Finite Group Databasebeta

← #34197499

Subgroups — #34197499

Order 20224.

418 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098634C211yesmaximal, characteristic
2#23765011C211yesmaximal, characteristic, complemented
3#23761339C212yesmaximal, complemented
4#23761323C212yesmaximal, complemented
5C2×C22⋊C16×C79C211yesmaximal, characteristic, complemented
6C22×#11929607C411yescharacteristic
7C2×C4×C22⋊D79C411yescharacteristic
8C2×#14579101—21no—
9#19098634—21no—
10#19098634—21no—
11#19098634—21no—
12#19098634—21no—
13C2×Dic79⋊C8C2211yescharacteristic
14C2×C22⋊(C79⋊C8)C411yescharacteristic
15C2×(C2×D79)⋊C8C2211yescharacteristic
16C2×#14579071C411yescharacteristic
17C22⋊(C79⋊C16)—21no—
18C22⋊(C79⋊C16)C2212yes—
19C23×C8×C79C2211yescharacteristic
20C22×C16×C79—21no—
21C22⋊C16×C79—21no—
22C22⋊C16×C79C2212yes—
23C22×C22⋊D79C811yescharacteristic
24C22×C4×D79C2×C411yescharacteristic
25C2×#11929607—21no—
26C2×#11929607—21no—
27C2×#11929607—42no—
28C22×C79⋊C8C2×C411yescharacteristic
29C22×C79⋊C8C2×C411yescharacteristic
30C4×C22⋊D79—21no—
31C4×C22⋊D79—21no—
32C4×C22⋊D79—21no—
33C4×C22⋊D79—21no—
34C2×Dic79⋊C4C811yescharacteristic
35C2×C22⋊Dic79C811yescharacteristic
36C2×(C2×D79)⋊C4C811yescharacteristic
37C2×C4×Dic79C2×C411yescharacteristic
38#14579101—42no—
39C2×C79⋊C16—22no—
40Dic79⋊C8—21no—
41Dic79⋊C8—21no—
42C22⋊(C79⋊C8)—21no—
43C22⋊(C79⋊C8)—21no—
44(C2×D79)⋊C8—21no—
45(C2×D79)⋊C8—21no—
46#14579071—21no—
47#14579071—21no—
48C23×C4×C79C2×C411yescharacteristic
49C22×C8×C79—42no—
50C22×C8×C79C2×C411yescharacteristic
51C22×C8×C79D411yescharacteristic
52C22×C8×C79D411yescharacteristic
53C22×C8×C79D411yescharacteristic
54C22×C8×C79D411yescharacteristic
55C22×C8×C79C2311yescharacteristic
56C2×C16×C79—22no—
57C2×C16×C79—42no—
58C23×D79C2×C811yescharacteristic
59C2×C22⋊D79—21no—
60C2×C22⋊D79—21no—
61C2×C22⋊D79—42no—
62C22×Dic79C2×C811yescharacteristic
63C22×Dic79C2×C811yescharacteristic
64C22×Dic79C2×C811yescharacteristic
65C2×C4×D79—21no—
66C2×C4×D79—21no—
67C2×C4×D79—41no—
68C2×C4×D79—41no—
69#11929607—42no—
70#11929607—42no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—21no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8—41no—
75Dic79⋊C4—21no—
76Dic79⋊C4—21no—
77C22⋊Dic79—21no—
78C22⋊Dic79—21no—
79(C2×D79)⋊C4—21no—
80(C2×D79)⋊C4—21no—
81C4×Dic79—21no—
82C4×Dic79—21no—
83C79⋊C16—42no—
84C24×C79C2×C811yescharacteristic
85C22×C4×C79—41no—
86C22×C4×C79—41no—
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C2×C811yescharacteristic
90C22×C4×C79M1611yescharacteristic
91C22×C4×C79M1611yescharacteristic
92C22×C4×C79C22×C411yescharacteristic
93C2×C8×C79—22no—
94C2×C8×C79—22no—
95C2×C8×C79—22no—
96C2×C8×C79—41no—
97C2×C8×C79—41no—
98C2×C8×C79—41no—
99C2×C8×C79C22⋊C411yescharacteristic
