Finite Group Databasebeta

← #34206913

Subgroups — #34206913

Order 20224.

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

normal subgroups only

This group has 484 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#23767717C211yesmaximal, characteristic
2#23763388C211yesmaximal, characteristic
3#23761519C214yesmaximal
4C4×#100×C79C211yesmaximal, characteristic
5C22×C4⋊Dic79C411yescharacteristic, complemented
6C2×#14579076C2212yes—
7#19098502—21no—
8#19098499—21no—
9#19098461—21no—
10#19098443C2212yes—
11#19098435C412yescomplemented
12#19098423—21no—
13#19098423—21no—
14#19098423C411yescharacteristic, complemented
15#19098330—21no—
16C2×C4×D4×C79C2211yescharacteristic
17C2×#100×C79C2212yes—
18C4×M16×C79—21no—
19#343×C79—21no—
20C23×Dic79C2×C412yes—
21C2×C4⋊Dic79—21no—
22C2×C4⋊Dic79—21no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—41no—
25C2×C4⋊Dic79—41no—
26C2×C4⋊Dic79—41no—
27C2×Dic79⋊C4—22no—
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79C2×C412yes—
31#14579076—24no—
32C79⋊#95—22no—
33C79⋊#95—21no—
34C79⋊#95—22no—
35C79⋊#95—41no—
36C22×D4×C79C2311yescharacteristic
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C2×M16×C79—22no—
40C4×D4×C79—41no—
41C4×D4×C79—41no—
42C2×C4⋊C4×C79D411yescharacteristic
43C2×C22⋊C4×C79C2×C411yescharacteristic
44C2×C22⋊C4×C79D411yescharacteristic
45C2×C42×C79D411yescharacteristic
46#100×C79—22no—
47#100×C79Q814yes—
48#97×C79—41no—
49C4×C8×C79—41no—
50C22×Dic79—22no—
51C22×Dic79—22no—
52C22×Dic79—42no—
53C22×Dic79—42no—
54C22×Dic79—82no—
55C4⋊Dic79—42no—
56C4⋊Dic79—42no—
57C4⋊Dic79—41no—
58C4⋊Dic79—41no—
59C4⋊Dic79—82no—
60Dic79⋊C4—82no—
61C22⋊Dic79—42no—
62C22⋊Dic79—42no—
63C22⋊Dic79—42no—
64C22⋊Dic79—82no—
65C24×C79C22×C411yescharacteristic
66C24×C79C2×D411yescharacteristic
67C2×D4×C79—41no—
68C2×D4×C79—41no—
69C2×D4×C79C2×Q812yes—
70C2×D4×C79#4112yes—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—42no—
76C22×C4×C79—42no—
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82M16×C79—24no—
83M16×C79—42no—
84C2×C8×C79—42no—
85C4⋊C4×C79—41no—
86C22⋊C4×C79—41no—
87C22⋊C4×C79—41no—
88C42×C79—21no—
89C42×C79—21no—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—41no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79C2×C22⋊C411yescharacteristic
99C23×C79C4×Q812yes—
100C23×C79#12411yescharacteristic
101C23×C79C4⋊Q811yescharacteristic
102D4×C79—42no—
103D4×C79—42no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—22no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—81no—
117C2×C4×C79#9911yescharacteristic
118C2×C4×C79#9911yescharacteristic
119C2×C4×C79C22⋊Q812yes—
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126C2×Dic79—84no—
127C2×Dic79—82no—
128C2×Dic79—82no—
129C2×Dic79—82no—
130C8×C79—44no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—22no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—42no—
137C22×C79—41no—
138C22×C79—41no—
139C22×C79—42no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—41no—
143C22×C79—41no—
144C22×C79—81no—
145C22×C79#39111yescharacteristic
146C22×C79C2×#9911yescharacteristic
147C22×C79#41011yescharacteristic
148C4×C79—21no—
149C4×C79—22no—
150C4×C79—41no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79#35212yes—
154Dic79—82no—
155Dic79—82no—
156Dic79—162no—
157#13292—791nomaximal, Sylow
158C2×C79—21no—
159C2×C79—21no—
