Finite Group Databasebeta

← #34192884

Subgroups — #34192884

Order 20224.

370 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764362C211yesmaximal, characteristic
2#23764163C211yesmaximal, characteristic
3#23764119C211yesmaximal, characteristic
4C2×#19098315C211yesmaximal, characteristic
5C2×#19098312C211yesmaximal, characteristic
6#23761284C211yesmaximal, characteristic
7C79×#1987C211yesmaximal, characteristic
8C2×#14579092—21no—
9C2×C4⋊(C79⋊C8)—21no—
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×C4⋊(C79⋊C8)C2211yescharacteristic
12C2×C4×C79⋊C8C2211yescharacteristic
13#19098433C2211yescharacteristic
14#19098325—21no—
15#19098325—21no—
16#19098325—21no—
17#19098318—21no—
18#19098315—22no—
19#19098312—22no—
20C2×C4⋊Q8×C79C2211yescharacteristic
21C2×C4⋊C8×C79C2211yescharacteristic
22C4×C4⋊C4×C79C2211yescharacteristic
23#340×C79—21no—
24#340×C79—21no—
25C22×C79⋊C8—21no—
26C22×C79⋊C8—21no—
27C22×C79⋊C8D411yescharacteristic
28C22×C79⋊C8D411yescharacteristic
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79—21no—
32#14579092—42no—
33C4⋊(C79⋊C8)—21no—
34C4⋊(C79⋊C8)—21no—
35C4⋊(C79⋊C8)—21no—
36C4⋊(C79⋊C8)—42no—
37C4⋊(C79⋊C8)D412yes—
38C4×C79⋊C8—21no—
39C4×C79⋊C8—21no—
40C22×Q8×C79—21no—
41C22×C8×C79—21no—
42C4⋊Q8×C79—21no—
43C4⋊Q8×C79—21no—
44C4⋊Q8×C79—21no—
45C4⋊Q8×C79Q812yes—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C42×C79—21no—
51C2×C42×C79C2311yescharacteristic
52C4⋊C8×C79—21no—
53C4⋊C8×C79—21no—
54#95×C79—21no—
55C22×Dic79—41no—
56C2×C79⋊C8—21no—
57C2×C79⋊C8—21no—
58C2×C79⋊C8—22no—
59C2×C79⋊C8—21no—
60C2×C79⋊C8—21no—
61C2×C79⋊C8—21no—
62C2×C79⋊C8—41no—
63C2×C79⋊C8—41no—
64C2×C79⋊C8—41no—
65C2×C79⋊C8—41no—
66C2×C79⋊C8—41no—
67C2×C79⋊C8SD1612yes—
68C4⋊Dic79—41no—
69C4⋊Dic79—41no—
70C4⋊Dic79—41no—
71C4⋊Dic79—41no—
72C4⋊Dic79—81no—
73C2×Q8×C79—22no—
74C2×Q8×C79—41no—
75C2×Q8×C79—81no—
76C2×Q8×C79—81no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—41no—
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79#4111yescharacteristic
83C2×C8×C79—41no—
84C2×C8×C79—41no—
85C2×C8×C79—41no—
86C4⋊C4×C79—21no—
87C4⋊C4×C79—22no—
88C4⋊C4×C79—22no—
89C4⋊C4×C79—21no—
90C4⋊C4×C79—21no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—41no—
93C4⋊C4×C79Q1612yes—
94C42×C79—41no—
95C42×C79—41no—
96C42×C79C2×D411yescharacteristic
97C42×C79C2×Q811yescharacteristic
98C42×C79#4111yescharacteristic
99C42×C79#4111yescharacteristic
100C23×C79#12111yescharacteristic
101Q8×C79—82no—
102Q8×C79—82no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—22no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—42no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—81no—
113C2×C4×C79—81no—
114C2×C4×C79#12111yescharacteristic
115C2×C4×C79#12111yescharacteristic
116C2×C4×C79C22⋊Q811yescharacteristic
117C2×C4×C79C22⋊Q811yescharacteristic
118C2×C4×C79C22⋊Q811yescharacteristic
119C2×C4×C79#12611yescharacteristic
120C2×C4×C79C2×SD1611yescharacteristic
121C2×C4×C79C2×Q1611yescharacteristic
122C2×C4×C79#13511yescharacteristic
123C2×C4×C79#13511yescharacteristic
124C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C79#13711yescharacteristic
126C2×C4×C79#13711yescharacteristic
127C2×C4×C79#13711yescharacteristic
128C2×Dic79—81no—
129C2×Dic79—81no—
130C2×Dic79—81no—
131C8×C79—81no—
132C79⋊C8—22no—
133C79⋊C8—42no—
134C79⋊C8—41no—
135C79⋊C8—81no—
136C22×C79#39611yescharacteristic
137C22×C79#46111yescharacteristic
138C22×C79#46411yescharacteristic
139C22×C79#46711yescharacteristic
140C22×C79#46811yescharacteristic
141C22×C79#47111yescharacteristic
142C22×C79#47211yescharacteristic
143C4×C79—21no—
144C4×C79—21no—
145C4×C79—42no—
146C4×C79—82no—
147C4×C79—81no—
148C4×C79#46211yescharacteristic
149C4×C79#46311yescharacteristic
150C4×C79#46511yescharacteristic
151C4×C79#47011yescharacteristic
152C4×C79#47511yescharacteristic
153C4×C79#47711yescharacteristic
154C4×C79#47911yescharacteristic
155C4×C79#47911yescharacteristic
156Dic79—161no—
157#10601—791nomaximal, Sylow
158C2×C79#167511yescharacteristic
159C2×C79#168411yescharacteristic
160C2×C79#169111yescharacteristic
