Finite Group Databasebeta

← #34198798

Subgroups — #34198798

Order 20224.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#23768780C211yesmaximal, characteristic
2#23761730C211yesmaximal, characteristic
3#23761429C212yesmaximal, complemented
4#23761405C212yesmaximal, complemented
5C2×#343×C79C211yesmaximal, characteristic
6C2×#14579140C411yescharacteristic
7#19098527—21no—
8#19098493C411yescharacteristic
9#19098493C2211yescharacteristic
10#19098483C411yescharacteristic
11#19098483C2211yescharacteristic
12#19098458—21no—
13#19098458—21no—
14#19098436—21no—
15#19098436—21no—
16C4×C22⋊Dic79C411yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C22×M16×C79—21no—
20C22×C4⋊C4×C79C2211yescharacteristic
21#343×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C2×#11929608—41no—
25#14579140—41no—
26#14579140—41no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C4×Dic79—21no—
36C22⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—41no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79C2×C411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C2×M16×C79—24no—
48C2×M16×C79—22no—
49C2×M16×C79—41no—
50C2×M16×C79—42no—
51C22×C8×C79—41no—
52C2×C4⋊C4×C79—41no—
53C2×C4⋊C4×C79—41no—
54C2×C4⋊C4×C79C2×C411yescharacteristic
55C2×C4⋊C4×C79D411yescharacteristic
56C2×C4⋊C4×C79D411yescharacteristic
57C2×C4⋊C4×C79C2311yescharacteristic
58C2×Dic158—41no—
59C2×Dic158—81no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64#11929608—82no—
65C2×C79⋊C8—42no—
66C4⋊Dic79—41no—
67Dic79⋊C4—42no—
68Dic79⋊C4—42no—
69Dic79⋊C4—41no—
70Dic79⋊C4—41no—
71Dic79⋊C4—81no—
72Dic79⋊C4—81no—
73C22⋊Dic79—24no—
74C22⋊Dic79—41no—
75C22⋊Dic79—41no—
76C4×Dic79—81no—
77C24×C79C22⋊C411yescharacteristic
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—22no—
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—81no—
88C22×C4×C79—81no—
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C22⋊C411yescharacteristic
91C22×C4×C79C22⋊C411yescharacteristic
92C22×C4×C79C22×C411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×D411yescharacteristic
95M16×C79—24no—
96M16×C79—44no—
97M16×C79—42no—
98C2×C8×C79—44no—
99C2×C8×C79—42no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79—41no—
104C23×C79—22no—
105C23×C79—21no—
106C23×C79—21no—
107C23×C79—41no—
108C23×C79#9911yescharacteristic
109C23×C79#9911yescharacteristic
110C23×C79C2×C22⋊C411yescharacteristic
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—22no—
115C2×C4×C79—21no—
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—21no—
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—81no—
130C2×C4×C79—81no—
131C2×C4×C79#10412yes—
132C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C79#13711yescharacteristic
134C2×Dic79—22no—
135C2×Dic79—22no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146Dic158—82no—
147C8×C79—44no—
148C79⋊C8—82no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79—21no—
156C22×C79—42no—
157C22×C79—42no—
158C22×C79#35612yes—
159C22×C79C2×#9911yescharacteristic
160C22×C79#41711yescharacteristic
161C22×C79C2×#10411yescharacteristic
162C4×C79—22no—
163C4×C79—21no—
164C4×C79—21no—
165C4×C79—41no—
166C4×C79—41no—
167C4×C79—41no—
168C4×C79—41no—
169Dic79—42no—
170Dic79—42no—
171Dic79—81no—
172Dic79—81no—
173#11768—791nomaximal, Sylow
174C2×C79—21no—
175C2×C79—21no—
176C2×C79—22no—
177C2×C79—41no—
178C2×C79#150411yescharacteristic
179C2×C79#212111yescharacteristic
180C2×C79C2×#35611yescharacteristic
