Finite Group Databasebeta

← #34197826

Subgroups — #34197826

Order 20224.

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

normal subgroups only

This group has 496 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#23767867C211yesmaximal, characteristic
2C2×#19098289C211yesmaximal, characteristic, complemented
3#23761345C212yesmaximal, complemented
4#23761331C212yesmaximal, complemented
5C79×#1463C211yesmaximal, characteristic, complemented
6C2×#14579144—21no—
7C2×#14579144—21no—
8C2×#14579135C411yescharacteristic
9C2×C22⋊(C79⋊C8)—21no—
10#19098498—21no—
11#19098498—21no—
12#19098464C411yescharacteristic
13#19098464C2211yescharacteristic
14#19098432C411yescharacteristic
15#19098432C2211yescharacteristic
16C4×C22⋊Dic79C411yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C22×C4⋊C4×C79C2211yescharacteristic
20#407×C79—21no—
21#324×C79—21no—
22#324×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C22⋊D79—21no—
25C22×C79⋊C8—41no—
26#14579144—44no—
27#14579144—44no—
28#14579135—41no—
29#14579135—41no—
30C2×C4⋊Dic79C2×C411yescharacteristic
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—21no—
34C2×Dic79⋊C4—21no—
35C2×C22⋊Dic79—21no—
36C2×(C2×D79)⋊C4—21no—
37C2×(C2×D79)⋊C4—21no—
38C2×(C2×D79)⋊C4—21no—
39C2×(C2×D79)⋊C4—21no—
40C2×(C2×D79)⋊C4—21no—
41C2×C4×Dic79C2×C411yescharacteristic
42C2×C4×Dic79C2×C411yescharacteristic
43C22⋊(C79⋊C8)—22no—
44C22⋊(C79⋊C8)—41no—
45C79⋊#95—21no—
46C23×C4×C79C2×C411yescharacteristic
47C23×C4×C79D411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C2×M16×C79—41no—
50C2×C4⋊C4×C79—41no—
51C2×C4⋊C4×C79—41no—
52C2×C4⋊C4×C79C2×C411yescharacteristic
53C2×C4⋊C4×C79D411yescharacteristic
54C2×C4⋊C4×C79D411yescharacteristic
55C2×C4⋊C4×C79C2311yescharacteristic
56C22⋊C8×C79—22no—
57C22⋊C8×C79—41no—
58C23×D79—21no—
59C2×D316—22no—
60C2×D316—41no—
61C2×D316—41no—
62C2×C22⋊D79—42no—
63C2×C22⋊D79—42no—
64C2×C22⋊D79—41no—
65C2×C22⋊D79—41no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69C2×C79⋊C8—42no—
70C2×C79⋊C8—82no—
71C4⋊Dic79—41no—
72Dic79⋊C4—42no—
73Dic79⋊C4—42no—
74Dic79⋊C4—42no—
75Dic79⋊C4—42no—
76C22⋊Dic79—24no—
77(C2×D79)⋊C4—42no—
78(C2×D79)⋊C4—42no—
79(C2×D79)⋊C4—42no—
80(C2×D79)⋊C4—42no—
81(C2×D79)⋊C4—42no—
82C4×Dic79—22no—
83C4×Dic79—41no—
84C24×C79C22⋊C411yescharacteristic
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—22no—
92C22×C4×C79—41no—
93C22×C4×C79—42no—
94C22×C4×C79—42no—
95C22×C4×C79C22⋊C411yescharacteristic
96C22×C4×C79C22⋊C411yescharacteristic
97C22×C4×C79C22⋊C411yescharacteristic
98C22×C4×C79C22×C411yescharacteristic
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79C2×D411yescharacteristic
101M16×C79—82no—
102C2×C8×C79—42no—
103C4⋊C4×C79—41no—
104C4⋊C4×C79—41no—
105C4⋊C4×C79—41no—
106C4⋊C4×C79—41no—
107C23×C79—22no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—41no—
111C23×C79#9911yescharacteristic
112C23×C79#9911yescharacteristic
113C23×C79C2×C22⋊C411yescharacteristic
114C22×D79—41no—
115C22×D79—41no—
116C22×D79—41no—
117C22×D79—41no—
118C22×D79—41no—
119C22×D79—41no—
120C22×D79—41no—
121C2×C4×C79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—42no—
131C2×C4×C79—42no—
132C2×C4×C79—42no—
133C2×C4×C79—42no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—81no—
139C2×C4×C79—81no—
140C2×C4×C79#13112yes—
141C22⋊D79—82no—
142C22⋊D79—82no—
143C2×Dic79—22no—
144C2×Dic79—22no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—41no—
149C2×Dic79—41no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152D316—82no—
153C8×C79—82no—
154C79⋊C8—82no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79—21no—
158C22×C79—21no—
159C22×C79—21no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—42no—
163C22×C79—42no—
164C22×C79#35312yes—
165C22×C79#40811yescharacteristic
166C22×C79C2×#9911yescharacteristic
167C22×C79#41311yescharacteristic
168C2×D79—81no—
169C2×D79—81no—
170C2×D79—81no—
171C2×D79—81no—
172C2×D79—81no—
173C2×D79—81no—
174C2×D79—81no—
175C4×C79—22no—
176C4×C79—41no—
177C4×C79—41no—
178C4×C79—41no—
179C4×C79—41no—
