Finite Group Databasebeta

← #34256721

Subgroups — #34256721

Order 20224.

1172 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772511C211yesmaximal, characteristic, complemented
2#23771968C211yesmaximal, characteristic, complemented
3#23762077C213yesmaximal, complemented
4#23762072C213yesmaximal, complemented
5#23762049C213yesmaximal, complemented
6C2×#19098309C213yesmaximal, complemented
7C79×#3499C211yesmaximal, characteristic
8C2×#14579168—21no—
9Q8×C79⋊C8—21no—
10D4×C79⋊C8—23no—
11#19098616C2213yes—
12#19098590—23no—
13#19098587C2213yes—
14C2×C4⋊(C79⋊C8)C2213yes—
15C2×C4⋊(C79⋊C8)C2213yes—
16#19098309C2213yes—
17#19098309C2216yes—
18#19098309C2213yes—
19#19098307C2211yescharacteristic
20#19098307C2213yes—
21C2×2−1+4×C79—21no—
22D4×Q8×C79—21no—
23D4×Q8×C79C412yes—
24#547×C79—23no—
25#547×C79C416yes—
26#536×C79—23no—
27#532×C79C2213yes—
28C2×C4⋊Q8×C79C2213yes—
29C4×#41×C79C2211yescharacteristic
30#14579168—44no—
31C2×#11929608—23no—
32C22×C79⋊C8—23no—
33C22×C79⋊C8—23no—
34C4⋊(C79⋊C8)—23no—
35C4⋊(C79⋊C8)—23no—
36C4⋊(C79⋊C8)—23no—
37C4⋊(C79⋊C8)C2311yescharacteristic
38C4⋊(C79⋊C8)C2313yes—
39C4⋊(C79⋊C8)C2313yes—
40C4⋊(C79⋊C8)C2311yescharacteristic
41C22⋊(C79⋊C8)—23no—
42C4×C79⋊C8—23no—
432−1+4×C79—44no—
442−1+4×C79—44no—
45C2×#41×C79—26no—
46C2×#41×C79—23no—
47C2×#41×C79D411yescharacteristic
48C22×Q8×C79—22no—
49C22×Q8×C79—23no—
50C4⋊Q8×C79—23no—
51C4⋊Q8×C79—23no—
52C4⋊Q8×C79C2×C416yes—
53C4⋊Q8×C79C2313yes—
54#127×C79C2311yescharacteristic
55#124×C79—23no—
56#123×C79—212no—
57C22⋊Q8×C79—26no—
58C22⋊Q8×C79—26no—
59#121×C79—23no—
60#121×C79C2×C416yes—
61C4×Q8×C79C2×C411yescharacteristic
62C4×Q8×C79D411yescharacteristic
63C4×D4×C79—23no—
64C4×D4×C79C2×C416yes—
65C4×D4×C79C2×C413yes—
66C4×D4×C79D413yes—
67#117×C79D413yes—
68C2×C4⋊C4×C79—23no—
69C2×C4⋊C4×C79C2×C416yes—
70C2×C42×C79C2313yes—
71#11929608—46no—
72C2×C79⋊C8—21no—
73C2×C79⋊C8—23no—
74C2×C79⋊C8—21no—
75C2×C79⋊C8—23no—
76C2×C79⋊C8—46no—
77C2×C79⋊C8—83no—
78#41×C79—424no—
79#41×C79—46no—
80#41×C79—46no—
81#41×C79D814yes—
82#41×C79SD1614yes—
83C2×Q8×C79—26no—
84C2×Q8×C79—23no—
85C2×Q8×C79—48no—
86C2×Q8×C79—46no—
87C2×Q8×C79—46no—
88C2×Q8×C79C22⋊C411yescharacteristic
89C2×D4×C79—26no—
90C2×D4×C79—21no—
91C2×D4×C79—21no—
92C2×D4×C79C22⋊C413yes—
93C2×D4×C79C22×C412yes—
94C22×C4×C79—26no—
95C22×C4×C79—26no—
96C22×C4×C79C22×C413yes—
97C22×C4×C79C2×D413yes—
98C22×C4×C79C2×D413yes—
99C4⋊C4×C79—26no—
100C4⋊C4×C79—26no—
101C4⋊C4×C79—23no—
102C4⋊C4×C79C22⋊C413yes—
103C4⋊C4×C79C22⋊C413yes—
104C4⋊C4×C79C22×C416yes—
105C22⋊C4×C79—26no—
106C22⋊C4×C79—26no—
107C22⋊C4×C79C22⋊C416yes—
