Finite Group Databasebeta

← #34181729

Subgroups — #34181729

Order 20224.

514 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23765757C211yesmaximal, characteristic
2C2×#19098289C211yesmaximal, characteristic
3#23761247C212yesmaximal
4#23761246C212yesmaximal
5C2×#324×C79C211yesmaximal, characteristic
6C2×#14579120C411yescharacteristic
7C2×#14579120C411yescharacteristic
8C2×C22⋊(C79⋊C8)C2212yes—
9C4⋊C4×Dic79—21no—
10C4⋊C4×Dic79—21no—
11C4⋊C4×Dic79—21no—
12C4⋊C4×Dic79—21no—
13C4×C22⋊Dic79C411yescharacteristic
14C4×C22⋊Dic79C411yescharacteristic
15C4×C22⋊Dic79C2211yescharacteristic
16C4×C22⋊Dic79C2211yescharacteristic
17#19098289—21no—
18#19098289—21no—
19C22×C4⋊C4×C79C2211yescharacteristic
20C2×C22⋊C8×C79C2212yes—
21#324×C79—21no—
22#324×C79—21no—
23C22×C79⋊C8—22no—
24#14579120—22no—
25#14579120—22no—
26#14579120—22no—
27#14579120C814yescomplemented
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×C22⋊Dic79C2×C411yescharacteristic
31C2×C22⋊Dic79C2×C411yescharacteristic
32C2×C22⋊Dic79C2×C411yescharacteristic
33C2×C22⋊Dic79C2×C411yescharacteristic
34C2×C4×Dic79—21no—
35C2×C4×Dic79—21no—
36C2×C4×Dic79—21no—
37C2×C4×Dic79—21no—
38C2×C4×Dic79—21no—
39C2×C4×Dic79—21no—
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)—22no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C23×C4×C79C2×C411yescharacteristic
47C23×C4×C79C2×C411yescharacteristic
48C23×C4×C79C2311yescharacteristic
49C22×C8×C79—22no—
50C2×C4⋊C4×C79—22no—
51C2×C4⋊C4×C79—22no—
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79D411yescharacteristic
54C2×C4⋊C4×C79D411yescharacteristic
55C2×C4⋊C4×C79D411yescharacteristic
56C22⋊C8×C79—22no—
57C22⋊C8×C79—22no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C2×C79⋊C8—42no—
63C2×C79⋊C8—82no—
64C4⋊Dic79—24no—
65C4⋊Dic79—24no—
66C22⋊Dic79—21no—
67C22⋊Dic79—21no—
68C22⋊Dic79—21no—
69C22⋊Dic79—21no—
70C22⋊Dic79C2×C812yes—
71C22⋊Dic79C2×C812yes—
72C22⋊Dic79M1612yes—
73C22⋊Dic79M1612yes—
74C4×Dic79—24no—
75C4×Dic79—24no—
76C4×Dic79—81no—
77C4×Dic79—81no—
78C4×Dic79—81no—
79C4×Dic79—81no—
80C24×C79C22×C411yescharacteristic
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—81no—
88C22×C4×C79—81no—
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C22⋊C411yescharacteristic
91C22×C4×C79C22⋊C411yescharacteristic
92C22×C4×C79C22⋊C411yescharacteristic
93C22×C4×C79C2×C812yes—
94C22×C4×C79M1612yes—
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79#4112yes—
98C22×C4×C79#4112yes—
99C2×C8×C79—42no—
100C2×C8×C79—82no—
101C4⋊C4×C79—22no—
102C4⋊C4×C79—22no—
103C4⋊C4×C79—22no—
104C4⋊C4×C79—22no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79C2×C22⋊C411yescharacteristic
108C23×C79#11711yescharacteristic
109C23×C79#11711yescharacteristic
110C23×C79C22×C811yescharacteristic
111C23×C79C2×M1611yescharacteristic
112C23×C79C2×M1611yescharacteristic
113C23×C79C2×M1611yescharacteristic
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—22no—
119C2×C4×C79—22no—
120C2×C4×C79—22no—
121C2×C4×C79—22no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—81no—
131C2×C4×C79—81no—
132C2×C4×C79C22⋊C812yes—
133C2×C4×C79C22⋊C812yes—
134C2×C4×C79#12312yes—
135C2×C4×C79#12312yes—
136C2×Dic79—22no—
137C2×Dic79—22no—
138C2×Dic79—22no—
139C2×Dic79—22no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C8×C79—82no—
149C79⋊C8—82no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79—21no—
156C22×C79—41no—
157C22×C79—41no—
158C22×C79—41no—
159C22×C79—41no—
160C22×C79#38111yescharacteristic
161C22×C79C2×C22⋊C811yescharacteristic
162C22×C79#40711yescharacteristic
163C22×C79#43111yescharacteristic
164C22×C79#43111yescharacteristic
165C22×C79#43211yescharacteristic
166C22×C79#43211yescharacteristic
167C4×C79—22no—
168C4×C79—22no—
169C4×C79—22no—
170C4×C79—22no—
171C4×C79—41no—
172C4×C79—41no—
173Dic79—42no—
174Dic79—42no—
175Dic79—42no—
176Dic79—42no—
177#8735—791nomaximal, Sylow
178C2×C79—21no—
179C2×C79—21no—
