Finite Group Databasebeta

← #34184029

Subgroups — #34184029

Order 20224.

630 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 147 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#23771739C211yesmaximal, characteristic
2#23763468C211yesmaximal, characteristic
3C2×#19098299C212yesmaximal
4#23761279C212yesmaximal
5C2×#382×C79C211yesmaximal, characteristic
6C2×#14579140C2211yescharacteristic
7C2×C22⋊(C79⋊C8)C2212yes—
8#19098469C412yescomplemented
9#19098461C412yescomplemented
10#19098447—22no—
11#19098436—21no—
12#19098426—21no—
13#19098423C2211yescharacteristic
14#19098318—22no—
15#19098299—24no—
16C22×C4⋊C4×C79—21no—
17C22×C42×C79C2211yescharacteristic
18#382×C79—24no—
19C2×#95×C79—21no—
20C2×#95×C79C2212yes—
21C22×Dic158C2×C411yescharacteristic
22C22×C79⋊C8—22no—
23#14579140—22no—
24#14579140—22no—
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×C4⋊Dic79C2×C412yes—
27C2×Dic79⋊C4—22no—
28C2×Dic79⋊C4—21no—
29C2×C22⋊Dic79—21no—
30C2×C22⋊Dic79—21no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C79⋊#95—21no—
37C23×C4×C79—22no—
38C23×C4×C79D411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C23×C4×C79C2311yescharacteristic
41C2×C4⋊C4×C79—24no—
42C2×C4⋊C4×C79—42no—
43C2×C4⋊C4×C79—42no—
44C2×C42×C79—22no—
45C2×C42×C79—22no—
46C2×C42×C79C2×C412yes—
47C2×C42×C79D412yes—
48#95×C79—24no—
49#95×C79—22no—
50#95×C79D414yes—
51C2×Dic158—22no—
52C2×Dic158—42no—
53C22×Dic79—21no—
54C22×Dic79—22no—
55C22×Dic79—21no—
56C2×C79⋊C8—44no—
57C2×C79⋊C8—42no—
58C4⋊Dic79—22no—
59C4⋊Dic79—42no—
60Dic79⋊C4—44no—
61Dic79⋊C4—42no—
62C22⋊Dic79—41no—
63C22⋊Dic79—41no—
64C22⋊Dic79—41no—
65C22⋊Dic79—41no—
66C24×C79C2×D411yescharacteristic
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—24no—
74C22×C4×C79—42no—
75C22×C4×C79—42no—
76C22×C4×C79—44no—
77C22×C4×C79—44no—
78C22×C4×C79—42no—
79C22×C4×C79—42no—
80C22×C4×C79C22⋊C412yes—
81C22×C4×C79C22⋊C412yes—
82C22×C4×C79SD1612yes—
83C22×C4×C79Q1612yes—
84C22×C4×C79C22×C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D412yes—
87C22×C4×C79#4112yes—
88C4⋊C4×C79—48no—
89C42×C79—24no—
90C42×C79—24no—
91C23×C79—22no—
92C23×C79—22no—
93C23×C79C2×C22⋊C411yescharacteristic
94C23×C79#12011yescharacteristic
95C23×C79#12311yescharacteristic
96C23×C79C2×SD1611yescharacteristic
97C23×C79C2×Q1611yescharacteristic
98C23×C79AGL(1,ℤ/8)11yescharacteristic
99C23×C79#13711yescharacteristic
100C2×C4×C79—22no—
101C2×C4×C79—22no—
102C2×C4×C79—22no—
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—44no—
116C2×C4×C79—44no—
117C2×C4×C79—44no—
118C2×C4×C79—84no—
119C2×C4×C79—84no—
120C2×C4×C79#10314yes—
121C2×C4×C79C4×D412yes—
122C2×C4×C79#12112yes—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—42no—
128C2×Dic79—42no—
129C2×Dic79—42no—
130Dic158—82no—
131C79⋊C8—82no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—22no—
137C22×C79—22no—
138C22×C79—22no—
139C22×C79—22no—
140C22×C79#38611yescharacteristic
141C22×C79#41012yes—
142C22×C79C2×#10311yescharacteristic
143C22×C79#41711yescharacteristic
144C22×C79#42112yes—
145C22×C79#45011yescharacteristic
146C22×C79#45111yescharacteristic
147C22×C79#48311yescharacteristic
148C22×C79#48411yescharacteristic
149C4×C79—24no—
150C4×C79—22no—
151C4×C79—22no—
152C4×C79—42no—
153C4×C79—84no—
154Dic79—81no—
155Dic79—81no—
156Dic79—82no—
157#9152—791nomaximal, Sylow
158C2×C79—21no—
159C2×C79—21no—
