Finite Group Databasebeta

← #34177139

Subgroups — #34177139

Order 20224.

504 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 95 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#23764728C211yesmaximal, characteristic, complemented
2#23764646C211yesmaximal, characteristic, complemented
3#23764536C211yesmaximal, characteristic
4#23761223C211yesmaximal, characteristic, complemented
5#23761223C211yesmaximal, characteristic, complemented
6#23761221C211yesmaximal, characteristic
7C79×#2683C211yesmaximal, characteristic
8C2×#14579161C2211yescharacteristic
9C2×#14579094—21no—
10C2×#14579084—21no—
11C2×#14579084—21no—
12C2×#14579082—21no—
13#19098325—21no—
14#19098325—21no—
15#19098318C2211yescharacteristic
16#19098318C2211yescharacteristic
17#19098318C2211yescharacteristic
18C2×C22⋊Q8×C79—21no—
19C2×C22⋊Q8×C79—21no—
20#396×C79—21no—
21#396×C79—21no—
22#395×C79—21no—
23#390×C79C2211yescharacteristic
24#389×C79C2211yescharacteristic
25#389×C79C2211yescharacteristic
26#386×C79—21no—
27C22×Dic158—21no—
28C22×C79⋊C8—21no—
29C22×C79⋊C8—21no—
30C22×C79⋊C8—21no—
31#14579161—42no—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34#14579094—42no—
35#14579084—42no—
36#14579084—42no—
37#14579082—42no—
38C22×Q8×C79—21no—
39C22×D4×C79D411yescharacteristic
40C23×C4×C79—21no—
41C22⋊Q8×C79—42no—
42C22⋊Q8×C79—42no—
43C22⋊Q8×C79—42no—
44C22⋊Q8×C79—42no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79—21no—
55C2×C22⋊C4×C79—21no—
56C2×C22⋊C4×C79—21no—
57C2×C22⋊C4×C79—21no—
58C2×C22⋊C4×C79—21no—
59C2×C42×C79C2311yescharacteristic
60#95×C79—21no—
61#95×C79—21no—
62#95×C79—21no—
63C2×Dic158—22no—
64C2×Dic158—41no—
65C2×Dic158—41no—
66C2×Dic158—41no—
67C2×Dic158—41no—
68C2×Dic158—41no—
69C22×Dic79—41no—
70C2×C79⋊C8—22no—
71C2×C79⋊C8—41no—
72C2×C79⋊C8—41no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8—41no—
75C2×C79⋊C8—41no—
76C2×C79⋊C8—41no—
77C2×C79⋊C8—41no—
78C2×C79⋊C8—41no—
79C4⋊Dic79—22no—
80C4⋊Dic79—22no—
81C4⋊Dic79—41no—
82C4⋊Dic79—41no—
83C24×C79—21no—
84C2×Q8×C79—42no—
85C2×Q8×C79—42no—
86C2×Q8×C79—81no—
87C2×D4×C79—21no—
88C2×D4×C79—41no—
89C2×D4×C79—41no—
90C2×D4×C79Q1612yes—
91C22×C4×C79—21no—
92C22×C4×C79—21no—
93C22×C4×C79—21no—
94C22×C4×C79—21no—
95C22×C4×C79—21no—
96C22×C4×C79—42no—
97C22×C4×C79—42no—
98C22×C4×C79—81no—
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C4⋊C4×C79—21no—
103C4⋊C4×C79—21no—
104C4⋊C4×C79—42no—
105C4⋊C4×C79—42no—
106C4⋊C4×C79—41no—
107C4⋊C4×C79—41no—
108C4⋊C4×C79—41no—
109C4⋊C4×C79—41no—
110C4⋊C4×C79—41no—
111C4⋊C4×C79—41no—
112C4⋊C4×C79—41no—
113C4⋊C4×C79D812yes—
114C4⋊C4×C79SD1612yes—
115C22⋊C4×C79—42no—
116C22⋊C4×C79—42no—
117C22⋊C4×C79—41no—
118C22⋊C4×C79—41no—
119C22⋊C4×C79—41no—
120C22⋊C4×C79—41no—
121C22⋊C4×C79—81no—
122C42×C79—41no—
123C23×C79—22no—
124C23×C79—41no—
125C23×C79—41no—
126C23×C79—41no—
127C23×C79—41no—
128C23×C79—41no—
129C23×C79—41no—
130C23×C79#12011yescharacteristic
131Q8×C79—84no—
132D4×C79—42no—
133D4×C79—81no—
134C2×C4×C79—21no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—22no—
141C2×C4×C79—22no—
142C2×C4×C79—41no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
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—84no—
156C2×C4×C79C2×D811yescharacteristic
157C2×C4×C79C2×SD1611yescharacteristic
158C2×C4×C79C2×Q1611yescharacteristic
159C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C79#13711yescharacteristic
162C2×Dic79—42no—
163C2×Dic79—81no—
164C2×Dic79—81no—
165Dic158—42no—
166Dic158—42no—
167Dic158—81no—
168Dic158—81no—
169C79⋊C8—42no—
170C79⋊C8—81no—
171C79⋊C8—81no—
172C22×C79—42no—
173C22×C79—42no—
174C22×C79—42no—
175C22×C79—81no—
176C22×C79—81no—
177C22×C79—81no—
178C22×C79—81no—
179C22×C79#44711yescharacteristic
180C22×C79#45011yescharacteristic
181C22×C79#45111yescharacteristic
182C22×C79#45411yescharacteristic
183C22×C79#45511yescharacteristic
184C22×C79#45611yescharacteristic
185C22×C79B(4,𝔽2)11yescharacteristic
186C4×C79—21no—
187C4×C79—21no—
188C4×C79—21no—
189C4×C79—21no—
190C4×C79—21no—
191C4×C79—21no—
192C4×C79—42no—
193C4×C79—42no—
