Finite Group Databasebeta

← #34306683

Subgroups — #34306683

Order 20224.

1062 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D632C212yesmaximal
2C2×C4×D632C211yesmaximal, characteristic
3#23766311C212yesmaximal, complemented
4#23764133C212yesmaximal, complemented
5#23763664C214yesmaximal, complemented
6#23763564C211yesmaximal, characteristic, complemented
7#23763411C212yesmaximal, complemented
8C8×C4⋊C4×C79C211yesmaximal, characteristic, complemented
9C22×D632C2211yescharacteristic
10C2×C4×D316C2214yes—
11C2×C4×D316C2212yes—
12C4×D632—28no—
13C4×D632C418yescomplemented
14C2×#14579095C2212yes—
15C2×#14579095C2211yescharacteristic
16C2×#14579093C2212yes—
17C2×#14579083C2214yes—
18C2×#14579083C2212yes—
19#19098461C2214yes—
20#19098434—24no—
21#19098429C2212yes—
22#19098422—22no—
23#95×D79—22no—
24#19098326C2214yes—
25C2×C4⋊C8×C79C2211yescharacteristic
26C2×C4×C8×C79C2212yes—
27C2×C4×C8×C79C2211yescharacteristic
28C4×C4⋊C4×C79C2211yescharacteristic
29#336×C79C2212yes—
30C22×D316C2312yes—
31C22×C4×D79—22no—
32C22×C4×D79—24no—
33C2×D632C2×C414yes—
34C2×D632D414yes—
35C2×D632Q814yes—
36C4×D316—24no—
37C4×D316—28no—
38C4×D316—44no—
39C4×D316C2×C418yescomplemented
40C2×C4⋊Dic79C2314yes—
41C2×C4⋊Dic79C2312yes—
42C2×Dic79⋊C4—24no—
43C2×(C2×D79)⋊C4—22no—
44C2×(C2×D79)⋊C4—24no—
45#14579095—24no—
46#14579095C2×C414yescomplemented
47#14579093—42no—
48#14579083—28no—
49#14579083C2×C418yescomplemented
50C79⋊#95—22no—
51C79⋊#95—24no—
52C79⋊#95—22no—
53C22×C8×C79C2312yes—
54C22×C8×C79C2311yescharacteristic
55C22×C8×C79C2311yescharacteristic
56C2×C4⋊C4×C79D412yes—
57C2×C42×C79C2312yes—
58C2×C42×C79C2311yescharacteristic
59C4⋊C8×C79—22no—
60C4×C8×C79—24no—
61C4×C8×C79C2×C414yescomplemented
62#95×C79D411yescharacteristic
63#95×C79D411yescharacteristic
64C23×D79—22no—
65C2×D316—22no—
66C2×D316—22no—
67C2×D316—22no—
68C2×D316C22×C414yes—
69C2×D316C2×D414yes—
70C2×D316C2×Q814yes—
71C22×Dic79—22no—
72C22×Dic79—24no—
73C2×C4×D79—28no—
74C2×C4×D79—48no—
75C2×C4×D79—48no—
76C2×C4×D79—84no—
77C2×C4×D79—84no—
78D632C4⋊C4116yescomplemented
79C4⋊Dic79—24no—
80C4⋊Dic79—22no—
81C4⋊Dic79—24no—
82C4⋊Dic79C22×C414yes—
83Dic79⋊C4—84no—
84(C2×D79)⋊C4—28no—
85(C2×D79)⋊C4—48no—
86C22×C4×C79C2×D412yes—
87C22×C4×C79C2×D411yescharacteristic
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79C2×D412yes—
90C22×C4×C79C2411yescharacteristic
91C2×C8×C79—22no—
92C2×C8×C79—22no—
93C2×C8×C79—21no—
94C2×C8×C79—21no—
95C2×C8×C79—22no—
96C2×C8×C79C22×C412yes—
97C2×C8×C79C22×C412yes—
98C2×C8×C79C2×D412yes—
99C2×C8×C79C2×Q812yes—
100C4⋊C4×C79—24no—
101C42×C79—22no—
102C42×C79—22no—
103C42×C79C22×C412yes—
104C42×C79#4112yes—
105C23×C79C22×D411yescharacteristic
106C22×D79—24no—
107C22×D79—24no—
108C22×D79—24no—
109C22×D79—44no—
