641C412yes—
13#19098616C2211yescharacteristic
14#19098587C2211yescharacteristic
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×C22⋊(C79⋊C8)—21no—
17C2×C4×C79⋊C8C2211yescharacteristic
18#19098327—21no—
19#19098327—21no—
20#19098315—21no—
21#19098313—21no—
22D4⋊(C79⋊C8)—22no—
23C2×C4×D4×C79C2211yescharacteristic
24#431×C79C2211yescharacteristic
25#423×C79C2211yescharacteristic
26D4⋊C8×C79—22no—
27C2×#11929608—22no—
28C2×#11929608—22no—
29C2×#11929608—21no—
30C2×#11929608—42no—
31C2×#11929608—42no—
32C22×C79⋊C8—21no—
33C22×C79⋊C8D411yescharacteristic
34C22×C79⋊C8D411yescharacteristic
35C4⋊(C79⋊C8)—21no—
36C4⋊(C79⋊C8)—21no—
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
48C22×D4×C79Q811yescharacteristic
49C23×C4×C79—21no—
50C2×M16×C79—21no—
51C4×D4×C79—21no—
52C4×D4×C79—21no—
53C4×D4×C79C2×C412yes—
54C4×D4×C79D412yes—
55C2×C4⋊C4×C79Q811yescharacteristic
56C2×C22⋊C4×C79—21no—
57C2×C42×C79C2311yescharacteristic
58C4⋊C8×C79—21no—
59C4⋊C8×C79—21no—
60C4⋊C8×C79—21no—
61C22⋊C8×C79—21no—
62C4×C8×C79—21no—
63#11929608—44no—
64#11929608—44no—
65#11929608—41no—
66#11929608—41no—
67C2×C79⋊C8—22no—
68C2×C79⋊C8—21no—
69C2×C79⋊C8—21no—
70C2×C79⋊C8—21no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—41no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8—41no—
75C2×C79⋊C8—41no—
76C2×C79⋊C8C22⋊C411yescharacteristic
77C2×C79⋊C8C22⋊C411yescharacteristic
78C2×C79⋊C8C22⋊C411yescharacteristic
79C2×C79⋊C8C22⋊C411yescharacteristic
80C24×C79—21no—
81C2×D4×C79—21no—
82C2×D4×C79—42no—
83C2×D4×C79C4⋊C412yes—
84C22×C4×C79—21no—
85C22×C4×C79—22no—
86C22×C4×C79—22no—
87C22×C4×C79—41no—
88C22×C4×C79—41no—
89C22×C4×C79—41no—
90C22×C4×C79—41no—
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×Q811yescharacteristic
93C22×C4×C79#4111yescharacteristic
94M16×C79—41no—
95M16×C79—41no—
96C2×C8×C79—41no—
97C2×C8×C79—41no—
98C4⋊C4×C79—21no—
99C4⋊C4×C79C4⋊C412yes—
100C22⋊C4×C79—22no—
101C22⋊C4×C79—22no—
102C42×C79C22×C411yescharacteristic
103C42×C79C2×D411yescharacteristic
104C42×C79C2×D411yescharacteristic
105C42×C79#4111yescharacteristic
106C23×C79—22no—
107C23×C79—42no—
108C23×C79—42no—
109C23×C79—41no—
110C23×C79—41no—
111C23×C79C22⋊Q811yescharacteristic
112D4×C79—24no—
113D4×C79—42no—
114C2×C4×C79—22no—
115C2×C4×C79—22no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—42no—
120C2×C4×C79—42no—
121C2×C4×C79—42no—
122C2×C4×C79—42no—
123C2×C4×C79C2×C22⋊C411yescharacteristic
124C2×C4×C79C2×C4⋊C411yescharacteristic
125C2×C4×C79#11711yescharacteristic
126C2×C4×C79#12111yescharacteristic
127C2×C4×C79#12111yescharacteristic
128C2×C4×C79C22⋊Q811yescharacteristic
129C2×C4×C79#13511yescharacteristic
130C2×C4×C79#13511yescharacteristic
131C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
133C8×C79—81no—
134C8×C79—81no—
135C79⋊C8—21no—
136C79⋊C8—21no—
137C79⋊C8—21no—
138C79⋊C8—21no—
139C79⋊C8—42no—
140C79⋊C8—81no—
141C79⋊C8—81no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—42no—
145C22×C79—42no—
146C22×C79—42no—
147C22×C79—82no—
148C22×C79—82no—
149C22×C79#38011yescharacteristic
150C22×C79#47811yescharacteristic
151C22×C79#47811yescharacteristic
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—21no—
155C4×C79—42no—
156C4×C79#41611yescharacteristic
157C4×C79#41811yescharacteristic
158C4×C79#43012yes—
159C4×C79#46311yescharacteristic
160C4×C79#46311yescharacteristic
161#8664—791nomaximal, Sylow
162C2×C79—21no—
163C2×C79—21no—
164C2×C79—82no—
165C2×C79#180411yescharacteristic
166C2×C79#181111yescharacteristic
167C2×C79#181411yescharacteristic
168C2×#436—791no—
169#1837—791no—
170#1564—791no—
171#1486—791no—
172#1486D7911yescharacteristic
