Finite Group Databasebeta

← #34262191

Subgroups — #34262191

Order 20224.

940 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 187 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×#19099194C211yesmaximal, characteristic
2#23774172C212yesmaximal
3#23762441C214yesmaximal, complemented
4#23762373C214yesmaximal, complemented
5#23761971C212yesmaximal
6C2×#19098289C211yesmaximal, characteristic
7C79×#2627C211yesmaximal, characteristic
8#19099194—24no—
9#19099194—22no—
10#19099194C2214yes—
11C2×#14579160—22no—
12C2×#14579152—22no—
13C2×#14579140—21no—
14C2×#14579140C2212yes—
15C2×#14579120—21no—
16#19098898C2214yes—
17#19098896—22no—
18#19098802—24no—
19#19098719C2214yes—
20#19098718—22no—
21C2×C22⋊(C79⋊C8)C2212yes—
22#19098299C2218yes—
23#19098289C2214yes—
24C2×C22⋊Q8×C79C2211yescharacteristic
25C22×C4⋊C4×C79C2211yescharacteristic
26#400×C79C2212yes—
27#398×C79—22no—
28#395×C79C2212yes—
29#385×C79—21no—
30#381×C79—21no—
31#380×C79C2211yescharacteristic
32C22×Dic158—21no—
33C22×Dic158D412yes—
34C2×#11929614—24no—
35C22×C79⋊C8—22no—
36#14579160—28no—
37#14579160—28no—
38#14579152—28no—
39#14579152—28no—
40#14579140—24no—
41#14579140—22no—
42#14579140—22no—
43#14579140—24no—
44#14579140D414yes—
45#14579140C2314yes—
46#14579120—24no—
47#14579120—24no—
48C2×C4⋊Dic79—21no—
49C2×C4⋊Dic79—21no—
50C2×C4⋊Dic79—21no—
51C2×C4⋊Dic79D412yes—
52C2×Dic79⋊C4—22no—
53C2×Dic79⋊C4—22no—
54C2×C22⋊Dic79—22no—
55C2×C4×Dic79—22no—
56#14579094—24no—
57#14579085—24no—
58#14579084—24no—
59C22⋊(C79⋊C8)—22no—
60C22⋊(C79⋊C8)C2314yes—
61C22×Q8×C79C2311yescharacteristic
62C23×C4×C79D411yescharacteristic
63C23×C4×C79D411yescharacteristic
64C23×C4×C79C2311yescharacteristic
65C22⋊Q8×C79—22no—
66C22⋊Q8×C79—22no—
67C2×C4⋊C4×C79—22no—
68C2×C4⋊C4×C79—22no—
69C2×C4⋊C4×C79—21no—
70C2×C4⋊C4×C79—21no—
71C2×C4⋊C4×C79—22no—
72C2×C4⋊C4×C79D412yes—
73C2×C4⋊C4×C79C2311yescharacteristic
74C2×C4⋊C4×C79C2312yes—
75C2×C22⋊C4×C79—21no—
76C2×C22⋊C4×C79—21no—
77#95×C79—22no—
78#95×C79—22no—
79#95×C79—22no—
80#95×C79C2312yes—
81C2×Dic158—24no—
82C2×Dic158—22no—
83C2×Dic158—82no—
84C2×Dic158—82no—
85C2×Dic158C2×D414yes—
86C22×Dic79—22no—
87C22×Dic79—22no—
88#11929614—84no—
89C2×C79⋊C8—24no—
90C2×C79⋊C8—82no—
91C4⋊Dic79—24no—
92C4⋊Dic79—24no—
93C4⋊Dic79—24no—
94C4⋊Dic79—22no—
95C4⋊Dic79C2×D414yes—
96Dic79⋊C4—28no—
97Dic79⋊C4—24no—
98Dic79⋊C4—24no—
99C22⋊Dic79—24no—
100C22⋊Dic79—24no—
101C4×Dic79—28no—
102C24×C79C2×D411yescharacteristic
103C2×Q8×C79—22no—
104C2×Q8×C79—81no—
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79—22no—
108C22×C4×C79—21no—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—22no—
112C22×C4×C79—21no—
113C22×C4×C79—81no—
114C22×C4×C79—81no—
115C22×C4×C79Q1614yescomplemented
116C22×C4×C79C2×D412yes—
117C22×C4×C79C2×D412yes—
