Finite Group Databasebeta

← #34199404

Subgroups — #34199404

Order 20224.

508 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 99 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#23768353C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761423C214yesmaximal, complemented
4C79×#1799C211yesmaximal, characteristic
5C2×#14579161C411yescharacteristic, complemented
6C2×#14579140—21no—
7C2×#14579140—21no—
8C2×C22⋊(C79⋊C8)—21no—
9#19098499C412yescomplemented
10#19098499C2212yes—
11#19098466—21no—
12#19098432—21no—
13C4×C4⋊Dic79C411yescharacteristic, complemented
14#19098289—21no—
15#19098287C2212yes—
16C2×C4×D4×C79C2211yescharacteristic
17C2×#99×C79—21no—
18#380×C79—21no—
19#342×C79C2212yes—
20C22×Dic158—21no—
21C22×C79⋊C8—41no—
22#14579161—41no—
23#14579161—41no—
24#14579140—44no—
25#14579140—44no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79C2×C412yes—
29C2×Dic79⋊C4—22no—
30C2×Dic79⋊C4—22no—
31C2×C22⋊Dic79—22no—
32C2×C22⋊Dic79—22no—
33C2×C4×Dic79—21no—
34C22⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—41no—
36C79⋊#95C2×C412yes—
37C22×D4×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C4×D4×C79—41no—
41C4×D4×C79—41no—
42C2×C4⋊C4×C79—41no—
43C2×C4⋊C4×C79D411yescharacteristic
44C2×C22⋊C4×C79—22no—
45C2×C22⋊C4×C79C2×C411yescharacteristic
46C2×C22⋊C4×C79C2311yescharacteristic
47C2×C42×C79C2×C411yescharacteristic
48#99×C79—42no—
49#95×C79—41no—
50C2×Dic158—42no—
51C2×Dic158—42no—
52C2×Dic158—41no—
53C2×Dic158—41no—
54C22×Dic79—22no—
55C22×Dic79—22no—
56C2×C79⋊C8—42no—
57C2×C79⋊C8—82no—
58C4⋊Dic79—24no—
59C4⋊Dic79—42no—
60C4⋊Dic79—42no—
61Dic79⋊C4—44no—
62Dic79⋊C4—44no—
63C22⋊Dic79—44no—
64C22⋊Dic79—44no—
65C4×Dic79—42no—
66C24×C79C22⋊C411yescharacteristic
67C24×C79C2×D411yescharacteristic
68C2×D4×C79—41no—
69C2×D4×C79—41no—
70C2×D4×C79SD1612yes—
71C2×D4×C79Q1612yes—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—42no—
77C22×C4×C79—42no—
78C22×C4×C79—42no—
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C4⋊C4×C79—41no—
85C4⋊C4×C79—82no—
86C22⋊C4×C79—41no—
87C22⋊C4×C79—41no—
88C22⋊C4×C79—42no—
89C22⋊C4×C79—42no—
90C42×C79—21no—
91C42×C79—21no—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79#10312yes—
101C23×C79C2×C22⋊C411yescharacteristic
102C23×C79C2×SD1611yescharacteristic
103C23×C79C2×Q1611yescharacteristic
104D4×C79—42no—
105D4×C79—42no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—22no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—81no—
118C2×C4×C79—81no—
119C2×C4×C79—82no—
120C2×C4×C79—82no—
121C2×C4×C79—82no—
122C2×C4×C79#9911yescharacteristic
123C2×C4×C79#9911yescharacteristic
124C2×C4×C79#10312yes—
125C2×Dic79—24no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129C2×Dic79—42no—
130C2×Dic79—42no—
131Dic158—82no—
132Dic158—82no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—22no—
138C22×C79—21no—
139C22×C79—42no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—41no—
143C22×C79—41no—
144C22×C79—41no—
145C22×C79—41no—
146C22×C79—81no—
147C22×C79—81no—
148C22×C79C2×#9911yescharacteristic
149C22×C79C2×#10311yescharacteristic
150C22×C79#42111yescharacteristic
151C4×C79—21no—
152C4×C79—22no—
153C4×C79—41no—
154C4×C79—41no—
155C4×C79—41no—
156C4×C79—162no—
157C4×C79#35312yes—
158Dic79—44no—
159Dic79—82no—
160#11901—791nomaximal, Sylow
161C2×C79—21no—
162C2×C79—21no—
163C2×C79—41no—
164C2×C79—42no—
165C2×C79—41no—
