Finite Group Databasebeta

← #34179621

Subgroups — #34179621

Order 20224.

810 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×#14579151C211yesmaximal, characteristic
2#23763466C212yesmaximal
3C2×#19098289C211yesmaximal, characteristic
4C2×#19098288C212yesmaximal
5C2×#324×C79C211yesmaximal, characteristic
6C2×#14579151—24no—
7C2×#14579151—24no—
8C2×#14579151—22no—
9C2×#14579151—22no—
10C22×C4⋊Dic79C411yescharacteristic
11C22×Dic79⋊C4C411yescharacteristic
12C22×Dic79⋊C4C411yescharacteristic
13C22×Dic79⋊C4C411yescharacteristic
14C22×Dic79⋊C4C2211yescharacteristic
15C22×C4×Dic79C2211yescharacteristic
16C2×C22⋊(C79⋊C8)C2212yes—
17#19098317—24no—
18#19098289—22no—
19#19098288—24no—
20C22×C4⋊C4×C79C2211yescharacteristic
21C2×C22⋊C8×C79C2212yes—
22#324×C79—22no—
23C23×Dic79C2×C411yescharacteristic
24C23×Dic79C2×C411yescharacteristic
25C23×Dic79C2×C411yescharacteristic
26C23×Dic79C2×C411yescharacteristic
27C22×C79⋊C8—22no—
28#14579151—216no—
29#14579151—216no—
30C2×C4⋊Dic79—22no—
31C2×C4⋊Dic79—22no—
32C2×C4⋊Dic79—22no—
33C2×Dic79⋊C4—22no—
34C2×Dic79⋊C4—22no—
35C2×Dic79⋊C4—22no—
36C2×Dic79⋊C4—22no—
37C2×Dic79⋊C4—21no—
38C2×Dic79⋊C4—21no—
39C2×Dic79⋊C4—22no—
40C2×Dic79⋊C4—22no—
41C2×Dic79⋊C4—22no—
42C2×Dic79⋊C4—22no—
43C2×Dic79⋊C4—22no—
44C2×Dic79⋊C4—22no—
45C2×Dic79⋊C4D414yes—
46C2×C4×Dic79—22no—
47C2×C4×Dic79—21no—
48C2×C4×Dic79—21no—
49C2×C4×Dic79D414yes—
50C22⋊(C79⋊C8)—22no—
51C22⋊(C79⋊C8)—22no—
52C23×C4×C79C2×C411yescharacteristic
53C23×C4×C79C2×C411yescharacteristic
54C23×C4×C79C2311yescharacteristic
55C22×C8×C79—22no—
56C2×C4⋊C4×C79—22no—
57C2×C4⋊C4×C79—22no—
58C2×C4⋊C4×C79D412yes—
59C2×C4⋊C4×C79D412yes—
60C22⋊C8×C79—22no—
61C22⋊C8×C79—22no—
62C22×Dic79—24no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69C22×Dic79—21no—
70C22×Dic79—21no—
71C22×Dic79—22no—
72C22×Dic79—22no—
73C22×Dic79—22no—
74C22×Dic79—22no—
75C22×Dic79—42no—
76C22×Dic79C22⋊C412yes—
77C22×Dic79C22⋊C412yes—
78C22×Dic79C22⋊C412yes—
79C22×Dic79C22⋊C412yes—
80C2×C79⋊C8—44no—
81C2×C79⋊C8—42no—
82C4⋊Dic79—28no—
83Dic79⋊C4—28no—
84Dic79⋊C4—28no—
85Dic79⋊C4—28no—
86Dic79⋊C4—24no—
87Dic79⋊C4—24no—
88C4×Dic79—24no—
89C4×Dic79—24no—
90C24×C79C22×C411yescharacteristic
91C22×C4×C79—21no—
92C22×C4×C79—21no—
93C22×C4×C79—21no—
94C22×C4×C79—21no—
95C22×C4×C79—22no—
96C22×C4×C79—22no—
97C22×C4×C79—42no—
98C22×C4×C79—42no—
99C22×C4×C79C22⋊C412yes—
100C22×C4×C79C22⋊C412yes—
101C22×C4×C79C2×D412yes—
102C22×C4×C79C2×D412yes—
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79C2×D411yescharacteristic
105C2×C8×C79—44no—
106C2×C8×C79—42no—
107C4⋊C4×C79—24no—
108C4⋊C4×C79—24no—
