Finite Group Databasebeta

← #13248263

Subgroups — #13248263

Order 1792.

476 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 155 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#11740675C211yesmaximal, characteristic
2C2×C7×#333C211yesmaximal, characteristic
3#11729539C211yesmaximal, characteristic
4#11728208C212yesmaximal
5#11725510C212yesmaximal
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×C4⋊C8C2212yes—
8#90002—22no—
9C2×#6409—22no—
10C2×#6391—22no—
11#89757—22no—
12C4⋊C4×Dic7—22no—
13#89707C2211yescharacteristic
14C4×C4⋊Dic7C2212yes—
15C2×C4×C7⋊C8C2211yescharacteristic
16C7×#333C414yescomplemented
17#89346—22no—
18#8928—71nomaximal, Sylow
19C22×C7×Q8D412yes—
20C22×C7×C8—22no—
21C7×C4⋊Q8—22no—
22C7×C4⋊Q8C2×C412yes—
23C7×C4⋊Q8Q812yes—
24C2×C7×C4⋊C4—22no—
25C2×C42×C7C2311yescharacteristic
26C2×C4⋊Dic7—22no—
27C2×C4⋊Dic7—21no—
28C2×C4⋊Dic7—22no—
29C2×C4⋊Dic7D412yes—
30C2×C4⋊Dic7D412yes—
31C2×C4×Dic7—22no—
32C2×C4×Dic7—22no—
33C22×C7⋊C8—21no—
34C7×C4⋊C8—22no—
35C7×C4⋊C8C2×C414yes—
36#6409—44no—
37C7⋊#95—24no—
38C7⋊#95—22no—
39#6391—44no—
40C4×C7⋊C8—22no—
41#2369—71no—
42#1956—71no—
43#1948—72no—
44C2×#333D711yescharacteristic
45#1279—72no—
46C2×C7×Q8—22no—
47C2×C7×Q8—42no—
48C2×C7×Q8—42no—
49C2×C7×Q8C4⋊C414yes—
50C22×C4×C7—22no—
51C22×C4×C7C2×D411yescharacteristic
52C22×C4×C7C2×D412yes—
53C22×Dic7—22no—
54C22×Dic7—22no—
55C22×Dic7—41no—
56C2×C7×C8—24no—
57C2×C7×C8—42no—
58C2×C7×C8—42no—
59C7×C4⋊C4—24no—
60C7×C4⋊C4—24no—
61C42×C7C22×C411yescharacteristic
62C42×C7C2×Q811yescharacteristic
63C42×C7#4112yes—
64C4⋊Dic7—22no—
65C4⋊Dic7—22no—
66C4⋊Dic7—22no—
67C4⋊Dic7—42no—
68C4⋊Dic7—42no—
69C4⋊Dic7—42no—
70C4⋊Dic7—82no—
71C4⋊Dic7SD1614yes—
72C4×Dic7—42no—
73C4×Dic7—42no—
74C4×Dic7—82no—
75C2×C7⋊C8—22no—
76C2×C7⋊C8—41no—
77C2×C7⋊C8—41no—
78C2×C4⋊Q8C2×D711yescharacteristic
79C2×#106—142no—
80C2×C4⋊C8C2×D712yes—
81C2×#103—142no—
82C2×C4×C8—71no—
83#391—142no—
84#389—142no—
85#384—71no—
86C4×C4⋊C4—72no—
87C4×C4⋊C4—142no—
88#340—142no—
89#333Dic714yescomplemented
90C23×C7#12011yescharacteristic
91C7×Q8—44no—
92C7×Q8—82no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—24no—
96C2×C4×C7—42no—
97C2×C4×C7—42no—
98C2×C4×C7C2×C4⋊C411yescharacteristic
99C2×C4×C7C4×D412yes—
100C2×C4×C7C22⋊Q812yes—
101C2×C4×C7#12311yescharacteristic
102C2×C4×C7C2×SD1612yes—
103C2×C4×C7#13512yes—
104C2×C4×C7#13712yes—
105C2×C4×C7#13712yes—
106C2×Dic7—24no—
107C2×Dic7—42no—
108C2×Dic7—42no—
109C2×Dic7—42no—
110C2×Dic7—42no—
111C2×Dic7—42no—
112C2×Dic7—82no—
113C2×Dic7—81no—
114C7×C8—44no—
115C7⋊C8—42no—
116C22×Q8D2812yes—
117C22×C8—22no—
118C22×C8—141no—
119C4⋊Q8—22no—
120C4⋊Q8Dic1412yes—
121C4⋊Q8C2×Dic712yes—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—72no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4—141no—
