Finite Group Databasebeta

← #34182240

Subgroups — #34182240

Order 20224.

532 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 123 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#23763693C211yesmaximal, characteristic
2#23763577C212yesmaximal, complemented
3C4×#14579076C211yesmaximal, characteristic
4#23761247C212yesmaximal, complemented
5C79×#2973C211yesmaximal, characteristic
6C23×C79⋊C8C2211yescharacteristic
7C2×Dic79⋊C8—22no—
8C2×#14579076C412yes—
9C2×C22⋊(C79⋊C8)—21no—
10C2×C22⋊(C79⋊C8)C2211yescharacteristic
11C4×C22⋊Dic79C2212yes—
12#19098330C411yescharacteristic
13#19098330C411yescharacteristic
14#19098318—21no—
15#19098317—22no—
16C2×C4×D4×C79C2211yescharacteristic
17#435×C79—22no—
18#435×C79—22no—
19#432×C79—21no—
20#407×C79—21no—
21C2×C22⋊C8×C79C2212yes—
22C22×C79⋊C8—21no—
23C22×C79⋊C8—22no—
24C22×C79⋊C8—21no—
25C22×C79⋊C8—42no—
26C2×C22⋊Dic79C2×C412yes—
27C2×C22⋊Dic79C2×C412yes—
28C2×C4×Dic79—22no—
29Dic79⋊C8—44no—
30#14579076—24no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—42no—
33C79⋊#95—22no—
34C22×D4×C79C2×C411yescharacteristic
35C23×C4×C79D411yescharacteristic
36C23×C4×C79C2311yescharacteristic
37C2×M16×C79—21no—
38C2×M16×C79—21no—
39C22×C8×C79D412yes—
40C22×C8×C79D412yes—
41C4×D4×C79—22no—
42C4×D4×C79—22no—
43C2×C4⋊C4×C79Q811yescharacteristic
44C2×C22⋊C4×C79C2×C411yescharacteristic
45C2×C22⋊C4×C79Q811yescharacteristic
46C2×C42×C79D411yescharacteristic
47C4⋊C8×C79—21no—
48C4⋊C8×C79—21no—
49C22⋊C8×C79—22no—
50C22⋊C8×C79—22no—
51C22⋊C8×C79—22no—
52#97×C79—22no—
53C22×Dic79—22no—
54C22×Dic79—22no—
55C2×C79⋊C8—22no—
56C2×C79⋊C8—24no—
57C2×C79⋊C8—42no—
58C2×C79⋊C8—41no—
59C2×C79⋊C8—41no—
60C2×C79⋊C8—41no—
61C2×C79⋊C8—41no—
62C2×C79⋊C8—41no—
63C22⋊Dic79—22no—
64C22⋊Dic79—22no—
65C22⋊Dic79—22no—
66C22⋊Dic79M1614yescomplemented
67C4×Dic79—42no—
68C4×Dic79—42no—
69C24×C79C4⋊C411yescharacteristic
70C24×C79C22×C411yescharacteristic
71C2×D4×C79—22no—
72C2×D4×C79—22no—
73C2×D4×C79—21no—
74C2×D4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—41no—
78C22×C4×C79—41no—
79C22×C4×C79—41no—
80C22×C4×C79—41no—
81C22×C4×C79C4⋊C411yescharacteristic
82C22×C4×C79C4⋊C411yescharacteristic
83C22×C4×C79C4⋊C411yescharacteristic
84C22×C4×C79C2×D412yes—
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×Q811yescharacteristic
87C22×C4×C79#4112yes—
88M16×C79—81no—
89M16×C79—81no—
90C2×C8×C79—22no—
91C2×C8×C79—22no—
92C2×C8×C79—42no—
93C2×C8×C79—42no—
94C4⋊C4×C79—22no—
95C22⋊C4×C79—22no—
96C22⋊C4×C79—22no—
97C42×C79—21no—
98C42×C79—21no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—41no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79C2×C4⋊C411yescharacteristic
107C23×C79C4×D412yes—
108C23×C79C2×M1612yes—
109C23×C79#13111yescharacteristic
110C23×C79#13111yescharacteristic
111D4×C79—42no—
112D4×C79—42no—
113C2×C4×C79—22no—
114C2×C4×C79—21no—
115C2×C4×C79—22no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—41no—
123C2×C4×C79—42no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—81no—
129C2×C4×C79#12011yescharacteristic
130C2×C4×C79C22⋊Q812yes—
131C2×C4×C79#12311yescharacteristic
132C2×Dic79—24no—
133C2×Dic79—42no—
134C2×Dic79—42no—
135C2×Dic79—42no—
136C2×Dic79—42no—
137C2×Dic79—42no—
138C8×C79—42no—
139C8×C79—42no—
140C79⋊C8—42no—
141C79⋊C8—81no—
142C22×C79—21no—
143C22×C79—22no—
144C22×C79—21no—
145C22×C79—22no—
146C22×C79—42no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—42no—
150C22×C79—41no—
151C22×C79—41no—
152C22×C79—41no—
153C22×C79—81no—
154C22×C79#38511yescharacteristic
155C22×C79#42311yescharacteristic
156C22×C79#42411yescharacteristic
157C22×C79#43512yes—
158C22×C79#43612yes—
159C4×C79—21no—
160C4×C79—22no—
161C4×C79—41no—
162C4×C79—41no—