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79C2×D412yes—
120C22×C4×C79C2×D412yes—
121C22×C4×C79C2411yescharacteristic
122C4⋊C4×C79—24no—
123C4⋊C4×C79—24no—
124C4⋊C4×C79—22no—
125C4⋊C4×C79—42no—
126C4⋊C4×C79—81no—
127C4⋊C4×C79—82no—
128C22⋊C4×C79—41no—
129C22⋊C4×C79—41no—
130C22⋊C4×C79—42no—
131C23×C79—41no—
132C23×C79—41no—
133C23×C79#12011yescharacteristic
134C23×C79#12011yescharacteristic
135C23×C79C2×Q1612yes—
136C23×C79#13712yes—
137C23×C79C22×D411yescharacteristic
138Q8×C79—44no—
139C2×C4×C79—22no—
140C2×C4×C79—22no—
141C2×C4×C79—24no—
142C2×C4×C79—24no—
143C2×C4×C79—22no—
144C2×C4×C79—42no—
145C2×C4×C79—42no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—42no—
149C2×C4×C79—42no—
150C2×C4×C79—42no—
151C2×C4×C79—42no—
152C2×C4×C79—42no—
153C2×C4×C79—42no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—42no—
157C2×C4×C79—81no—
158C2×C4×C79#12012yes—
159C2×C4×C79C2×Q1614yes—
160C2×C4×C79C22×D412yes—
161C2×Dic79—24no—
162C2×Dic79—24no—
163C2×Dic79—44no—
164C2×Dic79—44no—
165Dic158—44no—
166Dic158—82no—
167Dic158—84no—
168C79⋊C8—44no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79—42no—
176C22×C79—42no—
177C22×C79#45112yes—
178C22×C79#45112yes—
179C22×C79C2×#12011yescharacteristic
180C22×C79C22×Q1611yescharacteristic
181C22×C79C2×#13711yescharacteristic
182C4×C79—24no—
183C4×C79—22no—
184C4×C79—22no—
185C4×C79—42no—
186C4×C79—82no—
187C4×C79—81no—
188C4×C79—81no—
189Dic79—44no—
190Dic79—44no—
191#18295—791nomaximal, Sylow
192C2×C79—21no—
193C2×C79—21no—
194C2×C79—22no—
195C2×C79—22no—
196C2×C79C2×#45111yescharacteristic
197C2×C79#302011yescharacteristic
198C2×C79#302611yescharacteristic
199C2×#544—791no—
200#3189—792no—
201#2627D7911yescharacteristic
202#1617—794no—
203#1605—794no—
204#1520—792no—
205C2×#324—791no—
206C79#1829511yescharacteristic, complemented, Sylow
207#544—794no—
208#544—1584no—
209#544—1582no—
210C2×C4⋊Q8—1582no—
211C2×#125—1582no—
212C2×C22⋊Q8—792no—
213C2×C22⋊Q8—1581no—
214C2×C22⋊Q8C2×D7911yescharacteristic
215C2×#117—1581no—
216C22×C4⋊C4C2×D7911yescharacteristic
217#484—794no—
218#484—1582no—
219#467—1584no—
220#451—794no—
221#451—1582no—
222C2×C22⋊C8—792no—
223#400C2×D7912yes—
224#398—22no—
225#395C2×D7912yes—
226#385—21no—
227#381—21no—
228#380C2×D7911yescharacteristic
229#328—798no—
230#324—794no—
231C22×Q8—792no—
232C22×Q8—1581no—
233C22×Q8C22×D7911yescharacteristic
234C23×C4C22⋊D7911yescharacteristic
235C23×C4C22⋊D7911yescharacteristic
236C23×C4C22×D7911yescharacteristic
237C2×Q16—1584no—
238C22×C8—1582no—
239C4⋊Q8—1588no—
240C4⋊Q8—1588no—
241#125—1588no—
242#125—1588no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—794no—
246C22⋊Q8—794no—
247C22⋊Q8—1584no—
248C22⋊Q8—1582no—
249C22⋊Q8—1582no—
250C22⋊Q8—1584no—
