Finite Group Databasebeta

← #34198773

Subgroups — #34198773

Order 20224.

428 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 83 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#23768977C211yesmaximal, characteristic
2#23761734C211yesmaximal, characteristic
3#23761429C212yesmaximal, complemented
4#23761405C212yesmaximal, complemented
5C79×#1742C211yesmaximal, characteristic
6C2×C4×Dic158C411yescharacteristic
7#19098620—21no—
8#19098493C2211yescharacteristic
9#19098488C411yescharacteristic
10#19098483C2211yescharacteristic
11#19098475C411yescharacteristic
12#19098451—21no—
13#19098447—21no—
14#19098444—21no—
15#19098436—21no—
16#19098431C411yescharacteristic
17#19098289C2212yes—
18#19098287—21no—
19C22×M16×C79—21no—
20C2×#117×C79C2211yescharacteristic
21#343×C79—21no—
22#343×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C4×Dic158—41no—
25C4×Dic158—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—21no—
32C2×C4×Dic79—21no—
33C4⋊(C79⋊C8)—41no—
34C22⋊(C79⋊C8)—22no—
35#14579072—41no—
36C79⋊#95—21no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C2×M16×C79—24no—
46C2×M16×C79—22no—
47C2×M16×C79—41no—
48C2×M16×C79—42no—
49C22×C8×C79—41no—
50#117×C79—41no—
51#117×C79—41no—
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C4⋊C4×C79C2311yescharacteristic
54C2×C22⋊C4×C79C2×C411yescharacteristic
55C2×C22⋊C4×C79D411yescharacteristic
56C2×C42×C79C2×C411yescharacteristic
57C2×C42×C79D411yescharacteristic
58C2×Dic158—41no—
59C2×Dic158—42no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C2×C79⋊C8—42no—
65C4⋊Dic79—41no—
66Dic79⋊C4—42no—
67Dic79⋊C4—41no—
68Dic79⋊C4—41no—
69Dic79⋊C4—41no—
70Dic79⋊C4—41no—
71C22⋊Dic79—42no—
72C22⋊Dic79—42no—
73C4×Dic79—42no—
74C24×C79C22⋊C411yescharacteristic
75C22×C4×C79—22no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86M16×C79—24no—
87M16×C79—44no—
88M16×C79—42no—
89C2×C8×C79—44no—
90C2×C8×C79—42no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—41no—
93C22⋊C4×C79—41no—
94C22⋊C4×C79—41no—
95C42×C79—21no—
96C42×C79—21no—
97C42×C79—21no—
98C42×C79—21no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—41no—
102C23×C79—41no—
103C23×C79#10412yes—
104C23×C79C2×C22⋊C411yescharacteristic
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79#9911yescharacteristic
122C2×C4×C79#9911yescharacteristic
123C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C79#13711yescharacteristic
125C2×Dic79—22no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137Dic158—42no—
138Dic158—42no—
139C8×C79—44no—
140C79⋊C8—82no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—22no—
144C22×C79—42no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—81no—
148C22×C79C2×#9911yescharacteristic
149C22×C79#41711yescharacteristic
150C22×C79C2×#10411yescharacteristic
151C4×C79—21no—
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—21no—
155C4×C79—41no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79—41no—
159Dic79—42no—
160Dic79—81no—
161Dic79—81no—
162Dic79—81no—
163#11763—791nomaximal, Sylow
164C2×C79—21no—
165C2×C79—21no—
166C2×C79—42no—
167C2×C79#150411yescharacteristic
168C2×C79#212011yescharacteristic
169C2×C79#213511yescharacteristic
170#2599—791no—
171#1742D7911yescharacteristic
172#1464—791no—
173#1354—792no—
174#1346—792no—
175C79#1176311yescharacteristic, complemented, Sylow
176C22×M16—21no—
177C2×C4×Q8—791no—
178C2×#117C2×D7911yescharacteristic
179#432—1581no—
180#398—791no—
181#397—791no—
182#395—791no—
183#393—791no—
184#388—1581no—
185#387—1581no—
186#387—1581no—
187#385—1581no—
188#383—791no—
189#343—21no—
190#343C2×D7912yes—
191#324—792no—
192#323—1581no—
193C22×Q8—791no—
194C23×C4C22⋊D7911yescharacteristic
195C2×M16—24no—
196C2×M16—22no—
197C2×M16—41no—
198C2×M16—42no—
199C22×C8—41no—
200C4×Q8—3161no—
201C4×Q8—3161no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—791no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4C22⋊D7911yescharacteristic
209C2×C4⋊C4C22×D7911yescharacteristic
210C2×C22⋊C4—1581no—
211C2×C22⋊C4—1581no—
212C2×C22⋊C4C4×D7911yescharacteristic
213C2×C22⋊C4D31611yescharacteristic
214C2×C42—1581no—
215C2×C42C4×D7911yescharacteristic
216C2×C42D31611yescharacteristic
217C4⋊C8—3161no—
218C22⋊C8—1582no—
219#97—3161no—
220#95—791no—
221#95—791no—
222#95—1581no—
223#95—1581no—
224#95—1581no—
225#95—1581no—
226#95—1581no—
227#95—1581no—
228C24(C2×D79)⋊C411yescharacteristic
229C2×Q8—3161no—
230C2×Q8—3162no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1581no—
237C22×C4—1581no—
238C22×C4—1581no—
239C22×C4—1581no—
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4(C2×D79)⋊C411yescharacteristic
243C22×C4C2×C4×D7911yescharacteristic
244C22×C4C2×C22⋊D7911yescharacteristic
245C22×C4C2×D31611yescharacteristic
246M16—24no—
247M16—44no—
248M16—42no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—3162no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—3161no—
255C4⋊C4—3162no—
256C4⋊C4—3161no—
257C4⋊C4—3161no—
258C4⋊C4—3161no—
259C4⋊C4—3161no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—3162no—
263C22⋊C4—3162no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—3162no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23#1457908912yes—
274C23C2×(C2×D79)⋊C411yescharacteristic
275Q8—3162no—
276Q8—3162no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4—3161no—
297C2×C4—3161no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4—3161no—
303C2×C4—3161no—
304C2×C4—1582no—
305C2×C4#1457907511yescharacteristic
306C2×C4#1457907511yescharacteristic
307C2×C4#1457918711yescharacteristic
308C2×C4#1457919211yescharacteristic
309C8—44no—
310C8—6322no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1457907511yescharacteristic
319C22#1909856211yescharacteristic
320C22C2×#1457908911yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—3162no—
330C4—6321no—
331C4—6321no—
332C4—6321no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2376190211yescharacteristic
337C2#2376504211yescharacteristic
338C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1742C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1742D79
hypocenterC79#11763
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #23761429C2semidirect
4. #23761405C2semidirect
175. C79#11763semidirect

Maximal subgroups

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

subgroupindex
#237689772
#237617342
#237614292
#237614052
C79×#17422
#1176379