110C22×D79—44no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—22no—
115C2×C4×C79—22no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79C4×D412yes—
119C2×C4×C79C4×D412yes—
120C2×C4×C79#13514yes—
121C2×C4×C79C23×C411yescharacteristic
122C2×C4×C79C22×D411yescharacteristic
123C2×C4×C79C22×Q811yescharacteristic
124C2×C4×C79C2×#4111yescharacteristic
125C2×C4×C79H(5,𝔽2)11yescharacteristic
126C2×C4×C792−1+411yescharacteristic
127C2×Dic79—24no—
128C2×Dic79—44no—
129C2×Dic79—44no—
130C2×Dic79—44no—
131C2×Dic79—44no—
132D316—24no—
133D316—24no—
134D316—24no—
135D316C2×C4⋊C418yes—
136C4×D79—88no—
137C4×D79—816no—
138C8×C79—22no—
139C8×C79—42no—
140C8×C79C2×C4⋊C414yes—
141C22×C79C2×C4×D411yescharacteristic
142C22×C79D4211yescharacteristic
143C22×C79D4×Q811yescharacteristic
144C22×C79C2×#13512yes—
145C22×C79#57612yes—
146C2×D79—28no—
147C2×D79—48no—
148C2×D79—44no—
149C2×D79—44no—
150C2×D79—48no—
151C4×C79—22no—
152C4×C79—22no—
153C4×C79—24no—
154C4×C79—42no—
155C4×C79C22×C4⋊C411yescharacteristic
156C4×C79#52011yescharacteristic
157C4×C79#54711yescharacteristic
158C4×C79#55411yescharacteristic
159Dic79—44no—
160Dic79—84no—
161#22682—791nomaximal, Sylow
162C2×C79D4×C4⋊C411yescharacteristic
163C2×C79C4×#13511yescharacteristic
164C2×C79#294811yescharacteristic
165C2×C79#328012yes—
166C2×C79#338012yes—
167D79—48no—
168C2×C4×D8—791no—
169C2×C4×D8—792no—
170#2349—792no—
171#1942—792no—
172#1864—794no—
173#1846—791no—
174#1804—792no—
175C8×C4⋊C4D7911yescharacteristic, complemented
176C79#2268211yescharacteristic, complemented, Sylow
177C22×D8—791no—
178C2×C4×D4—794no—
179C2×C4×D4—792no—
180C4×D8—798no—
181C4×D8—1588no—
182C2×#107—791no—
183C2×#107—792no—
184C2×#106—792no—
185C2×C4⋊C8C2×D7911yescharacteristic
186C2×#102—792no—
187C2×#102—794no—
188C2×C4×C8C2×D7912yes—
189C2×C4×C8C2×D7911yescharacteristic
190#389—794no—
191#385—1584no—
192#382—792no—
193#380—1582no—
194C4×C4⋊C4C2×D7911yescharacteristic
195C2×#95—1582no—
196#340—794no—
197#336C2×D7912yes—
198C22×D4—792no—
199C23×C4—1584no—
200C23×C4—1582no—
201C2×D8—794no—
202C2×D8—794no—
203C2×D8—794no—
204C22×C8C22×D7912yes—
205C22×C8C22×D7911yescharacteristic
206C22×C8C22×D7911yescharacteristic
207C4×D4—3164no—
208C4×D4—798no—
209C4×D4—1588no—
210C4×D4—1584no—
211C2×C4⋊C4—792no—
212C2×C4⋊C4—794no—
213C2×C4⋊C4—1584no—
214C2×C4⋊C4D31612yes—
215C2×C22⋊C4—1584no—
216C2×C22⋊C4—1582no—
217C2×C42C22×D7912yes—
218C2×C42C22×D7911yescharacteristic
219#107—794no—
220#107—1584no—
221#106—3162no—
222C4⋊C8—22no—
223#102—798no—
224#102—1588no—
225C4×C8—24no—
226C4×C8C4×D7914yescomplemented
227#95—1582no—
228#95—1584no—
229#95—1582no—
230#95D31611yescharacteristic
231#95D31611yescharacteristic
232C24—1582no—
233C2×D4—794no—
234C2×D4—794no—
