Finite Group Databasebeta

← #34184012

Subgroups — #34184012

Order 20224.

756 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771731C211yesmaximal, characteristic
2C2×#19098409C211yesmaximal, characteristic, complemented
3C2×#19098298C212yesmaximal
4#23761280C212yesmaximal, complemented
5C79×#1817C211yesmaximal, characteristic, complemented
6C2×#14579144—21no—
7C2×#14579135C2211yescharacteristic
8C22×C4×Dic79—21no—
9#19098465C412yescomplemented
10#19098461C412yescomplemented
11#19098454—22no—
12#19098442—21no—
13#19098423C2211yescharacteristic
14#19098409—24no—
15C2×C79⋊#95—21no—
16C2×C79⋊#95C2212yes—
17#19098298—24no—
18C22×C42×C79C2211yescharacteristic
19C2×C22⋊C8×C79C2212yes—
20#336×C79—22no—
21C22×D316C2×C411yescharacteristic
22C22×C22⋊D79—21no—
23C23×Dic79—22no—
24#14579144—44no—
25#14579135—22no—
26#14579135—22no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C412yes—
29C2×Dic79⋊C4—22no—
30C2×C22⋊Dic79—21no—
31C2×(C2×D79)⋊C4—22no—
32C2×(C2×D79)⋊C4—22no—
33C2×(C2×D79)⋊C4—21no—
34C2×C4×Dic79—24no—
35C2×C4×Dic79—28no—
36C79⋊#95—24no—
37C79⋊#95—22no—
38C79⋊#95—22no—
39C79⋊#95—21no—
40C79⋊#95D414yes—
41C23×C4×C79D411yescharacteristic
42C23×C4×C79D411yescharacteristic
43C23×C4×C79C2311yescharacteristic
44C22×C8×C79—22no—
45C2×C42×C79—22no—
46C2×C42×C79—22no—
47C2×C42×C79C2×C412yes—
48C2×C42×C79D412yes—
49C22⋊C8×C79—22no—
50C22⋊C8×C79—22no—
51C23×D79—21no—
52C2×D316—22no—
53C2×D316—42no—
54C2×C22⋊D79—41no—
55C2×C22⋊D79—41no—
56C2×C22⋊D79—42no—
57C2×C22⋊D79—42no—
58C22×Dic79—28no—
59C22×Dic79—22no—
60C22×Dic79—21no—
61C22×Dic79—24no—
62C22×Dic79—44no—
63C22×Dic79—42no—
64C22×Dic79—42no—
65C4⋊Dic79—22no—
66C4⋊Dic79—42no—
67Dic79⋊C4—44no—
68C22⋊Dic79—42no—
69(C2×D79)⋊C4—44no—
70(C2×D79)⋊C4—44no—
71(C2×D79)⋊C4—42no—
72C4×Dic79—216no—
73C24×C79C2×D411yescharacteristic
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—24no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—22no—
80C22×C4×C79—22no—
81C22×C4×C79—42no—
82C22×C4×C79C22⋊C412yes—
83C22×C4×C79C22⋊C412yes—
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D412yes—
87C22×C4×C79#4112yes—
88C2×C8×C79—44no—
89C2×C8×C79—42no—
90C42×C79—24no—
91C42×C79—24no—
92C23×C79—22no—
93C23×C79—22no—
94C23×C79C2×C22⋊C411yescharacteristic
95C23×C79#12011yescharacteristic
96C23×C79#12311yescharacteristic
97C23×C79#13512yes—
98C23×C79AGL(1,ℤ/8)12yes—
99C22×D79—41no—
100C22×D79—42no—
101C22×D79—41no—
102C22×D79—41no—
103C22×D79—42no—
104C2×C4×C79—24no—
105C2×C4×C79—24no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79#10414yes—
118C2×C4×C79C4×D412yes—
119C2×C4×C79#12112yes—
120C22⋊D79—82no—
121C22⋊D79—82no—
122C2×Dic79—28no—
123C2×Dic79—44no—
124C2×Dic79—48no—
125C2×Dic79—48no—
126C2×Dic79—44no—
127C2×Dic79—41no—
128C2×Dic79—42no—
129C2×Dic79—41no—
130C2×Dic79—42no—
131C2×Dic79—41no—
132C2×Dic79—42no—
133D316—82no—
134C8×C79—82no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—24no—
140C22×C79—24no—
141C22×C79#38611yescharacteristic
142C22×C79#41012yes—
143C22×C79#41611yescharacteristic
144C22×C79#41811yescharacteristic
145C22×C79C2×#10412yes—
146C22×C79#44912yes—
147C22×C79#48212yes—
148C2×D79—81no—
149C2×D79—81no—
150C2×D79—81no—
151C2×D79—82no—
152C2×D79—82no—
153C4×C79—24no—
154C4×C79—22no—
155C4×C79—22no—
156C4×C79—42no—
157Dic79—48no—
158Dic79—81no—
159Dic79—82no—
160#9149—791nomaximal, Sylow
161C2×C79—22no—
162C2×C79—22no—
163C2×C79#150012yes—
164C2×C79#150512yes—
165C2×C79#187511yescharacteristic
166C2×C79#189611yescharacteristic
167C2×C79#189911yescharacteristic
