Finite Group Databasebeta

← #34451131

Subgroups — #34451131

Order 20224.

2410 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#23766581C212yesmaximal, complemented
2#23766577C212yesmaximal, complemented
3#23766512C214yesmaximal, complemented
4#23766329C218yesmaximal, complemented
5#23766167C218yesmaximal, complemented
6#23765992C214yesmaximal, complemented
7#23765798C212yesmaximal, complemented
8C2×#532×C79C211yesmaximal, characteristic
9C2×#14579120C2214yes—
10C22×C4⋊Dic79C2214yes—
11#19098470C2218yes—
12#19098466C2218yes—
13#19098456C22116yes—
14#19098449C22116yes—
15#19098443C22116yes—
16#19098443C2218yes—
17#19098443C2218yes—
18#19098431C2214yes—
19#19098429C2214yes—
20#19098426C2214yes—
21#19098423C2214yes—
22#19098423C2218yes—
23C4×C4⋊Dic79C2214yes—
24C22⋊C4×Dic79C22116yes—
25C4×C22⋊Dic79C2214yes—
26#19098409C2214yes—
27C22×#41×C79C2212yes—
28#532×C79C4116yescomplemented
29C2×C4⋊Q8×C79C2211yescharacteristic
30C2×#127×C79C2211yescharacteristic
31C2×#124×C79C2212yes—
32C2×#121×C79C2214yes—
33C2×C4×D4×C79C2214yes—
34C22×C42×C79C2211yescharacteristic
35C23×Dic79C2318yes—
36#14579120—48no—
37C2×C4⋊Dic79—48no—
38C2×C4⋊Dic79C2318yes—
39C2×C4⋊Dic79C2318yes—
40C2×C22⋊Dic79C2318yes—
41C2×C22⋊Dic79C23116yes—
42C2×C22⋊Dic79C2318yes—
43C2×C4×Dic79C2318yes—
44C2×C4×Dic79C2318yes—
45C79⋊#95C2318yes—
46C79⋊#95C2318yes—
47C79⋊#95C2318yes—
48C79⋊#95C23116yes—
49C79⋊#95C2318yes—
50C2×#41×C79—44no—
51C2×#41×C79—44no—
52C2×#41×C79C2×C4116yescomplemented
53C22×Q8×C79C2312yes—
54C22×D4×C79C2314yes—
55C22×D4×C79C2312yes—
56C22×D4×C79C2314yes—
57C23×C4×C79C2311yescharacteristic
58C23×C4×C79C2314yes—
59C23×C4×C79C2312yes—
60C4⋊Q8×C79C2×C418yescomplemented
61#127×C79C2×C418yes—
62#124×C79C2×C4116yescomplemented
63#121×C79C2×C4132yescomplemented
64C4×D4×C79C2×C4132yescomplemented
65C2×C4⋊C4×C79C2314yes—
66C2×C22⋊C4×C79C2318yes—
67C2×C42×C79C2×C418yescomplemented
68C2×C42×C79C2311yescharacteristic
69C2×C42×C79C2312yes—
70C2×C42×C79C2311yescharacteristic
71C22×Dic79—48no—
72C22×Dic79—416no—
73C22×Dic79C24116yes—
74C4⋊Dic79—216no—
75C4⋊Dic79—48no—
76C22⋊Dic79—48no—
77C22⋊Dic79—416no—
78C22⋊Dic79—48no—
79C4×Dic79—48no—
80C4×Dic79—48no—
81C24×C79C2414yes—
82C24×C79C2411yescharacteristic
83#41×C79—432no—
84C2×Q8×C79—42no—
85C2×Q8×C79—42no—
86C2×Q8×C79C22×C418yes—
87C2×D4×C79—44no—
88C2×D4×C79—44no—
89C2×D4×C79—44no—
90C2×D4×C79—44no—
91C2×D4×C79—42no—
92C2×D4×C79—42no—
93C2×D4×C79C22×C4116yes—
94C2×D4×C79C22×C4116yes—
95C2×D4×C79C22×C418yes—
96C22×C4×C79—24no—
97C22×C4×C79—28no—
98C22×C4×C79—28no—
99C22×C4×C79—44no—
100C22×C4×C79C22×C414yes—
101C22×C4×C79C22×C4116yes—
102C22×C4×C79C22×C418yes—
103C22×C4×C79C2412yes—
104C22×C4×C79C2412yes—
105C22×C4×C79C2411yescharacteristic
106C22×C4×C79C2414yes—