160C2×C79—21no—
161C2×C79—21no—
162C2×C79#151212yes—
163C2×C79#151812yes—
164C2×C79#187211yescharacteristic
165C2×C79#189711yescharacteristic
166C2×C79#189911yescharacteristic
167#2855—791no—
168C2×#382D7911yescharacteristic
169#1817—791no—
170C2×#328—792no—
171#1294—792no—
172C79#915211yescharacteristic, complemented, Sylow
173C2×C22⋊Q8—791no—
174C22×C4⋊C4—21no—
175C22×C42C2×D7911yescharacteristic
176C2×C22⋊C8—792no—
177#391—792no—
178#389—792no—
179#387—1582no—
180#385—1581no—
181#382—24no—
182#381—1581no—
183#380—791no—
184C2×#95—21no—
185C2×#95C2×D7912yes—
186#336—1582no—
187#328—1584no—
188C22×Q8—791no—
189C23×C4—22no—
190C23×C4C22⋊D7911yescharacteristic
191C23×C4C22⋊D7911yescharacteristic
192C23×C4C22×D7911yescharacteristic
193C22×C8—1582no—
194C22⋊Q8—1582no—
195C22⋊Q8—1582no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—792no—
200C2×C4⋊C4—791no—
201C2×C4⋊C4—1582no—
202C2×C4⋊C4—1581no—
203C2×C22⋊C4—1581no—
204C2×C22⋊C4—1581no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D7912yes—
208C2×C42D31612yes—
209C22⋊C8—1582no—
210C22⋊C8—1582no—
211#95—24no—
212#95—22no—
213#95—1582no—
214#95—1582no—
215#95—1582no—
216#95—1581no—
217#95C22⋊D7914yes—
218C24C2×C22⋊D7911yescharacteristic
219C2×Q8—3162no—
220C2×Q8—1582no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—1582no—
235C22×C4—1581no—
236C22×C4—1581no—
237C22×C4(C2×D79)⋊C412yes—
238C22×C4(C2×D79)⋊C412yes—
239C22×C4#1192961212yes—
240C22×C4#1192961412yes—
241C22×C4C2×C4×D7911yescharacteristic
242C22×C4C2×C22⋊D7912yes—
243C22×C4C2×D31611yescharacteristic
244C22×C4#1192962412yes—
245C2×C8—3164no—
246C2×C8—3162no—
247C4⋊C4—48no—
248C4⋊C4—3162no—
249C4⋊C4—3164no—
250C4⋊C4—3162no—
251C4⋊C4—1582no—
252C22⋊C4—3161no—
253C22⋊C4—3161no—
254C22⋊C4—3161no—
255C22⋊C4—3161no—
256C42—24no—
257C42—24no—
258C23—22no—
259C23—22no—
260C23C2×(C2×D79)⋊C411yescharacteristic
261C23#1457913011yescharacteristic
262C23#1457914411yescharacteristic
263C23C2×#1192961211yescharacteristic
264C23C2×#1192961411yescharacteristic
265C23#1457918711yescharacteristic
266C23#1457919211yescharacteristic
267Q8—6322no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—3162no—
291C2×C4—3162no—
292C2×C4—3162no—
293C2×C4—3162no—
294C2×C4—3162no—
295C2×C4#1457908414yes—
296C2×C4C4×C22⋊D7912yes—
297C2×C4#1457913512yes—
298C8—6322no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22#1909844211yescharacteristic
308C22#1909853612yes—
309C22C2×#1457908411yescharacteristic
310C22#1909856211yescharacteristic
311C22#1909858112yes—
312C22#1909871411yescharacteristic
313C22#1909871911yescharacteristic
314C22#1909889311yescharacteristic
315C22#1909889811yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—84no—
321C4—6322no—
322C4—6321no—
323C4—6321no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#2376193612yes—
329C2#2376196312yes—
330C2#2376370711yescharacteristic
331C2#2376384811yescharacteristic
332C2#2376385811yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#382×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenterC2×#382D79
hypocenterC79#9152
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
8. #19098469C4semidirect
9. #19098461C4semidirect
172. C79#9152semidirect

Maximal subgroups

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

subgroupindex
#237717392
#237634682
C2×#190982992
#237612792
C2×#382×C792
#915279