181#2583—791no—
182C2×#343D7911yescharacteristic
183#1463—791no—
184#1354—792no—
185#1346—792no—
186C79#1176811yescharacteristic, complemented, Sylow
187C22×M16—21no—
188C2×C22⋊Q8—791no—
189C22×C4⋊C4C2×D7911yescharacteristic
190#407—1581no—
191#398—791no—
192#398—791no—
193#395—791no—
194#395—791no—
195#389—1581no—
196#389—1581no—
197#385—1581no—
198#385—1581no—
199C4×C22⋊C4—791no—
200#343—21no—
201#343C2×D7912yes—
202#324—792no—
203#324—1581no—
204C22×Q8—791no—
205C23×C4C4×D7911yescharacteristic
206C23×C4D31611yescharacteristic
207C23×C4C22⋊D7911yescharacteristic
208C2×M16—24no—
209C2×M16—22no—
210C2×M16—41no—
211C2×M16—42no—
212C2×M16—3161no—
213C22×C8—41no—
214C22⋊Q8—3161no—
215C22⋊Q8—3161no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—791no—
219C2×C4⋊C4—1581no—
220C2×C4⋊C4—1581no—
221C2×C4⋊C4—1581no—
222C2×C4⋊C4—1581no—
223C2×C4⋊C4—1581no—
224C2×C4⋊C4C4×D7911yescharacteristic
225C2×C4⋊C4D31611yescharacteristic
226C2×C4⋊C4C22⋊D7911yescharacteristic
227C2×C4⋊C4C22×D7911yescharacteristic
228C2×C22⋊C4—1581no—
229C2×C22⋊C4—1581no—
230C2×C42—1581no—
231C22⋊C8—3161no—
232C22⋊C8—1582no—
233#95—791no—
234#95—791no—
235#95—1581no—
236#95—1581no—
237#95—1581no—
238#95—1581no—
239C24(C2×D79)⋊C411yescharacteristic
240C2×Q8—3161no—
241C2×Q8—6321no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—1581no—
254C22×C4—1581no—
255C22×C4—1581no—
256C22×C4—1581no—
257C22×C4(C2×D79)⋊C411yescharacteristic
258C22×C4(C2×D79)⋊C411yescharacteristic
259C22×C4(C2×D79)⋊C411yescharacteristic
260C22×C4C2×C4×D7911yescharacteristic
261C22×C4C2×C22⋊D7911yescharacteristic
262C22×C4C2×D31611yescharacteristic
263M16—24no—
264M16—44no—
265M16—42no—
266M16—6322no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—3162no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—3161no—
275C4⋊C4—3162no—
276C4⋊C4—3162no—
277C4⋊C4—3161no—
278C4⋊C4—3161no—
279C4⋊C4—6321no—
280C4⋊C4—6321no—
281C22⋊C4—3161no—
282C22⋊C4—3161no—
283C22⋊C4—1584no—
284C42—6321no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—41no—
289C23#1457907511yescharacteristic
290C23#1457907511yescharacteristic
291C23C2×(C2×D79)⋊C411yescharacteristic
292Q8—6322no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—3161no—
314C2×C4—3161no—
315C2×C4—3161no—
316C2×C4—3161no—
317C2×C4—3161no—
318C2×C4—3161no—
319C2×C4—3161no—
320C2×C4—3161no—
321C2×C4—3161no—
322C2×C4—3161no—
323C2×C4—1582no—
324C2×C4—1582no—
325C2×C4#1457908912yes—
326C2×C4#1457918711yescharacteristic
327C2×C4#1457919211yescharacteristic
328C8—44no—
329C8—6322no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22#1909836112yes—
340C22C2×#1457907511yescharacteristic
341C22#1909856211yescharacteristic
342C22C2×#1457908911yescharacteristic
343C4—21no—
344C4—22no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3162no—
351C4—3162no—
352C4—6321no—
353C4—6321no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—41no—
358C2#2376190211yescharacteristic
359C2#2376504711yescharacteristic
360C2C2×#1909836111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC2×#343×C79C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenterC2×#343D79
hypocenterC79#11768
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
3. #23761429C2semidirect
4. #23761405C2semidirect
186. C79#11768semidirect

Maximal subgroups

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

subgroupindex
#237687802
#237617302
#237614292
#237614052
C2×#343×C792
#1176879