Finite Group Databasebeta

← #34184580

Subgroups — #34184580

Order 20224.

452 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 93 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×#19098901C211yesmaximal, characteristic
2#23772060C211yesmaximal, characteristic
3#23763500C211yesmaximal, characteristic
4#23761764C211yesmaximal, characteristic
5#23761276C211yesmaximal, characteristic
6#23761248C211yesmaximal, characteristic
7C79×#1508C211yesmaximal, characteristic
8C2×#14579120C2211yescharacteristic
9#19098901—21no—
10#19098901—21no—
11#19098901—21no—
12#19098901—21no—
13#19098621—22no—
14#19098617—21no—
15#19098613—21no—
16C2×#14579094C411yescharacteristic, complemented
17C2×#14579092C411yescharacteristic, complemented
18#19098591—21no—
19C2×#14579085C411yescharacteristic, complemented
20C2×#14579085C411yescharacteristic, complemented
21#19098527C2211yescharacteristic
22C2×C22⋊(C79⋊C8)C2211yescharacteristic
23C2×C22⋊(C79⋊C8)C2211yescharacteristic
24#19098521—21no—
25#19098323—21no—
26#19098323—21no—
27#19098289—21no—
28#19098289—21no—
29C2×C22⋊Q8×C79C2211yescharacteristic
30#407×C79C2211yescharacteristic
31C4×C22⋊C4×C79C2211yescharacteristic
32#328×C79—22no—
33C2×#11929608—21no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8C2×C411yescharacteristic
36C22×C79⋊C8C2×C411yescharacteristic
37#14579120—21no—
38#14579120—21no—
39#14579120D411yescharacteristic
40#14579120D411yescharacteristic
41#14579120D411yescharacteristic
42#14579120D411yescharacteristic
43C2×C4⋊Dic79C2×C411yescharacteristic
44C2×C4⋊Dic79C2×C411yescharacteristic
45C2×C22⋊Dic79—21no—
46C2×C4×Dic79—21no—
47#14579094—21no—
48#14579094—21no—
49#14579092—21no—
50#14579092—21no—
51Dic79⋊C8—22no—
52Dic79⋊C8—22no—
53#14579085—21no—
54#14579085—21no—
55#14579085—21no—
56#14579085—21no—
57C22⋊(C79⋊C8)—21no—
58C22⋊(C79⋊C8)—21no—
59C22⋊(C79⋊C8)—21no—
60C22⋊(C79⋊C8)—21no—
61C22⋊(C79⋊C8)—21no—
62C22⋊(C79⋊C8)D411yescharacteristic
63C22⋊(C79⋊C8)D411yescharacteristic
64#14579071—22no—
65C8×Dic79—22no—
66C22×Q8×C79C2×C411yescharacteristic
67C23×C4×C79C2311yescharacteristic
68C2×M16×C79—21no—
69C22⋊Q8×C79—21no—
70C22⋊Q8×C79—21no—
71C22⋊Q8×C79—21no—
72C22⋊Q8×C79—21no—
73C2×C4⋊C4×C79—21no—
74C2×C4⋊C4×C79C2×C411yescharacteristic
75C2×C22⋊C4×C79—21no—
76C2×C22⋊C4×C79—21no—
77C2×C42×C79—21no—
78C22⋊C8×C79—21no—
79C22⋊C8×C79D412yes—
80#95×C79—21no—
81C22×Dic79—21no—
82C22×Dic79—21no—
83#11929608—41no—
84#11929608—41no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C4⋊Dic79—21no—
95C4⋊Dic79—21no—
96C4⋊Dic79C22⋊C411yescharacteristic
97C4⋊Dic79C22⋊C411yescharacteristic
98C4⋊Dic79C22⋊C411yescharacteristic
99C4⋊Dic79C22⋊C411yescharacteristic
100C22⋊Dic79—21no—
101C22⋊Dic79—21no—
102C22⋊Dic79—21no—
103C22⋊Dic79—21no—
104C4×Dic79—21no—
105C4×Dic79—21no—
106C4×Dic79—21no—
107C4×Dic79—21no—
108C24×C79#4111yescharacteristic
109C2×Q8×C79—21no—
110C2×Q8×C79—21no—
111C2×Q8×C79—41no—
112C2×Q8×C79—41no—
113C22×C4×C79—21no—
114C22×C4×C79—21no—
115C22×C4×C79—41no—
