Finite Group Databasebeta

← #13266258

Subgroups — #13266258

Order 1792.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#11736482C211yesmaximal, characteristic
2C7×#2056C211yesmaximal, characteristic
3#11729818C212yesmaximal
4#11725515C211yesmaximal, characteristic
5#11725292C212yesmaximal
6C22×C7×Q16C2211yescharacteristic
7#90161—21no—
8C2×C7×C4⋊C8C2211yescharacteristic
9C2×C7×#103—21no—
10C7×#391—21no—
11C2×#6403—22no—
12#89888—21no—
13#89888C2212yes—
14#89622—22no—
15#89621—22no—
16C7×#340C2211yescharacteristic
17#89359—22no—
18C22⋊(C7⋊C16)C2212yes—
19#12162—71nomaximal, Sylow
20C22×C7×Q8—21no—
21C2×C7×Q16—22no—
22C2×C7×Q16—42no—
23C22×C7×C8D411yescharacteristic
24C22×C7×C8C2311yescharacteristic
25C2×C7×C4⋊C4—21no—
26C2×C42×C7D411yescharacteristic
27#6489—41no—
28#6487—22no—
29#6487—41no—
30C2×Dic28—21no—
31C2×Dic28D412yes—
32#6433—41no—
33#6432—21no—
34#6432—21no—
35C4×Dic14—22no—
36C4×Dic14—41no—
37C7×C4⋊C8—22no—
38C7×#103—42no—
39C7×#95—41no—
40C7×#95—41no—
41#6403—44no—
42#6392—21no—
43#6392D412yes—
44#6390—22no—
45#6390—22no—
46C2×C7⋊C16—22no—
47#3249—71no—
48#2188—72no—
49#2056D711yescharacteristic
50#1348—72no—
51#1322—71no—
52C2×C7×Q8—22no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8—41no—
58C22×C4×C7—41no—
59C22×C4×C7—41no—
60C22×C4×C7D811yescharacteristic
61C22×C4×C7D811yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C2×Dic14—22no—
64C2×Dic14—41no—
65C7×Q16—24no—
66C7×Q16—42no—
67C2×C7×C8—21no—
68C2×C7×C8—41no—
69C2×C7×C8—41no—
70C2×C7×C8C2×D412yes—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—41no—
73C42×C7—21no—
74C42×C7—21no—
75C22⋊Dic7—42no—
76C4⋊Dic7—22no—
77C4⋊Dic7—41no—
78C4⋊Dic7—41no—
79C4⋊Dic7—41no—
80Dic7⋊C4—42no—
81Dic7⋊C4—42no—
82C4×Dic7—42no—
83Dic28—42no—
84Dic28—81no—
85C7⋊C16—42no—
86C22×Q16C2×D711yescharacteristic
87#533—141no—
88C2×Q32—142no—
89#477—142no—
90#467—72no—
91#467—141no—
92#462—142no—
93C2×C4⋊C8C2×D711yescharacteristic
94C2×#103—21no—
95#391—21no—
96#358—142no—
97C22⋊C16—72no—
98#340C2×D711yescharacteristic
99C23×C7C2×D811yescharacteristic
100C7×Q8—42no—
101C7×Q8—42no—
102C7×Q8—81no—
103C7×Q8—81no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—41no—
110C2×C4×C7—42no—
111C2×C4×C7—81no—
112C2×C4×C7—81no—
113C2×C4×C7—81no—
114C2×C4×C7—81no—
115C2×C4×C7—81no—
116C2×C4×C7Q3212yescomplemented
117C2×C4×C7#12011yescharacteristic
118C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
119C2×Dic7—42no—
120C2×Dic7—42no—
121Dic14—42no—
122Dic14—42no—
123Dic14—82no—
124C7×C8—22no—
125C7×C8—41no—
126C22×Q8—21no—
127C2×Q16—22no—
128C2×Q16—42no—
129C2×Q16—72no—
130C2×Q16—141no—
131C22×C8C22⋊D711yescharacteristic
132C22×C8C22×D711yescharacteristic
133C4⋊Q8—141no—
134C4⋊Q8—141no—
135#125—281no—
136C22⋊Q8—142no—
137C22⋊Q8—281no—
138C4×Q8—142no—
139C4×Q8—281no—
140#117—281no—
141C2×C4⋊C4—21no—
142C2×C42C22⋊D711yescharacteristic
143Q32—284no—
144C2×C16—142no—
145#107—72no—
146#107—141no—
147C4⋊C8—22no—
148#103—42no—
149#103—142no—
150#103—142no—
151#95—41no—
152#95—41no—
153C22×C7—21no—
154C22×C7—21no—
155C22×C7#44711yescharacteristic
156C22×C7C2×Q3211yescharacteristic
157C22×C7#51011yescharacteristic
158C4×C7—21no—
159C4×C7—22no—
160C4×C7—41no—
161C4×C7—82no—
162C4×C7—161no—
163C4×C7#45411yescharacteristic
164C4×C7B(4,𝔽2)11yescharacteristic
165Dic7—82no—
166Dic7—82no—
167C2×Q8—22no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—142no—
174C2×Q8—281no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4#114111yescharacteristic
178C22×C4#114111yescharacteristic
179C22×C4C2×C22⋊D711yescharacteristic
180Q16—24no—
181Q16—42no—
182Q16—282no—
183Q16—561no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8D4×D712yes—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—142no—
191C4⋊C4—282no—
192C4⋊C4—281no—
193C4⋊C4—281no—
194C4⋊C4—281no—
195C4⋊C4—282no—
196C22⋊C4—282no—
197C42—21no—
198C42—21no—
199C42—282no—
200C16—282no—
201C2×C7—21no—
202C2×C7—21no—
203C2×C7#159811yescharacteristic
204C2×C7#218811yescharacteristic
205C2×C7#219111yescharacteristic
206C23C2×#114111yescharacteristic
207Q8—42no—
208Q8—42no—
209Q8—81no—
210Q8—81no—
211Q8—282no—
212Q8—282no—
213Q8—562no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—282no—
227C2×C4—282no—
228C2×C4#640312yescomplemented
229C2×C4#650011yescharacteristic
230C2×C4#651211yescharacteristic
231C8—22no—
232C8—41no—
233C7#1216211yescharacteristic, complemented, Sylow
234C22—21no—
235C22—21no—
236C22#8980811yescharacteristic
237C22C2×#640311yescharacteristic
238C22#8996711yescharacteristic
239C4—21no—
240C4—22no—
241C4—41no—
242C4—82no—
243C4—161no—
244C4—562no—
245C4—562no—
246C4#8980411yescharacteristic
247C4#8983411yescharacteristic
248C2—21no—
249C2—21no—
250C2#1172745711yescharacteristic
251C2#1172884111yescharacteristic
252C2#1172884511yescharacteristic
subgroupG/H
centerC22#89808
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#2056C2
socleC22×C7#447
prime radicalC23×C7C2×D8
hypercenter#2056D7
hypocenterC7#12162
LCSG ▷ C22×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2056

Complements

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

NH ≅ G/Ntype
116. C2×C4×C7Q32semidirect
228. C2×C4#6403semidirect
233. C7#12162semidirect

Maximal subgroups

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

subgroupindex
#117364822
C7×#20562
#117298182
#117255152
#117252922
#121627