Finite Group Databasebeta

← #13267879

Subgroups — #13267879

Order 1792.

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

normal subgroups only

This group has 284 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#11737375C211yesmaximal, characteristic
2#11728699C211yesmaximal, characteristic, complemented
3#11727044C211yesmaximal, characteristic
4#11726798C211yesmaximal, characteristic, complemented
5C7×Q8⋊C16C211yesmaximal, characteristic, complemented
6#11725371C211yesmaximal, characteristic, complemented
7#11725310C211yesmaximal, characteristic
8#90270—21no—
9C7×C8×Q8C2211yescharacteristic
10#89793—21no—
11#89619—22no—
12#89614—21no—
13C4×D56—21no—
14C4×D56C2211yescharacteristic
15#89455—21no—
16#89454C2211yescharacteristic
17#89454C2211yescharacteristic
18C7×C4⋊C16C2211yescharacteristic
19C4×C7×C16C2211yescharacteristic
20#89304—21no—
21#89300—21no—
22#89299—22no—
23#89276—21no—
24#89253C2211yescharacteristic
25#12411—71nomaximal, Sylow
26C4×C7×Q8D411yescharacteristic
27#6509—41no—
28C2×D56D411yescharacteristic
29#6459—42no—
30D7×C4⋊C4—41no—
31C4×D28—22no—
32C4×D28—21no—
33#6433—22no—
34#6432—21no—
35#6432—21no—
36C2×C7×C16—21no—
37C2×C7×C16—21no—
38C7×C4⋊C8—21no—
39C7×C4⋊C8D411yescharacteristic
40C4×C7×C8—21no—
41C4×C7×C8C2311yescharacteristic
42#6409—41no—
43#6395—22no—
44#6395—21no—
45#6392—21no—
46#6392—21no—
47#6392—21no—
48#6392D412yes—
49#6392D411yescharacteristic
50#6391—22no—
51#6381—41no—
52C4⋊(C7⋊C8)—21no—
53#3380—71no—
54#2252—71no—
55#2226—71no—
56#1676—71no—
57#1365—71no—
58Q8⋊C16D711yescharacteristic, complemented
59#1331—71no—
60C2×C7×Q8D811yescharacteristic
61C2×D28—21no—
62C2×C4×D7—41no—
63C2×C4×D7—42no—
64C2×Dic14—41no—
65C2×Dic14—41no—
66C2×C7×C8—21no—
67C2×C7×C8C2×D411yescharacteristic
68C2×C7×C8C2×D411yescharacteristic
69C7×C4⋊C4—21no—
70C7×C4⋊C4D811yescharacteristic
71C42×C7—21no—
72C42×C7C2×D411yescharacteristic
73(C2×D7)⋊C4—42no—
74(C2×D7)⋊C4—41no—
75C4⋊Dic7—22no—
76C4⋊Dic7—21no—
77C4⋊Dic7—41no—
78C4⋊Dic7—41no—
79C4⋊Dic7—41no—
80Dic7⋊C4—42no—
81Dic7⋊C4—42no—
82C2×C7⋊C8—41no—
83D56—21no—
84D56Q1612yes—
85C7×C16—22no—
86C7×C16—41no—
87#554—141no—
88#500—71no—
89#500—71no—
90#499—141no—
91#478—142no—
92#477—141no—
93#476—141no—
94C8×Q8C2×D711yescharacteristic
95C4×D8—71no—
96C4×D8—141no—
97#367—141no—
98#366—142no—
99C4⋊C16C2×D711yescharacteristic
100#357—141no—
101C4×C16C2×D711yescharacteristic
102#335—71no—
103#333—141no—
104C22×D7—41no—
105C7×Q8—21no—
106C7×Q8—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7#12011yescharacteristic
110C2×C4×C7C2×D811yescharacteristic
111C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
112C2×Dic7—41no—
113C2×Dic7—42no—
114C2×Dic7—81no—
115D28—22no—
116D28—41no—
117C4×D7—81no—
118C4×D7—81no—
119C4×D7—162no—
120Dic14—161no—
121Dic14—161no—
122C7×C8—21no—
123C7×C8—41no—
124C7×C8C2×Q1611yescharacteristic
125C7×C8#13711yescharacteristic
126C7⋊C8—81no—