116C22×C4×C79—41no—
117C22×C4×C79—41no—
118C22×C4×C79C22×C411yescharacteristic
119C22×C4×C79C2×D411yescharacteristic
120C22×C4×C79C2×D411yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79C2×D411yescharacteristic
123C22×C4×C79#4111yescharacteristic
124M16×C79—41no—
125M16×C79—41no—
126C2×C8×C79—22no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79—41no—
130C4⋊C4×C79—41no—
131C22⋊C4×C79—22no—
132C22⋊C4×C79—41no—
133C22⋊C4×C79—41no—
134C22⋊C4×C79—41no—
135C42×C79—41no—
136C42×C79—41no—
137C23×C79—21no—
138C23×C79—21no—
139C23×C79—41no—
140C23×C79—41no—
141C23×C79C4×D411yescharacteristic
142C23×C79#12111yescharacteristic
143C23×C79#12311yescharacteristic
144Q8×C79—41no—
145Q8×C79—41no—
146Q8×C79—81no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—42no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—81no—
162C2×C4×C79—81no—
163C2×C4×C79—81no—
164C2×C4×C79C2×C22⋊C411yescharacteristic
165C2×C4×C79C4×D411yescharacteristic
166C2×C4×C79#12011yescharacteristic
167C2×C4×C79#12111yescharacteristic
168C2×Dic79—21no—
169C2×Dic79—21no—
170C2×Dic79—21no—
171C2×Dic79—21no—
172C2×Dic79—41no—
173C2×Dic79—41no—
174C2×Dic79—81no—
175C8×C79—42no—
176C79⋊C8—41no—
177C79⋊C8—41no—
178C79⋊C8—41no—
179C79⋊C8—41no—
180C79⋊C8—81no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—21no—
187C22×C79—21no—
188C22×C79—21no—
189C22×C79—81no—
190C22×C79—81no—
191C22×C79#38611yescharacteristic
192C22×C79#44311yescharacteristic
193C22×C79#44411yescharacteristic
194C4×C79—21no—
195C4×C79—21no—
196C4×C79—21no—
197C4×C79—82no—
198C4×C79—81no—
199C4×C79—81no—
200C4×C79#41011yescharacteristic
201C4×C79#41011yescharacteristic
202Dic79—41no—
203Dic79—41no—
204Dic79—41no—
205Dic79—41no—
206#9240—791nomaximal, Sylow
207C2×C79—21no—
208C2×C79—21no—
209C2×C79—41no—
210C2×C79—41no—
211C2×C79#187611yescharacteristic
212C2×C79#189911yescharacteristic
213C2×C79#190211yescharacteristic
214C2×#485—791no—
215#2924—791no—
216#1826—791no—
217#1508D7911yescharacteristic
218#1472—791no—
219#1293—791no—
220#1282—791no—
221C79#924011yescharacteristic, complemented, Sylow
222C2×C22⋊Q8C2×D7911yescharacteristic
223C2×#117—791no—
224#485—1581no—
225#485—1581no—
226#485—1581no—
227#485—1581no—
228#433—1582no—
229#432—1581no—
230#431—1581no—
231C2×#107—791no—
232C2×#106—791no—
233#424—1581no—
234C2×#103—791no—
235C2×#103—791no—
236#407—791no—
237#407C2×D7911yescharacteristic
238C2×C22⋊C8—791no—
239C2×C22⋊C8—791no—
240#405—1581no—
241C4×C22⋊C4C2×D7911yescharacteristic
242#339—1581no—
243#339—1581no—
244#328—22no—
245#324—1581no—
246#324—1581no—
247C22×Q8C4×D7911yescharacteristic
248C23×C4C22×D7911yescharacteristic
249C2×M16—21no—
250C2×M16—1581no—
251C22×C8—791no—
252C22×C8—791no—
253C22×C8—1581no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258#117—791no—
259#117—791no—
260#117—791no—
261#117—791no—
262#117—1581no—
263#117—1581no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—791no—
266C2×C4⋊C4—791no—
