Finite Group Databasebeta

← #34186337

Subgroups — #34186337

Order 20224.

482 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 99 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#23773344C211yesmaximal, characteristic
2#23763500C211yesmaximal, characteristic
3#23763475C211yesmaximal, characteristic
4#23761933C211yesmaximal, characteristic
5#23761308C211yesmaximal, characteristic
6#23761253C211yesmaximal, characteristic
7C79×#2332C211yesmaximal, characteristic
8C2×#14579120C2211yescharacteristic
9#19098901—22no—
10#19098896C412yescomplemented
11#19098891C412yescomplemented
12#19098603—21no—
13#19098572—21no—
14#19098527C2211yescharacteristic
15#19098527C2211yescharacteristic
16#19098426C2211yescharacteristic
17#19098329—21no—
18#19098329—21no—
19#19098323—21no—
20#19098323—21no—
21#19098300—22no—
22C2×C22⋊Q8×C79C2211yescharacteristic
23#391×C79—21no—
24#388×C79—21no—
25#388×C79—21no—
26#387×C79—21no—
27#387×C79—21no—
28#381×C79C2211yescharacteristic
29C4×C4⋊C4×C79—21no—
30C4×C22⋊C4×C79C2211yescharacteristic
31C2×#11929608—21no—
32C2×#11929608—21no—
33#14579120—21no—
34#14579120—21no—
35#14579120—21no—
36#14579120—21no—
37C2×C4⋊Dic79C2×C411yescharacteristic
38C2×C4⋊Dic79C2×C411yescharacteristic
39C2×C22⋊Dic79—21no—
40C2×C22⋊Dic79—21no—
41C2×C4×Dic79—21no—
42#14579094—22no—
43#14579092—22no—
44#14579085—22no—
45#14579085—22no—
46C22⋊(C79⋊C8)—21no—
47C22⋊(C79⋊C8)—21no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)C2×C412yes—
50C79⋊#95—21no—
51C79⋊#95—21no—
52C22×Q8×C79D411yescharacteristic
53C23×C4×C79C2311yescharacteristic
54C22⋊Q8×C79—21no—
55C22⋊Q8×C79—21no—
56C22⋊Q8×C79C2×C412yes—
57C22⋊Q8×C79D412yes—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79D411yescharacteristic
62C2×C22⋊C4×C79—21no—
63C2×C22⋊C4×C79—21no—
64C2×C22⋊C4×C79D411yescharacteristic
65C2×C22⋊C4×C79D411yescharacteristic
66C2×C42×C79—21no—
67C2×C42×C79—21no—
68C2×C42×C79D411yescharacteristic
69C2×C42×C79D411yescharacteristic
70#95×C79—21no—
71#95×C79—21no—
72#95×C79—21no—
73#95×C79—21no—
74#95×C79—21no—
75#95×C79—21no—
76#95×C79—21no—
77C22×Dic79—21no—
78C22×Dic79—21no—
79C22×Dic79—41no—
80#11929608—41no—
81#11929608—41no—
82#11929608—41no—
83#11929608—41no—
84C2×C79⋊C8—22no—
85C2×C79⋊C8—41no—
86C4⋊Dic79—21no—
87C4⋊Dic79—21no—
88C4⋊Dic79—21no—
89C4⋊Dic79—21no—
90C22⋊Dic79—41no—
91C22⋊Dic79—41no—
92C22⋊Dic79—41no—
93C22⋊Dic79—41no—
94C4×Dic79—41no—
95C4×Dic79—41no—
96C24×C79C2×D411yescharacteristic
97C2×Q8×C79—21no—
98C2×Q8×C79—41no—
99C2×Q8×C79—41no—
100C2×Q8×C79C22⋊C412yes—
101C22×C4×C79—21no—
102C22×C4×C79—21no—
103C22×C4×C79—21no—
104C22×C4×C79—21no—
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79—41no—
108C22×C4×C79—41no—
109C22×C4×C79C22×C411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79#4111yescharacteristic
114C22×C4×C79#4111yescharacteristic
115C4⋊C4×C79—22no—
116C4⋊C4×C79—22no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—42no—
119C4⋊C4×C79—81no—
120C4⋊C4×C79C22⋊C412yes—
121C22⋊C4×C79—22no—
122C22⋊C4×C79—22no—
123C22⋊C4×C79—42no—
124C22⋊C4×C79—41no—
125C22⋊C4×C79—41no—
126C42×C79—42no—
127C42×C79—41no—
128C42×C79—41no—
129C42×C79—81no—
130C23×C79—21no—
131C23×C79—21no—
132C23×C79—41no—
133C23×C79—41no—
134C23×C79C4×D411yescharacteristic
135C23×C79#12011yescharacteristic
136C23×C79#12111yescharacteristic
137Q8×C79—42no—
138Q8×C79—81no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—22no—
144C2×C4×C79—22no—
145C2×C4×C79—22no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—41no—
165C2×C4×C79C2×C22⋊C411yescharacteristic
166C2×C4×C79C4×D411yescharacteristic
167C2×C4×C79#12111yescharacteristic
168C2×C4×C79#12311yescharacteristic
169C2×Dic79—41no—
170C2×Dic79—41no—
171C2×Dic79—81no—
172C2×Dic79—81no—
173C2×Dic79—81no—
174C2×Dic79—81no—
175C2×Dic79—81no—
176C79⋊C8—42no—
177C79⋊C8—81no—
178C22×C79—21no—
179C22×C79—21no—
180C22×C79—21no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—81no—
187C22×C79—81no—
188C22×C79#38611yescharacteristic
189C22×C79#45811yescharacteristic
190C22×C79#45811yescharacteristic
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79—21no—
195C4×C79—42no—