108C42×C79C22×C413yes—
109C42×C79C2×D413yes—
110C42×C79C2×D413yes—
111C42×C79C2411yescharacteristic
112C23×C79—22no—
113C23×C79C2×C22⋊C413yes—
114Q8×C79—43no—
115Q8×C79—43no—
116Q8×C79—412no—
117Q8×C79#10214yes—
118D4×C79—42no—
119D4×C79—412no—
120D4×C79—41no—
121D4×C79—41no—
122D4×C79—41no—
123D4×C79—41no—
124D4×C79#102112yes—
125C2×C4×C79—23no—
126C2×C4×C79—23no—
127C2×C4×C79—26no—
128C2×C4×C79—43no—
129C2×C4×C79—412no—
130C2×C4×C79—412no—
131C2×C4×C79C2×C22⋊C413yes—
132C2×C4×C79C2×C22⋊C413yes—
133C2×C4×C79C2×C22⋊C413yes—
134C2×C4×C79C2×D816yes—
135C2×C4×C79C2×SD1616yes—
136C2×C4×C79C23×C411yescharacteristic
137C2×C4×C79C22×D411yescharacteristic
138C2×C4×C79C22×D411yescharacteristic
139C79⋊C8—41no—
140C79⋊C8—43no—
141C79⋊C8—43no—
142C79⋊C8—41no—
143C22×C79—23no—
144C22×C79—23no—
145C22×C79—42no—
146C22×C79—42no—
147C22×C79—42no—
148C22×C79C2×#10216yes—
149C22×C79C22×C22⋊C411yescharacteristic
150C4×C79—21no—
151C4×C79—23no—
152C4×C79—23no—
153C4×C79—46no—
154C4×C79C2×#10216yes—
155C4×C79C22×D811yescharacteristic
156C4×C79C22×SD1611yescharacteristic
157C4×C79#57711yescharacteristic
158C4×C79#57811yescharacteristic
159#17721—791nomaximal, Sylow
160C2×C79—26no—
161C2×C79—82no—
162C2×C79#288911yescharacteristic
163C2×C79C22×#10211yescharacteristic
164C2×C79#289811yescharacteristic
165#3499D7911yescharacteristic
166#2988—791no—
167#2906—791no—
168#1547—793no—
169#1546—793no—
170#1540—793no—
171C2×#332—793no—
172C79#1772111yescharacteristic, complemented, Sylow
173C2×2−1+4—21no—
174C2×#131—1581no—
175D4×Q8—21no—
176D4×Q8Dic7912yes—
177#547—23no—
178#547Dic7916yes—
179#536—23no—
180#532C2×D7913yes—
181C2×C4⋊Q8C2×D7913yes—
182C4×#41C2×D7911yescharacteristic
183C8×Q8—1581no—
184C8×D4—1583no—
185#431—793no—
186#424—1583no—
187#423—793no—
188C2×C4⋊C8—793no—
189C2×C4⋊C8—793no—
190#332—793no—
191#332—796no—
192#332—793no—
193#331—791no—
194#331—793no—
1952−1+4—44no—
1962−1+4—44no—
197C2×#41—26no—
198C2×#41—23no—
199C2×#41C22⋊D7911yescharacteristic
200C22×Q8—22no—
201C22×Q8—23no—
202#131—3164no—
203C2×M16—1583no—
204C22×C8—1583no—
205C22×C8—1583no—
206C4⋊Q8—23no—
207C4⋊Q8—23no—
208C4⋊Q8C2×Dic7916yes—
209C4⋊Q8C22×D7913yes—
210#127C22×D7911yescharacteristic
211#124—23no—
212#123—212no—
213C22⋊Q8—26no—
214C22⋊Q8—26no—
215#121—23no—
216#121C2×Dic7916yes—
217C4×Q8C2×Dic7911yescharacteristic
218C4×Q8C22⋊D7911yescharacteristic
219C4×D4—23no—
220C4×D4C2×Dic7916yes—
221C4×D4C2×Dic7913yes—
222C4×D4C22⋊D7913yes—
223#117C22⋊D7913yes—
224C2×C4⋊C4—23no—
225C2×C4⋊C4C2×Dic7916yes—
226C2×C42C22×D7913yes—
227C4⋊C8—791no—
228C4⋊C8—793no—