100C2×C8×C79C22⋊C411yescharacteristic
101C2×C8×C79C2×D411yescharacteristic
102C2×C8×C79C2×D411yescharacteristic
103C16×C79—42no—
104C23×C79—22no—
105C23×C79—22no—
106C23×C79—41no—
107C23×C79C22⋊C811yescharacteristic
108C23×C79C22⋊C811yescharacteristic
109C23×C79C22×C811yescharacteristic
110C22×D79—21no—
111C22×D79—22no—
112C22×D79—41no—
113C22×D79—41no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79C22⋊C811yescharacteristic
124C2×C4×C79C22⋊C811yescharacteristic
125C2×C4×C79C2×C22⋊C411yescharacteristic
126C2×C4×C79C2×M1611yescharacteristic
127C22⋊D79—42no—
128C22⋊D79—42no—
129C2×Dic79—21no—
130C2×Dic79—21no—
131C2×Dic79—21no—
132C2×Dic79—22no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C4×D79—42no—
136C4×D79—41no—
137C4×D79—41no—
138C8×C79—21no—
139C8×C79—21no—
140C8×C79—21no—
141C8×C79#9912yes—
142C79⋊C8—41no—
143C79⋊C8—41no—
144C22×C79—21no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—21no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—44no—
152C22×C79#34912yes—
153C22×C79C2×C22⋊C811yescharacteristic
154C22×C79#50412yes—
155C2×D79—41no—
156C2×D79—41no—
157C2×D79—41no—
158C2×D79—82no—
159C4×C79—21no—
160C4×C79—21no—
161C4×C79—21no—
162C4×C79C2×#9911yescharacteristic
163C4×C79#41311yescharacteristic
164Dic79—41no—
165Dic79—41no—
166Dic79—41no—
167#11467—791nomaximal, Sylow
168C2×C79—22no—
169C2×C79—22no—
170C2×C79—41no—
171C2×C79#145911yescharacteristic
172C2×C79#211311yescharacteristic
173C2×C79C2×#34911yescharacteristic
174D79—81no—
175C2×#435—791no—
176#2112—791no—
177C2×C22⋊C16D7911yescharacteristic, complemented
178#1322—792no—
179#1314—792no—
180C79#1146711yescharacteristic, complemented, Sylow
181C22×M16—791no—
182C23×C8C2×D7911yescharacteristic
183C2×C4×D4—791no—
184C2×M32—1581no—
185C22×C16—21no—
186#435—1581no—
187#435—1581no—
188#435—1581no—
189#435—1581no—
190C2×C4⋊C8—791no—
191C2×C22⋊C8—791no—
192C2×C22⋊C8—791no—
193C2×#97—791no—
194C22⋊C16—21no—
195C22⋊C16—792no—
196C22⋊C16—1581no—
197C22⋊C16C2×D7912yes—
198C22×D4—791no—
199C23×C4—791no—
200C23×C4C4×D7911yescharacteristic
201C2×M16—3162no—
202C2×M16—1581no—
203C2×M16—1581no—
204C22×C8—42no—
205C22×C8—791no—
206C22×C8—791no—
207C22×C8C4×D7911yescharacteristic
208C22×C8D31611yescharacteristic
209C22×C8D31611yescharacteristic
210C22×C8C22⋊D7911yescharacteristic
211C22×C8C22⋊D7911yescharacteristic
212C22×C8C22×D7911yescharacteristic
213C4×D4—1581no—
214C4×D4—1581no—
215C4×D4—1581no—
216C4×D4—1581no—
217C2×C4⋊C4—791no—
218C2×C22⋊C4—791no—
219C2×C22⋊C4—791no—
220C2×C42—791no—
221M32—3162no—
222C2×C16—22no—
223C2×C16—42no—
224C2×C16—1582no—
225C4⋊C8—1581no—
226C4⋊C8—1581no—
227C22⋊C8—1581no—
228C22⋊C8—1581no—
229C22⋊C8—1581no—
230C22⋊C8—1581no—
231#97—1581no—
232#97—1581no—
233C24—791no—
