Finite Group Databasebeta

← #34186813

Subgroups — #34186813

Order 20224.

382 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764118C211yesmaximal, characteristic
2C4×#14579094C211yesmaximal, characteristic
3C2×#19098313C211yesmaximal, characteristic
4C2×#19098285C211yesmaximal, characteristic
5#23761241C211yesmaximal, characteristic
6#23761239C211yesmaximal, characteristic
7C79×#1939C211yesmaximal, characteristic
8C2×#14579094C411yescharacteristic
9C2×#14579094C411yescharacteristic
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×Dic79⋊C8—21no—
12C2×C4×C79⋊C8C2211yescharacteristic
13C2×C4×C79⋊C8C2211yescharacteristic
14C4×C4⋊Dic79C2211yescharacteristic
15#19098325C411yescharacteristic
16#19098325C411yescharacteristic
17#19098317—21no—
18#19098313—22no—
19#19098285—22no—
20C2×C4⋊C8×C79—21no—
21C2×C4⋊C8×C79C2211yescharacteristic
22C2×C4×C8×C79C2211yescharacteristic
23C4×C4⋊C4×C79C2211yescharacteristic
24#336×C79—21no—
25C22×C79⋊C8—21no—
26C22×C79⋊C8—21no—
27C22×C79⋊C8C2×C411yescharacteristic
28C22×C79⋊C8C2×C411yescharacteristic
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×C4⋊Dic79C2×C411yescharacteristic
31C2×C4×Dic79—21no—
32#14579094—22no—
33#14579094C814yescomplemented
34C4⋊(C79⋊C8)—21no—
35C4⋊(C79⋊C8)—21no—
36Dic79⋊C8—42no—
37C4×C79⋊C8—21no—
38C4×C79⋊C8D412yes—
39C4×C79⋊C8Q812yes—
40C4×C79⋊C8Q812yes—
41C79⋊#95—21no—
42C22×C8×C79—21no—
43C22×C8×C79D411yescharacteristic
44C22×C8×C79D411yescharacteristic
45C2×C4⋊C4×C79C2×C411yescharacteristic
46C2×C4⋊C4×C79C2×C411yescharacteristic
47C2×C42×C79—21no—
48C2×C42×C79C2311yescharacteristic
49C4⋊C8×C79—21no—
50C4⋊C8×C79—21no—
51C4⋊C8×C79—42no—
52C4×C8×C79—21no—
53C4×C8×C79—21no—
54#95×C79—21no—
55C22×Dic79—21no—
56C22×Dic79—21no—
57C2×C79⋊C8—22no—
58C2×C79⋊C8—21no—
59C2×C79⋊C8—21no—
60C2×C79⋊C8—41no—
61C2×C79⋊C8—41no—
62C2×C79⋊C8—41no—
63C2×C79⋊C8—41no—
64C2×C79⋊C8—41no—
65C2×C79⋊C8C4⋊C412yes—
66C2×C79⋊C8C4⋊C412yes—
67C2×C79⋊C8C2×C812yes—
68C2×C79⋊C8M1612yes—
69C4⋊Dic79—21no—
70C4⋊Dic79—21no—
71C4⋊Dic79—21no—
72C4⋊Dic79C2×C812yes—
73C4×Dic79—41no—
74C4×Dic79—41no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79C22×C411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79#4111yescharacteristic
80C2×C8×C79—21no—
81C2×C8×C79—21no—
82C2×C8×C79—21no—
83C2×C8×C79—21no—
84C2×C8×C79—21no—
85C2×C8×C79—21no—
86C2×C8×C79—41no—
87C2×C8×C79—41no—
88C2×C8×C79—41no—
89C4⋊C4×C79—21no—
90C4⋊C4×C79—21no—
91C4⋊C4×C79—21no—
92C4⋊C4×C79C2×C812yes—
93C42×C79—41no—
94C42×C79—41no—
95C42×C79C2×D411yescharacteristic
96C42×C79C2×Q811yescharacteristic
97C42×C79C2×Q811yescharacteristic
98C42×C79#4111yescharacteristic
99C23×C79C4×D411yescharacteristic
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—22no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79C2×C4⋊C411yescharacteristic
109C2×C4×C79C4×Q811yescharacteristic
110C2×C4×C79#12111yescharacteristic
111C2×C4×C79C22⋊Q811yescharacteristic
112C2×C4×C79#12511yescharacteristic
113C2×C4×C79C4⋊Q811yescharacteristic
114C2×C4×C79C22×C811yescharacteristic
115C2×C4×C79C2×M1611yescharacteristic
116C2×C4×C79#13111yescharacteristic
117C2×C4×C79#13111yescharacteristic
118C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C79#13711yescharacteristic
121C2×C4×C79#13711yescharacteristic
122C2×Dic79—22no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C8×C79—41no—
129C8×C79—41no—
130C8×C79—81no—
131C79⋊C8—22no—
132C79⋊C8—42no—
133C79⋊C8—81no—
134C79⋊C8C4⋊C814yescomplemented
135C22×C79#38911yescharacteristic
136C22×C79C8×D411yescharacteristic
137C22×C79#43611yescharacteristic
138C22×C79#44111yescharacteristic
139C22×C79#44211yescharacteristic
140C22×C79#44411yescharacteristic
141C22×C79#44411yescharacteristic
142C4×C79—21no—
143C4×C79—21no—
144C4×C79—42no—
145C4×C79—81no—
146C4×C79C2×C4⋊C811yescharacteristic
147C4×C79#42411yescharacteristic
148C4×C79C8×Q811yescharacteristic
149C4×C79#44611yescharacteristic
150C4×C79#46811yescharacteristic
151C4×C79#46811yescharacteristic
152C4×C79#50111yescharacteristic
153C4×C79#50111yescharacteristic
154Dic79—42no—
155Dic79—81no—
156#9595—791nomaximal, Sylow
157C2×C79#157911yescharacteristic
158C2×C79#158011yescharacteristic
159C2×C79#159311yescharacteristic
160C2×C79#159411yescharacteristic
161C2×C79#193911yescharacteristic
