Finite Group Databasebeta

← #13453631

Subgroups — #13453631

Order 1792.

754 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
1C4×C7×#124C211yesmaximal, characteristic
2#11730752C211yesmaximal, characteristic, complemented
3#11729479C212yesmaximal
4#11729380C212yesmaximal, complemented
5#11729358C212yesmaximal
6#11727423C212yesmaximal, complemented
7#11727393C212yesmaximal
8#11727301C212yesmaximal, complemented
9C4×C4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
10C2×C7×#124C2211yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13C2×#6433C2212yes—
14C7×#391C2211yescharacteristic
15C7×#390C2211yescharacteristic
16C7×#388—22no—
17C7×#382C2211yescharacteristic
18C4×C7×C22⋊C4—22no—
19C43×C7C2211yescharacteristic
20C2×#6409C2214yes—
21C2×#6406C2214yes—
22#89708C2212yes—
23#89707C2212yes—
24#89700—22no—
25#89700—22no—
26C2×C4⋊(C7⋊C8)C2212yes—
27C2×C4×C7⋊C8C2212yes—
28#89324C2214yes—
29#89324C2214yes—
30#89319C2212yes—
31#29140—71nomaximal, Sylow
32C22×C7×Q8C2311yescharacteristic
33C22×C7×D4C2311yescharacteristic
34C23×C4×C7—21no—
35C7×#124—22no—
36C7×#124Q814yes—
37C4×C7×Q8—24no—
38C4×C7×D4—24no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4C2311yescharacteristic
41C2×C7×C4⋊C4C2311yescharacteristic
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—21no—
45C2×C42×C7—21no—
46C2×C42×C7D412yes—
47C2×C42×C7D412yes—
48C2×C42×C7C2311yescharacteristic
49C2×C42×C7C2311yescharacteristic
50C2×C42×C7C2311yescharacteristic
51C2×C4⋊Dic7—22no—
52C2×C4⋊Dic7—22no—
53C2×C4⋊Dic7C2314yes—
54C2×Dic7⋊C4—24no—
55C22×C7⋊C8C2314yes—
56#6433—44no—
57C7×#95—22no—
58C7×#95—22no—
59#6409—44no—
60#6406—44no—
61C7⋊#95—24no—
62C4⋊(C7⋊C8)—24no—
63C4×C7⋊C8—24no—
64#2859—71no—
65C4×#124D711yescharacteristic
66#2026—72no—
67#2025—72no—
68#1958—72no—
69#1835—72no—
70#1803—72no—
71#1802—72no—
72C4×C4⋊C8—71no—
73C24×C7—21no—
74C2×C7×Q8—21no—
75C2×C7×Q8—22no—
76C2×C7×Q8—22no—
77C2×C7×Q8C2×Q812yes—
78C2×C7×D4—22no—
79C2×C7×D4—21no—
80C2×C7×D4—22no—
81C2×C7×D4C2×Q812yes—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—42no—
86C22×C4×C7—44no—
87C22×C4×C7C2×D412yes—
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2411yescharacteristic
92C22×Dic7—24no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—42no—
96C7×C22⋊C4—28no—
97C7×C22⋊C4—44no—
98C7×C22⋊C4—42no—
99C42×C7—24no—
100C42×C7—22no—
101C42×C7—42no—
102C42×C7—42no—
103C42×C7C2×Q812yes—
104C42×C7#4112yes—
105C42×C7#4114yes—
106C4⋊Dic7—24no—
107C4⋊Dic7—24no—
108C4⋊Dic7—44no—
109C4⋊Dic7—44no—
110Dic7⋊C4—48no—
111C2×C7⋊C8—24no—
112C2×C7⋊C8—24no—
113C2×C7⋊C8—24no—
114C2×#125—72no—
115C2×#124C2×D711yescharacteristic
116C2×C4×Q8C2×D711yescharacteristic
117C2×C4×D4C2×D711yescharacteristic
118C2×C4⋊C8—72no—
119C2×#103—74no—
120C2×#102—74no—
121C2×C4×C8—72no—
122#391C2×D711yescharacteristic
123#390C2×D711yescharacteristic
124#389—142no—
125#389—142no—
126#388—22no—
127#384—72no—
128#382—72no—
129#382C2×D711yescharacteristic
130C4×C22⋊C4—22no—
131C43C2×D711yescharacteristic
132#340—74no—
133#340—74no—
134#336—72no—
135C23×C7—22no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—42no—
139C23×C7—42no—
140C23×C7C22×D411yescharacteristic
141C7×Q8—24no—
142C7×Q8—42no—
143C7×D4—24no—
144C7×D4—42no—
145C2×C4×C7—22no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—22no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—42no—
157C2×C4×C7—42no—
158C2×C4×C7—84no—
159C2×C4×C7C22⋊Q814yes—
160C2×C4×C7#12412yes—
161C2×C4×C7#12412yes—
162C2×C4×C7#13514yes—
163C2×C4×C7C22×Q811yescharacteristic