109C23×C79—22no—
110C23×C79—22no—
111C23×C79C2×C22⋊C411yescharacteristic
112C23×C79C2×C22⋊C411yescharacteristic
113C23×C79C2×C22⋊C411yescharacteristic
114C23×C79#13112yes—
115C23×C79#13112yes—
116C2×C4×C79—24no—
117C2×C4×C79—22no—
118C2×C4×C79—22no—
119C2×C4×C79—22no—
120C2×C4×C79—22no—
121C2×C4×C79—42no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—42no—
125C2×C4×C79—42no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79#12012yes—
130C2×C4×C79#12012yes—
131C2×Dic79—24no—
132C2×Dic79—24no—
133C2×Dic79—22no—
134C2×Dic79—22no—
135C2×Dic79—22no—
136C2×Dic79—22no—
137C2×Dic79—22no—
138C2×Dic79—24no—
139C2×Dic79—24no—
140C2×Dic79—42no—
141C2×Dic79—42no—
142C2×Dic79—42no—
143C2×Dic79—41no—
144C2×Dic79—41no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—42no—
148C2×Dic79—42no—
149C2×Dic79#10114yes—
150C8×C79—82no—
151C79⋊C8—82no—
152C22×C79—24no—
153C22×C79—24no—
154C22×C79—21no—
155C22×C79—21no—
156C22×C79—21no—
157C22×C79—21no—
158C22×C79#37911yescharacteristic
159C22×C79#40812yes—
160C22×C79#40812yes—
161C22×C79#40811yescharacteristic
162C22×C79#40811yescharacteristic
163C22×C79#41012yes—
164C22×C79C2×#10112yes—
165C4×C79—22no—
166C4×C79—22no—
167C4×C79—42no—
168C4×C79—42no—
169Dic79—24no—
170Dic79—42no—
171Dic79—42no—
172Dic79—42no—
173#8306—791nomaximal, Sylow
174C2×C79—22no—
175C2×C79—21no—
176C2×C79—21no—
177C2×C79#147014yes—
178C2×C79#178011yescharacteristic
179C2×C79#178211yescharacteristic
180C2×C79#178411yescharacteristic
181C22×#125—791no—
182#1817—792no—
183C2×#324—792no—
184C2×#324—791no—
185C2×#324D7911yescharacteristic
186C79#830611yescharacteristic, complemented, Sylow
187C2×#125—1584no—
188C2×#125—1584no—
189C2×#125—1582no—
190C2×#125—1582no—
191C22×C4⋊C4—791no—
192C22×C4⋊C4—791no—
193C22×C4⋊C4—791no—
194C22×C4⋊C4—791no—
195C22×C4⋊C4—791no—
196C22×C4⋊C4C2×D7911yescharacteristic
197C22×C42—791no—
198C2×C22⋊C8—792no—
199C2×C22⋊C8C2×D7912yes—
200#336—1584no—
201#324—22no—
202#324—1584no—
203#324—1582no—
204C23×C4—791no—
205C23×C4—791no—
206C23×C4—791no—
207C23×C4—791no—
208C23×C4C4×D7911yescharacteristic
209C23×C4C4×D7911yescharacteristic
210C23×C4C22×D7911yescharacteristic
211C22×C8—22no—
212C22×C8—1582no—
213#125—15816no—
214#125—15816no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—794no—
218C2×C4⋊C4—1582no—
219C2×C4⋊C4—1582no—
220C2×C4⋊C4—1582no—
221C2×C4⋊C4—1582no—
222C2×C4⋊C4—1582no—
223C2×C4⋊C4—1582no—
224C2×C4⋊C4—1582no—
225C2×C4⋊C4—1582no—
226C2×C4⋊C4—1582no—
227C2×C4⋊C4—1582no—
228C2×C4⋊C4—1582no—
229C2×C4⋊C4—1582no—
230C2×C4⋊C4—1582no—
231C2×C4⋊C4—1581no—
232C2×C4⋊C4—1581no—
233C2×C4⋊C4D31612yes—
