Finite Group Databasebeta

← #13267520

Subgroups — #13267520

Order 1792.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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#11737019C211yesmaximal, characteristic
2C7×#2208C211yesmaximal, characteristic
3#11729064C211yesmaximal, characteristic, complemented
4#11729062C211yesmaximal, characteristic, complemented
5#11725569C211yesmaximal, characteristic
6#11725371C211yesmaximal, characteristic, complemented
7#11725364C211yesmaximal, characteristic, complemented
8#90234—21no—
9C2×C7×SD32—21no—
10C7×#494C2211yescharacteristic
11C4×C7×D8C2211yescharacteristic
12D8×Dic7—21no—
13#89866—21no—
14#89780—21no—
15#89650—21no—
16#89538—22no—
17#89454C2211yescharacteristic
18C8×Dic14C2211yescharacteristic
19C7×C4⋊C16C2211yescharacteristic
20C7×#358—21no—
21#89359—22no—
22#89342—21no—
23C4×C7⋊C16C2211yescharacteristic
24#89253C2211yescharacteristic
25#12359—71nomaximal, Sylow
26C2×C7×Q16—21no—
27C2×C7×Q16—21no—
28C2×C7×D8D411yescharacteristic
29C7×C4⋊Q8—21no—
30C4×C7×D4—21no—
31D4×Dic7—22no—
32C2×C4⋊Dic7—41no—
33#6487—41no—
34#6453—22no—
35#6441—42no—
36#6432—21no—
37C4×Dic14D411yescharacteristic
38C7×SD32—42no—
39C2×C7×C16—21no—
40C7×#107D411yescharacteristic
41C7×#102—21no—
42C4×C7×C8C2311yescharacteristic
43#6406—22no—
44#6392—21no—
45#6392D412yes—
46Dic7⋊C8—21no—
47C8×Dic7—21no—
48C2×C7⋊C16—21no—
49#6384—41no—
50#6382—22no—
51#6382—41no—
52C4⋊(C7⋊C8)—21no—
53C4⋊(C7⋊C8)D411yescharacteristic
54#3380—71no—
55#2241—71no—
56#2208D711yescharacteristic
57#1670—71no—
58#1365—71no—
59Q8⋊C16—71no—
60#1331—71no—
61C2×C7×Q8—41no—
62C2×C7×D4—21no—
63C22×C4×C7—41no—
64C22×Dic7—42no—
65C2×Dic14—41no—
66C2×Dic14D811yescharacteristic
67C7×Q16—42no—
68C7×Q16—41no—
69C7×Q16—41no—
70C7×D8—21no—
71C7×D8SD1612yes—
72C2×C7×C8C2×D411yescharacteristic
73C2×C7×C8C2×D411yescharacteristic
74C7×C4⋊C4—21no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
77C7×C22⋊C4—41no—
78C42×C7C2×D411yescharacteristic
79C22⋊Dic7—42no—
80C4⋊Dic7—22no—
81C4⋊Dic7—42no—
82C4⋊Dic7—41no—
83C4⋊Dic7D811yescharacteristic
84Dic7⋊C4—21no—
85Dic7⋊C4—42no—
86C4×Dic7—21no—
87C2×C7⋊C8—21no—
88C2×C7⋊C8—41no—
89C7×C16—41no—
90C7⋊C16—41no—
91#554—141no—
92C2×SD32—21no—
93#500—71no—
94#499—141no—
95#494C2×D711yescharacteristic
96#478—142no—
97#476—141no—
98#475—141no—
99C8×Q8—71no—
100C4×D8—141no—
101C4×D8C2×D711yescharacteristic
102C4⋊C16C2×D711yescharacteristic
103#358—21no—
104#358—142no—
105C4×C16—71no—
106#335—71no—
107#332—141no—
108C23×C7—41no—
109C7×Q8—81no—
110C7×Q8—81no—
111C7×D4—22no—
112C7×D4—41no—
113C2×C4×C7—41no—
114C2×C4×C7—81no—
115C2×C4×C7—81no—
116C2×C4×C7—81no—
117C2×C4×C7#12011yescharacteristic
118C2×C4×C7C2×D811yescharacteristic
119C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
120C2×Dic7—21no—
121C2×Dic7—21no—
122C2×Dic7—42no—
123C2×Dic7—162no—
124Dic14—21no—
125Dic14—21no—
126Dic14—161no—
127C7×C8—21no—
128C7×C8C2×SD1611yescharacteristic