194C4×C79—81no—
195C4×C79—81no—
196C4×C79—81no—
197Dic79—82no—
198#7764—791nomaximal, Sylow
199C2×C79—82no—
200C2×C79#159811yescharacteristic
201C2×C79#160211yescharacteristic
202C2×C79#160411yescharacteristic
203C2×C79#163711yescharacteristic
204C2×C79#163911yescharacteristic
205C2×C79#164911yescharacteristic
206C2×C79#166411yescharacteristic
207#2683D7911yescharacteristic
208#2057—791no—
209#2043—791no—
210#2023—791no—
211#1271—791no—
212#1271—791no—
213#1270—791no—
214C79#776411yescharacteristic, complemented, Sylow
215C2×C4⋊Q8—791no—
216C2×C22⋊Q8—21no—
217C2×C22⋊Q8—21no—
218C2×#107—1581no—
219C2×#103—1581no—
220C2×#103—1581no—
221C2×#102—1581no—
222#396—21no—
223#396—21no—
224#395—21no—
225#390C2×D7911yescharacteristic
226#389C2×D7911yescharacteristic
227#389C2×D7911yescharacteristic
228#386—21no—
229#340—1581no—
230#340—1581no—
231#336—791no—
232#336—791no—
233#336—791no—
234C22×Q8—21no—
235C22×Q8—1581no—
236C22×D4C22⋊D7911yescharacteristic
237C23×C4—21no—
238C22×C8—1581no—
239C22×C8—1581no—
240C22×C8—1581no—
241C4⋊Q8—3162no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244C22⋊Q8—42no—
245C22⋊Q8—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—1581no—
251C2×C4⋊C4—1581no—
252C2×C4⋊C4C22⋊D7911yescharacteristic
253C2×C4⋊C4C22⋊D7911yescharacteristic
254C2×C4⋊C4C22⋊D7911yescharacteristic
255C2×C4⋊C4C22⋊D7911yescharacteristic
256C2×C4⋊C4C22⋊D7911yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C42C22×D7911yescharacteristic
263#107—3162no—
264#103—3162no—
265#103—3162no—
266#102—3162no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—81no—
274C2×Q8—3161no—
275C2×Q8—3161no—
276C2×Q8—3161no—
277C2×Q8—3161no—
278C2×Q8—3161no—
279C2×Q8—1582no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4#1192961412yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—81no—
292C22×C4—3161no—
293C22×C4C2×C22⋊D7911yescharacteristic
294C22×C4C2×C22⋊D7911yescharacteristic
295C22×C4C2×C22⋊D7911yescharacteristic
296C2×C8—3161no—
297C2×C8—3161no—
298C2×C8—3161no—
299C2×C8—3161no—
300C2×C8—3161no—
301C2×C8—3161no—
302C2×C8—3161no—
303C2×C8—3161no—
304C2×C8—1582no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—3161no—
317C4⋊C4—3161no—
318C4⋊C4—1582no—
319C4⋊C4—1582no—
320C4⋊C4#1192960912yes—
321C4⋊C4#1192961212yes—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—81no—
329C42—41no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#1457913011yescharacteristic
338Q8—84no—
339Q8—3162no—
340Q8—3162no—
341Q8—6321no—
342Q8—6321no—
343D4—42no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
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—84no—
367C2×C4—3162no—
368C2×C4—6321no—
369C2×C4—6321no—
370C2×C4C2×#1192960911yescharacteristic
371C2×C4C2×#1192961211yescharacteristic
372C2×C4C2×#1192961411yescharacteristic
373C2×C4#1457918711yescharacteristic
374C2×C4#1457918711yescharacteristic
375C2×C4#1457919211yescharacteristic
376C8—3162no—
377C8—6321no—
378C8—6321no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1909869711yescharacteristic
387C22#1909871411yescharacteristic
388C22#1909871911yescharacteristic
389C22#1909873611yescharacteristic
390C22#1909874111yescharacteristic
391C22#1909874611yescharacteristic
392C22#1909875011yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—42no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—6322no—
405C2—82no—
406C2#2376233511yescharacteristic
407C2#2376235711yescharacteristic
408C2#2376236911yescharacteristic
409C2#2376256511yescharacteristic
410C2#2376257911yescharacteristic
411C2#2376264911yescharacteristic
412C2#2376273411yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2683C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#2683D79
hypocenterC79#7764
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. #23764728C2semidirect
2. #23764646C2semidirect
4. #23761223C2semidirect
5. #23761223C2semidirect
214. C79#7764semidirect

Maximal subgroups

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

subgroupindex
#237647282
#237646462
#237645362
#237612232
#237612212
C79×#26832
#776479