Finite Group Databasebeta

← #34206552

Subgroups — #34206552

Order 20224.

536 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 107 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#23767708C211yesmaximal, characteristic
2C2×#19098354C214yesmaximal
3#23763388C211yesmaximal, characteristic
4C4×#100×C79C211yesmaximal, characteristic
5C2×#14579142C2212yes—
6C22×C4⋊Dic79C411yescharacteristic, complemented
7C2×#14579076C2212yes—
8#19098493—21no—
9#19098488—21no—
10#19098461—21no—
11#19098424C412yescomplemented
12#19098423—21no—
13#19098423—21no—
14#19098423C411yescharacteristic, complemented
15#19098354—28no—
16#19098330—21no—
17C2×C4×D4×C79C2211yescharacteristic
18C2×#100×C79C2212yes—
19C4×M16×C79—21no—
20#343×C79—21no—
21C23×Dic79C2×C412yes—
22#14579142—24no—
23#14579142—24no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—41no—
28C2×C4⋊Dic79—41no—
29C2×C4⋊Dic79—41no—
30C2×Dic79⋊C4—22no—
31C2×C22⋊Dic79—22no—
32C2×C22⋊Dic79—22no—
33C2×C22⋊Dic79C2×C412yes—
34#14579076—24no—
35C79⋊#95—22no—
36C79⋊#95—21no—
37C79⋊#95—22no—
38C79⋊#95—41no—
39C22×D4×C79C2311yescharacteristic
40C23×C4×C79C2×C411yescharacteristic
41C23×C4×C79D411yescharacteristic
42C2×M16×C79—22no—
43C4×D4×C79—41no—
44C4×D4×C79—41no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C22⋊C4×C79C2×C411yescharacteristic
47C2×C22⋊C4×C79D411yescharacteristic
48C2×C42×C79D411yescharacteristic
49#100×C79—22no—
50#100×C79D414yes—
51#97×C79—41no—
52C4×C8×C79—41no—
53C22×Dic79—22no—
54C22×Dic79—22no—
55C22×Dic79—24no—
56C22×Dic79—42no—
57C22×Dic79—42no—
58C22×Dic79—42no—
59C4⋊Dic79—42no—
60C4⋊Dic79—42no—
61C4⋊Dic79—41no—
62C4⋊Dic79—41no—
63C4⋊Dic79—82no—
64Dic79⋊C4—44no—
65C22⋊Dic79—24no—
66C22⋊Dic79—42no—
67C22⋊Dic79—42no—
68C22⋊Dic79—42no—
69C22⋊Dic79—42no—
70C24×C79C22×C411yescharacteristic
71C24×C79C2×D411yescharacteristic
72C2×D4×C79—41no—
73C2×D4×C79—41no—
74C2×D4×C79C2×D412yes—
75C2×D4×C79#4112yes—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—42no—
81C22×C4×C79—42no—
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87M16×C79—24no—
88M16×C79—42no—
89C2×C8×C79—42no—
90C4⋊C4×C79—41no—
91C22⋊C4×C79—41no—
92C22⋊C4×C79—41no—
93C42×C79—21no—
94C42×C79—21no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—21no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79—41no—
102C23×C79—41no—
103C23×C79C2×C22⋊C411yescharacteristic
104C23×C79C4×D412yes—
105C23×C79#12411yescharacteristic
106C23×C79#12711yescharacteristic
107D4×C79—42no—
108D4×C79—42no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—22no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—81no—
122C2×C4×C79#9911yescharacteristic
123C2×C4×C79#9911yescharacteristic
124C2×C4×C79#12112yes—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129C2×Dic79—42no—
130C2×Dic79—42no—
131C2×Dic79—84no—
132C2×Dic79—82no—
133C2×Dic79—82no—
134C2×Dic79—82no—
135C8×C79—44no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—22no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—42no—
142C22×C79—41no—
143C22×C79—41no—
144C22×C79—42no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—81no—
150C22×C79#39011yescharacteristic
151C22×C79C2×#9911yescharacteristic
152C22×C79#41011yescharacteristic
153C4×C79—21no—
154C4×C79—22no—
155C4×C79—41no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79#35212yescomplemented
159Dic79—82no—
160Dic79—82no—
161Dic79—162no—
162#13217—791nomaximal, Sylow
163C2×C79—21no—
164C2×C79—21no—
165C2×C79—41no—
