Finite Group Databasebeta

← #34184804

Subgroups — #34184804

Order 20224.

428 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774544C211yesmaximal, characteristic
2#23772343C211yesmaximal, characteristic
3#23764245C211yesmaximal, characteristic
4#23762088C211yesmaximal, characteristic
5#23761297C211yesmaximal, characteristic
6#23761250C211yesmaximal, characteristic
7C2×#332×C79C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098959—21no—
10#19098915—21no—
11#19098893C414yes—
12#19098845—22no—
13#19098692—22no—
14#19098690—21no—
15#19098687—21no—
16#19098616C2211yescharacteristic
17#19098589—21no—
18#19098587C2211yescharacteristic
19C2×#14579080—21no—
20C2×#14579080—21no—
21#19098518—21no—
22C4×#11929608C2211yescharacteristic
23#19098311—21no—
24#19098310—21no—
25C2×C4⋊Q8×C79C2211yescharacteristic
26C2×C4⋊C8×C79C2211yescharacteristic
27C2×C4⋊C8×C79C2211yescharacteristic
28#332×C79—21no—
29#332×C79—21no—
30C2×#11929608—21no—
31C2×#11929608D411yescharacteristic
32C2×#11929608D411yescharacteristic
33#14579161D411yescharacteristic
34#14579161D411yescharacteristic
35#14579153—21no—
36#14579140—21no—
37#14579140C2×C412yes—
38C4×Dic158—21no—
39C4×Dic158D412yes—
40#14579120—21no—
41#14579092—24no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45Dic79⋊C8—22no—
46Dic79⋊C8—22no—
47#14579084—24no—
48#14579080—42no—
49#14579080—42no—
50C22⋊(C79⋊C8)C2×C412yes—
51#14579072—21no—
52#14579071—22no—
53C4×C79⋊C8—21no—
54C4×C79⋊C8—21no—
55C8×Dic79—22no—
56C22×Q8×C79—21no—
57C22×C8×C79—21no—
58C22×C8×C79—21no—
59C4⋊Q8×C79—21no—
60C4⋊Q8×C79—21no—
61C4⋊Q8×C79—21no—
62C4⋊Q8×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C4⋊C4×C79C2×C412yes—
65C2×C42×C79C2311yescharacteristic
66C4⋊C8×C79—21no—
67C4⋊C8×C79—21no—
68C4⋊C8×C79—21no—
69C4⋊C8×C79D411yescharacteristic
70C4⋊C8×C79D411yescharacteristic
71C2×Dic158—21no—
72C2×Dic158C22⋊C412yes—
73#11929608—21no—
74#11929608—21no—
75#11929608—21no—
76#11929608—41no—
77#11929608—41no—
78#11929608SD1612yes—
79C2×C79⋊C8—22no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—41no—
83C4⋊Dic79—21no—
84C4⋊Dic79—21no—
85C4⋊Dic79—21no—
86C4⋊Dic79C22⋊C412yes—
87Dic79⋊C4—22no—
88Dic79⋊C4—22no—
89C22⋊Dic79—22no—
90C4×Dic79—22no—
91C2×Q8×C79—41no—
92C2×Q8×C79—41no—
93C2×Q8×C79—41no—
94C2×Q8×C79—41no—
95C2×Q8×C79—41no—
96C2×Q8×C79—41no—
97C22×C4×C79—22no—
98C22×C4×C79C22×C411yescharacteristic
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79#4111yescharacteristic
101C2×C8×C79—21no—
102C2×C8×C79—21no—
103C2×C8×C79—41no—
104C2×C8×C79—41no—
105C2×C8×C79—41no—
106C2×C8×C79—81no—
107C4⋊C4×C79—22no—
108C4⋊C4×C79—22no—
109C4⋊C4×C79—41no—
110C4⋊C4×C79—41no—
111C42×C79C2×D411yescharacteristic
112C42×C79C2×D411yescharacteristic
113C42×C79C2×D411yescharacteristic
114C42×C79#4111yescharacteristic
115C23×C79C4×D411yescharacteristic
116Q8×C79—81no—
117Q8×C79—81no—
118Q8×C79—81no—
119Q8×C79—81no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—41no—
123C2×C4×C79—42no—
124C2×C4×C79—42no—
125C2×C4×C79—42no—
126C2×C4×C79C2×C22⋊C411yescharacteristic
127C2×C4×C79C4×D411yescharacteristic
128C2×C4×C79#12011yescharacteristic
129C2×C4×C79#12111yescharacteristic
130C2×C4×C79#12111yescharacteristic
131C2×C4×C79#12311yescharacteristic
132C2×C4×C79C2×SD1611yescharacteristic
133C2×C4×C79#13511yescharacteristic
134C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C79#13711yescharacteristic
136C2×Dic79—22no—
137C2×Dic79—22no—
138Dic158—41no—
139Dic158—41no—
140Dic158—42no—
141C8×C79—41no—
142C8×C79—41no—
143C8×C79—81no—
144C79⋊C8—22no—
145C79⋊C8—42no—
146C79⋊C8—41no—
147C79⋊C8—81no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79#38611yescharacteristic
151C22×C79C4×SD1611yescharacteristic
152C22×C79#44011yescharacteristic
153C4×C79—21no—
154C4×C79—21no—
155C4×C79—21no—
156C4×C79—82no—
157C4×C79#44811yescharacteristic
158C4×C79#44911yescharacteristic
159C4×C79#46011yescharacteristic
160C4×C79#46411yescharacteristic
161C4×C79#47111yescharacteristic
