Finite Group Databasebeta

← #34183334

Subgroups — #34183334

Order 20224.

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

normal subgroups only

This group has 696 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#23765998C211yesmaximal, characteristic
2#23763486C212yesmaximal, complemented
3C2×#19098287C211yesmaximal, characteristic
4#23761279C212yesmaximal, complemented
5C2×#397×C79C211yesmaximal, characteristic
6C2×C22⋊(C79⋊C8)C2212yes—
7#19098456—24no—
8#19098423C2212yes—
9#19098419—21no—
10C4×C22⋊Dic79—21no—
11#19098325—24no—
12#19098287—22no—
13C22×C4⋊C4×C79C2211yescharacteristic
14C22×C22⋊C4×C79—21no—
15C22×C22⋊C4×C79C2211yescharacteristic
16#397×C79—22no—
17#397×C79C414yescomplemented
18C2×#95×C79—21no—
19C2×#95×C79C2211yescharacteristic
20C22×C79⋊C8—22no—
21C2×C4⋊Dic79D414yes—
22C2×C22⋊Dic79—22no—
23C2×C22⋊Dic79—22no—
24C2×C22⋊Dic79—22no—
25C2×C4×Dic79—22no—
26C22⋊(C79⋊C8)—22no—
27C22⋊(C79⋊C8)—22no—
28C79⋊#95—22no—
29C79⋊#95—22no—
30C25×C79D411yescharacteristic
31C23×C4×C79—21no—
32C23×C4×C79—21no—
33C23×C4×C79D411yescharacteristic
34C23×C4×C79C2311yescharacteristic
35C2×C4⋊C4×C79—22no—
36C2×C4⋊C4×C79—22no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79C2×C412yes—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79—24no—
42C2×C22⋊C4×C79—44no—
43C2×C22⋊C4×C79C2×C412yes—
44C2×C22⋊C4×C79D412yes—
45#95×C79—24no—
46#95×C79—21no—
47#95×C79C2×C412yes—
48C22×Dic79—24no—
49C2×C79⋊C8—44no—
50C2×C79⋊C8—42no—
51C4⋊Dic79—44no—
52C22⋊Dic79—42no—
53C22⋊Dic79—42no—
54C22⋊Dic79—82no—
55C22⋊Dic79—82no—
56C4×Dic79—82no—
57C24×C79—21no—
58C24×C79—22no—
59C24×C79—21no—
60C24×C79—22no—
61C24×C79—42no—
62C24×C79—42no—
63C24×C79C22⋊C412yes—
64C24×C79C2×D411yescharacteristic
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—22no—
68C22×C4×C79—22no—
69C22×C4×C79—42no—
70C22×C4×C79—42no—
71C22×C4×C79—42no—
72C22×C4×C79—42no—
73C22×C4×C79—41no—
74C22×C4×C79—41no—
75C22×C4×C79—42no—
76C22×C4×C79—42no—
77C22×C4×C79—42no—
78C22×C4×C79C22⋊C412yes—
79C22×C4×C79C22×C411yescharacteristic
80C22×C4×C79C2×D412yes—
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79#4112yes—
83C4⋊C4×C79—24no—
84C4⋊C4×C79—42no—
85C22⋊C4×C79—24no—
86C22⋊C4×C79—24no—
87C22⋊C4×C79—48no—
88C23×C79—22no—
89C23×C79—22no—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79—22no—
93C23×C79—42no—
94C23×C79—42no—
95C23×C79—48no—
96C23×C79—42no—
97C23×C79—42no—
98C23×C79—42no—
99C23×C79—42no—
100C23×C79—81no—
101C23×C79—81no—
102C23×C79—84no—
103C23×C79C2×C22⋊C411yescharacteristic
104C23×C79#12011yescharacteristic
105C23×C79#12311yescharacteristic
106C23×C79#13512yes—
107C23×C79AGL(1,ℤ/8)12yes—
108C2×C4×C79—24no—
109C2×C4×C79—22no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—42no—
119C2×C4×C79—42no—
120C2×C4×C79—42no—
121C2×C4×C79—81no—
122C2×C4×C79—84no—
123C2×C4×C79—82no—
124C2×C4×C79—82no—
125C2×C4×C79C4×D412yes—
126C2×C4×C79#12112yes—
127C2×Dic79—44no—
128C2×Dic79—44no—
129C2×Dic79—44no—
130C79⋊C8—82no—
131C22×C79—24no—
132C22×C79—24no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—44no—
138C22×C79—42no—
139C22×C79—42no—
140C22×C79—42no—
141C22×C79—82no—
142C22×C79—84no—
143C22×C79—84no—
144C22×C79#38611yescharacteristic
145C22×C79#41611yescharacteristic
146C22×C79#41811yescharacteristic
147C22×C79#44912yes—
148C22×C79#45512yes—
149C22×C79#45812yes—
150C22×C79#48212yes—
151C4×C79—22no—
152C4×C79—22no—
153C4×C79—42no—
