Finite Group Databasebeta

← #34183361

Subgroups — #34183361

Order 20224.

482 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 111 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#23772038C211yesmaximal, characteristic
2#23766668C211yesmaximal, characteristic
3#23761830C212yesmaximal, complemented
4#23761302C212yesmaximal, complemented
5C79×#1458C211yesmaximal, characteristic
6C2×C4×Dic158C2211yescharacteristic
7#19098624—24no—
8#19098587C2212yes—
9#19098527—21no—
10C4×#11929608—21no—
11#19098502C411yescharacteristic
12#19098499C411yescharacteristic
13#19098496—21no—
14#19098486—21no—
15#19098483C411yescharacteristic
16#19098475C411yescharacteristic
17C4×Dic79⋊C4—21no—
18C4×C4⋊Dic79C2211yescharacteristic
19C4×C22⋊Dic79—21no—
20#19098295—24no—
21C2×#117×C79C2211yescharacteristic
22#431×C79C2212yes—
23#324×C79—21no—
24#323×C79—21no—
25C22×Dic158C2×C411yescharacteristic
26C2×#11929608—22no—
27C4×Dic158—22no—
28C4×Dic158—22no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79C2×C411yescharacteristic
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—21no—
36C2×C4×Dic79—21no—
37C2×C4×Dic79—21no—
38C4⋊(C79⋊C8)—22no—
39C4⋊(C79⋊C8)D414yes—
40C22⋊(C79⋊C8)—24no—
41#14579072—22no—
42C4×C79⋊C8—22no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95C2×C411yescharacteristic
48C79⋊#95C2×C411yescharacteristic
49C23×C4×C79D411yescharacteristic
50#117×C79—22no—
51#117×C79—22no—
52C2×C4⋊C4×C79C2×C411yescharacteristic
53C2×C4⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79C2×C411yescharacteristic
55C2×C22⋊C4×C79D411yescharacteristic
56C2×C42×C79D411yescharacteristic
57C2×C42×C79C2311yescharacteristic
58C4⋊C8×C79—22no—
59C22⋊C8×C79—22no—
60C4×C8×C79—22no—
61C2×Dic158—22no—
62C2×Dic158—42no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67#11929608—42no—
68#11929608—42no—
69C2×C79⋊C8—24no—
70C4⋊Dic79—22no—
71C4⋊Dic79—42no—
72Dic79⋊C4—42no—
73Dic79⋊C4—42no—
74Dic79⋊C4—42no—
75C22⋊Dic79—42no—
76C22⋊Dic79—42no—
77C4×Dic79—41no—
78C4×Dic79—41no—
79C4×Dic79—42no—
80C24×C79C22⋊C411yescharacteristic
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—22no—
84C22×C4×C79—22no—
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C22⋊C411yescharacteristic
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C22×C411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C2×C8×C79—42no—
92C4⋊C4×C79—22no—
93C4⋊C4×C79—22no—
94C22⋊C4×C79—22no—
95C22⋊C4×C79—22no—
96C42×C79—21no—
97C42×C79—21no—
98C42×C79C2×D412yes—
99C42×C79#4112yes—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79C2×C22⋊C411yescharacteristic
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79C4×D412yes—
118C2×C4×C79#12011yescharacteristic
119C2×C4×C79#12112yes—
120C2×C4×C79#12311yescharacteristic
121C2×C4×C79#13512yes—
122C2×C4×C79#13712yes—
123C2×Dic79—22no—
124C2×Dic79—22no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135Dic158—24no—
136Dic158—42no—
137C8×C79—82no—
138C79⋊C8—44no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—42no—
142C22×C79—41no—
143C22×C79—41no—
144C22×C79—41no—
145C22×C79—81no—
146C22×C79#38611yescharacteristic
147C22×C79#41611yescharacteristic
148C22×C79#41911yescharacteristic
149C4×C79—22no—
150C4×C79—22no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79—41no—
154C4×C79#45212yes—
155C4×C79#48512yes—
156Dic79—42no—
157Dic79—42no—
158Dic79—81no—
159Dic79—81no—
160#9034—791nomaximal, Sylow
161C2×C79—21no—
162C2×C79—21no—
163C2×C79—81no—
164C2×C79#187311yescharacteristic
165C2×C79#188411yescharacteristic
