Finite Group Databasebeta

← #34180819

Subgroups — #34180819

Order 20224.

648 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 131 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#23764366C211yesmaximal, characteristic, complemented
2#23762027C211yesmaximal, characteristic, complemented
3#23761938C212yesmaximal, complemented
4#23761860C212yesmaximal, complemented
5C79×#3553C211yesmaximal, characteristic
6#19098620C2212yes—
7C2×C4×C79⋊C8C2211yescharacteristic
8#19098429C2211yescharacteristic
9#19098325—22no—
10#19098305—22no—
11#19098300—22no—
12#19098297—22no—
13Q8⋊(C79⋊C8)—22no—
14D4⋊(C79⋊C8)—22no—
15#564×C79C411yescharacteristic, complemented
16#563×C79C412yescomplemented
17#562×C79C411yescharacteristic, complemented
18#554×C79—22no—
19#553×C79—22no—
20#552×C79—22no—
21#541×C79—22no—
22#539×C79—22no—
23#539×C79—22no—
24#529×C79C2212yes—
25C2×#125×C79C2211yescharacteristic
26C22×C79⋊C8—21no—
27C2×C4⋊Dic79—21no—
28C4⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30#14579072—22no—
31C4×C79⋊C8—22no—
32C79⋊#95—22no—
33C4⋊Q8×C79—22no—
34#126×C79—22no—
35#126×C79—22no—
36#126×C79—22no—
37#126×C79—22no—
38#126×C79—22no—
39#125×C79—22no—
40#125×C79—22no—
41#125×C79—22no—
42#125×C79—22no—
43#125×C79—22no—
44#125×C79—22no—
45#125×C79C2×C411yescharacteristic
46#125×C79C2×C411yescharacteristic
47#125×C79C2×C412yes—
48#125×C79D412yes—
49#124×C79C2×C412yes—
50#123×C79—22no—
51#123×C79—22no—
52#123×C79—22no—
53#123×C79—22no—
54#123×C79—22no—
55C22⋊Q8×C79—22no—
56C22⋊Q8×C79—22no—
57C22⋊Q8×C79—22no—
58C22⋊Q8×C79—22no—
59C22⋊Q8×C79—22no—
60C22⋊Q8×C79—22no—
61C22⋊Q8×C79D412yes—
62#121×C79—22no—
63#121×C79—22no—
64#121×C79D412yes—
65C4×Q8×C79—22no—
66C4×Q8×C79D412yes—
67C4×D4×C79—22no—
68C4×D4×C79—22no—
69C4×D4×C79—22no—
70C4×D4×C79—22no—
71C4×D4×C79D412yes—
72#117×C79—22no—
73#117×C79—22no—
74C2×C4⋊C4×C79—22no—
75C2×C4⋊C4×C79—22no—
76C2×C4⋊C4×C79D412yes—
77C2×C42×C79C2311yescharacteristic
78C22×Dic79—41no—
79C2×C79⋊C8—22no—
80C2×C79⋊C8—41no—
81C2×C79⋊C8—41no—
82C2×C79⋊C8—42no—
83C4⋊Dic79—42no—
84C2×Q8×C79—22no—
85C2×Q8×C79C22⋊C412yes—
86C2×D4×C79—22no—
87C2×D4×C79—22no—
88C2×D4×C79C22⋊C412yes—
89C22×C4×C79—22no—
90C22×C4×C79—22no—
91C22×C4×C79—22no—
92C22×C4×C79—22no—
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×D412yes—
95C4⋊C4×C79—22no—
96C4⋊C4×C79—22no—
97C4⋊C4×C79—22no—
98C4⋊C4×C79—22no—
99C4⋊C4×C79—22no—
100C4⋊C4×C79—22no—
101C4⋊C4×C79—22no—
102C4⋊C4×C79—22no—
103C4⋊C4×C79—22no—
104C4⋊C4×C79—22no—
105C4⋊C4×C79—22no—
106C4⋊C4×C79—22no—
107C4⋊C4×C79—22no—
108C4⋊C4×C79—22no—
109C4⋊C4×C79—22no—
110C4⋊C4×C79C22⋊C412yes—
111C4⋊C4×C79C22⋊C412yes—
112C4⋊C4×C79C22⋊C412yes—
113C4⋊C4×C79C22⋊C412yes—
114C22⋊C4×C79—22no—
115C22⋊C4×C79—22no—
116C22⋊C4×C79—22no—
117C22⋊C4×C79—22no—
118C22⋊C4×C79—22no—
119C22⋊C4×C79—22no—
120C22⋊C4×C79—22no—
121C22⋊C4×C79—22no—
122C22⋊C4×C79—22no—
123C42×C79—22no—
124C42×C79—22no—
125C42×C79—22no—
126C42×C79C22×C411yescharacteristic
127C42×C79C2×D411yescharacteristic
128C42×C79C2×D412yes—
129C23×C79—22no—
130C23×C79#12011yescharacteristic
131Q8×C79—42no—
132Q8×C79—82no—
133D4×C79—42no—
134D4×C79—42no—
135D4×C79—42no—
136D4×C79—82no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—22no—
140C2×C4×C79—22no—
141C2×C4×C79—22no—
142C2×C4×C79—22no—
143C2×C4×C79—22no—
144C2×C4×C79—42no—
145C2×C4×C79—82no—
146C2×C4×C79—82no—
147C2×C4×C79—82no—
148C2×C4×C79—82no—
149C2×C4×C79C2×C22⋊C411yescharacteristic
150C2×C4×C79C2×C22⋊C412yes—
151C2×C4×C79#12012yes—
152C2×C4×C79#12011yescharacteristic
153C2×Dic79—82no—
154C2×Dic79—81no—
155C79⋊C8—42no—
156C79⋊C8—82no—
157C22×C79—22no—
158C22×C79—42no—
159C22×C79—42no—
160C22×C79—42no—
161C22×C79#37911yescharacteristic
