Finite Group Databasebeta

← #34297721

Subgroups — #34297721

Order 20224.

1130 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C2×#19099189C211yesmaximal, characteristic
2#23765996C212yesmaximal
3#23764383C218yesmaximal, complemented
4#23763474C211yesmaximal, characteristic
5#23763472C212yesmaximal
6C79×#1819C211yesmaximal, characteristic
7#19099189—24no—
8#19099189—24no—
9C2×#14579140C414yescomplemented
10C2×#14579138C414yescomplemented
11C2×#14579138C2214yes—
12C22×C22⋊Dic79C2211yescharacteristic
13C22×C22⋊Dic79C2211yescharacteristic
14#19098457—24no—
15#19098451—24no—
16#19098421C2214yes—
17#19098419C2212yes—
18C22⋊C4×Dic79C2214yes—
19#19098330C2214yes—
20#19098329C2218yes—
21C22×C22⋊C4×C79C2211yescharacteristic
22#379×C79C2212yes—
23#342×C79C2214yes—
24C22×Dic158C2×C414yes—
25C23×Dic79C2×C412yes—
26C23×Dic79C2312yes—
27#14579140—28no—
28#14579140—28no—
29#14579138—28no—
30#14579138—28no—
31#14579138—28no—
32#14579138—28no—
33C2×C4⋊Dic79C2×C414yes—
34C2×Dic79⋊C4—24no—
35C2×Dic79⋊C4C2×C418yes—
36C2×C22⋊Dic79—22no—
37C2×C22⋊Dic79—22no—
38C2×C22⋊Dic79—22no—
39C2×C22⋊Dic79—22no—
40C2×C22⋊Dic79C2×C412yes—
41C2×C22⋊Dic79C2×C412yes—
42C2×C22⋊Dic79C2×C412yes—
43C2×C22⋊Dic79D414yes—
44C2×C22⋊Dic79D414yes—
45C2×C22⋊Dic79C2314yes—
46C2×C22⋊Dic79C2312yes—
47C2×C4×Dic79—24no—
48C79⋊#95—24no—
49C79⋊#95—24no—
50C25×C79C2311yescharacteristic
51C23×C4×C79C2×C412yes—
52C2×C22⋊C4×C79—22no—
53C2×C22⋊C4×C79—22no—
54C2×C22⋊C4×C79—22no—
55C2×C22⋊C4×C79—22no—
56C2×C22⋊C4×C79C2×C412yes—
57C2×C22⋊C4×C79C2314yes—
58C2×C22⋊C4×C79C2312yes—
59C2×Dic158—28no—
60C2×Dic158—84no—
61C22×Dic79—22no—
62C22×Dic79—22no—
63C22×Dic79—24no—
64C22×Dic79—82no—
65C22×Dic79—82no—
66C22×Dic79C22×C414yes—
67C22×Dic79C22×C414yes—
68C22×Dic79C2×D414yes—
69C22×Dic79C2×D414yes—
70C4⋊Dic79—28no—
71Dic79⋊C4—216no—
72Dic79⋊C4—84no—
73C22⋊Dic79—24no—
74C22⋊Dic79—24no—
75C22⋊Dic79—28no—
76C22⋊Dic79—82no—
77C22⋊Dic79—82no—
78C22⋊Dic79C22⋊C414yes—
79C22⋊Dic79C22⋊C414yes—
80C22⋊Dic79C22⋊C418yes—
81C4×Dic79—84no—
82C24×C79—24no—
83C24×C79—82no—
84C24×C79C22×C411yescharacteristic
85C24×C79C22×C411yescharacteristic
86C24×C79C2×D412yes—
87C24×C79C2×D412yes—
88C24×C79C2411yescharacteristic
89C22×C4×C79—22no—
90C22×C4×C79—22no—
91C22×C4×C79—24no—
92C22×C4×C79—82no—
93C22×C4×C79C22×C414yes—
94C22⋊C4×C79—22no—
95C22⋊C4×C79—22no—
96C22⋊C4×C79—24no—
97C22⋊C4×C79—28no—
98C22⋊C4×C79—82no—
99C22⋊C4×C79—82no—
100C23×C79—22no—
101C23×C79—22no—
102C23×C79—24no—
103C23×C79—24no—
104C23×C79—42no—
105C23×C79—42no—
106C23×C79—42no—
107C23×C79—42no—
108C23×C79—42no—
109C23×C79—42no—
110C23×C79—42no—
111C23×C79—88no—
112C23×C79C2×C22⋊C412yes—
113C23×C79C2×C22⋊C411yescharacteristic
114C23×C79C2×C22⋊C411yescharacteristic
115C23×C79C23×C411yescharacteristic
116C23×C79C22×D412yes—
117C23×C792−1+414yes—
118C2×C4×C79—24no—
119C2×C4×C79—44no—
120C2×C4×C79—44no—
121C2×C4×C79—44no—
122C2×C4×C79—44no—
123C2×C4×C79—82no—
124C2×C4×C79—82no—
125C2×Dic79—24no—
126C2×Dic79—44no—
127C2×Dic79—44no—
128C2×Dic79—44no—
129C2×Dic79—44no—
130C2×Dic79—44no—
131C2×Dic79—44no—
132C2×Dic79—44no—
133C2×Dic79—82no—
134C2×Dic79—82no—
135C2×Dic79C2×C22⋊C418yes—
136Dic158—88no—
137C22×C79—22no—
138C22×C79—22no—
139C22×C79—22no—
140C22×C79—22no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—22no—
144C22×C79—22no—
145C22×C79—22no—
146C22×C79—22no—
147C22×C79—44no—
148C22×C79—44no—
149C22×C79—44no—
150C22×C79—84no—
151C22×C79—84no—
152C22×C79C22×C22⋊C411yescharacteristic
153C22×C79#51912yes—
154C22×C79#53614yes—
155C4×C79—44no—
156C4×C79—84no—
157Dic79—28no—
158Dic79—44no—