129C7×C8#13711yescharacteristic
130C7⋊C8—41no—
131C7⋊C8—81no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×D8C22⋊D711yescharacteristic
135C4⋊Q8—21no—
136C4⋊Q8—141no—
137#125—142no—
138C22⋊Q8—282no—
139C22⋊Q8—281no—
140C4×Q8—71no—
141C4×D4—21no—
142C4×D4—142no—
143C2×C4⋊C4—281no—
144SD32—42no—
145C2×C16—21no—
146C2×C16—141no—
147#107—72no—
148#107—141no—
149#107C22⋊D711yescharacteristic
150#106—142no—
151#106—281no—
152C4⋊C8—71no—
153C4⋊C8—141no—
154C4⋊C8—141no—
155#103—281no—
156#102—21no—
157#102—142no—
158C4×C8—141no—
159C4×C8C22×D711yescharacteristic
160C22×C7—42no—
161C22×C7—81no—
162C22×C7—81no—
163C22×C7#44711yescharacteristic
164C4×C7—21no—
165C4×C7—81no—
166C4×C7—161no—
167C4×C7#44811yescharacteristic
168C4×C7#45411yescharacteristic
169C4×C7#50811yescharacteristic
170C4×C7#51011yescharacteristic
171Dic7—41no—
172Dic7—41no—
173Dic7—82no—
174C2×Q8—41no—
175C2×Q8—71no—
176C2×Q8—281no—
177C2×D4—21no—
178C22×C4—41no—
179C22×C4—282no—
180Q16—42no—
181Q16—41no—
182Q16—41no—
183D8—21no—
184D8#114212yes—
185C2×C8—141no—
186C2×C8—281no—
187C2×C8D4×D711yescharacteristic
188C2×C8C2×C22⋊D711yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—71no—
193C4⋊C4—141no—
194C4⋊C4—142no—
195C4⋊C4—282no—
196C4⋊C4—282no—
197C4⋊C4—281no—
198C22⋊C4—41no—
199C22⋊C4—282no—
200C42—141no—
201C42D4×D711yescharacteristic
202C16—41no—
203C16—281no—
204C2×C7—82no—
205C2×C7#162711yescharacteristic
206C2×C7#218911yescharacteristic
207C2×C7#219411yescharacteristic
208C23—41no—
209Q8—81no—
210Q8—81no—
211Q8—141no—
212Q8—141no—
213Q8—1121no—
214D4—22no—
215D4—41no—
216C2×C4—41no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—141no—
221C2×C4—141no—
222C2×C4—282no—
223C2×C4—1122no—
224C2×C4D7×D811yescharacteristic
225C2×C4#647411yescharacteristic
226C2×C4#650011yescharacteristic
227C8—21no—
228C8—281no—
229C8—561no—
230C8C2×#114211yescharacteristic
231C8#651411yescharacteristic
232C7#1235911yescharacteristic, complemented, Sylow
233C22—42no—
234C22—81no—
235C22—81no—
236C22#8992811yescharacteristic
237C4—21no—
238C4—81no—
239C4—161no—
240C4—281no—
241C4—281no—
242C4—562no—
243C4#8968711yescharacteristic
244C4#8968811yescharacteristic
245C4#8981211yescharacteristic
246C4#8997311yescharacteristic
247C2—82no—
248C2#1172876811yescharacteristic
249C2#1172924011yescharacteristic
250C2#1172927711yescharacteristic
subgroupG/H
centerC22#89928
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#2208C2
socleC22×C7#447
prime radicalC2×C7×D8D4
hypercenter#2208D7
hypocenterC7#12359
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2208

Complements

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

NH ≅ G/Ntype
3. #11729064C2semidirect
4. #11729062C2semidirect
6. #11725371C2semidirect
7. #11725364C2semidirect
232. C7#12359semidirect

Maximal subgroups

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

subgroupindex
#117370192
C7×#22082
#117290642
#117290622
#117255692
#117253712
#117253642
#123597