267C2×C4⋊C4C4×D7911yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—1581no—
271C2×C42—21no—
272C2×C42—1581no—
273#107—1581no—
274#107—1581no—
275#106—1581no—
276#106—1581no—
277C4⋊C8—1582no—
278C4⋊C8—1582no—
279#103—1581no—
280#103—1581no—
281#103—1581no—
282#103—1581no—
283C22⋊C8—21no—
284C22⋊C8—791no—
285C22⋊C8—791no—
286C22⋊C8—1581no—
287C22⋊C8—1581no—
288C22⋊C8—1581no—
289C22⋊C8—1581no—
290C22⋊C8—1581no—
291C22⋊C8C22⋊D7912yes—
292#97—1582no—
293C4×C8—1582no—
294#95—21no—
295C24#1192962311yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—1581no—
306C22×C4—1581no—
307C22×C4C2×C4×D7911yescharacteristic
308C22×C4D4×D7911yescharacteristic
309C22×C4D4×D7911yescharacteristic
310C22×C4D4×D7911yescharacteristic
311C22×C4C2×C22⋊D7911yescharacteristic
312C22×C4#1192962411yescharacteristic
313M16—41no—
314M16—41no—
315M16—3161no—
316M16—3161no—
317C2×C8—22no—
318C2×C8—3161no—
319C2×C8—3161no—
320C2×C8—3161no—
321C2×C8—3161no—
322C2×C8—3161no—
323C2×C8—1581no—
324C2×C8—1581no—
325C2×C8—1581no—
326C2×C8—1581no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—791no—
332C4⋊C4—791no—
333C4⋊C4—791no—
334C4⋊C4—791no—
335C4⋊C4—1581no—
336C4⋊C4—1581no—
337C22⋊C4—22no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—1581no—
342C22⋊C4—1581no—
343C22⋊C4—1581no—
344C22⋊C4—1581no—
345C42—41no—
346C42—41no—
347C42—1581no—
348C42—1581no—
349C42—1581no—
350C42—1581no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23(C22⋊D79)⋊C411yescharacteristic
356C23#1457913211yescharacteristic
357C23#1457914311yescharacteristic
358Q8—41no—
359Q8—41no—
360Q8—81no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—3161no—
379C2×C4—3161no—
380C2×C4—6321no—
381C2×C4—1581no—
382C2×C4—1581no—
383C2×C4—1581no—
384C2×C4—1581no—
385C2×C4C22⋊C4×D7911yescharacteristic
386C2×C4C4×C22⋊D7911yescharacteristic
387C2×C4#1457912911yescharacteristic
388C2×C4#1457913411yescharacteristic
389C8—42no—
390C8—3161no—
391C8—3161no—
392C8—3161no—
393C8—3161no—
394C8—6321no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22#1909843911yescharacteristic
406C22#1909867711yescharacteristic
407C22#1909868211yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—3161no—
415C4—3161no—
416C4—3161no—
417C4—3161no—
418C4#1909853511yescharacteristic
419C4#1909853511yescharacteristic
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2#2376372411yescharacteristic
425C2#2376385511yescharacteristic
426C2#2376387011yescharacteristic
subgroupG/H
centerC22#19098439
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1508C2
socleC22×C79#386
prime radicalC24×C79#41
cosocleC23×C4×C79C23
hypercenter#1508D79
hypocenterC79#9240
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
16. C2×#14579094C4semidirect
17. C2×#14579092C4semidirect
19. C2×#14579085C4semidirect
20. C2×#14579085C4semidirect
221. C79#9240semidirect

Maximal subgroups

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

subgroupindex
C2×#190989012
#237720602
#237635002
#237617642
#237612762
#237612482
C79×#15082
#924079