164C2×C4×C7C2×#4111yescharacteristic
165C2×C4×C7C2×#4112yes—
166C2×C4×C7H(5,𝔽2)12yes—
167C2×Dic7—44no—
168C2×Dic7—44no—
169C2×Dic7—44no—
170C7⋊C8—44no—
171C22×Q8C22×D711yescharacteristic
172C22×D4C22×D711yescharacteristic
173C23×C4—21no—
174C22×C8—74no—
175#125—284no—
176#124—22no—
177#124Dic1414yes—
178C4×Q8—24no—
179C4×D4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—74no—
182C2×C4⋊C4—144no—
183C2×C4⋊C4—142no—
184C2×C4⋊C4—142no—
185C2×C4⋊C4C22×D711yescharacteristic
186C2×C4⋊C4C22×D711yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C42—21no—
191C2×C42C22⋊D712yes—
192C2×C42C22⋊D712yes—
193C2×C42C22×D711yescharacteristic
194C2×C42C22×D711yescharacteristic
195C2×C42C22×D711yescharacteristic
196C4⋊C8—144no—
197#103—284no—
198#102—284no—
199C4×C8—144no—
200#95—22no—
201#95—22no—
202#95—144no—
203C22×C7—42no—
204C22×C7—44no—
205C22×C7—81no—
206C22×C7—81no—
207C22×C7—82no—
208C22×C7C2×C22⋊Q811yescharacteristic
209C22×C7C2×#12411yescharacteristic
210C22×C7#53511yescharacteristic
211C22×C7C2×#13512yes—
212C22×C7#57512yes—
213C4×C7—24no—
214C4×C7—22no—
215C4×C7—42no—
216C4×C7—42no—
217C4×C7—81no—
218C4×C7—81no—
219C4×C7#48814yes—
220C4×C7#55412yes—
221C4×C7#55512yes—
222Dic7—84no—
223C24—21no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8C2×Dic1412yes—
228C2×D4—21no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4C2×Dic1412yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—144no—
238C22×C4C2×C22⋊D712yes—
239C22×C4C2×C22⋊D712yes—
240C22×C4C2×C22⋊D711yescharacteristic
241C22×C4C2×C22⋊D711yescharacteristic
242C22×C4C23×D711yescharacteristic
243C2×C8—144no—
244C2×C8—144no—
245C2×C8—144no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—144no—
250C4⋊C4—144no—
251C4⋊C4—288no—
252C4⋊C4—284no—
253C4⋊C4—284no—
254C22⋊C4—28no—
255C22⋊C4—44no—
256C22⋊C4—42no—
257C42—24no—
258C42—22no—
259C42—42no—
260C42—42no—
261C42C2×Dic1412yes—
262C42#115712yes—
263C42#115914yes—
264C2×C7—82no—
265C2×C7#256311yescharacteristic
266C2×C7#307612yes—
267C2×C7C2×#48811yescharacteristic
268C2×C7#314211yescharacteristic
269C2×C7#321212yes—
270C23—22no—
271C23—41no—
272C23—42no—
273C23—42no—
274C23—41no—
275C23C22×C22⋊D711yescharacteristic
276Q8—24no—
277Q8—42no—
278D4—24no—
279D4—42no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—84no—
294C2×C4—284no—
295C2×C4—284no—
296C2×C4—284no—
297C2×C4#648714yes—
298C2×C4#649912yes—
299C2×C4#649912yes—
300C2×C4#651314yes—
301C2×C4C22×Dic1411yescharacteristic
302C2×C4C2×#115711yescharacteristic
303C2×C4C2×#115912yes—
304C2×C4#655312yes—
305C8—284no—
306C7#2914011yescharacteristic, complemented, Sylow
307C22—44no—
308C22—42no—
309C22—81no—
310C22—82no—
311C22—81no—
312C22C2×#648711yescharacteristic
313C22C2×#649911yescharacteristic
314C22C2×#651312yes—
315C22#9051312yes—
316C22#9052111yescharacteristic
317C4—24no—
318C4—22no—
319C4—42no—
320C4—42no—
321C4—81no—
322C4—81no—
323C4—564no—
324C4#8982814yes—
325C4#9023412yes—
326C4#9025412yes—
327C2—82no—
328C2#1173982911yescharacteristic
329C2#1173995212yes—
330C2C2×#8982811yescharacteristic
331C2#1174059012yes—
332C2#1174072911yescharacteristic
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#29140
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. #11730752C2semidirect
4. #11729380C2semidirect
6. #11727423C2semidirect
8. #11727301C2semidirect
9. C4×C4⋊(C7⋊C8)C2semidirect
306. C7#29140semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1242
#117307522
#117294792
#117293802
#117293582
#117274232
#117273932
#117273012
C4×C4⋊(C7⋊C8)2
#291407