107C22×C4×C79C2411yescharacteristic
108C4⋊C4×C79C22×C4116yes—
109C22⋊C4×C79C22×C4132yes—
110C42×C79C22×C418yes—
111C42×C79C22×C414yes—
112C42×C79C22×C414yes—
113C23×C79—24no—
114C23×C79—24no—
115C23×C79—22no—
116C23×C79—24no—
117C23×C79—28no—
118C23×C79—28no—
119C23×C79C23×C418yes—
120C23×C79C23×C412yes—
121C23×C79C2511yescharacteristic
122Q8×C79—48no—
123D4×C79—416no—
124D4×C79—48no—
125D4×C79—416no—
126C2×C4×C79—24no—
127C2×C4×C79—24no—
128C2×C4×C79—22no—
129C2×C4×C79—24no—
130C2×C4×C79—24no—
131C2×C4×C79—232no—
132C2×C4×C79—28no—
133C2×C4×C79—48no—
134C2×C4×C79C23×C414yes—
135C2×C4×C79C23×C414yes—
136C2×C4×C79C23×C412yes—
137C2×C4×C79C23×C418yes—
138C2×C4×C79C23×C412yes—
139C2×C4×C79H(5,𝔽2)14yes—
140C2×Dic79—216no—
141C2×Dic79—216no—
142C2×Dic79—216no—
143C2×Dic79—816no—
144C22×C79—22no—
145C22×C79—28no—
146C22×C79—24no—
147C22×C79—216no—
148C22×C79—48no—
149C22×C79—44no—
150C22×C79—44no—
151C22×C79—44no—
152C22×C79C24×C411yescharacteristic
153C22×C79C2×H(5,𝔽2)12yes—
154C22×C79#58514yes—
155C4×C79—22no—
156C4×C79—28no—
157C4×C79—24no—
158C4×C79—24no—
159C4×C79#51814yes—
160Dic79—416no—
161#34369—791nomaximal, Sylow
162C2×C79—42no—
163C2×C79—48no—
164C2×C79C2×#51811yescharacteristic
165C2×C79#342812yes—
166C2×C79#355114yes—
167C2×#532D7911yescharacteristic
168#2381—792no—
169#2380—792no—
170#2372—794no—
171#2352—798no—
172#2333—798no—
173#2311—794no—
174#2283—792no—
175C79#3436911yescharacteristic, complemented, Sylow
176C22×#41C2×D7912yes—
177#532Dic79116yescomplemented
178C2×C4⋊Q8C2×D7911yescharacteristic
179C2×#127C2×D7911yescharacteristic
180C2×#124C2×D7912yes—
181C2×#121C2×D7914yes—
182C2×C4×D4C2×D7914yes—
183C2×#117—794no—
184C22×C4⋊C4—794no—
185C22×C42C2×D7911yescharacteristic
186#391—798no—
187#390—798no—
188#388—7916no—
189#387—7916no—
190#386—798no—
191#386—798no—
192#386—7916no—
193#383—794no—
194#382—794no—
195#381—794no—
196#380—794no—
197#380—798no—
198C4×C4⋊C4—794no—
199C4×C22⋊C4—794no—
200C4×C22⋊C4—7916no—
201#376—794no—
202C2×#41—44no—
203C2×#41—44no—
204C2×#41C2×Dic79116yescomplemented
205C22×Q8C22×D7912yes—
206C22×D4C22×D7914yes—
207C22×D4C22×D7914yes—
208C22×D4C22×D7912yes—
209C23×C4—798no—
210C23×C4C22×D7914yes—
211C23×C4C22×D7911yescharacteristic
212C23×C4C22×D7912yes—
213C4⋊Q8C2×Dic7918yescomplemented
214#127C2×Dic7918yes—
215#124C2×Dic79116yescomplemented
216#121C2×Dic79132yescomplemented
217C4×D4C2×Dic79132yescomplemented
218#117—3168no—
219C2×C4⋊C4—3168no—
220C2×C4⋊C4—798no—
221C2×C4⋊C4—798no—
222C2×C4⋊C4C22×D7914yes—
223C2×C22⋊C4—798no—
224C2×C22⋊C4—798no—
225C2×C22⋊C4—7916no—
226C2×C22⋊C4C22×D7918yes—
227C2×C42—798no—
228C2×C42—798no—
229C2×C42C2×Dic7918yescomplemented
230C2×C42C22×D7912yes—
231C2×C42C22×D7911yescharacteristic
232C2×C42C22×D7911yescharacteristic
