Finite Group Databasebeta

← #34187786

Subgroups — #34187786

Order 20224.

522 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#23764386C212yesmaximal
2#23763429C211yesmaximal, characteristic
3C8×C22⋊Dic79C211yesmaximal, characteristic
4#23761972C212yesmaximal
5C2×#467×C79C211yesmaximal, characteristic
6C2×#14579095—21no—
7C2×C22⋊(C79⋊C8)C2211yescharacteristic
8C2×C8×Dic79—21no—
9#19098423C2212yes—
10C4×C22⋊Dic79C2211yescharacteristic
11#19098330—24no—
12#19098326—21no—
13#19098317—21no—
14#19098300—24no—
15C22×Q16×C79C411yescharacteristic, complemented
16C23×C8×C79C2211yescharacteristic
17C2×C22⋊Q8×C79C2212yes—
18#467×C79—21no—
19#467×C79—21no—
20#467×C79—21no—
21#467×C79—21no—
22C2×#107×C79C411yescharacteristic, complemented
23C2×#103×C79C412yescomplemented
24C22×C79⋊C8—21no—
25C2×C4⋊Dic79—22no—
26C2×C22⋊Dic79—22no—
27C2×C22⋊Dic79D412yes—
28C2×C4×Dic79—21no—
29#14579095—42no—
30C22⋊(C79⋊C8)—21no—
31C22⋊(C79⋊C8)D412yes—
32C8×Dic79—42no—
33C79⋊#95—21no—
34C79⋊#95—22no—
35C22×Q8×C79C2×C412yes—
36C23×C4×C79C2311yescharacteristic
37C2×Q16×C79—21no—
38C2×Q16×C79—21no—
39C2×Q16×C79—42no—
40C22×C8×C79—42no—
41C22×C8×C79C2×C411yescharacteristic
42C22×C8×C79C2×C411yescharacteristic
43C22×C8×C79D411yescharacteristic
44C22×C8×C79D411yescharacteristic
45C22×C8×C79D411yescharacteristic
46C22×C8×C79D411yescharacteristic
47C22⋊Q8×C79—22no—
48C22⋊Q8×C79—22no—
49C22⋊Q8×C79—22no—
50C22⋊Q8×C79—22no—
51C2×C4⋊C4×C79—22no—
52C2×C4⋊C4×C79C2×C412yes—
53C2×C22⋊C4×C79—22no—
54#107×C79—21no—
55#107×C79—21no—
56#103×C79—22no—
57#103×C79—22no—
58C22×Dic79—22no—
59C22×Dic79—42no—
60C2×C79⋊C8—22no—
61C2×C79⋊C8—41no—
62C2×C79⋊C8—41no—
63C4⋊Dic79—42no—
64C4⋊Dic79—42no—
65C22⋊Dic79—22no—
66C22⋊Dic79—44no—
67C22⋊Dic79Q1614yescomplemented
68C4×Dic79—41no—
69C4×Dic79—41no—
70C24×C79C2×D411yescharacteristic
71C2×Q8×C79—22no—
72C2×Q8×C79—22no—
73C2×Q8×C79—42no—
74C2×Q8×C79—42no—
75C22×C4×C79—22no—
76C22×C4×C79—22no—
77C22×C4×C79—41no—
78C22×C4×C79—41no—
79C22×C4×C79C22×C411yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79#4111yescharacteristic
84C22×C4×C79#4111yescharacteristic
85Q16×C79—42no—
86Q16×C79—42no—
87C2×C8×C79—22no—
88C2×C8×C79—22no—
89C2×C8×C79—22no—
90C2×C8×C79—41no—
91C2×C8×C79—41no—
92C2×C8×C79—41no—
93C2×C8×C79C22⋊C411yescharacteristic
94C2×C8×C79C22⋊C411yescharacteristic
95C2×C8×C79C22⋊C411yescharacteristic
96C2×C8×C79C22⋊C411yescharacteristic
97C4⋊C4×C79—22no—
98C4⋊C4×C79—22no—
99C4⋊C4×C79—42no—
100C4⋊C4×C79—42no—
101C22⋊C4×C79—42no—
102C22⋊C4×C79—42no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79C4×D411yescharacteristic
106C23×C79#12111yescharacteristic
107C23×C79#12711yescharacteristic
108C23×C79C2×Q1612yes—
109C23×C79#13512yes—
110Q8×C79—42no—
111Q8×C79—42no—
112Q8×C79—82no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—42no—
123C2×C4×C79—42no—
124C2×C4×C79—82no—
125C2×C4×C79—82no—
126C2×C4×C79C2×C22⋊C411yescharacteristic
127C2×C4×C79C4×D411yescharacteristic
128C2×C4×C79#12111yescharacteristic
129C2×C4×C79#12411yescharacteristic
130C2×Dic79—24no—
131C2×Dic79—42no—
132C2×Dic79—42no—
133C2×Dic79—82no—
134C2×Dic79—82no—
135C2×Dic79—82no—
136C8×C79—21no—
137C8×C79—21no—
138C8×C79—21no—
139C8×C79#9912yes—
140C79⋊C8—42no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—42no—
148C22×C79—42no—
149C22×C79#39011yescharacteristic
150C22×C79C4×Q1612yes—
151C22×C79#46712yes—
152C22×C79#49411yescharacteristic
153C22×C79#49511yescharacteristic
154C4×C79—21no—
155C4×C79—21no—
156C4×C79—21no—
157C4×C79—82no—
158C4×C79—82no—
159C4×C79C2×#9911yescharacteristic
160C4×C79#41011yescharacteristic
161Dic79—44no—
162Dic79—162no—
163#9782—791nomaximal, Sylow
164C2×C79—22no—
165C2×C79—21no—
166C2×C79—22no—
167C2×C79—21no—
168C2×C79#196411yescharacteristic
169C2×C79#196911yescharacteristic
170C2×C79#197611yescharacteristic
171C2×#467D7911yescharacteristic