166C2×C79#189111yescharacteristic
167#2921—791no—
168#2396—791no—
169#1487—792no—
170#1458D7911yescharacteristic
171#1303—792no—
172C79#903411yescharacteristic, complemented, Sylow
173C2×C4×Q8—791no—
174C2×#117C2×D7911yescharacteristic
175#433—1584no—
176#431C2×D7912yes—
177#423—792no—
178#407—1581no—
179C4×M16—1581no—
180#400—791no—
181#399—791no—
182#398—1581no—
183#396—1581no—
184#395—791no—
185#393—791no—
186C4×C4⋊C4—791no—
187C4×C4⋊C4—1581no—
188C4×C22⋊C4—1581no—
189Q8⋊C8—1584no—
190#324—21no—
191#323—21no—
192C22×Q8—791no—
193C23×C4D31611yescharacteristic
194C2×M16—1582no—
195C4×Q8—1582no—
196C4×Q8—1582no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—791no—
200C2×C4⋊C4—1581no—
201C2×C4⋊C4—1581no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4C4×D7911yescharacteristic
205C2×C4⋊C4C22⋊D7911yescharacteristic
206C2×C22⋊C4—1581no—
207C2×C22⋊C4—1581no—
208C2×C22⋊C4C4×D7911yescharacteristic
209C2×C22⋊C4C22⋊D7911yescharacteristic
210C2×C42—1581no—
211C2×C42—1581no—
212C2×C42D31611yescharacteristic
213C2×C42C22×D7911yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—794no—
216C4⋊C8—1582no—
217C22⋊C8—22no—
218C22⋊C8—1584no—
219#97—1582no—
220C4×C8—22no—
221C4×C8—1582no—
222#95—791no—
223#95—791no—
224#95—1581no—
225#95—1581no—
226#95—1581no—
227#95—1581no—
228C24(C2×D79)⋊C411yescharacteristic
229C2×Q8—3162no—
230C2×Q8—1582no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—1581no—
236C22×C4—1581no—
237C22×C4—1581no—
238C22×C4—1581no—
239C22×C4(C2×D79)⋊C411yescharacteristic
240C22×C4(C2×D79)⋊C411yescharacteristic
241C22×C4(C2×D79)⋊C411yescharacteristic
242C22×C4C2×C4×D7911yescharacteristic
243C22×C4C2×C22⋊D7911yescharacteristic
244C22×C4C2×D31611yescharacteristic
245M16—3162no—
246M16—3162no—
247C2×C8—42no—
248C2×C8—1584no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—3162no—
252C4⋊C4—3162no—
253C4⋊C4—3162no—
254C4⋊C4—3162no—
255C4⋊C4—1582no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—3162no—
259C22⋊C4—3162no—
260C42—21no—
261C42—21no—
262C42—3161no—
263C42—3161no—
264C42—3162no—
265C42D4×D7912yes—
266C42#1192962312yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23C2×(C2×D79)⋊C411yescharacteristic
273Q8—3162no—
274Q8—1584no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—3161no—
291C2×C4—3161no—
292C2×C4—3161no—
293C2×C4—3161no—
294C2×C4—3161no—
295C2×C4—3161no—
296C2×C4—1582no—
297C2×C4—1582no—
298C2×C4(C22⋊D79)⋊C412yes—
299C2×C4#1457913111yescharacteristic
300C2×C4#1457913312yes—
301C2×C4#1457914511yescharacteristic
302C2×C4#1457918312yes—
303C2×C4#1457919412yes—
304C8—82no—
305C8—3164no—
306C22—21no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—81no—
313C22#1909844011yescharacteristic
314C22#1909856011yescharacteristic
315C22#1909857311yescharacteristic
316C4—22no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—3162no—
322C4—3162no—
323C4—6321no—
324C4—6321no—
325C4#1909872712yes—
326C4#1909890512yes—
327C2—21no—
328C2—21no—
329C2—81no—
330C2#2376371011yescharacteristic
331C2#2376378111yescharacteristic
332C2#2376381611yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1458C2
socleC22×C79#386
prime radicalC24×C79C22⋊C4
cosocleC2×C42×C79C23
hypercenter#1458D79
hypocenterC79#9034
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #23761830C2semidirect
4. #23761302C2semidirect
172. C79#9034semidirect

Maximal subgroups

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

subgroupindex
#237720382
#237666682
#237618302
#237613022
C79×#14582
#903479