Finite Group Databasebeta

← #34201249

Subgroups — #34201249

Order 20224.

398 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764753C211yesmaximal, characteristic
2#23762390C211yesmaximal, characteristic
3#23762390C211yesmaximal, characteristic
4#23761400C211yesmaximal, characteristic
5#23761400C211yesmaximal, characteristic
6#23761348C211yesmaximal, characteristic
7C79×#2722C211yesmaximal, characteristic
8#19099189C2211yescharacteristic
9#19098893—21no—
10#19098893—21no—
11#19098436—21no—
12#19098330C2211yescharacteristic
13#19098329—21no—
14#19098299—21no—
15#19098299—21no—
16#19098287C2211yescharacteristic
17#19098287C2211yescharacteristic
18C2×#126×C79—21no—
19#398×C79—21no—
20#398×C79—21no—
21#397×C79C2211yescharacteristic
22#397×C79C2211yescharacteristic
23#389×C79—21no—
24#387×C79—21no—
25#385×C79C2211yescharacteristic
26#380×C79—21no—
27#14579140—21no—
28#14579140—21no—
29#14579138—41no—
30#14579138—41no—
31C2×Dic79⋊C4—41no—
32C2×C22⋊Dic79—21no—
33C2×C22⋊Dic79—21no—
34#14579092—41no—
35#14579092—41no—
36#14579084—41no—
37#14579084—41no—
38C22⋊(C79⋊C8)—21no—
39C22⋊(C79⋊C8)—21no—
40C79⋊#95—41no—
41C23×C4×C79D411yescharacteristic
42#126×C79—42no—
43#126×C79—42no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79D411yescharacteristic
55C2×C22⋊C4×C79C2311yescharacteristic
56C2×C42×C79—21no—
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79—21no—
61#95×C79—21no—
62#95×C79D411yescharacteristic
63#95×C79D411yescharacteristic
64C2×Dic158—21no—
65C2×Dic158—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—41no—
69C2×C79⋊C8—41no—
70C2×C79⋊C8—41no—
71C4⋊Dic79—21no—
72C4⋊Dic79—21no—
73Dic79⋊C4—41no—
74Dic79⋊C4—41no—
75Dic79⋊C4—82no—
76C22⋊Dic79—22no—
77C22⋊Dic79—41no—
78C22⋊Dic79—41no—
79C22⋊Dic79—41no—
80C24×C79C2×D411yescharacteristic
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—42no—
88C22×C4×C79—41no—
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C4⋊C4×C79—42no—
92C4⋊C4×C79—42no—
93C4⋊C4×C79—42no—
94C4⋊C4×C79—42no—
95C4⋊C4×C79—42no—
96C4⋊C4×C79—41no—
97C4⋊C4×C79—41no—
98C4⋊C4×C79—81no—
99C22⋊C4×C79—21no—
100C22⋊C4×C79—41no—
101C22⋊C4×C79—41no—
102C22⋊C4×C79—41no—
103C22⋊C4×C79—41no—
104C22⋊C4×C79—41no—
105C22⋊C4×C79SD1612yes—
106C42×C79—42no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—41no—
110C23×C79—41no—
111C23×C79#12011yescharacteristic
112C23×C79C2×SD1611yescharacteristic
113C23×C79AGL(1,ℤ/8)11yescharacteristic
114C2×C4×C79—22no—
115C2×C4×C79—22no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—42no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—81no—
138C2×Dic79—42no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—81no—
142C2×Dic79—81no—
143C2×Dic79—81no—
144C2×Dic79—81no—
145Dic158—81no—
146Dic158—81no—
147C79⋊C8—81no—
148C79⋊C8—81no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—41no—
154C22×C79—41no—
155C22×C79—41no—
156C22×C79—41no—
157C22×C79—81no—
158C22×C79#45011yescharacteristic
159C22×C79#45611yescharacteristic
160C22×C79B(4,𝔽2)11yescharacteristic
161C4×C79—42no—
162C4×C79—42no—
163C4×C79—41no—
164C4×C79—41no—
165C4×C79—81no—
166C4×C79—81no—
167C4×C79—81no—
168C4×C79—81no—
169Dic79—82no—
170Dic79—81no—
171Dic79—81no—
172#12244—791nomaximal, Sylow