235C2×D4—794no—
236C2×D4—1582no—
237C2×D4—1582no—
238C2×D4—1582no—
239C22×C4—3168no—
240C22×C4—3168no—
241C22×C4—6324no—
242C22×C4—6324no—
243C22×C4—1584no—
244C22×C4—1582no—
245C22×C4—1588no—
246C22×C4C2×D31612yes—
247C22×C4C2×D31611yescharacteristic
248C22×C4C2×D31611yescharacteristic
249C22×C4C2×D31612yes—
250C22×C4C23×D7911yescharacteristic
251D8—7916no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8C2×C4×D7912yes—
258C2×C8C2×C4×D7912yes—
259C2×C8D4×D7912yes—
260C2×C8Q8×D7912yes—
261C4⋊C4—24no—
262C4⋊C4—794no—
263C4⋊C4—6324no—
264C4⋊C4—1584no—
265C4⋊C4—1584no—
266C4⋊C4—1582no—
267C22⋊C4—3168no—
268C22⋊C4—1588no—
269C42—22no—
270C42—22no—
271C42C2×C4×D7912yes—
272C42#1192962412yes—
273C23—3164no—
274C23—3164no—
275C23—1584no—
276C23—1584no—
277C23—1584no—
278C23C22×D31611yescharacteristic
279D4—798no—
280D4—1584no—
281D4—1584no—
282D4—1584no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—3164no—
291C2×C4—3164no—
292C2×C4—3164no—
293C2×C4—3164no—
294C2×C4—6328no—
295C2×C4—63216no—
296C2×C4—1584no—
297C2×C4C4×D31612yes—
298C2×C4C4×D31612yes—
299C2×C4#1457918314yes—
300C2×C4C22×C4×D7911yescharacteristic
301C2×C4C2×D4×D7911yescharacteristic
302C2×C4C2×Q8×D7911yescharacteristic
303C2×C4C2×#1192962411yescharacteristic
304C2×C4#1457920711yescharacteristic
305C2×C4#1457920811yescharacteristic
306C8—22no—
307C8—42no—
308C8C4⋊C4×D7914yes—
309C22—3168no—
310C22—3164no—
311C22—3164no—
312C22—3168no—
313C22—1588no—
314C22C2×C4×D31611yescharacteristic
315C22D4×D31611yescharacteristic
316C22Q8×D31611yescharacteristic
317C22C2×#1457918312yes—
318C22#1909936812yes—
319C4—22no—
320C4—22no—
321C4—24no—
322C4—42no—
323C4—3164no—
324C4—6324no—
325C4C2×C4⋊C4×D7911yescharacteristic
326C4#1909904311yescharacteristic
327C4#1909921811yescharacteristic
328C4#1909927311yescharacteristic
329C2—3168no—
330C2C4⋊C4×D31611yescharacteristic
331C2C4×#1457918311yescharacteristic
332C2#2377225311yescharacteristic
333C2#2377508712yes—
334C2#2377626812yes—
subgroupG/H
centerC23C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC8×C4⋊C4×C79C2
socleC23×C79C22×D4
prime radicalC22×D632C22
cosocleC22×C4×C79C24
hypercenterC8×C4⋊C4D79
hypocenterC79#22682
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #23766311C2semidirect
4. #23764133C2semidirect
5. #23763664C2semidirect
6. #23763564C2semidirect
7. #23763411C2semidirect
8. C8×C4⋊C4×C79C2semidirect
13. C4×D632C4semidirect
39. C4×D316C2×C4semidirect
46. #14579095C2×C4semidirect
49. #14579083C2×C4semidirect
61. C4×C8×C79C2×C4semidirect
78. D632C4⋊C4semidirect
175. C8×C4⋊C4D79semidirect
176. C79#22682semidirect
226. C4×C8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D6322
#237663112
#237641332
#237636642
#237635642
#237634112
C8×C4⋊C4×C792
#2268279