162C22×C79#45411yescharacteristic
163C22×C79#45611yescharacteristic
164C4×C79—21no—
165C4×C79—21no—
166C4×C79—22no—
167C4×C79—22no—
168C4×C79—42no—
169C4×C79—42no—
170C4×C79—42no—
171C4×C79—42no—
172C4×C79—42no—
173Dic79—161no—
174#8529—791nomaximal, Sylow
175C2×C79—21no—
176C2×C79—21no—
177C2×C79—82no—
178C2×C79#179311yescharacteristic
179C2×C79#179412yes—
180#3553D7911yescharacteristic
181#1991—791no—
182#1534—791no—
183#1512—792no—
184#1494—792no—
185C79#852911yescharacteristic, complemented, Sylow
186#564Dic7911yescharacteristic, complemented
187#563Dic7912yescomplemented
188#562Dic7911yescharacteristic, complemented
189#554—22no—
190#553—22no—
191#552—22no—
192#541—22no—
193#539—22no—
194#539—22no—
195#529C2×D7912yes—
196C2×#125C2×D7911yescharacteristic
197#432—792no—
198C2×C4×C8—791no—
199#382—791no—
200#340—1582no—
201#330—1582no—
202#328—1582no—
203#327—1582no—
204Q8⋊C8—1582no—
205D4⋊C8—1582no—
206C22×C8—1581no—
207C4⋊Q8—22no—
208#126—22no—
209#126—22no—
210#126—22no—
211#126—22no—
212#126—22no—
213#125—22no—
214#125—22no—
215#125—22no—
216#125—22no—
217#125—22no—
218#125—22no—
219#125C2×Dic7911yescharacteristic
220#125C2×Dic7911yescharacteristic
221#125C2×Dic7912yes—
222#125C22⋊D7912yes—
223#124C2×Dic7912yes—
224#123—22no—
225#123—22no—
226#123—22no—
227#123—22no—
228#123—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8C22⋊D7912yes—
236#121—22no—
237#121—22no—
238#121C22⋊D7912yes—
239C4×Q8—22no—
240C4×Q8C22⋊D7912yes—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4C22⋊D7912yes—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—1581no—
251C2×C4⋊C4C22⋊D7912yes—
252C2×C42C22×D7911yescharacteristic
253C4⋊C8—1582no—
254C22⋊C8—1582no—
255#97—1582no—
256C4×C8—1582no—
257#95—1582no—
258C2×Q8—22no—
259C2×Q8C22⋊Dic7912yes—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4C22⋊Dic7912yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—3161no—
268C22×C4C2×C22⋊D7911yescharacteristic
269C22×C4C2×C22⋊D7912yes—
270C2×C8—3161no—
271C2×C8—3161no—
272C2×C8—3162no—
273C2×C8—1582no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—3162no—
290C4⋊C4C22⋊Dic7912yes—
291C4⋊C4C22⋊Dic7912yes—
292C4⋊C4C22⋊Dic7912yes—
293C4⋊C4C22⋊Dic7912yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42C22×Dic7911yescharacteristic
307C42C2×C22⋊D7912yes—
308C42C2×C22⋊D7911yescharacteristic
309C23—22no—
310C23#1457913011yescharacteristic
311Q8—42no—
312Q8—82no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—6322no—
330C2×C4—6321no—
331C2×C4C2×C22⋊Dic7911yescharacteristic
332C2×C4C2×C22⋊Dic7912yes—
333C2×C4#1457913012yes—
334C2×C4#1457913011yescharacteristic
335C8—3162no—
336C8—6322no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22#1909841911yescharacteristic
342C22#1909873611yescharacteristic
343C22#1909874611yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—12641no—
354C2—21no—
355C2—21no—
356C2—82no—
357C2#2376335911yescharacteristic
358C2#2376336412yes—
subgroupG/H
centerC22#19098419
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3553C2
socleC22×C79#379
cosocleC2×C42×C79C23
hypercenter#3553D79
hypocenterC79#8529
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3553

Complements

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

NH ≅ G/Ntype
1. #23764366C2semidirect
2. #23762027C2semidirect
3. #23761938C2semidirect
4. #23761860C2semidirect
15. #564×C79C4semidirect
16. #563×C79C4semidirect
17. #562×C79C4semidirect
185. C79#8529semidirect
186. #564Dic79semidirect
187. #563Dic79semidirect
188. #562Dic79semidirect

Maximal subgroups

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

subgroupindex
#237643662
#237620272
#237619382
#237618602
C79×#35532
#852979