166C2×C79—42no—
167C2×C79—41no—
168C2×C79#196411yescharacteristic
169C2×C79C2×#35211yescharacteristic
170C2×C79#212311yescharacteristic
171#2486—791no—
172C2×#352—794no—
173#1799—791no—
174C4×#100D7911yescharacteristic
175C79#1321711yescharacteristic, complemented, Sylow
176C2×#123—792no—
177C2×C4×D4C2×D7911yescharacteristic
178C22×C4⋊C4—791no—
179C2×#100C2×D7912yes—
180C2×#99—792no—
181C4×M16—21no—
182#398—1581no—
183#397—1581no—
184#389—1581no—
185#381—792no—
186#380—791no—
187#380—1581no—
188#380—1581no—
189#352—1588no—
190#343—21no—
191#342—1581no—
192C22×D4C22×D7911yescharacteristic
193C23×C4—792no—
194C23×C4C4×D7911yescharacteristic
195C23×C4D31611yescharacteristic
196C2×M16—22no—
197#123—1584no—
198#123—1584no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—3161no—
202C2×C4⋊C4—3161no—
203C2×C4⋊C4—3161no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1582no—
208C2×C4⋊C4C22⋊D7911yescharacteristic
209C2×C22⋊C4—792no—
210C2×C22⋊C4—1582no—
211C2×C22⋊C4—1582no—
212C2×C22⋊C4C4×D7911yescharacteristic
213C2×C22⋊C4D31611yescharacteristic
214C2×C42C22⋊D7911yescharacteristic
215#100—22no—
216#100D31614yes—
217#99—1584no—
218#97—41no—
219C4×C8—41no—
220#95—3161no—
221#95—1581no—
222#95—1582no—
223#95—1582no—
224C24C2×C4×D7911yescharacteristic
225C24C2×D31611yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4C2×D31612yes—
229C2×D4#1192962412yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—3162no—
237C22×C4—3162no—
238C22×C4—3162no—
239C22×C4—1582no—
240C22×C4—1582no—
241C22×C4—1584no—
242C22×C4(C2×D79)⋊C411yescharacteristic
243C22×C4(C2×D79)⋊C411yescharacteristic
244C22×C4(C2×D79)⋊C411yescharacteristic
245C22×C4(C2×D79)⋊C411yescharacteristic
246C22×C4C2×C22⋊D7911yescharacteristic
247M16—24no—
248M16—42no—
249C2×C8—42no—
250C4⋊C4—41no—
251C4⋊C4—3162no—
252C4⋊C4—3162no—
253C4⋊C4—3161no—
254C4⋊C4—3161no—
255C4⋊C4—3164no—
256C4⋊C4—6322no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—3162no—
260C22⋊C4—3162no—
261C22⋊C4—3162no—
262C22⋊C4—3162no—
263C22⋊C4—1584no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×(C2×D79)⋊C411yescharacteristic
275C23C4×D31612yes—
276C23#1457914911yescharacteristic
277C23#1457915811yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—3162no—
294C2×C4—3162no—
295C2×C4—3162no—
296C2×C4—3162no—
297C2×C4—3162no—
298C2×C4—3162no—
299C2×C4—6324no—
300C2×C4—6322no—
301C2×C4—6322no—
302C2×C4—6322no—
303C2×C4#1457907511yescharacteristic
304C2×C4#1457907511yescharacteristic
305C2×C4#1457913512yes—
306C8—44no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1909846511yescharacteristic
322C22C2×#1457907511yescharacteristic
323C22#1909853611yescharacteristic
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—6322no—
330C4—6322no—
331C4—12642no—
332C4#1909835412yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#2376423611yescharacteristic
339C2C2×#1909835411yescharacteristic
340C2#2376505511yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC24×C79C22×C4
FrattiniC22×D4C22×D79
FittingC4×#100×C79C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenterC4×#100D79
hypocenterC79#13217
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊Dic79C4semidirect
11. #19098424C4semidirect
14. #19098423C4semidirect
158. C4×C79#352semidirect
175. C79#13217semidirect
332. C4#19098354semidirect

Maximal subgroups

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

subgroupindex
#237677082
C2×#190983542
#237633882
C4×#100×C792
#1321779