Finite Group Databasebeta

← #34187376

Subgroups — #34187376

Order 20224.

486 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771777C211yesmaximal, characteristic
2#23763551C211yesmaximal, characteristic
3#23763255C211yesmaximal, characteristic
4C2×#19098310C211yesmaximal, characteristic
5C2×#19098309C211yesmaximal, characteristic
6C2×#19098308C211yesmaximal, characteristic
7C2×#332×C79C211yesmaximal, characteristic
8C2×#14579161C2211yescharacteristic
9C2×C4⋊(C79⋊C8)C2211yescharacteristic
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×Dic79⋊C8—22no—
12C2×#14579071—21no—
13#19098470—21no—
14#19098470—21no—
15#19098468—22no—
16#19098428C414yes—
17C42×Dic79C2211yescharacteristic
18#19098317—21no—
19#19098310—22no—
20#19098309—22no—
21#19098308—22no—
22C2×C4⋊Q8×C79C2211yescharacteristic
23C2×C4⋊C8×C79C2211yescharacteristic
24C2×C4⋊C8×C79C2211yescharacteristic
25#332×C79—22no—
26C22×Dic158—21no—
27C22×C79⋊C8—21no—
28C22×C79⋊C8—21no—
29#14579161—21no—
30#14579161—21no—
31#14579161—21no—
32#14579161—21no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79C2×C412yes—
35C2×C4×Dic79C2×C412yes—
36C2×C4×Dic79D412yes—
37C2×C4×Dic79D411yescharacteristic
38C2×C4×Dic79D411yescharacteristic
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)D412yes—
43Dic79⋊C8—44no—
44#14579071—42no—
45C79⋊#95—24no—
46C79⋊#95—24no—
47C22×Q8×C79—21no—
48C22×C8×C79—21no—
49C22×C8×C79—21no—
50C4⋊Q8×C79—21no—
51C4⋊Q8×C79—21no—
52C4⋊Q8×C79—21no—
53C4⋊Q8×C79—21no—
54C2×C4⋊C4×C79—21no—
55C2×C4⋊C4×C79C2×C412yes—
56C2×C42×C79C2311yescharacteristic
57C4⋊C8×C79—21no—
58C4⋊C8×C79—21no—
59C4⋊C8×C79—21no—
60C4⋊C8×C79D412yes—
61C2×Dic158—41no—
62C2×Dic158—41no—
63C2×Dic158—42no—
64C2×Dic158—42no—
65C22×Dic79—22no—
66C22×Dic79C22⋊C412yes—
67C22×Dic79C22⋊C412yes—
68C2×C79⋊C8—22no—
69C2×C79⋊C8—41no—
70C2×C79⋊C8—41no—
71C2×C79⋊C8—41no—
72C2×C79⋊C8—42no—
73C4⋊Dic79—22no—
74C4⋊Dic79—22no—
75C4⋊Dic79—41no—
76C4⋊Dic79—41no—
77C4×Dic79—24no—
78C4×Dic79—21no—
79C4×Dic79—21no—
80C4×Dic79—21no—
81C4×Dic79—22no—
82C4×Dic79—21no—
83C4×Dic79—22no—
84C2×Q8×C79—41no—
85C2×Q8×C79—41no—
86C2×Q8×C79—41no—
87C2×Q8×C79—41no—
88C2×Q8×C79—41no—
89C2×Q8×C79—41no—
90C22×C4×C79—22no—
91C22×C4×C79C22×C411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94C2×C8×C79—22no—
95C2×C8×C79—41no—
96C2×C8×C79—41no—
97C2×C8×C79—41no—
98C2×C8×C79—42no—
99C4⋊C4×C79—22no—
100C4⋊C4×C79—22no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—41no—
103C42×C79C2×D411yescharacteristic
104C42×C79C2×D411yescharacteristic
105C42×C79#4111yescharacteristic
106C42×C79#4111yescharacteristic
107C23×C79C2×C22⋊C411yescharacteristic
108Q8×C79—81no—
109Q8×C79—81no—
110Q8×C79—81no—
111Q8×C79—81no—
112C2×C4×C79—22no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79C4×D411yescharacteristic
117C2×C4×C79C4×D411yescharacteristic
118C2×C4×C79#12111yescharacteristic
119C2×C4×C79#12111yescharacteristic
120C2×C4×C79#12411yescharacteristic
121C2×C4×C79#12711yescharacteristic
122C2×C4×C79#13511yescharacteristic
123C2×C4×C79#13511yescharacteristic
124C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C79AGL(1,ℤ/8)12yes—
126C2×C4×C79#13711yescharacteristic
127C2×C4×C79#13712yes—
128C2×Dic79—22no—
129C2×Dic79—22no—
130C2×Dic79—22no—
131C2×Dic79—22no—
132C2×Dic79—22no—
133C2×Dic79—42no—
134C2×Dic79—44no—
135C2×Dic79#10114yes—
136Dic158—82no—
137Dic158—81no—
138Dic158—81no—
139C8×C79—42no—
140C8×C79—81no—
141C79⋊C8—42no—
142C79⋊C8—81no—
143C22×C79#39011yescharacteristic
144C22×C79C2×#10112yes—
145C22×C79#41611yescharacteristic
146C22×C79#41711yescharacteristic
147C22×C79#41712yes—
148C4×C79—21no—
149C4×C79—21no—
150C4×C79—82no—
151C4×C79#46311yescharacteristic
152C4×C79#46311yescharacteristic
153C4×C79#46411yescharacteristic
154C4×C79#46411yescharacteristic
155C4×C79#48912yes—
156C4×C79#49611yescharacteristic
157C4×C79#49711yescharacteristic
158Dic79—24no—
159Dic79—42no—
160Dic79—82no—
161#9708—791nomaximal, Sylow
162C2×C79#155212yes—