160C2×C79—41no—
161C2×C79—42no—
162C2×C79—41no—
163C2×C79#199611yescharacteristic
164C2×C79C2×#35211yescharacteristic
165C2×C79#212211yescharacteristic
166#2487—791no—
167#1799—791no—
168C4×#100D7911yescharacteristic
169#1390—794no—
170C79#1329211yescharacteristic, complemented, Sylow
171C2×C4×D4C2×D7911yescharacteristic
172C22×C4⋊C4—791no—
173C2×#100C2×D7912yes—
174C2×#99—792no—
175C4×M16—21no—
176#400—1581no—
177#399—1581no—
178#389—1581no—
179#386—792no—
180#385—792no—
181#380—791no—
182#380—1581no—
183#380—1581no—
184#343—21no—
185#342—1581no—
186C22×D4C22×D7911yescharacteristic
187C23×C4—792no—
188C23×C4C4×D7911yescharacteristic
189C23×C4D31611yescharacteristic
190C2×M16—22no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—3161no—
194C2×C4⋊C4—3161no—
195C2×C4⋊C4—3161no—
196C2×C4⋊C4—1581no—
197C2×C4⋊C4—1581no—
198C2×C4⋊C4—1581no—
199C2×C4⋊C4—1582no—
200C2×C4⋊C4C22⋊D7911yescharacteristic
201C2×C22⋊C4—792no—
202C2×C22⋊C4—1582no—
203C2×C22⋊C4—1582no—
204C2×C22⋊C4C4×D7911yescharacteristic
205C2×C22⋊C4D31611yescharacteristic
206C2×C42C22⋊D7911yescharacteristic
207#100—22no—
208#100Dic15814yes—
209#99—1584no—
210#97—41no—
211C4×C8—41no—
212#95—3161no—
213#95—1581no—
214#95—1582no—
215#95—1582no—
216C24C2×C4×D7911yescharacteristic
217C24C2×D31611yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4C2×Dic15812yes—
221C2×D4#1192962412yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—3162no—
229C22×C4—3162no—
230C22×C4—6322no—
231C22×C4—1582no—
232C22×C4—1582no—
233C22×C4(C2×D79)⋊C411yescharacteristic
234C22×C4(C2×D79)⋊C411yescharacteristic
235C22×C4(C2×D79)⋊C411yescharacteristic
236C22×C4(C2×D79)⋊C411yescharacteristic
237C22×C4C2×C22⋊D7911yescharacteristic
238M16—24no—
239M16—42no—
240C2×C8—42no—
241C4⋊C4—41no—
242C4⋊C4—3162no—
243C4⋊C4—3162no—
244C4⋊C4—3161no—
245C4⋊C4—3161no—
246C4⋊C4—6322no—
247C4⋊C4—6322no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—3162no—
251C22⋊C4—3162no—
252C22⋊C4—3162no—
253C22⋊C4—6322no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23C2×(C2×D79)⋊C411yescharacteristic
265C23C4×Dic15812yes—
266C23#1457914911yescharacteristic
267C23#1457916111yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—3162no—
284C2×C4—3162no—
285C2×C4—3162no—
286C2×C4—3162no—
287C2×C4—3162no—
288C2×C4—3162no—
289C2×C4—6324no—
290C2×C4—6322no—
291C2×C4—6322no—
292C2×C4—6322no—
293C2×C4#1457907511yescharacteristic
294C2×C4#1457907511yescharacteristic
295C2×C4#1457914012yes—
296C8—44no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22#1909846911yescharacteristic
312C22C2×#1457907511yescharacteristic
313C22#1909853611yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—6322no—
320C4—6322no—
321C4—12642no—
322C4#1909835412yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2376438511yescharacteristic
329C2C2×#1909835411yescharacteristic
330C2#2376505111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC24×C79C22×C4
FrattiniC22×D4C22×D79
FittingC4×#100×C79C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenterC4×#100D79
hypocenterC79#13292
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
5. C22×C4⋊Dic79C4semidirect
11. #19098435C4semidirect
14. #19098423C4semidirect
170. C79#13292semidirect

Maximal subgroups

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

subgroupindex
#237677172
#237633882
#237615192
C4×#100×C792
#1329279