Finite Group Databasebeta

← #13453647

Subgroups — #13453647

Order 1792.

770 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×#124C211yesmaximal, characteristic
2#11730753C211yesmaximal, characteristic, complemented
3#11729479C212yesmaximal
4#11729381C212yesmaximal, complemented
5#11729359C212yesmaximal
6#11727422C212yesmaximal, complemented
7#11727392C212yesmaximal
8#11727301C212yesmaximal, complemented
9C4×C4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
10C2×C7×#124C2211yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13C2×#6432C2212yes—
14C7×#391C2211yescharacteristic
15C7×#390C2211yescharacteristic
16C7×#388—22no—
17C7×#382C2211yescharacteristic
18C4×C7×C22⋊C4—22no—
19C43×C7C2211yescharacteristic
20C2×#6409C2214yes—
21C2×#6406C2214yes—
22#89708C2212yes—
23#89708C2212yes—
24#89701—22no—
25#89701—22no—
26C2×C4⋊(C7⋊C8)C2212yes—
27C2×C4×C7⋊C8C2212yes—
28#89324C2214yes—
29#89324C2214yes—
30#89319C2212yes—
31#29142—71nomaximal, Sylow
32C22×C7×Q8C2311yescharacteristic
33C22×C7×D4C2311yescharacteristic
34C23×C4×C7—21no—
35C22×Dic14—22no—
36C7×#124—22no—
37C7×#124Q814yescomplemented
38C4×C7×Q8—24no—
39C4×C7×D4—24no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4C2311yescharacteristic
42C2×C7×C4⋊C4C2311yescharacteristic
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—22no—
45C2×C7×C22⋊C4—21no—
46C2×C42×C7—21no—
47C2×C42×C7D412yes—
48C2×C42×C7D412yes—
49C2×C42×C7C2311yescharacteristic
50C2×C42×C7C2311yescharacteristic
51C2×C42×C7C2311yescharacteristic
52C2×C4⋊Dic7—22no—
53C2×C4⋊Dic7C2314yes—
54C22×C7⋊C8C2314yes—
55#6432—44no—
56C7×#95—22no—
57C7×#95—22no—
58#6409—44no—
59#6406—44no—
60C7⋊#95—24no—
61C7⋊#95—22no—
62C7⋊#95—22no—
63C4⋊(C7⋊C8)—24no—
64C4×C7⋊C8—24no—
65#2862—71no—
66C4×#124D711yescharacteristic
67#2024—72no—
68#2023—72no—
69#1958—72no—
70#1835—72no—
71#1805—72no—
72#1804—72no—
73C4×C4⋊C8—71no—
74C24×C7—21no—
75C2×C7×Q8—21no—
76C2×C7×Q8—22no—
77C2×C7×Q8—22no—
78C2×C7×Q8C2×Q812yes—
79C2×C7×D4—22no—
80C2×C7×D4—21no—
81C2×C7×D4—22no—
82C2×C7×D4C2×Q812yes—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—44no—
87C22×C4×C7—42no—
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D412yes—
92C22×C4×C7C2411yescharacteristic
93C22×Dic7—24no—
94C2×Dic14—44no—
95C2×Dic14—44no—
96C2×Dic14—44no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—42no—
100C7×C22⋊C4—28no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—44no—
103C42×C7—24no—
104C42×C7—22no—
105C42×C7—42no—
106C42×C7—42no—
107C42×C7C2×Q812yes—
108C42×C7#4112yes—
109C42×C7#4114yes—
110C4⋊Dic7—24no—
111C4⋊Dic7—24no—
112C4⋊Dic7—44no—
113C2×C7⋊C8—24no—
114C2×C7⋊C8—24no—
115C2×C7⋊C8—24no—
116C2×C4⋊Q8—72no—
117C2×#124C2×D711yescharacteristic
118C2×C4×Q8C2×D711yescharacteristic
119C2×C4×D4C2×D711yescharacteristic
120C2×C4⋊C8—72no—
121C2×#103—74no—
122C2×#102—74no—
123C2×C4×C8—72no—
124#391—142no—
125#391—142no—
126#391C2×D711yescharacteristic
127#390C2×D711yescharacteristic
128#388—22no—
129#382—72no—
130#382—72no—
131#382C2×D711yescharacteristic
132C4×C22⋊C4—22no—
133C43C2×D711yescharacteristic
134#340—74no—
135#340—74no—
136#336—72no—
137C23×C7—22no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—42no—
141C23×C7—42no—
142C23×C7C22×D411yescharacteristic
143C7×Q8—24no—
144C7×Q8—42no—
145C7×D4—24no—
146C7×D4—42no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—22no—
152C2×C4×C7—22no—
153C2×C4×C7—22no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—84no—
161C2×C4×C7C22⋊Q814yes—
162C2×C4×C7#12412yes—
163C2×C4×C7#12412yes—
164C2×C4×C7#13514yes—
165C2×C4×C7C22×Q811yescharacteristic
166C2×C4×C7C2×#4111yescharacteristic
167C2×C4×C7C2×#4112yes—
168C2×C4×C7H(5,𝔽2)12yes—