196C4×C79—42no—
197C4×C79—42no—
198C4×C79—81no—
199C4×C79—81no—
200C4×C79—81no—
201Dic79—81no—
202Dic79—81no—
203Dic79—161no—
204#9526—791nomaximal, Sylow
205C2×C79—21no—
206C2×C79—21no—
207C2×C79—41no—
208C2×C79—41no—
209C2×C79#189011yescharacteristic
210C2×C79#190511yescharacteristic
211C2×C79#190611yescharacteristic
212#3095—791no—
213#2332D7911yescharacteristic
214#1826—791no—
215#1820—791no—
216#1511—791no—
217#1306—791no—
218#1284—791no—
219C79#952611yescharacteristic, complemented, Sylow
220C2×C22⋊Q8C2×D7911yescharacteristic
221C2×#117—791no—
222#485—1582no—
223#484—792no—
224#483—792no—
225#428—1581no—
226#419—1581no—
227#407—791no—
228#407—791no—
229#391—21no—
230#388—21no—
231#388—21no—
232#387—21no—
233#387—21no—
234#381—791no—
235#381C2×D7911yescharacteristic
236C4×C4⋊C4—21no—
237C4×C22⋊C4C2×D7911yescharacteristic
238#342—1581no—
239#342—1581no—
240#339—1581no—
241#339—1581no—
242#328—1582no—
243C22×Q8C22⋊D7911yescharacteristic
244C23×C4C22×D7911yescharacteristic
245C2×M16—1581no—
246C2×M16—1581no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8C4×D7912yes—
250C22⋊Q8D31612yes—
251#117—1581no—
252#117—1581no—
253#117—1581no—
254#117—1581no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—791no—
259C2×C4⋊C4—791no—
260C2×C4⋊C4C22⋊D7911yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—1581no—
264C2×C22⋊C4—1581no—
265C2×C22⋊C4C22⋊D7911yescharacteristic
266C2×C22⋊C4C22⋊D7911yescharacteristic
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—1581no—
270C2×C42C22⋊D7911yescharacteristic
271C2×C42C22⋊D7911yescharacteristic
272#107—1582no—
273#106—1582no—
274#103—1582no—
275#103—1582no—
276C22⋊C8—792no—
277C22⋊C8—1581no—
278C22⋊C8—1581no—
279C22⋊C8—1581no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—1581no—
288#95—1581no—
289C24C2×C22⋊D7911yescharacteristic
290C2×Q8—21no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8(C2×D79)⋊C412yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—3161no—
303C22×C4—1581no—
304C22×C4—1581no—
305C22×C4C2×C4×D7911yescharacteristic
306C22×C4C2×C22⋊D7911yescharacteristic
307C22×C4C2×C22⋊D7911yescharacteristic
308C22×C4C2×D31611yescharacteristic
309C22×C4#1192962411yescharacteristic
310C22×C4#1192962411yescharacteristic
311M16—3161no—
312M16—3161no—
313M16—3161no—
314M16—3161no—
315C2×C8—3161no—
316C2×C8—1582no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—42no—
321C4⋊C4—81no—
322C4⋊C4—1581no—
323C4⋊C4—1581no—
324C4⋊C4—1581no—
325C4⋊C4—1581no—
326C4⋊C4(C2×D79)⋊C412yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—3161no—
333C22⋊C4—3161no—
334C22⋊C4—3161no—
335C22⋊C4—3161no—
336C42—41no—
337C42—41no—
338C42—42no—
339C42—81no—
340C42—3161no—
341C42—3161no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23C4×C22⋊D7911yescharacteristic
347C23#1457913011yescharacteristic
348C23#1457913511yescharacteristic
349Q8—42no—
350Q8—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—3161no—
378C2×C4—3161no—
379C2×C4—6321no—
380C2×C4—6321no—
381C2×C4—6321no—
382C2×C4—6321no—
383C2×C4—6321no—
384C2×C4C2×(C2×D79)⋊C411yescharacteristic
385C2×C4C4×C22⋊D7911yescharacteristic
386C2×C4#1457913511yescharacteristic
387C2×C4#1457914411yescharacteristic
388C8—3162no—
389C8—6321no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22#1909844211yescharacteristic
401C22#1909875311yescharacteristic
402C22#1909875311yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—6321no—
414C4—6321no—
415C4—12641no—
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#2376381311yescharacteristic
421C2#2376389411yescharacteristic
422C2#2376390111yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2332C2
socleC22×C79#386
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#2332D79
hypocenterC79#9526
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2332

Complements

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

NH ≅ G/Ntype
10. #19098896C4semidirect
11. #19098891C4semidirect
219. C79#9526semidirect

Maximal subgroups

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

subgroupindex
#237733442
#237635002
#237634752
#237619332
#237613082
#237612532
C79×#23322
#952679