163C4×C79—41no—
164C4×C79B(4,𝔽2)11yescharacteristic
165C4×C79#45811yescharacteristic
166Dic79—44no—
167Dic79—82no—
168#8852—791nomaximal, Sylow
169C2×C79—22no—
170C2×C79—22no—
171C2×C79—21no—
172C2×C79—21no—
173C2×C79—81no—
174C2×C79#185111yescharacteristic
175C2×C79#186711yescharacteristic
176C2×C79#186811yescharacteristic
177#2973D7911yescharacteristic
178#1869—791no—
179#1850—792no—
180C4×#99—791no—
181#1282—792no—
182C79#885211yescharacteristic, complemented, Sylow
183C23×C8—791no—
184C2×C4×D4C2×D7911yescharacteristic
185#435—22no—
186#435—22no—
187#432—21no—
188C2×C4⋊C8—1582no—
189C2×#99—792no—
190#407—21no—
191C2×C22⋊C8—791no—
192C2×C22⋊C8—1581no—
193C2×C22⋊C8C2×D7912yes—
194C4×C22⋊C4—792no—
195#342—791no—
196#342—791no—
197#336—1582no—
198#336—1581no—
199C22×D4C4×D7911yescharacteristic
200C23×C4C22⋊D7911yescharacteristic
201C23×C4C22×D7911yescharacteristic
202C2×M16—21no—
203C2×M16—21no—
204C22×C8—3162no—
205C22×C8—1581no—
206C22×C8—1581no—
207C22×C8—1582no—
208C22×C8C22⋊D7912yes—
209C22×C8C22⋊D7912yes—
210C4×D4—22no—
211C4×D4—22no—
212C2×C4⋊C4Dic15811yescharacteristic
213C2×C22⋊C4—792no—
214C2×C22⋊C4—792no—
215C2×C22⋊C4Dic15811yescharacteristic
216C2×C22⋊C4C4×D7911yescharacteristic
217C2×C42—1582no—
218C2×C42C22⋊D7911yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—3164no—
222#99—1584no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—3162no—
227C22⋊C8—1582no—
228#97—22no—
229#95—1582no—
230C24Dic79⋊C411yescharacteristic
231C24C2×C4×D7911yescharacteristic
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—1582no—
243C22×C4—1582no—
244C22×C4Dic79⋊C411yescharacteristic
245C22×C4Dic79⋊C411yescharacteristic
246C22×C4Dic79⋊C411yescharacteristic
247C22×C4C2×C22⋊D7912yes—
248C22×C4C2×C22⋊D7911yescharacteristic
249C22×C4C2×Dic15811yescharacteristic
250C22×C4#1192962412yes—
251M16—81no—
252M16—81no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—3162no—
258C2×C8—3161no—
259C2×C8—3161no—
260C2×C8—3161no—
261C2×C8—3161no—
262C2×C8—3161no—
263C2×C8—1582no—
264C2×C8—1584no—
265C4⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—794no—
269C22⋊C4—1582no—
270C22⋊C4—1582no—
271C22⋊C4—1582no—
272C42—21no—
273C42—21no—
274C42—3162no—
275C42—3162no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×Dic79⋊C411yescharacteristic
284C23C4×C22⋊D7912yes—
285C23M16×D7912yes—
286C23#1457916911yescharacteristic
287C23#1457916911yescharacteristic
288D4—42no—
289D4—42no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—3162no—
307C2×C4—3162no—
308C2×C4—3162no—
309C2×C4—3162no—
310C2×C4—3162no—
311C2×C4—1584no—
312C2×C4#1457913011yescharacteristic
313C2×C4#1457914012yes—
314C2×C4#1457914411yescharacteristic
315C8—42no—
316C8—42no—
317C8—3162no—
318C8—6321no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1909843611yescharacteristic
332C22#1909858611yescharacteristic
333C22#1909859111yescharacteristic
334C22#1909863212yes—
335C22#1909863912yes—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—3164no—
342C4—6322no—
343C4#1909875011yescharacteristic
344C4#1909875311yescharacteristic
345C2—22no—
346C2—22no—
347C2—21no—
348C2—21no—
349C2—81no—
350C2#2376358311yescharacteristic
351C2#2376368111yescharacteristic
352C2#2376368611yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2973C2
socleC22×C79#385
prime radicalC22×D4×C79C2×C4
cosocleC23×C4×C79C23
hypercenter#2973D79
hypocenterC79#8852
LCSG ▷ C24×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #23763577C2semidirect
4. #23761247C2semidirect
66. C22⋊Dic79M16semidirect
182. C79#8852semidirect

Maximal subgroups

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

subgroupindex
#237636932
#237635772
C4×#145790762
#237612472
C79×#29732
#885279