Finite Group Databasebeta

← #34182557

Subgroups — #34182557

Order 20224.

374 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#23772525C211yesmaximal, characteristic
2#23763538C211yesmaximal, characteristic, complemented
3#23762189C211yesmaximal, characteristic, complemented
4#23761839C211yesmaximal, characteristic
5#23761291C211yesmaximal, characteristic, complemented
6#23761237C211yesmaximal, characteristic, complemented
7C79×#1476C211yesmaximal, characteristic
8#19098694—21no—
9#19098635—22no—
10D4×C79⋊C8—21no—
11#19098620—21no—
12#19098616C2211yescharacteristic
13#19098587C2211yescharacteristic
14#19098587C2211yescharacteristic
15C2×C4⋊(C79⋊C8)—21no—
16C4×#11929608C2211yescharacteristic
17#19098328—22no—
18#19098316—21no—
19#19098314—21no—
20Q8⋊(C79⋊C8)—21no—
21D4⋊(C79⋊C8)—21no—
22C4×#41×C79C2211yescharacteristic
23#431×C79C2211yescharacteristic
24C2×C4×C8×C79C2211yescharacteristic
25Q8⋊C8×C79C411yescharacteristic
26Q8⋊C8×C79C411yescharacteristic
27D4⋊C8×C79C411yescharacteristic
28D4⋊C8×C79C411yescharacteristic
29C2×#11929608—21no—
30C2×#11929608—21no—
31C2×#11929608D412yes—
32C22×C79⋊C8—22no—
33C4⋊(C79⋊C8)—22no—
34C4⋊(C79⋊C8)—21no—
35C4⋊(C79⋊C8)—21no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)D411yescharacteristic
39C4⋊(C79⋊C8)D411yescharacteristic
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)—21no—
42#14579072—22no—
43#14579072—21no—
44C4×C79⋊C8—21no—
45C4×C79⋊C8—21no—
46C2×#41×C79Q811yescharacteristic
47C22×C8×C79—21no—
48C4×Q8×C79C2×C411yescharacteristic
49C4×Q8×C79D411yescharacteristic
50C4×D4×C79—21no—
51C4×D4×C79—21no—
52C4×D4×C79C2×C411yescharacteristic
53C4×D4×C79D411yescharacteristic
54#117×C79—21no—
55#117×C79Q811yescharacteristic
56C2×C42×C79—21no—
57C2×C42×C79C2311yescharacteristic
58C4⋊C8×C79C2×C411yescharacteristic
59C4⋊C8×C79C2×C411yescharacteristic
60C22⋊C8×C79—21no—
61C4×C8×C79—21no—
62C4×C8×C79—21no—
63C4×C8×C79C2×C411yescharacteristic
64C4×C8×C79C2×C411yescharacteristic
65#11929608—22no—
66#11929608—42no—
67#11929608—41no—
68#11929608—42no—
69C2×C79⋊C8—21no—
70C2×C79⋊C8—21no—
71C2×C79⋊C8—22no—
72C2×C79⋊C8—41no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8—82no—
75#41×C79—22no—
76#41×C79—41no—
77C2×Q8×C79C4⋊C411yescharacteristic
78C2×D4×C79—21no—
79C2×D4×C79C4⋊C411yescharacteristic
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79C2×Q811yescharacteristic
85C22×C4×C79#4111yescharacteristic
86C2×C8×C79—21no—
87C2×C8×C79—21no—
88C2×C8×C79—21no—
89C2×C8×C79—21no—
90C2×C8×C79—41no—
91C2×C8×C79—41no—
92C4⋊C4×C79—21no—
93C4⋊C4×C79—21no—
94C4⋊C4×C79C4⋊C411yescharacteristic
95C4⋊C4×C79C4⋊C411yescharacteristic
96C22⋊C4×C79—21no—
97C22⋊C4×C79—21no—
98C22⋊C4×C79—21no—
99C42×C79—21no—
100C42×C79—21no—
101C42×C79—21no—
102C42×C79C22×C411yescharacteristic
103C42×C79C2×D411yescharacteristic
104C42×C79C2×D411yescharacteristic
105C42×C79#4111yescharacteristic
106C23×C79—21no—
107C23×C79C22⋊Q811yescharacteristic
108Q8×C79—21no—
109Q8×C79—21no—
110D4×C79—21no—
111D4×C79—42no—
112D4×C79#10812yes—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—42no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79C2×C4⋊C411yescharacteristic
128C2×C4×C79C4×D411yescharacteristic
129C2×C4×C79C4×D411yescharacteristic
130C2×C4×C79#12011yescharacteristic
131C2×C4×C79C22⋊Q811yescharacteristic
132C2×C4×C79#12311yescharacteristic
133C8×C79—41no—
134C8×C79—41no—
135C8×C79—41no—
136C8×C79—41no—
137C79⋊C8—42no—
138C79⋊C8—41no—
139C79⋊C8—41no—
140C79⋊C8—42no—
141C79⋊C8—81no—
142C22×C79—21no—
143C22×C79—22no—
144C22×C79—21no—
145C22×C79—21no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79#38511yescharacteristic
149C4×C79—21no—
150C4×C79—21no—
151C4×C79—21no—
152C4×C79—41no—
153C4×C79—41no—
154C4×C79—41no—
155C4×C79C2×#10811yescharacteristic
156C4×C79#43011yescharacteristic
157C4×C79#44311yescharacteristic
158C4×C79#44411yescharacteristic