161C2×C79#169811yescharacteristic
162C2×C79#206311yescharacteristic
163C2×C79#206611yescharacteristic
164C2×C79#206811yescharacteristic
165#1990—791no—
166#1987D7911yescharacteristic
167#1948—791no—
168#1939—791no—
169C2×#335—791no—
170C2×#333—791no—
171#1296—791no—
172C79#1060111yescharacteristic, complemented, Sylow
173C2×C4⋊Q8C2×D7911yescharacteristic
174C2×#106—1581no—
175C2×C4⋊C8—791no—
176C2×C4⋊C8—791no—
177C2×C4⋊C8—1581no—
178C2×C4⋊C8C2×D7911yescharacteristic
179C2×C4×C8—791no—
180#384—791no—
181C4×C4⋊C4C2×D7911yescharacteristic
182#340—21no—
183#340—21no—
184#340—1581no—
185#340—1581no—
186#340—1581no—
187#336—1581no—
188#335—1582no—
189#333—1582no—
190C22×Q8—21no—
191C22×C8—21no—
192C22×C8—791no—
193C22×C8—791no—
194C22×C8—1581no—
195C22×C8—1581no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8Dic15812yes—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—1581no—
203C2×C4⋊C4—1581no—
204C2×C4⋊C4—1581no—
205C2×C4⋊C4C22⋊D7911yescharacteristic
206C2×C4⋊C4C22⋊D7911yescharacteristic
207C2×C42—21no—
208C2×C42C22×D7911yescharacteristic
209#106—3162no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—3162no—
213C4⋊C8—792no—
214C4⋊C8—1581no—
215C4⋊C8—1581no—
216C4⋊C8—1581no—
217C4×C8—1581no—
218C4×C8—1581no—
219#95—21no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—81no—
223C2×Q8—81no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—3161no—
228C22×C4D4×D7911yescharacteristic
229C22×C4C2×C22⋊D7911yescharacteristic
230C22×C4#1192962311yescharacteristic
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—3161no—
235C2×C8—3161no—
236C2×C8—3161no—
237C2×C8—3161no—
238C2×C8—3161no—
239C2×C8—792no—
240C2×C8—1582no—
241C2×C8—1581no—
242C2×C8—1581no—
243C2×C8—1581no—
244C2×C8—1581no—
245C2×C8—1581no—
246C4⋊C4—21no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—3161no—
254C4⋊C4—3161no—
255C4⋊C4—3161no—
256C4⋊C4—3161no—
257C4⋊C4—6321no—
258C4⋊C4#1192961412yes—
259C42—41no—
260C42—41no—
261C42D4×D7911yescharacteristic
262C42C2×Dic15811yescharacteristic
263C42#1192962311yescharacteristic
264C42#1192962411yescharacteristic
265C23#1457913311yescharacteristic
266Q8—82no—
267Q8—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—6321no—
280C2×C4—6321no—
281C2×C4—6321no—
282C2×C4#1457913211yescharacteristic
283C2×C4#1457913411yescharacteristic
284C2×C4#1457913811yescharacteristic
285C2×C4#1457913811yescharacteristic
286C2×C4#1457914011yescharacteristic
287C2×C4#1457915411yescharacteristic
288C2×C4SD16×D7911yescharacteristic
289C2×C4C2×#1192961411yescharacteristic
290C2×C4#1457918111yescharacteristic
291C2×C4#1457918411yescharacteristic
292C2×C4#1457918611yescharacteristic
293C2×C4#1457919011yescharacteristic
294C2×C4#1457919211yescharacteristic
295C2×C4#1457919311yescharacteristic
296C8—81no—
297C8—3162no—
298C8—3161no—
299C8—6321no—
300C8—1582no—
301C22#1909848611yescharacteristic
302C22#1909876611yescharacteristic
303C22#1909878711yescharacteristic
304C22#1909880211yescharacteristic
305C22#1909881111yescharacteristic
306C22#1909882411yescharacteristic
307C22#1909883111yescharacteristic
308C4—21no—
309C4—21no—
310C4—42no—
311C4—82no—
312C4—81no—
313C4—12641no—
314C4#1909877011yescharacteristic
315C4#1909877511yescharacteristic
316C4#1909879511yescharacteristic
317C4#1909882011yescharacteristic
318C4#1909884611yescharacteristic
319C4#1909885811yescharacteristic
320C4#1909887011yescharacteristic
321C4#1909887211yescharacteristic
322C2#2376279711yescharacteristic
323C2#2376286011yescharacteristic
324C2#2376290111yescharacteristic
325C2#2376294211yescharacteristic
326C2#2376476011yescharacteristic
327C2#2376477711yescharacteristic
328C2#2376479511yescharacteristic
subgroupG/H
centerC23#14579133
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1987C2
socle, prime radicalC23×C79#121
hypercenter#1987D79
hypocenterC79#10601
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
172. C79#10601semidirect

Maximal subgroups

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

subgroupindex
#237643622
#237641632
#237641192
C2×#190983152
C2×#190983122
#237612842
C79×#19872
#1060179