162C4×C79#47311yescharacteristic
163Dic79—42no—
164Dic79—42no—
165#9272—791nomaximal, Sylow
166C2×C79—21no—
167C2×C79—21no—
168C2×C79#188011yescharacteristic
169C2×C79#190811yescharacteristic
170C2×C79#191511yescharacteristic
171#3229—791no—
172#2962—791no—
173#1966—791no—
174#1550—791no—
175C2×#332D7911yescharacteristic
176#1301—791no—
177#1283—791no—
178C79#927211yescharacteristic, complemented, Sylow
179#533—791no—
180C2×C4⋊Q8C2×D7911yescharacteristic
181#498—1581no—
182#487—1581no—
183#483—794no—
184#475—1582no—
185#446—1582no—
186#446—1581no—
187C8×Q8—1581no—
188#431—791no—
189#424—1581no—
190#423—791no—
191C2×C4⋊C8C2×D7911yescharacteristic
192C2×C4⋊C8C2×D7911yescharacteristic
193C2×#101—1581no—
194C2×#101—1581no—
195#405—1581no—
196C4×M16—791no—
197#333—1581no—
198#332—21no—
199#332—21no—
200#332—1581no—
201C22×Q8—21no—
202C2×M16—791no—
203C2×M16—791no—
204C2×M16—1581no—
205C22×C8—21no—
206C22×C8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—791no—
212C4⋊Q8—791no—
213#125—1581no—
214C22⋊Q8—792no—
215C22⋊Q8—1581no—
216C4×Q8—792no—
217C4×Q8—1581no—
218#117—1581no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C4×D7912yes—
221C2×C42C22×D7911yescharacteristic
222#106—1584no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—1582no—
227C4⋊C8—1582no—
228C4⋊C8—1581no—
229C4⋊C8—1581no—
230C4⋊C8—1581no—
231C4⋊C8C22⋊D7911yescharacteristic
232C4⋊C8C22⋊D7911yescharacteristic
233#103—1584no—
234#101—3162no—
235#101—3162no—
236C22⋊C8—792no—
237#97—1582no—
238#97—1581no—
239C4×C8—1582no—
240C4×C8—1581no—
241C4×C8—1581no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—792no—
249C2×Q8—1581no—
250C22×C4—22no—
251C22×C4C2×C4×D7911yescharacteristic
252C22×C4D4×D7911yescharacteristic
253C22×C4#1192962311yescharacteristic
254M16—3161no—
255M16—3161no—
256M16—792no—
257M16—1581no—
258M16—1581no—
259M16—1581no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—81no—
266C2×C8—3161no—
267C2×C8—1582no—
268C2×C8—1581no—
269C2×C8—1581no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—792no—
275C4⋊C4—1582no—
276C4⋊C4—1581no—
277C4⋊C4—1581no—
278C4⋊C4—1581no—
279C4⋊C4—1582no—
280C22⋊C4—1582no—
281C42—1582no—
282C42D4×D7911yescharacteristic
283C42D4×D7911yescharacteristic
284C42C2×C22⋊D7911yescharacteristic
285C42#1192962411yescharacteristic
286C23(C22⋊D79)⋊C411yescharacteristic
287Q8—81no—
288Q8—81no—
289Q8—81no—
290Q8—81no—
291Q8—3162no—
292Q8—3161no—
293Q8—3161no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—1582no—
301C2×C4—1582no—
302C2×C4C22⋊C4×D7911yescharacteristic
303C2×C4C4×C22⋊D7911yescharacteristic
304C2×C4#1457912911yescharacteristic
305C2×C4#1457913211yescharacteristic
306C2×C4#1457913411yescharacteristic
307C2×C4#1457914311yescharacteristic
308C2×C4SD16×D7911yescharacteristic
309C2×C4#1457918011yescharacteristic
310C2×C4#1457918511yescharacteristic
311C2×C4#1457919011yescharacteristic
312C8—41no—
313C8—41no—
314C8—81no—
315C8—3162no—
316C8—3161no—
317C8—6321no—
318C8—1582no—
319C22—21no—
320C22—21no—
321C22#1909843911yescharacteristic
322C22#1909865411yescharacteristic
323C22#1909866011yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—82no—
328C4—3162no—
329C4—3162no—
330C4#1909870011yescharacteristic
331C4#1909870511yescharacteristic
332C4#1909876211yescharacteristic
333C4#1909878411yescharacteristic
334C4#1909882311yescharacteristic
335C4#1909883511yescharacteristic
336C2—21no—
337C2—21no—
338C2#2376375211yescharacteristic
339C2#2376391811yescharacteristic
340C2#2376394911yescharacteristic
subgroupG/H
centerC22#19098439
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC2×#332×C79C2
socleC22×C79#386
prime radicalC23×C79C4×D4
cosocleC2×C42×C79C23
hypercenterC2×#332D79
hypocenterC79#9272
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
178. C79#9272semidirect

Maximal subgroups

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

subgroupindex
#237745442
#237723432
#237642452
#237620882
#237612972
#237612502
C2×#332×C792
#927279