229C4⋊C8—793no—
230C4⋊C8—791no—
231C4⋊C8—1583no—
232C4⋊C8—1583no—
233C4⋊C8—1583no—
234C22⋊C8—1583no—
235C4×C8—1583no—
236#41—424no—
237#41—46no—
238#41—46no—
239#41#1192960914yes—
240#41#1192961214yes—
241C2×Q8—26no—
242C2×Q8—23no—
243C2×Q8—48no—
244C2×Q8—46no—
245C2×Q8—46no—
246C2×Q8C22⋊Dic7911yescharacteristic
247C2×D4—26no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4C22⋊Dic7913yes—
251C2×D4C22×Dic7912yes—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4C22×Dic7913yes—
255C22×C4C2×C22⋊D7913yes—
256C22×C4C2×C22⋊D7913yes—
257M16—3166no—
258C2×C8—3166no—
259C2×C8—6323no—
260C2×C8—1581no—
261C2×C8—1583no—
262C2×C8—1581no—
263C2×C8—1583no—
264C4⋊C4—26no—
265C4⋊C4—26no—
266C4⋊C4—23no—
267C4⋊C4C22⋊Dic7913yes—
268C4⋊C4C22⋊Dic7913yes—
269C4⋊C4C22×Dic7916yes—
270C22⋊C4—26no—
271C22⋊C4—26no—
272C22⋊C4C22⋊Dic7916yes—
273C42C22×Dic7913yes—
274C42C2×C22⋊D7913yes—
275C42C2×C22⋊D7913yes—
276C42C23×D7911yescharacteristic
277C23—22no—
278C23C2×C22⋊Dic7913yes—
279Q8—412no—
280Q8—43no—
281Q8—43no—
282Q8#1457908214yes—
283D4—42no—
284D4—412no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4#14579082112yes—
290C2×C4—23no—
291C2×C4—23no—
292C2×C4—26no—
293C2×C4—43no—
294C2×C4—412no—
295C2×C4—412no—
296C2×C4C2×C22⋊Dic7913yes—
297C2×C4C2×C22⋊Dic7913yes—
298C2×C4C2×C22⋊Dic7913yes—
299C2×C4C2×#1192960916yes—
300C2×C4C2×#1192961216yes—
301C2×C4C23×Dic7911yescharacteristic
302C2×C4C22×C22⋊D7911yescharacteristic
303C2×C4C22×C22⋊D7911yescharacteristic
304C8—3161no—
305C8—3163no—
306C8—3163no—
307C8—3161no—
308C22—23no—
309C22—23no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22C2×#1457908216yes—
314C22C22×C22⋊Dic7911yescharacteristic
315C4—23no—
316C4—21no—
317C4—23no—
318C4—46no—
319C4C2×#1457908216yes—
320C4C22×#1192960911yescharacteristic
321C4C22×#1192961211yescharacteristic
322C4#1909937111yescharacteristic
323C4#1909937811yescharacteristic
324C2—26no—
325C2—82no—
326C2#2377188311yescharacteristic
327C2C22×#1457908211yescharacteristic
328C2#2377193311yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
Fitting, prime radicalC79×#3499C2
socleC22×C79C22×C22⋊C4
cosocleC42×C79C24
hypercenter#3499D79
hypocenterC79#17721
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #23772511C2semidirect
2. #23771968C2semidirect
3. #23762077C2semidirect
4. #23762072C2semidirect
5. #23762049C2semidirect
6. C2×#19098309C2semidirect
172. C79#17721semidirect

Maximal subgroups

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

subgroupindex
#237725112
#237719682
#237620772
#237620722
#237620492
C2×#190983092
C79×#34992
#1772179