180C2×C79—21no—
181C2×C79—21no—
182C2×C79—21no—
183C2×C79—21no—
184C2×C79#181711yescharacteristic
185C2×C79#182811yescharacteristic
186C2×C79#183011yescharacteristic
187#2274—791no—
188C2×#324—791no—
189C2×#324D7911yescharacteristic
190#1282—792no—
191#1282—792no—
192C79#873511yescharacteristic, complemented, Sylow
193C2×#117—791no—
194C2×#117—791no—
195C22×C4⋊C4C2×D7911yescharacteristic
196C2×C22⋊C8—792no—
197C2×C22⋊C8C2×D7912yes—
198C4×C4⋊C4—1581no—
199C4×C4⋊C4—1581no—
200C4×C4⋊C4—1581no—
201C4×C4⋊C4—1581no—
202C4×C22⋊C4—791no—
203C4×C22⋊C4—791no—
204C4×C22⋊C4—791no—
205C4×C22⋊C4—791no—
206#324—21no—
207#324—21no—
208#324—1581no—
209#324—1581no—
210C23×C4C4×D7911yescharacteristic
211C23×C4C4×D7911yescharacteristic
212C23×C4C22×D7911yescharacteristic
213C22×C8—22no—
214C22×C8—1582no—
215#117—794no—
216#117—1582no—
217#117—1582no—
218#117—1582no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—1581no—
222C2×C4⋊C4—1581no—
223C2×C4⋊C4D31611yescharacteristic
224C2×C4⋊C4D31611yescharacteristic
225C2×C4⋊C4C22⋊D7911yescharacteristic
226C2×C4⋊C4C22⋊D7911yescharacteristic
227C2×C22⋊C4—791no—
228C2×C22⋊C4—791no—
229C2×C22⋊C4—791no—
230C2×C22⋊C4—791no—
231C2×C42—1581no—
232C2×C42—1581no—
233C2×C42—1581no—
234C2×C42—1581no—
235C2×C42—1581no—
236C2×C42—1581no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—1582no—
240C22⋊C8—1582no—
241#95—1581no—
242#95—1581no—
243#95—1581no—
244#95—1581no—
245C24C2×C4×D7911yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—1581no—
255C22×C4—1581no—
256C22×C4—1581no—
257C22×C4—1581no—
258C22×C4(C2×D79)⋊C411yescharacteristic
259C22×C4(C2×D79)⋊C411yescharacteristic
260C22×C4(C2×D79)⋊C411yescharacteristic
261C22×C4(C2×D79)⋊C411yescharacteristic
262C22×C4C8×D7912yes—
263C22×C4#1192960712yes—
264C22×C4C2×C22⋊D7911yescharacteristic
265C22×C4C2×D31611yescharacteristic
266C22×C4#1192962312yes—
267C22×C4#1192962312yes—
268C2×C8—42no—
269C2×C8—82no—
270C2×C8—3162no—
271C2×C8—6322no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—1584no—
277C4⋊C4—1584no—
278C22⋊C4—792no—
279C22⋊C4—792no—
280C22⋊C4—792no—
281C22⋊C4—792no—
282C22⋊C4—1581no—
283C22⋊C4—1581no—
284C22⋊C4—1581no—
285C22⋊C4—1581no—
286C42—6321no—
287C42—6321no—
288C42—6321no—
289C42—6321no—
290C42—1584no—
291C42—1584no—
292C23—41no—
293C23—41no—
294C23C2×(C2×D79)⋊C411yescharacteristic
295C23#1457911711yescharacteristic
296C23#1457911711yescharacteristic
297C23C2×C8×D7911yescharacteristic
298C23M16×D7911yescharacteristic
299C23M16×D7911yescharacteristic
300C23C2×#1192960711yescharacteristic
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—3161no—
320C2×C4—3161no—
321C2×C4—3161no—
322C2×C4—3161no—
323C2×C4—3161no—
324C2×C4—3161no—
325C2×C4—3161no—
326C2×C4—3161no—
327C2×C4—1582no—
328C2×C4—1582no—
329C2×C4—1582no—
330C2×C4—1582no—
331C2×C4(C2×D79)⋊C812yes—
332C2×C4(C2×D79)⋊C812yes—
333C2×C4#1457914212yes—
334C2×C4#1457914512yes—
335C8—82no—
336C8—6322no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22#1909842411yescharacteristic
348C22C2×(C2×D79)⋊C811yescharacteristic
349C22#1909852611yescharacteristic
350C22#1909861311yescharacteristic
351C22#1909861311yescharacteristic
352C22#1909861711yescharacteristic
353C22#1909861711yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—3162no—
361C4—3162no—
362C4—3162no—
363C4—3162no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2376346611yescharacteristic
371C2#2376350611yescharacteristic
372C2#2376351211yescharacteristic
subgroupG/H
centerC22#19098424
commutatorC23×C79C22×C8
FrattiniC23×C4C22×D79
FittingC2×#324×C79C2
socleC22×C79#381
prime radicalC24×C79C22×C4
cosocleC23×C4×C79C23
hypercenterC2×#324D79
hypocenterC79#8735
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
27. #14579120C8semidirect
192. C79#8735semidirect

Maximal subgroups

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

subgroupindex
#237657572
C2×#190982892
#237612472
#237612462
C2×#324×C792
#873579