233#95—798no—
234#95—798no—
235#95—7916no—
236#95—798no—
237#95—798no—
238C24C23×D7911yescharacteristic
239C24C23×D7914yes—
240#41—432no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8C22×Dic7918yes—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4C22×Dic79116yes—
251C2×D4C22×Dic79116yes—
252C2×D4C22×Dic7918yes—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—3168no—
258C22×C4—31616no—
259C22×C4—7916no—
260C22×C4C22×Dic7914yes—
261C22×C4C22×Dic79116yes—
262C22×C4C22×Dic7918yes—
263C22×C4C23×D7912yes—
264C22×C4C23×D7911yescharacteristic
265C22×C4C23×D7912yes—
266C22×C4C23×D7914yes—
267C22×C4C23×D7911yescharacteristic
268C4⋊C4—3168no—
269C4⋊C4—15816no—
270C4⋊C4C22×Dic79116yes—
271C22⋊C4—31616no—
272C22⋊C4—3168no—
273C22⋊C4—3168no—
274C22⋊C4C22×Dic79132yes—
275C42—3168no—
276C42—3168no—
277C42C22×Dic7918yes—
278C42C22×Dic7914yes—
279C42C22×Dic7914yes—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—24no—
284C23—28no—
285C23—28no—
286C23C23×Dic7918yes—
287C23C23×Dic7912yes—
288C23C24×D7911yescharacteristic
289Q8—48no—
290D4—416no—
291D4—48no—
292D4—416no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—232no—
299C2×C4—28no—
300C2×C4—48no—
301C2×C4—63216no—
302C2×C4—15816no—
303C2×C4—15816no—
304C2×C4—15816no—
305C2×C4C23×Dic7914yes—
306C2×C4C23×Dic7914yes—
307C2×C4C23×Dic7912yes—
308C2×C4C23×Dic7918yes—
309C2×C4C23×Dic7912yes—
310C2×C4#1457920614yes—
311C22—22no—
312C22—28no—
313C22—24no—
314C22—216no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—44no—
319C22C24×Dic7911yescharacteristic
320C22C2×#1457920612yes—
321C22#1909939914yes—
322C4—22no—
323C4—28no—
324C4—24no—
325C4—24no—
326C4—31616no—
327C4#1909903814yes—
328C2—42no—
329C2—48no—
330C2C2×#1909903811yescharacteristic
331C2#2377660312yes—
332C2#2377782714yes—
subgroupG/H
center, FrattiniC23C24×D79
commutatorC22×C79C24×C4
Fitting, prime radicalC2×#532×C79C2
socle, cosocleC23×C79C25
hypercenterC2×#532D79
hypocenterC79#34369
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. #23766581C2semidirect
2. #23766577C2semidirect
3. #23766512C2semidirect
4. #23766329C2semidirect
5. #23766167C2semidirect
6. #23765992C2semidirect
7. #23765798C2semidirect
28. #532×C79C4semidirect
52. C2×#41×C79C2×C4semidirect
60. C4⋊Q8×C79C2×C4semidirect
62. #124×C79C2×C4semidirect
63. #121×C79C2×C4semidirect
64. C4×D4×C79C2×C4semidirect
67. C2×C42×C79C2×C4semidirect
175. C79#34369semidirect
177. #532Dic79semidirect
204. C2×#41C2×Dic79semidirect
213. C4⋊Q8C2×Dic79semidirect
215. #124C2×Dic79semidirect
216. #121C2×Dic79semidirect
217. C4×D4C2×Dic79semidirect
229. C2×C42C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237665812
#237665772
#237665122
#237663292
#237661672
#237659922
#237657982
C2×#532×C792
#3436979