251#117—1584no—
252#117—1584no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—792no—
259C2×C4⋊C4—1581no—
260C2×C4⋊C4—1581no—
261C2×C4⋊C4—1581no—
262C2×C4⋊C4—1582no—
263C2×C4⋊C4—1582no—
264C2×C4⋊C4C22⋊D7912yes—
265C2×C4⋊C4C22×D7911yescharacteristic
266C2×C4⋊C4C22×D7912yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—1582no—
270C2×C42—1582no—
271#107—1584no—
272#103—1584no—
273#103—1584no—
274C22⋊C8—794no—
275C22⋊C8—1582no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95C22×D7912yes—
280C24C2×C22⋊D7911yescharacteristic
281C2×Q8—22no—
282C2×Q8—81no—
283C2×Q8—794no—
284C2×Q8—6322no—
285C2×Q8—6322no—
286C2×Q8—1584no—
287C2×Q8—1582no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—1582no—
299C22×C4—1582no—
300C22×C4#1192961414yescomplemented
301C22×C4D4×D7912yes—
302C22×C4D4×D7912yes—
303C22×C4C2×C22⋊D7912yes—
304C22×C4C2×C22⋊D7912yes—
305C22×C4C2×C22⋊D7911yescharacteristic
306C22×C4C23×D7911yescharacteristic
307Q16—6324no—
308C2×C8—6322no—
309C2×C8—1584no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—81no—
315C4⋊C4—82no—
316C4⋊C4—794no—
317C4⋊C4—1588no—
318C4⋊C4—1584no—
319C4⋊C4—1584no—
320C4⋊C4—1584no—
321C4⋊C4—1584no—
322C4⋊C4—1582no—
323C4⋊C4—1584no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—1584no—
328C22⋊C4—1584no—
329C42—1588no—
330C23—41no—
331C23—41no—
332C23#1457912911yescharacteristic
333C23#1457912911yescharacteristic
334C23C2×#1192961412yes—
335C23#1457919112yes—
336C23C22×C22⋊D7911yescharacteristic
337Q8—44no—
338Q8—3164no—
339Q8—6322no—
340Q8—6324no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—3164no—
361C2×C4—3164no—
362C2×C4—1584no—
363C2×C4—1584no—
364C2×C4#1457912912yes—
365C2×C4C2×#1192961414yes—
366C2×C4C2×D4×D7912yes—
367C8—3164no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1909871812yes—
377C22#1909871812yes—
378C22C2×#1457912911yescharacteristic
379C22C22×#1192961411yescharacteristic
380C22C2×#1457919111yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—42no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4—3164no—
389C4—3164no—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2C2×#1909871811yescharacteristic
395C2#2377276311yescharacteristic
396C2#2377279611yescharacteristic
subgroupG/H
centerC22C2×#14579129
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2627C2
socleC22×C79C2×#120
prime radicalC24×C79C2×D4
hypercenter#2627D79
hypocenterC79#18295
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
3. #23762441C2semidirect
4. #23762373C2semidirect
115. C22×C4×C79Q16semidirect
206. C79#18295semidirect
300. C22×C4#11929614semidirect

Maximal subgroups

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

subgroupindex
C2×#190991942
#237741722
#237624412
#237623732
#237619712
C2×#190982892
C79×#26272
#1829579