126C2×C4⋊C4—142no—
127C2×C4⋊C4—142no—
128C2×C42—142no—
129C2×C42—142no—
130C2×C42C22×D711yescharacteristic
131#106—284no—
132C4⋊C8—22no—
133C4⋊C8C2×Dic714yes—
134#103—284no—
135C4×C8—142no—
136#95—144no—
137#95—142no—
138C22×C7#38511yescharacteristic
139C22×C7#44812yes—
140C22×C7#45212yes—
141C22×C7#45311yescharacteristic
142C22×C7#45511yescharacteristic
143C4×C7—21no—
144C4×C7—21no—
145C4×C7—44no—
146C4×C7C4×SD1612yes—
147C4×C7#44112yes—
148C4×C7#47512yes—
149C4×C7#47912yes—
150Dic7—44no—
151Dic7—82no—
152Dic7—161no—
153C2×Q8—22no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8C4⋊Dic714yes—
157C22×C4—22no—
158C22×C4—142no—
159C22×C4—142no—
160C22×C4—281no—
161C22×C4C2×D2811yescharacteristic
162C22×C4D4×D712yes—
163C2×C8—24no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—142no—
167C2×C8—281no—
168C2×C8—281no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—74no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—282no—
176C4⋊C4—282no—
177C4⋊C4—282no—
178C4⋊C4—562no—
179C42—282no—
180C42—282no—
181C42—562no—
182C42C2×Dic1411yescharacteristic
183C42#115912yes—
184C42C22×Dic711yescharacteristic
185C2×C7#162311yescharacteristic
186C2×C7#163411yescharacteristic
187C2×C7#164312yes—
188C2×C7#185611yescharacteristic
189C2×C7#186512yes—
190C23#644511yescharacteristic
191Q8—44no—
192Q8—82no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—144no—
199C2×C4—282no—
200C2×C4—282no—
201C2×C4—282no—
202C2×C4—282no—
203C2×C4—282no—
204C2×C4—561no—
205C2×C4—562no—
206C2×C4#644112yes—
207C2×C4#644911yescharacteristic
208C2×C4D7×SD1612yes—
209C2×C4#647912yes—
210C2×C4#648112yes—
211C2×C4#648112yes—
212C2×C4C2×C4⋊Dic711yescharacteristic
213C2×C4D4×Dic712yes—
214C8—44no—
215C8—282no—
216C7#892811yescharacteristic, complemented, Sylow
217C22#8951811yescharacteristic
218C22#8951911yescharacteristic
219C22#8957812yes—
220C22#8958212yes—
221C22#8972211yescharacteristic
222C4—21no—
223C4—21no—
224C4—44no—
225C4—284no—
226C4—562no—
227C4—1121no—
228C4#8956312yes—
229C4#8957112yes—
230C4SD16×Dic712yes—
231C4#8995612yes—
232C2#1172624711yescharacteristic
233C2#1172624812yes—
234C2#1172625811yescharacteristic
235C2#1172782211yescharacteristic
236C2#1172799212yes—
subgroupG/H
centerC23#6445
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC2×C7×#333C2
socle, prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenterC2×#333D7
hypocenterC7#8928
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
16. C7×#333C4semidirect
89. #333Dic7semidirect
216. C7#8928semidirect

Maximal subgroups

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

subgroupindex
#117406752
C2×C7×#3332
#117295392
#117282082
#117255102
#89287