127C2×D8—71no—
128C4⋊Q8—141no—
129C4⋊Q8—141no—
130#125—142no—
131C22⋊Q8—282no—
132C22⋊Q8—281no—
133C4×Q8D2811yescharacteristic
134C4×D4—142no—
135C4×D4—141no—
136C2×C4⋊C4—281no—
137C2×C16—21no—
138C2×C16—21no—
139#107—72no—
140#107—71no—
141#107—141no—
142#107—141no—
143#107—141no—
144#107—281no—
145#106—142no—
146C4⋊C8—21no—
147C4⋊C8—141no—
148C4⋊C8D2811yescharacteristic
149#103—281no—
150#102—142no—
151#102—141no—
152C4×C8—21no—
153C4×C8C22×D711yescharacteristic
154C22×C7#44711yescharacteristic
155C2×D7—42no—
156C2×D7—81no—
157C2×D7—81no—
158C4×C7—21no—
159C4×C7—41no—
160C4×C7—41no—
161C4×C7#45111yescharacteristic
162C4×C7#45611yescharacteristic
163C4×C7#50811yescharacteristic
164C4×C7#51011yescharacteristic
165Dic7—81no—
166Dic7—82no—
167Dic7—161no—
168C2×Q8—281no—
169C2×Q8—281no—
170C2×Q8D5611yescharacteristic
171C2×D4—141no—
172C22×C4—282no—
173C22×C4—281no—
174D8—72no—
175D8—141no—
176C2×C8—21no—
177C2×C8—281no—
178C2×C8D4×D711yescharacteristic
179C2×C8D4×D711yescharacteristic
180C4⋊C4—21no—
181C4⋊C4—141no—
182C4⋊C4—142no—
183C4⋊C4—282no—
184C4⋊C4—282no—
185C4⋊C4—281no—
186C4⋊C4—281no—
187C4⋊C4—281no—
188C4⋊C4D5611yescharacteristic
189C22⋊C4—282no—
190C22⋊C4—281no—
191C42—21no—
192C42C2×D2811yescharacteristic
193C16—22no—
194C16—41no—
195C2×C7#164911yescharacteristic
196C2×C7#218911yescharacteristic
197C2×C7#219211yescharacteristic
198D7—82no—
199C23—281no—
200Q8—21no—
201Q8—21no—
202Q8—1121no—
203Q8—1121no—
204D4—142no—
205D4—281no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—281no—
209C2×C4—282no—
210C2×C4—561no—
211C2×C4—561no—
212C2×C4—561no—
213C2×C4—1122no—
214C2×C4#644511yescharacteristic
215C2×C4C2×D5611yescharacteristic
216C2×C4#647111yescharacteristic
217C8—21no—
218C8—41no—
219C8—561no—
220C8#647811yescharacteristic
221C8D7×Q1611yescharacteristic
222C7#1241111yescharacteristic, complemented, Sylow
223C22—282no—
224C22—561no—
225C22—561no—
226C22#8950411yescharacteristic
227C4—21no—
228C4—41no—
229C4—41no—
230C4—561no—
231C4—562no—
232C4—1121no—
233C4#8958011yescharacteristic
234C4#8959811yescharacteristic
235C4#8967611yescharacteristic
236C4#8968111yescharacteristic
237C2—562no—
238C2#1172602711yescharacteristic
239C2#1172686511yescharacteristic
240C2#1172688011yescharacteristic
subgroupG/H
centerC22#89504
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×Q8⋊C16C2
socleC22×C7#447
prime radicalC2×D56D4
hypercenterQ8⋊C16D7
hypocenterC7#12411
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ Q8⋊C16

Complements

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

NH ≅ G/Ntype
2. #11728699C2semidirect
4. #11726798C2semidirect
5. C7×Q8⋊C16C2semidirect
6. #11725371C2semidirect
58. Q8⋊C16D7semidirect
222. C7#12411semidirect

Maximal subgroups

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

subgroupindex
#117373752
#117286992
#117270442
#117267982
C7×Q8⋊C162
#117253712
#117253102
#124117