Finite Group Databasebeta

← #13246431

Subgroups — #13246431

Order 1792.

1154 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6510C211yesmaximal, characteristic
2C7×#1784C211yesmaximal, characteristic, complemented
3#11729887C211yesmaximal, characteristic, complemented
4#11727721C214yesmaximal, complemented
5C23×C7×Q8C2211yescharacteristic
6C23×D28C411yescharacteristic
7C2×#6510—22no—
8C2×#6510—22no—
9C2×#6510—28no—
10C22×(C2×D7)⋊C4C412yes—
11C22×(C2×D7)⋊C4C2212yes—
12C22×C4×Dic7C411yescharacteristic
13C2×C7×#101—22no—
14C7×#407C2212yes—
15C2×#6410—22no—
16#89979C2212yes—
17#89263—28no—
18#8552—71nomaximal, Sylow
19C24×D7C2×C412yes—
20C22×C7×Q8—24no—
21C22×C7×Q8—44no—
22C22×C7×Q8D412yes—
23C22×C7×Q8D412yes—
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C23×Dic7C2×C412yes—
28C22×D28—24no—
29C22×D28—22no—
30C22×D28—44no—
31C2×C7×M16—22no—
32#6510—216no—
33#6510—216no—
34C2×(C2×D7)⋊C4—24no—
35C2×(C2×D7)⋊C4—24no—
36C2×(C2×D7)⋊C4—22no—
37C2×(C2×D7)⋊C4—24no—
38C2×(C2×D7)⋊C4—22no—
39C2×(C2×D7)⋊C4—24no—
40C2×(C2×D7)⋊C4D418yes—
41C2×C4×Dic7—22no—
42C2×C4×Dic7—24no—
43C2×#1136—22no—
44C7×#101—44no—
45C7×C22⋊C8—24no—
46#6410—44no—
47C22⋊(C7⋊C8)—24no—
48C22×#124—71no—
49#1784—71no—
50#1784D711yescharacteristic, complemented
51#1462—74no—
52C24×C7C22×C411yescharacteristic
53C23×D7—22no—
54C23×D7—22no—
55C23×D7—44no—
56C23×D7—44no—
57C23×D7—44no—
58C23×D7C22⋊C414yes—
59C2×C7×Q8—24no—
60C2×C7×Q8—24no—
61C2×C7×Q8—24no—
62C2×C7×Q8—24no—
63C2×C7×Q8—42no—
64C2×C7×Q8—42no—
65C2×C7×Q8—416no—
66C22×C4×C7—22no—
67C22×C4×C7—24no—
68C22×C4×C7—22no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D414yes—
76C22×Dic7—22no—
77C22×Dic7—22no—
78C22×Dic7—44no—
79C22×Dic7C22⋊C414yes—
80C2×D28—28no—
81C2×D28—416no—
82C2×D28—44no—
83C2×D28—44no—
84C7×M16—44no—
85C2×C7×C8—42no—
86(C2×D7)⋊C4—216no—
87(C2×D7)⋊C4—28no—
88(C2×D7)⋊C4—28no—
89C4×Dic7—28no—
90#1136—44no—
91C2×C7⋊C8—42no—
92C23×Q8C2×D711yescharacteristic
93C23×D4—71no—
94C2×#124—148no—
95C2×#124—142no—
96C2×#124—142no—
97C22×C22⋊C4—72no—
98C22×C22⋊C4—72no—
99C22×C42—71no—
100C2×#101—22no—
101C2×#101—142no—
102#407—72no—
103#407C2×D712yes—
104#323—148no—
105C23×C7—22no—
106C23×C7—24no—
107C23×C7C2×C22⋊C412yes—
108C23×C7C2×C22⋊C411yescharacteristic
109C22×D7—24no—
110C22×D7—28no—
111C22×D7—44no—
112C22×D7—44no—
113C22×D7—44no—
114C22×D7—44no—
115C22×D7—44no—
116C22×D7—44no—
117C22×D7—44no—
118C22×D7—44no—
119C22×D7—816no—
120C7×Q8—48no—
121C7×Q8—48no—
122C2×C4×C7—28no—
123C2×C4×C7—22no—
124C2×C4×C7—22no—
125C2×C4×C7—22no—
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—44no—
131C2×C4×C7—44no—
132C2×C4×C7#9914yes—
133C2×C4×C7#12012yes—
134C2×C4×C7#12012yes—
135C2×Dic7—24no—
136C2×Dic7—44no—
137C2×Dic7—44no—
138C2×Dic7—44no—
139D28—416no—
140C7×C8—82no—
141C7⋊C8—82no—
142C25—72no—
143C22×Q8—24no—
144C22×Q8—44no—
145C22×Q8D2812yes—
146C22×Q8C22⋊D712yes—
147C22×D4—144no—