162C2×C79#194311yescharacteristic
163C2×C79#194611yescharacteristic
164#1939—791no—
165#1939D7911yescharacteristic
166C4×#107—791no—
167C2×#334—791no—
168C2×#322—791no—
169#1279—791no—
170#1279—791no—
171C79#959511yescharacteristic, complemented, Sylow
172C2×#107—791no—
173C2×#107—791no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8—791no—
176C2×C4⋊C8—1581no—
177C2×C4⋊C8C2×D7911yescharacteristic
178C2×C4×C8—791no—
179C2×C4×C8—791no—
180C2×C4×C8C2×D7911yescharacteristic
181C4×C4⋊C4—791no—
182C4×C4⋊C4C2×D7911yescharacteristic
183#340—791no—
184#340—791no—
185#336—21no—
186#336—1581no—
187#334—1582no—
188#322—1582no—
189C22×C8—21no—
190C22×C8—791no—
191C22×C8—791no—
192C22×C8—1581no—
193C22×C8—1581no—
194C22×C8C22⋊D7911yescharacteristic
195C22×C8C22⋊D7911yescharacteristic
196C2×C4⋊C4—791no—
197C2×C4⋊C4—791no—
198C2×C4⋊C4C4×D7911yescharacteristic
199C2×C4⋊C4C4×D7911yescharacteristic
200C2×C42—21no—
201C2×C42—1581no—
202C2×C42C22×D7911yescharacteristic
203#107—794no—
204#107—1582no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—42no—
208C4⋊C8—3162no—
209C4⋊C8—1581no—
210C4⋊C8—1581no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—792no—
214C4×C8—792no—
215C4×C8—792no—
216C4×C8—1581no—
217#95—21no—
218#95—1581no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—1581no—
222C22×C4—1581no—
223C22×C4C2×C4×D7911yescharacteristic
224C22×C4C2×C22⋊D7911yescharacteristic
225C22×C4#1192962411yescharacteristic
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—3161no—
236C2×C8—3161no—
237C2×C8—3161no—
238C2×C8—3161no—
239C2×C8—3161no—
240C2×C8—792no—
241C2×C8—792no—
242C2×C8—792no—
243C2×C8—792no—
244C2×C8—1582no—
245C2×C8—1581no—
246C2×C8—1581no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—792no—
251C4⋊C4—1581no—
252C4⋊C4—1581no—
253C4⋊C4—1581no—
254C4⋊C4C8×D7912yes—
255C42—41no—
256C42—41no—
257C42—3161no—
258C42—3161no—
259C42D4×D7911yescharacteristic
260C42Q8×D7911yescharacteristic
261C42Q8×D7911yescharacteristic
262C42#1192962311yescharacteristic
263C23C4×C22⋊D7911yescharacteristic
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—3161no—
273C2×C4—3161no—
274C2×C4—3161no—
275C2×C4—3161no—
276C2×C4—3161no—
277C2×C4—1582no—
278C2×C4C4⋊C4×D7911yescharacteristic
279C2×C4#1457912611yescharacteristic
280C2×C4#1457913311yescharacteristic
281C2×C4#1457913911yescharacteristic
282C2×C4#1457915111yescharacteristic
283C2×C4#1457915911yescharacteristic
284C2×C4C2×C8×D7911yescharacteristic
285C2×C4M16×D7911yescharacteristic
286C2×C4#1457916711yescharacteristic
287C2×C4#1457916911yescharacteristic
288C2×C4#1457918611yescharacteristic
289C2×C4#1457918711yescharacteristic
290C2×C4#1457919111yescharacteristic
291C2×C4#1457919211yescharacteristic
292C8—41no—
293C8—41no—
294C8—81no—
295C8—3162no—
296C8—794no—
297C8—6321no—
298C8—1582no—
299C22#1909845811yescharacteristic
300C22C8×C22⋊D7911yescharacteristic
301C22#1909863911yescharacteristic
302C22#1909866811yescharacteristic
303C22#1909867311yescharacteristic
304C22#1909868311yescharacteristic
305C22#1909868311yescharacteristic
306C4—21no—
307C4—21no—
308C4—42no—
309C4—81no—
310C4—3162no—
311C4—6321no—
312C4C4⋊C8×D7911yescharacteristic
313C4#1909858811yescharacteristic
314C4#1909868711yescharacteristic
315C4#1909869211yescharacteristic
316C4#1909880711yescharacteristic
317C4#1909881111yescharacteristic
318C4#1909896911yescharacteristic
319C4#1909897311yescharacteristic
320C2#2376221711yescharacteristic
321C2#2376222211yescharacteristic
322C2#2376230711yescharacteristic
323C2#2376231211yescharacteristic
324C2#2376411511yescharacteristic
325C2#2376413511yescharacteristic
326C2#2376415011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D79
commutatorC2×C4×C79C22×C8
FrattiniC2×C42C22×D79
FittingC79×#1939C2
socle, prime radicalC23×C79C4×D4
cosocleC2×C42×C79C23
hypercenter#1939D79
hypocenterC79#9595
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1939

Complements

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

NH ≅ G/Ntype
33. #14579094C8semidirect
134. C79⋊C8C4⋊C8semidirect
171. C79#9595semidirect

Maximal subgroups

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

subgroupindex
#237641182
C4×#145790942
C2×#190983132
C2×#190982852
#237612412
#237612392
C79×#19392
#959579