154C4×C79—82no—
155C4×C79—82no—
156Dic79—84no—
157#9028—791nomaximal, Sylow
158C2×C79—21no—
159C2×C79—21no—
160C2×C79—22no—
161C2×C79—82no—
162C2×C79#161214yes—
163C2×C79#187511yescharacteristic
164C2×C79#188511yescharacteristic
165C2×C79#189011yescharacteristic
166C2×#397D7911yescharacteristic
167#2312—791no—
168#1823—792no—
169C2×#323—791no—
170#1294—792no—
171C79#902811yescharacteristic, complemented, Sylow
172C22×C4⋊C4C2×D7911yescharacteristic
173C22×C22⋊C4—21no—
174C22×C22⋊C4C2×D7911yescharacteristic
175C2×C22⋊C8—792no—
176#397—22no—
177#397Dic7914yescomplemented
178#388—1584no—
179#380—792no—
180#379—1581no—
181C4×C22⋊C4—1581no—
182C2×#95—21no—
183C2×#95C2×D7911yescharacteristic
184#340—1584no—
185#323—1582no—
186C25C22⋊D7911yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C22⋊D7911yescharacteristic
190C23×C4C22×D7911yescharacteristic
191C22×C8—1582no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—794no—
196C2×C4⋊C4C2×Dic7912yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—1582no—
202C2×C22⋊C4—1582no—
203C2×C22⋊C4—1582no—
204C2×C22⋊C4C2×Dic7912yes—
205C2×C22⋊C4C22⋊D7912yes—
206C2×C42—1582no—
207C22⋊C8—1582no—
208C22⋊C8—1582no—
209#95—24no—
210#95—21no—
211#95—1582no—
212#95—1582no—
213#95C2×Dic7912yes—
214C24—21no—
215C24—22no—
216C24—21no—
217C24—22no—
218C24—42no—
219C24—42no—
220C24C22⋊Dic7912yes—
221C24C2×C22⋊D7911yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1584no—
236C22×C4C22⋊Dic7912yes—
237C22×C4C22×Dic7911yescharacteristic
238C22×C4D4×D7912yes—
239C22×C4C2×C22⋊D7911yescharacteristic
240C22×C4#1192962312yes—
241C2×C8—3164no—
242C2×C8—3162no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—3164no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—48no—
249C22⋊C4—3162no—
250C22⋊C4—3162no—
251C22⋊C4—6322no—
252C22⋊C4—6322no—
253C42—6322no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—48no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—81no—
267C23—81no—
268C23—84no—
269C23C2×C22⋊Dic7911yescharacteristic
270C23#1457912911yescharacteristic
271C23#1457914211yescharacteristic
272C23#1457918112yes—
273C23#1457918712yes—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—84no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—3164no—
292C2×C4—3164no—
293C2×C4—3164no—
294C2×C4D4×Dic7912yes—
295C2×C4#1457913312yes—
296C8—6322no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—24no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—84no—
309C22—84no—
310C22#1909844311yescharacteristic
311C22#1909855911yescharacteristic
312C22#1909856811yescharacteristic
313C22#1909870612yes—
314C22#1909874012yes—
315C22#1909875212yes—
316C22#1909888412yes—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—82no—
322C4—6324no—
323C2—21no—
324C2—22no—
325C2—21no—
326C2—82no—
327C2#2376241214yes—
328C2#2376372311yescharacteristic
329C2#2376378911yescharacteristic
330C2#2376381411yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#397×C79C2
socleC23×C79#120
prime radicalC25×C79D4
cosocleC23×C4×C79C23
hypercenterC2×#397D79
hypocenterC79#9028
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. #23763486C2semidirect
4. #23761279C2semidirect
17. #397×C79C4semidirect
171. C79#9028semidirect
177. #397Dic79semidirect

Maximal subgroups

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

subgroupindex
#237659982
#237634862
C2×#190982872
#237612792
C2×#397×C792
#902879