169C2×Dic7—44no—
170C2×Dic7—44no—
171C2×Dic7—44no—
172Dic14—88no—
173C7⋊C8—44no—
174C22×Q8—142no—
175C22×Q8C22×D711yescharacteristic
176C22×D4C22×D711yescharacteristic
177C23×C4—21no—
178C22×C8—74no—
179C4⋊Q8—284no—
180#124—22no—
181#124Dic1414yescomplemented
182C4×Q8—24no—
183C4×D4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—74no—
186C2×C4⋊C4—142no—
187C2×C4⋊C4C22×D711yescharacteristic
188C2×C4⋊C4C22×D711yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C42—21no—
193C2×C42C22⋊D712yes—
194C2×C42C22⋊D712yes—
195C2×C42C22×D711yescharacteristic
196C2×C42C22×D711yescharacteristic
197C2×C42C22×D711yescharacteristic
198C4⋊C8—144no—
199#103—284no—
200#102—284no—
201C4×C8—144no—
202#95—22no—
203#95—22no—
204#95—144no—
205#95—142no—
206#95—142no—
207C22×C7—42no—
208C22×C7—44no—
209C22×C7—81no—
210C22×C7—81no—
211C22×C7—82no—
212C22×C7C2×C22⋊Q811yescharacteristic
213C22×C7C2×#12411yescharacteristic
214C22×C7#53511yescharacteristic
215C22×C7C2×#13512yes—
216C22×C7#57512yes—
217C4×C7—24no—
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—24no—
221C4×C7—42no—
222C4×C7—81no—
223C4×C7—81no—
224C4×C7#55412yes—
225C4×C7#55512yes—
226Dic7—84no—
227C24—21no—
228C2×Q8—21no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—284no—
232C2×Q8—284no—
233C2×Q8—284no—
234C2×Q8C2×Dic1412yes—
235C2×D4—21no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4C2×Dic1412yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—144no—
245C22×C4C2×C22⋊D712yes—
246C22×C4C2×C22⋊D711yescharacteristic
247C22×C4C2×C22⋊D711yescharacteristic
248C22×C4C2×C22⋊D712yes—
249C22×C4C23×D711yescharacteristic
250C2×C8—144no—
251C2×C8—144no—
252C2×C8—144no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—42no—
256C4⋊C4—144no—
257C4⋊C4—144no—
258C4⋊C4—284no—
259C22⋊C4—28no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C42—24no—
263C42—22no—
264C42—42no—
265C42—42no—
266C42C2×Dic1412yes—
267C42#115712yes—
268C42#115914yes—
269C2×C7—82no—
270C2×C7#256311yescharacteristic
271C2×C7#307612yes—
272C2×C7#313111yescharacteristic
273C2×C7#314011yescharacteristic
274C2×C7#321112yes—
275C23—22no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23C22×C22⋊D711yescharacteristic
281Q8—24no—
282Q8—42no—
283Q8—568no—
284D4—24no—
285D4—42no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—84no—
300C2×C4—284no—
301C2×C4—284no—
302C2×C4—284no—
303C2×C4#648714yes—
304C2×C4#649912yes—
305C2×C4#649912yes—
306C2×C4#651314yes—
307C2×C4C22×Dic1411yescharacteristic
308C2×C4C2×#115711yescharacteristic
309C2×C4C2×#115912yes—
310C2×C4#655312yes—
311C8—284no—
312C7#2914211yescharacteristic, complemented, Sylow
313C22—44no—
314C22—42no—
315C22—81no—
316C22—82no—
317C22—81no—
318C22C2×#648711yescharacteristic
319C22C2×#649911yescharacteristic
320C22C2×#651312yes—
321C22#9051312yes—
322C22#9052111yescharacteristic
323C4—24no—
324C4—22no—
325C4—24no—
326C4—22no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—564no—
331C4#9023412yes—
332C4#9025412yes—
333C2—82no—
334C2#1173982911yescharacteristic
335C2#1173995212yes—
336C2#1174038411yescharacteristic
337C2#1174058912yes—
338C2#1174072611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC4×C7×#124C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC4×#124D7
hypocenterC7#29142
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
2. #11730753C2semidirect
4. #11729381C2semidirect
6. #11727422C2semidirect
8. #11727301C2semidirect
9. C4×C4⋊(C7⋊C8)C2semidirect
37. C7×#124Q8semidirect
181. #124Dic14semidirect
312. C7#29142semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1242
#117307532
#117294792
#117293812
#117293592
#117274222
#117273922
#117273012
C4×C4⋊(C7⋊C8)2
#291427