148C22×D4—142no—
149C22×D4—284no—
150C23×C4—72no—
151C23×C4C4×D711yescharacteristic
152C23×C4C4×D711yescharacteristic
153C23×C4C22×D711yescharacteristic
154C2×M16—22no—
155C2×M16—142no—
156#124—1416no—
157#124—1416no—
158C2×C22⋊C4—78no—
159C2×C22⋊C4—144no—
160C2×C22⋊C4—144no—
161C2×C22⋊C4—144no—
162C2×C22⋊C4—142no—
163C2×C22⋊C4—144no—
164C2×C22⋊C4—142no—
165C2×C42—144no—
166C2×C42—142no—
167#101—44no—
168#101—284no—
169C22⋊C8—24no—
170C22⋊C8—144no—
171C22×C7—28no—
172C22×C7—22no—
173C22×C7—22no—
174C22×C7—22no—
175C22×C7#37911yescharacteristic
176C22×C7C2×#9912yes—
177C22×C7#41314yes—
178C2×D7—44no—
179C2×D7—44no—
180C2×D7—44no—
181C2×D7—88no—
182C2×D7—88no—
183C2×D7—88no—
184C4×C7—22no—
185C4×C7—22no—
186C4×C7—24no—
187C4×C7—42no—
188Dic7—44no—
189C24—74no—
190C24—142no—
191C24—142no—
192C24—284no—
193C24—284no—
194C24—284no—
195C24C2×C4×D711yescharacteristic
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—416no—
203C2×D4—148no—
204C2×D4—2816no—
205C2×D4—284no—
206C2×D4—284no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—74no—
213C22×C4—142no—
214C22×C4—142no—
215C22×C4—284no—
216C22×C4(C2×D7)⋊C412yes—
217C22×C4(C2×D7)⋊C412yes—
218C22×C4C2×D2811yescharacteristic
219C22×C4D4×D714yes—
220C22×C4C2×C22⋊D711yescharacteristic
221M16—44no—
222M16—284no—
223C2×C8—42no—
224C2×C8—282no—
225C22⋊C4—1416no—
226C22⋊C4—148no—
227C22⋊C4—148no—
228C42—148no—
229C2×C7—22no—
230C2×C7—24no—
231C2×C7#178111yescharacteristic
232C2×C7#178512yes—
233D7—84no—
234C23—22no—
235C23—24no—
236C23—144no—
237C23—148no—
238C23—284no—
239C23—284no—
240C23—284no—
241C23—284no—
242C23—284no—
243C23—284no—
244C23—284no—
245C23—284no—
246C23—5616no—
247C23D7×C22⋊C412yes—
248C23C2×(C2×D7)⋊C411yescharacteristic
249Q8—48no—
250Q8—48no—
251D4—2816no—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—144no—
263C2×C4—284no—
264C2×C4—284no—
265C2×C4—284no—
266C2×C4#638014yes—
267C2×C4#644512yes—
268C2×C4#650012yes—
269C8—82no—
270C8—562no—
271C7#855211yescharacteristic, complemented, Sylow
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—284no—
277C22—284no—
278C22—284no—
279C22—568no—
280C22—568no—
281C22—568no—
282C22#8952314yes—
283C22C2×#638012yes—
284C22#8972711yescharacteristic
285C4—22no—
286C4—22no—
287C4—24no—
288C4—42no—
289C4—284no—
290C2—22no—
291C2—24no—
292C2—564no—
293C2#1172725211yescharacteristic
294C2#1172786812yes—
subgroupG/H
centerC22#89727
commutatorC24×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1784C2
socleC22×C7#379
prime radicalC23×D28C4
cosocleC23×C4×C7C23
hypercenter#1784D7
hypocenterC7#8552
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. C7×#1784C2semidirect
3. #11729887C2semidirect
4. #11727721C2semidirect
50. #1784D7semidirect
271. C7#8552semidirect

Maximal subgroups

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

subgroupindex
C22×#65102
C7×#17842
#117298872
#117277212
#85527