163C2×C79#156112yes—
164C2×C79#195911yescharacteristic
165C2×C79#196611yescharacteristic
166C2×C79#198311yescharacteristic
167#2862—791no—
168#1842—791no—
169#1767—791no—
170C2×#332—791no—
171C2×#332—791no—
172C2×#332—791no—
173C2×#332D7911yescharacteristic
174C79#970811yescharacteristic, complemented, Sylow
175C2×C4⋊Q8—791no—
176C2×C4⋊Q8C2×D7911yescharacteristic
177C2×C4⋊C8—791no—
178C2×C4⋊C8—791no—
179C2×C4⋊C8—1582no—
180C2×C4⋊C8C2×D7911yescharacteristic
181C2×C4⋊C8C2×D7911yescharacteristic
182C2×#97—1581no—
183#391—1582no—
184#391—1581no—
185#391—1581no—
186#382—794no—
187C43—791no—
188#336—1581no—
189#332—22no—
190#332—1582no—
191#332—1582no—
192#332—1582no—
193C22×Q8—21no—
194C22×Q8—1581no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—1581no—
198C22×C8—1581no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—1581no—
204C4⋊Q8—1581no—
205C4⋊Q8—1581no—
206C4⋊Q8—1581no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—792no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4C4×D7912yes—
211C2×C42—792no—
212C2×C42—791no—
213C2×C42—791no—
214C2×C42—792no—
215C2×C42C22×D7911yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—3164no—
220C4⋊C8—792no—
221C4⋊C8—1581no—
222C4⋊C8—1581no—
223C4⋊C8—1581no—
224C4⋊C8C22⋊D7912yes—
225#97—3162no—
226#95—1584no—
227#95—1584no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—3161no—
235C2×Q8—3161no—
236C2×Q8—3162no—
237C2×Q8—3162no—
238C22×C4—22no—
239C22×C4—792no—
240C22×C4—792no—
241C22×C4—1582no—
242C22×C4C2×C4×D7911yescharacteristic
243C22×C4D4×D7911yescharacteristic
244C22×C4D4×D7911yescharacteristic
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—3161no—
251C2×C8—3161no—
252C2×C8—3161no—
253C2×C8—3162no—
254C2×C8—1582no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—3161no—
260C4⋊C4—3161no—
261C4⋊C4—1582no—
262C4⋊C4—1582no—
263C42—1584no—
264C42—1581no—
265C42—1581no—
266C42—1581no—
267C42—1582no—
268C42—1581no—
269C42—1582no—
270C42D4×D7911yescharacteristic
271C42C2×C22⋊D7911yescharacteristic
272C42#1192962311yescharacteristic
273C42#1192962411yescharacteristic
274C23C22⋊C4×D7911yescharacteristic
275Q8—81no—
276Q8—81no—
277Q8—81no—
278Q8—81no—
279Q8—6322no—
280Q8—6321no—
281Q8—6321no—
282C2×C4—22no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—3162no—
287C2×C4—3164no—
288C2×C4—794no—
289C2×C4—1582no—
290C2×C4—1582no—
291C2×C4—1582no—
292C2×C4—1582no—
293C2×C4—1582no—
294C2×C4(C22⋊D79)⋊C411yescharacteristic
295C2×C4C4×C22⋊D7911yescharacteristic
296C2×C4#1457913311yescharacteristic
297C2×C4#1457913411yescharacteristic
298C2×C4#1457914711yescharacteristic
299C2×C4#1457915711yescharacteristic
300C2×C4#1457918011yescharacteristic
301C2×C4#1457918211yescharacteristic
302C2×C4#1457918511yescharacteristic
303C2×C4#1457918812yes—
304C2×C4#1457919011yescharacteristic
305C2×C4#1457919312yes—
306C8—42no—
307C8—81no—
308C8—3162no—
309C8—6321no—
310C22#1909846311yescharacteristic
311C22#101×D7912yes—
312C22#1909855611yescharacteristic
313C22#1909856111yescharacteristic
314C22#1909856312yes—
315C4—21no—
316C4—21no—
317C4—82no—
318C4—3162no—
319C4—6322no—
320C4—1584no—
321C4#1909877611yescharacteristic
322C4#1909877711yescharacteristic
323C4#1909878311yescharacteristic
324C4#1909878411yescharacteristic
325C4#1909892412yes—
326C4#1909894911yescharacteristic
327C4#1909895611yescharacteristic
328C2#2376209712yes—
329C2#2376213612yes—
330C2#2376421311yescharacteristic
331C2#2376424211yescharacteristic
332C2#2376433111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D79
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC2×#332×C79C2
socle, prime radicalC23×C79C2×C22⋊C4
cosocleC2×C42×C79C23
hypercenterC2×#332D79
hypocenterC79#9708
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
174. C79#9708semidirect

Maximal subgroups

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

subgroupindex
#237717772
#237635512
#237632552
C2×#190983102
C2×#190983092
C2×#190983082
C2×#332×C792
#970879