159C4×C79#45411yescharacteristic
160C4×C79#45511yescharacteristic
161#8900—791nomaximal, Sylow
162C2×C79—41no—
163C2×C79—42no—
164C2×C79#185511yescharacteristic
165C2×C79#186011yescharacteristic
166C2×C79#186611yescharacteristic
167#2990—791no—
168#1837—791no—
169#1574—791no—
170#1489—791no—
171#1476D7911yescharacteristic
172#1299—791no—
173#1278—791no—
174C79#890011yescharacteristic, complemented, Sylow
175C4×#41C2×D7911yescharacteristic
176#446—1581no—
177#435—1582no—
178C8×D4—1581no—
179#432—1581no—
180#431—791no—
181#431C2×D7911yescharacteristic
182#423—791no—
183#423—791no—
184C2×C4⋊C8—1581no—
185C4×M16—791no—
186C2×C4×C8C2×D7911yescharacteristic
187#341—1582no—
188#335—1581no—
189#334—1581no—
190Q8⋊C8—1581no—
191Q8⋊C8Dic7911yescharacteristic
192Q8⋊C8Dic7911yescharacteristic
193D4⋊C8—1581no—
194D4⋊C8Dic7911yescharacteristic
195D4⋊C8Dic7911yescharacteristic
196C2×#41Dic15811yescharacteristic
197C2×M16—792no—
198C2×M16—1581no—
199C2×M16—1581no—
200C22×C8—21no—
201C22×C8—1582no—
202C4×Q8D31611yescharacteristic
203C4×Q8C2×Dic7911yescharacteristic
204C4×D4—21no—
205C4×D4—21no—
206C4×D4D31611yescharacteristic
207C4×D4C2×Dic7911yescharacteristic
208#117—21no—
209#117Dic15811yescharacteristic
210C2×C42—21no—
211C2×C42C22×D7911yescharacteristic
212C4⋊C8—791no—
213C4⋊C8—791no—
214C4⋊C8—1582no—
215C4⋊C8—1581no—
216C4⋊C8—1581no—
217C4⋊C8—1581no—
218C4⋊C8—1581no—
219C4⋊C8C2×Dic7911yescharacteristic
220C4⋊C8C2×Dic7911yescharacteristic
221C22⋊C8—21no—
222C22⋊C8—1582no—
223C22⋊C8—1581no—
224#97—1582no—
225#97—1581no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—1581no—
229C4×C8—1581no—
230C4×C8C2×Dic7911yescharacteristic
231C4×C8C2×Dic7911yescharacteristic
232#41—22no—
233#41—41no—
234C2×Q8C4⋊Dic7911yescharacteristic
235C2×D4—21no—
236C2×D4C4⋊Dic7911yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4D4×D7911yescharacteristic
241C22×C4C2×Dic15811yescharacteristic
242C22×C4#1192962311yescharacteristic
243M16—3161no—
244M16—3162no—
245M16—3162no—
246M16—1582no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—3161no—
254C2×C8—3161no—
255C2×C8—6322no—
256C2×C8—1581no—
257C2×C8—1581no—
258C2×C8—1582no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4C4⋊Dic7911yescharacteristic
262C4⋊C4C4⋊Dic7911yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42C22×Dic7911yescharacteristic
270C42D4×D7911yescharacteristic
271C42C2×D31611yescharacteristic
272C42#1192962311yescharacteristic
273C23—21no—
274C23#1457913811yescharacteristic
275Q8—21no—
276Q8—21no—
277D4—21no—
278D4—42no—
279D4#1457909712yes—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4C2×C4⋊Dic7911yescharacteristic
295C2×C4D4×Dic7911yescharacteristic
296C2×C4D4×Dic7911yescharacteristic
297C2×C4#1457913111yescharacteristic
298C2×C4#1457913811yescharacteristic
299C2×C4#1457914511yescharacteristic
300C8—41no—
301C8—41no—
302C8—41no—
303C8—41no—
304C8—3162no—
305C8—3161no—
306C8—3161no—
307C8—3162no—
308C8—6321no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22#1909843811yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4C2×#1457909711yescharacteristic
323C4#1909861211yescharacteristic
324C4#1909868011yescharacteristic
325C4#1909868511yescharacteristic
326C4#1909873711yescharacteristic
327C4#1909874211yescharacteristic
328C2—41no—
329C2—42no—
330C2#2376360711yescharacteristic
331C2#2376363511yescharacteristic
332C2#2376367711yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1476C2
socleC22×C79#385
prime radicalC2×#41×C79Q8
cosocleC2×C42×C79C23
hypercenter#1476D79
hypocenterC79#8900
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #23763538C2semidirect
3. #23762189C2semidirect
5. #23761291C2semidirect
6. #23761237C2semidirect
174. C79#8900semidirect

Maximal subgroups

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

subgroupindex
#237725252
#237635382
#237621892
#237618392
#237612912
#237612372
C79×#14762
#890079