159Dic79—84no—
160#21827—791nomaximal, Sylow
161C2×C79—24no—
162C2×C79—22no—
163C2×C79—42no—
164C2×C79—41no—
165C2×C79—41no—
166C2×C79#231011yescharacteristic
167C2×C79#288312yes—
168C2×#543—791no—
169#2312—792no—
170#1996—798no—
171#1819—792no—
172#1819—791no—
173#1819D7911yescharacteristic
174C79#2182711yescharacteristic, complemented, Sylow
175#543—1584no—
176#543—1584no—
177C2×C22⋊Q8—794no—
178C2×C22⋊Q8—794no—
179C2×C22⋊Q8—794no—
180C22×C22⋊C4—791no—
181C22×C22⋊C4—791no—
182C22×C22⋊C4C2×D7911yescharacteristic
183#388—1584no—
184#388—1584no—
185#380—794no—
186#379—792no—
187#379C2×D7912yes—
188C4×C22⋊C4—794no—
189#342—798no—
190#342—794no—
191#342C2×D7914yes—
192C25C22×D7911yescharacteristic
193C22×Q8—794no—
194C23×C4—792no—
195C23×C4—792no—
196C23×C4C4×D7912yes—
197C22⋊Q8—1588no—
198C22⋊Q8—1588no—
199C22⋊Q8—1588no—
200C22⋊Q8—1588no—
201C22⋊Q8—1588no—
202C22⋊Q8—1588no—
203C2×C4⋊C4—798no—
204C2×C4⋊C4—794no—
205C2×C4⋊C4—1584no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—792no—
211C2×C22⋊C4—792no—
212C2×C22⋊C4—794no—
213C2×C22⋊C4—794no—
214C2×C22⋊C4—792no—
215C2×C22⋊C4—792no—
216C2×C22⋊C4—794no—
217C2×C22⋊C4—1582no—
218C2×C22⋊C4—1582no—
219C2×C22⋊C4—1582no—
220C2×C22⋊C4—1582no—
221C2×C22⋊C4C4×D7912yes—
222C2×C22⋊C4C22×D7914yes—
223C2×C22⋊C4C22×D7912yes—
224C2×C42—1584no—
225#95—1584no—
226#95—1584no—
227C24—24no—
228C24—82no—
229C24C2×C4×D7911yescharacteristic
230C24C2×C4×D7911yescharacteristic
231C24D4×D7912yes—
232C24D4×D7912yes—
233C24C23×D7911yescharacteristic
234C2×Q8—6324no—
235C2×Q8—1588no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—82no—
240C22×C4—794no—
241C22×C4—794no—
242C22×C4—794no—
243C22×C4—794no—
244C22×C4—6322no—
245C22×C4—6322no—
246C22×C4—1584no—
247C22×C4—1582no—
248C22×C4—1582no—
249C22×C4C2×C4×D7914yes—
250C4⋊C4—6324no—
251C4⋊C4—15816no—
252C4⋊C4—1588no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—24no—
256C22⋊C4—28no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—794no—
260C22⋊C4—794no—
261C22⋊C4—798no—
262C22⋊C4—6322no—
263C22⋊C4—6322no—
264C22⋊C4—1588no—
265C22⋊C4—1584no—
266C22⋊C4—1584no—
267C42—6324no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—24no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—88no—
280C23C22⋊C4×D7911yescharacteristic
281C23C22⋊C4×D7911yescharacteristic
282C23C22⋊C4×D7912yes—
283C23C22×C4×D7911yescharacteristic
284C23C2×D4×D7912yes—
285C23#1457920814yes—
286Q8—6328no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—3164no—
295C2×C4—3164no—
296C2×C4—3164no—
297C2×C4—3164no—
298C2×C4—3164no—
299C2×C4—3164no—
300C2×C4—3164no—
301C2×C4—798no—
302C2×C4—6322no—
303C2×C4—6322no—
304C2×C4—1584no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—44no—
316C22—44no—
317C22—44no—
318C22—84no—
319C22—84no—
320C22C2×C22⋊C4×D7911yescharacteristic
321C22#1909904012yes—
322C22#1909913514yes—
323C4—44no—
324C4—84no—
325C4—3164no—
326C4—6324no—
327C4—1588no—
328C2—24no—
329C2—22no—
330C2—42no—
331C2—41no—
332C2—41no—
333C2#2376598511yescharacteristic
334C2#2377185812yes—
subgroupG/H
centerC22C2×C22⋊C4×D79
commutatorC23×C79C23×C4
FrattiniC24C23×D79
FittingC79×#1819C2
socleC22×C79C22×C22⋊C4
prime radicalC25×C79C23
cosocleC24×C79C24
hypercenter#1819D79
hypocenterC79#21827
LCSG ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #23764383C2semidirect
9. C2×#14579140C4semidirect
10. C2×#14579138C4semidirect
174. C79#21827semidirect

Maximal subgroups

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

subgroupindex
C2×#190991892
#237659962
#237643832
#237634742
#237634722
C79×#18192
#2182779