234C2×C4⋊C4C22⋊D7912yes—
235C2×C42—794no—
236C2×C42—1582no—
237C2×C42—1581no—
238C2×C42—1581no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—1582no—
242C22⋊C8—1582no—
243C24C2×C4×D7911yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—3162no—
253C22×C4—792no—
254C22×C4—792no—
255C22×C4—792no—
256C22×C4—792no—
257C22×C4—1581no—
258C22×C4—1581no—
259C22×C4—1581no—
260C22×C4—1581no—
261C22×C4—1581no—
262C22×C4—1581no—
263C22×C4—1582no—
264C22×C4—1582no—
265C22×C4—1582no—
266C22×C4—1582no—
267C22×C4—1584no—
268C22×C4—1581no—
269C22×C4—1581no—
270C22×C4(C2×D79)⋊C412yes—
271C22×C4(C2×D79)⋊C412yes—
272C22×C4D4×D7912yes—
273C22×C4D4×D7912yes—
274C22×C4C2×C22⋊D7911yescharacteristic
275C22×C4C2×D31611yescharacteristic
276C2×C8—44no—
277C2×C8—42no—
278C2×C8—3164no—
279C2×C8—3162no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—1588no—
283C4⋊C4—1588no—
284C4⋊C4—1588no—
285C4⋊C4—1588no—
286C4⋊C4—1584no—
287C4⋊C4—1584no—
288C42—1584no—
289C42—1584no—
290C23—22no—
291C23—22no—
292C23C22⋊C4×D7911yescharacteristic
293C23C22⋊C4×D7911yescharacteristic
294C23C2×(C2×D79)⋊C411yescharacteristic
295C23#1457916712yes—
296C23#1457916912yes—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—3162no—
311C2×C4—3162no—
312C2×C4—3162no—
313C2×C4—3161no—
314C2×C4—3161no—
315C2×C4—3161no—
316C2×C4—3161no—
317C2×C4—3162no—
318C2×C4—3162no—
319C2×C4—794no—
320C2×C4—1582no—
321C2×C4—1582no—
322C2×C4—1582no—
323C2×C4—1584no—
324C2×C4—1584no—
325C2×C4—1584no—
326C2×C4—1584no—
327C2×C4—1582no—
328C2×C4—1582no—
329C2×C4#1457912912yes—
330C2×C4#1457913112yes—
331C8—82no—
332C8—6322no—
333C22—24no—
334C22—24no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22#1909841811yescharacteristic
340C22#1909852812yes—
341C22#1909852812yes—
342C22#1909852911yescharacteristic
343C22#1909853111yescharacteristic
344C22#1909853512yes—
345C22#101×D7912yes—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—3162no—
351C4—3162no—
352C4—3162no—
353C4—1584no—
354C2—22no—
355C2—21no—
356C2—21no—
357C2#2376175414yes—
358C2#2376331111yescharacteristic
359C2#2376331611yescharacteristic
360C2#2376332211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D79
commutator, prime radicalC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#324×C79C2
socleC23×C79C2×C22⋊C4
cosocleC23×C4×C79C23
hypercenterC2×#324D79
hypocenterC79#8306
LCSG ▷ C24×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
186. C79#8306semidirect

Maximal subgroups

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

subgroupindex
C22×#145791512
#237634662
C2×#190982892
C2×#190982882
C2×#324×C792
#830679