168D79—161no—
169#2854—791no—
170C2×#376—791no—
171#1817D7911yescharacteristic, complemented
172C2×#327—792no—
173#1294—792no—
174C79#914911yescharacteristic, complemented, Sylow
175C2×#123—1581no—
176C2×#121—791no—
177C22×C42—1581no—
178C22×C42C2×D7911yescharacteristic
179C2×C22⋊C8C2×D7912yes—
180#390—792no—
181#389—792no—
182#388—1582no—
183#386—1581no—
184#380—791no—
185#376—1584no—
186C2×#95—792no—
187C2×#95—1581no—
188#336—22no—
189#327—1584no—
190C22×D4—791no—
191C22×D4—1581no—
192C23×C4—1582no—
193C23×C4D31611yescharacteristic
194C23×C4D31611yescharacteristic
195C23×C4C22×D7911yescharacteristic
196C22×C8—22no—
197#123—3164no—
198#121—1582no—
199#121—1582no—
200C2×C4⋊C4—792no—
201C2×C4⋊C4—791no—
202C2×C4⋊C4—1582no—
203C2×C22⋊C4—1581no—
204C2×C22⋊C4—1582no—
205C2×C22⋊C4—1582no—
206C2×C22⋊C4—1581no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—1584no—
210C2×C42—1588no—
211C2×C42C4×D7912yes—
212C2×C42C22⋊D7912yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—794no—
216#95—1584no—
217#95—1582no—
218#95—1582no—
219#95—1581no—
220C24—1581no—
221C24C2×D31611yescharacteristic
222C2×D4—3162no—
223C2×D4—3161no—
224C2×D4—3161no—
225C2×D4—3162no—
226C2×D4—3162no—
227C2×D4—1582no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—3164no—
237C22×C4—3162no—
238C22×C4—3162no—
239C22×C4—1582no—
240C22×C4—1581no—
241C22×C4—1588no—
242C22×C4—1584no—
243C22×C4(C2×D79)⋊C412yes—
244C22×C4(C2×D79)⋊C412yes—
245C22×C4C2×C4×D7911yescharacteristic
246C22×C4D4×D7912yes—
247C22×C4C2×C22⋊D7911yescharacteristic
248C22×C4#1192962312yes—
249C2×C8—44no—
250C2×C8—42no—
251C4⋊C4—3162no—
252C4⋊C4—3164no—
253C4⋊C4—1582no—
254C22⋊C4—3164no—
255C22⋊C4—3162no—
256C22⋊C4—3164no—
257C22⋊C4—3162no—
258C42—24no—
259C42—24no—
260C42—15816no—
261C23—22no—
262C23—22no—
263C23—3161no—
264C23—3162no—
265C23—3161no—
266C23—3161no—
267C23—3162no—
268C23C2×(C2×D79)⋊C411yescharacteristic
269C23#1457913111yescharacteristic
270C23#1457914511yescharacteristic
271C23#1457918312yes—
272C23#1457918912yes—
273D4—6322no—
274D4—6322no—
275D4—6322no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—3164no—
290C2×C4—3168no—
291C2×C4—3168no—
292C2×C4—3164no—
293C2×C4—3161no—
294C2×C4—3162no—
295C2×C4—3161no—
296C2×C4—3162no—
297C2×C4—3161no—
298C2×C4—3162no—
299C2×C4—1588no—
300C2×C4#1457908714yes—
301C2×C4(C22⋊D79)⋊C412yes—
302C2×C4#1457913312yes—
303C8—82no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—24no—
310C22—6321no—
311C22—6321no—
312C22—6321no—
313C22—6322no—
314C22—6322no—
315C22#1909844011yescharacteristic
316C22#1909853612yes—
317C22#1909856011yescharacteristic
318C22#1909856911yescharacteristic
319C22C2×#1457908712yes—
320C22#1909871012yes—
321C22#1909888812yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—3168no—
327C4—6321no—
328C4—6322no—
329C2—22no—
330C2—22no—
331C2—12641no—
332C2#2376188712yes—
333C2#2376190912yes—
334C2#2376372011yescharacteristic
335C2#2376384111yescharacteristic
336C2#2376385611yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1817C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenter#1817D79
hypocenterC79#9149
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#19098409C2semidirect
4. #23761280C2semidirect
5. C79×#1817C2semidirect
9. #19098465C4semidirect
10. #19098461C4semidirect
171. #1817D79semidirect
174. C79#9149semidirect

Maximal subgroups

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

subgroupindex
#237717312
C2×#190984092
C2×#190982982
#237612802
C79×#18172
#914979