173#1301—791no—
174#1301—791no—
175C79#866411yescharacteristic, complemented, Sylow
176C22×M16—1581no—
177C2×C4×D4C2×D7911yescharacteristic
178#436—792no—
179#436—792no—
180#436—1581no—
181#436—1581no—
182#431—791no—
183#431C2×D7911yescharacteristic
184#423—791no—
185#423C2×D7911yescharacteristic
186C2×C4⋊C8—791no—
187C2×C22⋊C8—1581no—
188C2×C4×C8—791no—
189#341—1581no—
190#341—1581no—
191#335—1581no—
192#334—1581no—
193D4⋊C8—22no—
194D4⋊C8—1582no—
195C22×D4Dic15811yescharacteristic
196C23×C4—21no—
197C2×M16—21no—
198C2×M16—3162no—
199C2×M16—3162no—
200C2×M16—1581no—
201C2×M16—1582no—
202C2×M16—1582no—
203C22×C8—791no—
204C22×C8—791no—
205C22×C8—1581no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4C4×D7912yes—
209C4×D4C22⋊D7912yes—
210C2×C4⋊C4Dic15811yescharacteristic
211C2×C22⋊C4—21no—
212C2×C42C22×D7911yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—792no—
217C4⋊C8—1581no—
218C4⋊C8—1581no—
219C4⋊C8—1581no—
220C4⋊C8—1581no—
221C22⋊C8—21no—
222C22⋊C8—1581no—
223C22⋊C8—1582no—
224C22⋊C8—1582no—
225C4×C8—21no—
226C4×C8—791no—
227C4×C8—791no—
228C4×C8—791no—
229C4×C8—791no—
230C4×C8—1581no—
231C24—21no—
232C2×D4—21no—
233C2×D4—42no—
234C2×D4Dic79⋊C412yes—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4D4×D7911yescharacteristic
243C22×C4C2×Dic15811yescharacteristic
244C22×C4#1192962311yescharacteristic
245M16—41no—
246M16—41no—
247M16—3164no—
248M16—3164no—
249M16—3161no—
250M16—3161no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—3161no—
254C2×C8—3161no—
255C2×C8—3161no—
256C2×C8—3161no—
257C2×C8—791no—
258C2×C8—791no—
259C2×C8—791no—
260C2×C8—791no—
261C2×C8—1582no—
262C2×C8—1581no—
263C2×C8—1581no—
264C2×C8—1581no—
265C2×C8—1581no—
266C4⋊C4—21no—
267C4⋊C4Dic79⋊C412yes—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C42C2×C4×D7911yescharacteristic
271C42D4×D7911yescharacteristic
272C42C2×C22⋊D7911yescharacteristic
273C42#1192962311yescharacteristic
274C23—22no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—41no—
279C23#1457913811yescharacteristic
280D4—24no—
281D4—42no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4C22⋊C4×D7911yescharacteristic
292C2×C4C2×Dic79⋊C411yescharacteristic
293C2×C4#1457911711yescharacteristic
294C2×C4#1457913311yescharacteristic
295C2×C4#1457913311yescharacteristic
296C2×C4#1457913811yescharacteristic
297C2×C4#1457918011yescharacteristic
298C2×C4#1457918211yescharacteristic
299C2×C4#1457918511yescharacteristic
300C2×C4#1457918811yescharacteristic
301C8—81no—
302C8—81no—
303C8—3162no—
304C8—6321no—
305C8—6321no—
306C8—1581no—
307C8—1581no—
308C8—1581no—
309C8—1581no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—82no—
316C22—82no—
317C22#1909842111yescharacteristic
318C22#1909886111yescharacteristic
319C22#1909886511yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4#1909855611yescharacteristic
325C4#1909856611yescharacteristic
326C4#1909861112yes—
327C4#1909877611yescharacteristic
328C4#1909878011yescharacteristic
329C2—21no—
330C2—21no—
331C2—82no—
332C2#2376340811yescharacteristic
333C2#2376344011yescharacteristic
334C2#2376345211yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1486C2
socleC22×C79#380
prime radicalC22×D4×C79Q8
cosocleC2×C42×C79C23
hypercenter#1486D79
hypocenterC79#8664
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #23763537C2semidirect
3. #23762146C2semidirect
5. #23761296C2semidirect
6. #23761295C2semidirect
175. C79#8664semidirect

Maximal subgroups

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

subgroupindex
C2×#190986412
#237635372
#237621462
#237618252
#237612962
#237612952
C79×#14862
#866479