172#1996—792no—
173#1808—791no—
174C8×C22⋊C4—791no—
175#1520—792no—
176C79#978211yescharacteristic, complemented, Sylow
177C22×Q16Dic7911yescharacteristic, complemented
178C23×C8C2×D7911yescharacteristic
179C2×C22⋊Q8C2×D7912yes—
180#467—21no—
181#467—21no—
182#467—21no—
183#467—21no—
184C2×#107—1581no—
185C2×#107Dic7911yescharacteristic, complemented
186C2×#103Dic7912yescomplemented
187C2×C22⋊C8—791no—
188C2×C4×C8—1581no—
189#380—792no—
190C4×C22⋊C4—791no—
191#342—1584no—
192#340—1581no—
193#336—1581no—
194#328—1584no—
195C22×Q8C2×Dic7912yes—
196C23×C4C22×D7911yescharacteristic
197C2×Q16—21no—
198C2×Q16—21no—
199C2×Q16—42no—
200C22×C8—42no—
201C22×C8—1581no—
202C22×C8C2×Dic7911yescharacteristic
203C22×C8C2×Dic7911yescharacteristic
204C22×C8C22⋊D7911yescharacteristic
205C22×C8C22⋊D7911yescharacteristic
206C22×C8C22⋊D7911yescharacteristic
207C22×C8C22⋊D7911yescharacteristic
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—1582no—
214C2×C4⋊C4C2×Dic7912yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—792no—
217C2×C22⋊C4—1582no—
218C2×C42—1581no—
219#107—21no—
220#107—21no—
221#107—3162no—
222#103—22no—
223#103—22no—
224C22⋊C8—792no—
225C22⋊C8—1581no—
226C4×C8—3162no—
227#95—1582no—
228#95—1581no—
229C24D4×D7911yescharacteristic
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—42no—
233C2×Q8—42no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—3162no—
239C22×C4—1582no—
240C22×C4C22×Dic7911yescharacteristic
241C22×C4D4×D7911yescharacteristic
242C22×C4C2×C22⋊D7911yescharacteristic
243C22×C4C2×C22⋊D7911yescharacteristic
244C22×C4#1192962311yescharacteristic
245C22×C4#1192962311yescharacteristic
246Q16—42no—
247Q16—42no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—3161no—
255C2×C8—3161no—
256C2×C8—1582no—
257C2×C8C22⋊Dic7911yescharacteristic
258C2×C8C22⋊Dic7911yescharacteristic
259C2×C8C22⋊Dic7911yescharacteristic
260C2×C8C22⋊Dic7911yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—3162no—
266C4⋊C4—3162no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—3164no—
270C22⋊C4—794no—
271C22⋊C4—1582no—
272C42—3161no—
273C42—3161no—
274C23—41no—
275C23—41no—
276C23D4×Dic7911yescharacteristic
277C23#1457913211yescharacteristic
278C23#1457915711yescharacteristic
279C23Q16×D7912yes—
280C23#1457918412yes—
281Q8—42no—
282Q8—42no—
283Q8—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—3162no—
298C2×C4—3162no—
299C2×C4—6322no—
300C2×C4—6322no—
301C2×C4—6322no—
302C2×C4—1584no—
303C2×C4C2×C22⋊Dic7911yescharacteristic
304C2×C4D4×Dic7911yescharacteristic
305C2×C4#1457913211yescharacteristic
306C2×C4#1457914811yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—3162no—
311C8#1457907612yes—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1909846611yescharacteristic
321C22Q16×Dic7912yes—
322C22#1909880112yes—
323C22#1909894111yescharacteristic
324C22#1909894811yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—82no—
329C4—82no—
330C4—3164no—
331C4—12642no—
332C4C2×#1457907611yescharacteristic
333C4#1909853711yescharacteristic
334C2—22no—
335C2—21no—
336C2—22no—
337C2—21no—
338C2#2376423711yescharacteristic
339C2#2376426211yescharacteristic
340C2#2376429911yescharacteristic
subgroupG/H
centerC22#19098466
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#467×C79C2
socleC22×C79#390
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenterC2×#467D79
hypocenterC79#9782
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
15. C22×Q16×C79C4semidirect
22. C2×#107×C79C4semidirect
23. C2×#103×C79C4semidirect
67. C22⋊Dic79Q16semidirect
176. C79#9782semidirect
177. C22×Q16Dic79semidirect
185. C2×#107Dic79semidirect
186. C2×#103Dic79semidirect

Maximal subgroups

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

subgroupindex
#237643862
#237634292
C8×C22⋊Dic792
#237619722
C2×#467×C792
#978279