Finite Group Databasebeta

← #13248172

Subgroups — #13248172

Order 1792.

428 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 119 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#11740472C211yesmaximal, characteristic
2C2×C7×#330C211yesmaximal, characteristic
3#11729495C211yesmaximal, characteristic
4#11727570C212yesmaximal
5#11725465C212yesmaximal
6C2×C7×#125C2211yescharacteristic
7C2×C7×#97C2212yes—
8C2×#6392—22no—
9C2×#6391—22no—
10#89757—22no—
11#89753—22no—
12#89753—22no—
13#89708C2212yes—
14#89707C2211yescharacteristic
15C2×#6377C2211yescharacteristic
16C7×#330C414yescomplemented
17#89324—22no—
18#8913—71nomaximal, Sylow
19C22×C7×C8—22no—
20C7×#125—22no—
21C7×#125C2×C412yes—
22C7×#125Q812yes—
23C2×C7×C4⋊C4—22no—
24C2×C7×C4⋊C4D412yes—
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—
32C22×C7⋊C8—21no—
33C7×#97—22no—
34C7×#97C2×C414yes—
35C7⋊#95—22no—
36C7⋊#95—22no—
37C7⋊#95—22no—
38C7⋊#95—22no—
39#6392—44no—
40#6391—44no—
41#6377—22no—
42#2382—71no—
43#1992—71no—
44#1847—72no—
45C2×#330D711yescharacteristic
46#1295—72no—
47C22×C4×C7—22no—
48C22×C4×C7C2×D411yescharacteristic
49C22×C4×C7C2×D412yes—
50C22×Dic7—22no—
51C22×Dic7—22no—
52C22×Dic7—41no—
53C2×C7×C8—24no—
54C2×C7×C8—24no—
55C2×C7×C8—24no—
56C7×C4⋊C4—24no—
57C7×C4⋊C4—24no—
58C7×C4⋊C4—22no—
59C7×C4⋊C4C4⋊C414yes—
60C42×C7C22×C411yescharacteristic
61C42×C7C2×Q811yescharacteristic
62C42×C7#4112yes—
63C4⋊Dic7—42no—
64C4⋊Dic7—42no—
65C4⋊Dic7—42no—
66C4⋊Dic7—42no—
67C4⋊Dic7—42no—
68C4⋊Dic7—82no—
69C4×Dic7—82no—
70C2×C7⋊C8—22no—
71C2×C7⋊C8—41no—
72C2×C7⋊C8—41no—
73C2×#125C2×D711yescharacteristic
74C2×#107—142no—
75C2×#106—142no—
76C2×#97—71no—
77C2×#97C2×D712yes—
78#389—142no—
79#387—142no—
80#387—142no—
81#384—71no—
82#382—72no—
83#340—142no—
84#330Dic714yescomplemented
85C23×C7#12011yescharacteristic
86C2×C4×C7—22no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7—24no—
91C2×C4×C7—42no—
92C2×C4×C7—42no—
93C2×C4×C7C2×C4⋊C411yescharacteristic
94C2×C4×C7C4×D412yes—
95C2×C4×C7C22⋊Q812yes—
96C2×C4×C7#12311yescharacteristic
97C2×Dic7—42no—
98C2×Dic7—42no—
99C2×Dic7—42no—
100C2×Dic7—42no—
101C2×Dic7—42no—
102C2×Dic7—42no—
103C2×Dic7—82no—
104C2×Dic7—81no—
105C7×C8—28no—
106C7⋊C8—42no—
107C22×C8—22no—
108C22×C8—141no—
109#125—22no—
110#125Dic1412yes—
111#125C2×Dic712yes—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4—72no—
115C2×C4⋊C4—141no—
116C2×C4⋊C4—142no—
117C2×C4⋊C4—142no—
118C2×C4⋊C4D2812yes—
119C2×C42—142no—
120C2×C42C22×D711yescharacteristic
121#107—284no—
122#106—284no—
123#97—22no—
124#97—142no—
125#97C2×Dic714yes—
126#95—142no—
127#95—142no—
128#95—142no—
129#95—142no—
130C22×C7#38511yescharacteristic
131C22×C7#45312yes—
132C22×C7#45411yescharacteristic
133C22×C7#45611yescharacteristic
134C22×C7#45612yes—
135C4×C7—21no—
136C4×C7—21no—
137C4×C7—22no—
138C4×C7—22no—
139C4×C7—44no—
140Dic7—82no—
141Dic7—82no—
142Dic7—161no—
143C22×C4—22no—
144C22×C4—142no—
145C22×C4—142no—
146C22×C4—281no—
147C22×C4C2×D2811yescharacteristic
148C22×C4D4×D712yes—
149C2×C8—24no—
150C2×C8—24no—
151C2×C8—24no—
152C2×C8—142no—
153C2×C8—281no—
154C2×C8—281no—
155C4⋊C4—24no—
156C4⋊C4—22no—
157C4⋊C4—24no—
158C4⋊C4—282no—
159C4⋊C4—282no—
160C4⋊C4—282no—
161C4⋊C4—282no—
162C4⋊C4—282no—
163C4⋊C4—562no—
164C4⋊C4C4⋊Dic714yes—
165C42—562no—
166C42C2×Dic1411yescharacteristic
167C42#115912yes—
168C42C22×Dic711yescharacteristic
169C2×C7#165611yescharacteristic
170C2×C7#166012yes—
171C2×C7#166311yescharacteristic
172C2×C7#185711yescharacteristic
173C2×C7#185912yes—
174C23#644511yescharacteristic
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—282no—
183C2×C4—282no—
184C2×C4—282no—
185C2×C4—282no—
186C2×C4—282no—
187C2×C4—282no—
188C2×C4—561no—
189C2×C4—562no—
190C2×C4#644112yes—
191C2×C4#644911yescharacteristic
192C2×C4C2×C4⋊Dic711yescharacteristic
193C2×C4D4×Dic712yes—
194C8—28no—
195C8—282no—
196C7#891311yescharacteristic, complemented, Sylow
197C22#8952411yescharacteristic
198C22#8952511yescharacteristic
199C22#8959412yes—
200C22#8959812yes—
201C22#8972211yescharacteristic
202C4—21no—
203C4—21no—
204C4—22no—
205C4—22no—
206C4—44no—
207C4—562no—
208C4—562no—
209C4—1121no—
210C2#1172612411yescharacteristic
211C2#1172612612yes—
212C2#1172613111yescharacteristic
213C2#1172785611yescharacteristic
214C2#1172811712yes—
subgroupG/H
centerC23#6445
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC2×C7×#330C2
socle, prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenterC2×#330D7
hypocenterC7#8913
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
16. C7×#330C4semidirect
84. #330Dic7semidirect
196. C7#8913semidirect

Maximal subgroups

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

subgroupindex
#117404722
C2×C7×#3302
#117294952
#117275702
#117254652
#89137