234C24C8×D7911yescharacteristic
235C2×D4—3162no—
236C2×D4—1581no—
237C2×D4—1581no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—3161no—
241C22×C4—3161no—
242C22×C4—791no—
243C22×C4—791no—
244C22×C4—791no—
245C22×C4—1581no—
246C22×C4—1581no—
247C22×C4(C2×D79)⋊C411yescharacteristic
248C22×C4(C2×D79)⋊C411yescharacteristic
249C22×C4C8×D7911yescharacteristic
250C22×C4#1192960711yescharacteristic
251C22×C4#1192960711yescharacteristic
252C22×C4C2×C4×D7911yescharacteristic
253M16—3162no—
254M16—3162no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—3161no—
262C2×C8—3161no—
263C2×C8—1581no—
264C2×C8—1581no—
265C2×C8(C2×D79)⋊C411yescharacteristic
266C2×C8(C2×D79)⋊C411yescharacteristic
267C2×C8C2×C22⋊D7911yescharacteristic
268C2×C8C2×D31611yescharacteristic
269C4⋊C4—1581no—
270C4⋊C4—1581no—
271C22⋊C4—1581no—
272C22⋊C4—1581no—
273C22⋊C4—1581no—
274C22⋊C4—1581no—
275C42—1581no—
276C42—1581no—
277C16—42no—
278C16—3162no—
279C23—22no—
280C23—22no—
281C23—41no—
282C23—3161no—
283C23—3161no—
284C23—1581no—
285C23—1582no—
286C23(C2×D79)⋊C811yescharacteristic
287C23(C2×D79)⋊C811yescharacteristic
288C23C2×C8×D7911yescharacteristic
289D4—3162no—
290D4—3162no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—3162no—
301C2×C4—3161no—
302C2×C4—3161no—
303C2×C4—3161no—
304C2×C4—3161no—
305C2×C4—1582no—
306C2×C4—1581no—
307C2×C4—1581no—
308C2×C4—1581no—
309C2×C4(C2×D79)⋊C811yescharacteristic
310C2×C4(C2×D79)⋊C811yescharacteristic
311C2×C4C2×(C2×D79)⋊C411yescharacteristic
312C2×C4C2×#1192960711yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—3161no—
317C8—3161no—
318C8#1457907512yes—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—44no—
327C22—3161no—
328C22—3161no—
329C22—3161no—
330C22—6322no—
331C22#1909834412yes—
332C22C2×(C2×D79)⋊C811yescharacteristic
333C22#1909897912yes—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—3161no—
338C4—3161no—
339C4—3161no—
340C4C2×#1457907511yescharacteristic
341C4#1909854511yescharacteristic
342C2—22no—
343C2—22no—
344C2—41no—
345C2—6321no—
346C2#2376171611yescharacteristic
347C2#2376501311yescharacteristic
348C2C2×#1909834411yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D79)⋊C4
commutatorC23×C79C22×C8
FrattiniC22×C8C22×D79
FittingC2×C22⋊C16×C79C2
socleC22×C79C2×C22⋊C8
prime radicalC22×C22⋊D79C8
cosocleC22×C8×C79C23
hypercenterC2×C22⋊C16D79
hypocenterC79#11467
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. #23765011C2semidirect
3. #23761339C2semidirect
4. #23761323C2semidirect
5. C2×C22⋊C16×C79C2semidirect
177. C2×C22⋊C16D79semidirect
180. C79#11467semidirect

Maximal subgroups

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

subgroupindex
C2×#190986342
#237650112
#237613392
#237613232
C2×C22⋊C16×C792
#1146779