180C4×C79—41no—
181Dic79—42no—
182Dic79—42no—
183Dic79—81no—
184#11549—791nomaximal, Sylow
185C2×C79—21no—
186C2×C79—21no—
187C2×C79—22no—
188C2×C79—41no—
189C2×C79#146811yescharacteristic
190C2×C79C2×#35311yescharacteristic
191C2×C79#213311yescharacteristic
192D79—161no—
193#2501—791no—
194#1463D7911yescharacteristic, complemented
195C2×#324—791no—
196#1325—792no—
197#1318—792no—
198C79#1154911yescharacteristic, complemented, Sylow
199C2×#123—1581no—
200C2×#123—1581no—
201C2×#121—791no—
202C22×C4⋊C4C2×D7911yescharacteristic
203#407—21no—
204C2×C22⋊C8—1581no—
205#399—1581no—
206#399—1581no—
207#390—791no—
208#390—791no—
209#384—791no—
210#384—791no—
211C4×C22⋊C4—791no—
212#324—21no—
213#324—792no—
214#324—1581no—
215#324C2×D7912yes—
216C22×D4—791no—
217C22×D4—1581no—
218C23×C4C4×D7911yescharacteristic
219C23×C4D31611yescharacteristic
220C23×C4C22⋊D7911yescharacteristic
221C2×M16—41no—
222C22×C8—3161no—
223#123—3164no—
224#123—3164no—
225#121—3161no—
226#121—3161no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—791no—
230C2×C4⋊C4—1581no—
231C2×C4⋊C4—1581no—
232C2×C4⋊C4—1581no—
233C2×C4⋊C4—1581no—
234C2×C4⋊C4C4×D7911yescharacteristic
235C2×C4⋊C4D31611yescharacteristic
236C2×C4⋊C4C22⋊D7911yescharacteristic
237C2×C4⋊C4C22×D7911yescharacteristic
238C2×C22⋊C4—1581no—
239C2×C22⋊C4—1581no—
240C2×C22⋊C4—1581no—
241C2×C22⋊C4—1581no—
242C2×C22⋊C4—1581no—
243C2×C22⋊C4—1581no—
244C2×C42—791no—
245C2×C42—791no—
246C22⋊C8—22no—
247C22⋊C8—41no—
248C22⋊C8—3161no—
249C22⋊C8—1582no—
250#95—1581no—
251C24—1581no—
252C24(C2×D79)⋊C411yescharacteristic
253C2×D4—3161no—
254C2×D4—3161no—
255C2×D4—3162no—
256C2×D4—3162no—
257C2×D4—3161no—
258C2×D4—3161no—
259C2×D4—1582no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—1581no—
271C22×C4—1581no—
272C22×C4—1581no—
273C22×C4(C2×D79)⋊C411yescharacteristic
274C22×C4(C2×D79)⋊C411yescharacteristic
275C22×C4(C2×D79)⋊C411yescharacteristic
276C22×C4C2×C4×D7911yescharacteristic
277C22×C4C2×C22⋊D7911yescharacteristic
278C22×C4C2×D31611yescharacteristic
279M16—82no—
280C2×C8—42no—
281C2×C8—3162no—
282C2×C8—6322no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—3161no—
288C4⋊C4—3162no—
289C4⋊C4—3162no—
290C4⋊C4—3162no—
291C4⋊C4—3162no—
292C22⋊C4—3162no—
293C22⋊C4—3162no—
294C22⋊C4—3162no—
295C22⋊C4—3162no—
296C22⋊C4—3162no—
297C22⋊C4—1584no—
298C42—3161no—
299C42—1582no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—41no—
304C23—3161no—
305C23—3161no—
306C23—3161no—
307C23—3161no—
308C23—3161no—
309C23—3161no—
310C23—3161no—
311C23#1457907511yescharacteristic
312C23#1457907511yescharacteristic
313C23C2×(C2×D79)⋊C411yescharacteristic
314D4—6322no—
315D4—6322no—
316D4—6322no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—3161no—
337C2×C4—3161no—
338C2×C4—3161no—
339C2×C4—3161no—
340C2×C4—3161no—
341C2×C4—3161no—
342C2×C4—3161no—
343C2×C4—1582no—
344C2×C4—1582no—
345C2×C4#1457916912yes—
346C8—82no—
347C8—6322no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—42no—
357C22—6321no—
358C22—6321no—
359C22—6321no—
360C22—6321no—
361C22—6321no—
362C22—6321no—
363C22—6321no—
364C22#1909835512yes—
365C22#1909853111yescharacteristic
366C22C2×#1457907511yescharacteristic
367C22#1909854511yescharacteristic
368C4—22no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—3162no—
375C4—3162no—
376C4—6321no—
377C2—21no—
378C2—21no—
379C2—22no—
380C2—41no—
381C2—12641no—
382C2#2376174911yescharacteristic
383C2C2×#1909835511yescharacteristic
384C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1463C2
socleC22×C79C2×#99
prime radicalC2×#14579135C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1463D79
hypocenterC79#11549
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. C2×#19098289C2semidirect
3. #23761345C2semidirect
4. #23761331C2semidirect
5. C79×#1463C2semidirect
194. #1463D79semidirect
198. C79#11549semidirect

Maximal subgroups

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

subgroupindex
#237678672
C2×#190982892
#237613452
#237613312
C79×#14632
#1154979