173C2×C79—21no—
174C2×C79—21no—
175C2×C79—41no—
176C2×C79—41no—
177C2×C79#160411yescharacteristic
178C2×C79#219711yescharacteristic
179C2×C79#220111yescharacteristic
180#2722D7911yescharacteristic
181#2062—791no—
182#1608—791no—
183#1608—791no—
184#1344—791no—
185#1344—791no—
186#1326—791no—
187C79#1224411yescharacteristic, complemented, Sylow
188#543—791no—
189C2×#126—21no—
190#483—1581no—
191#483—1581no—
192#398—21no—
193#398—21no—
194#397C2×D7911yescharacteristic
195#397C2×D7911yescharacteristic
196#389—21no—
197#387—21no—
198#385—1581no—
199#385C2×D7911yescharacteristic
200#380—21no—
201#342—791no—
202#342—1581no—
203#328—1581no—
204#328—1581no—
205#323—791no—
206#323—791no—
207C23×C4C22⋊D7911yescharacteristic
208#126—42no—
209#126—42no—
210C22⋊Q8—3161no—
211C22⋊Q8—3161no—
212C22⋊Q8—1581no—
213C22⋊Q8—1581no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—3161no—
221C2×C4⋊C4C22⋊D7911yescharacteristic
222C2×C4⋊C4C22⋊D7911yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—1581no—
226C2×C22⋊C4—1581no—
227C2×C22⋊C4C22⋊D7911yescharacteristic
228C2×C22⋊C4C22×D7911yescharacteristic
229C2×C42—21no—
230#106—3161no—
231#106—3161no—
232#103—3161no—
233#103—3161no—
234C22⋊C8—1581no—
235C22⋊C8—1581no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—3161no—
242#95C22⋊D7911yescharacteristic
243#95C22⋊D7911yescharacteristic
244C24C2×C22⋊D7911yescharacteristic
245C2×Q8—1581no—
246C2×Q8—1581no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—3161no—
256C22×C4—1581no—
257C22×C4—1581no—
258C22×C4C2×C22⋊D7911yescharacteristic
259C22×C4C2×C22⋊D7911yescharacteristic
260C2×C8—3161no—
261C2×C8—3161no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—81no—
270C4⋊C4—3161no—
271C4⋊C4—3161no—
272C4⋊C4—6322no—
273C4⋊C4—1581no—
274C4⋊C4—1581no—
275C22⋊C4—21no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—3161no—
282C22⋊C4—3161no—
283C22⋊C4—3161no—
284C22⋊C4—1582no—
285C22⋊C4#1192961212yes—
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#1457913011yescharacteristic
292C23C2×#1192961211yescharacteristic
293C23#1457918711yescharacteristic
294Q8—6321no—
295Q8—6321no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—3162no—
321C2×C4—3161no—
322C2×C4—3161no—
323C2×C4—6321no—
324C2×C4—6321no—
325C2×C4—6321no—
326C2×C4—6321no—
327C8—6321no—
328C8—6321no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22#1909871411yescharacteristic
339C22#1909874611yescharacteristic
340C22#1909875011yescharacteristic
341C4—42no—
342C4—41no—
343C4—42no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—6322no—
350C4—6321no—
351C4—6321no—
352C2—21no—
353C2—21no—
354C2—41no—
355C2—41no—
356C2#2376236911yescharacteristic
357C2#2376539011yescharacteristic
358C2#2376541311yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2722C2
socleC22×C79B(4,𝔽2)
prime radicalC24×C79C2×D4
hypercenter#2722D79
hypocenterC79#12244
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2722

Complements

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

NH ≅ G/Ntype
187. C79#12244semidirect

Maximal subgroups

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

subgroupindex
#237647532
#237623902
#237614002
#237613482
C79×#27222
#1224479