166C2×C79#152011yescharacteristic
167C2×C79#212311yescharacteristic
168C2×C79C2×#35311yescharacteristic
169#2542—791no—
170#1799D7911yescharacteristic
171#1459—791no—
172#1352—794no—
173C79#1190111yescharacteristic, complemented, Sylow
174C2×C4⋊Q8—791no—
175C2×C22⋊Q8—1581no—
176C2×C22⋊Q8—1581no—
177C2×C4×D4C2×D7911yescharacteristic
178C2×#99—21no—
179C2×C22⋊C8—1581no—
180#399—792no—
181#399—792no—
182#390—1581no—
183#384—1581no—
184#380—21no—
185C4×C4⋊C4—791no—
186#342C2×D7912yes—
187#324—1581no—
188#323—792no—
189C22×Q8—1581no—
190C22×D4C22⋊D7911yescharacteristic
191C23×C4D31611yescharacteristic
192C23×C4C22⋊D7911yescharacteristic
193C22×C8—3161no—
194C4⋊Q8—3161no—
195C4⋊Q8—3161no—
196C22⋊Q8—3164no—
197C22⋊Q8—3164no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—792no—
202C2×C4⋊C4—1582no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4—1582no—
205C2×C4⋊C4—1581no—
206C2×C4⋊C4D31611yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—1582no—
209C2×C22⋊C4—1582no—
210C2×C22⋊C4C4×D7911yescharacteristic
211C2×C22⋊C4C22×D7911yescharacteristic
212C2×C42—1581no—
213C2×C42C4×D7911yescharacteristic
214#99—42no—
215C22⋊C8—3161no—
216C22⋊C8—1582no—
217#95—41no—
218#95—792no—
219C24(C2×D79)⋊C411yescharacteristic
220C24C2×C22⋊D7911yescharacteristic
221C2×Q8—3162no—
222C2×Q8—3162no—
223C2×Q8—3161no—
224C2×Q8—3161no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#1192961212yes—
228C2×D4#1192961412yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—1582no—
237C22×C4—1582no—
238C22×C4(C2×D79)⋊C411yescharacteristic
239C22×C4(C2×D79)⋊C411yescharacteristic
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4C2×C4×D7911yescharacteristic
242C22×C4C2×D31611yescharacteristic
243C2×C8—3162no—
244C2×C8—6322no—
245C4⋊C4—41no—
246C4⋊C4—82no—
247C4⋊C4—3162no—
248C4⋊C4—3162no—
249C4⋊C4—3164no—
250C4⋊C4—3164no—
251C4⋊C4—1584no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—3164no—
257C22⋊C4—3164no—
258C42—21no—
259C42—21no—
260C42—3162no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1457908412yes—
270C23C2×(C2×D79)⋊C411yescharacteristic
271C23C2×#1192961211yescharacteristic
272C23C2×#1192961411yescharacteristic
273Q8—6322no—
274Q8—6322no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—3162no—
294C2×C4—3162no—
295C2×C4—3162no—
296C2×C4—3162no—
297C2×C4—3162no—
298C2×C4—1584no—
299C2×C4#1457907511yescharacteristic
300C2×C4#1457907511yescharacteristic
301C2×C4#1457908412yes—
302C8—6322no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#1457907511yescharacteristic
318C22C2×#1457908411yescharacteristic
319C22#1909858111yescharacteristic
320C4—22no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—162no—
326C4—3164no—
327C4—6322no—
328C4#1909835512yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#2376197111yescharacteristic
335C2#2376505311yescharacteristic
336C2C2×#1909835511yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1799C2
socleC22×C79C2×#99
prime radicalC22×D4×C79D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1799D79
hypocenterC79#11901
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #23761423C2semidirect
5. C2×#14579161C4semidirect
9. #19098499C4semidirect
13. C4×C4⋊Dic79C4semidirect
173. C79#11901semidirect

Maximal subgroups

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

subgroupindex
#237683532
#237617182
#237614232
C79×#17992
#1190179