Finite Group Databasebeta

← #13251194

Subgroups — #13251194

Order 1792.

466 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 115 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#11739923C211yesmaximal, characteristic
2#11739881C211yesmaximal, characteristic
3C7×#1801C211yesmaximal, characteristic
4#11728261C212yesmaximal
5#11725521C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6432C2211yescharacteristic
8C7×#423C2212yes—
9C2×C7×#101—21no—
10#90002—21no—
11#89796—24no—
12#89751C414yes—
13#89748—21no—
14C4×C4⋊Dic7C2211yescharacteristic
15#89700—22no—
16#89698—22no—
17#89697C2212yes—
18C7×#343—21no—
19#89276—24no—
20#9400—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×Dic14C2×C412yes—
23C2×C7×M16—22no—
24C4×C7×Q8—22no—
25C4×C7×Q8—22no—
26C2×C7×C4⋊C4C2×C412yes—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C42×C7D412yes—
29C2×C42×C7C2311yescharacteristic
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×Dic7⋊C4—24no—
34C2×C4×Dic7—21no—
35#6432—22no—
36#6432—22no—
37C7×C4⋊C8—22no—
38C7×C4⋊C8—22no—
39C7×#101—42no—
40C7⋊#95—24no—
41C7⋊#95C2×C412yes—
42C22⋊(C7⋊C8)—22no—
43C4⋊(C7⋊C8)—24no—
44C4⋊(C7⋊C8)D414yes—
45#6377—24no—
46C4×C7⋊C8—22no—
47#2983—71no—
48#2567—71no—
49#1801D711yescharacteristic
50#1548—72no—
51#1303—72no—
52C2×C7×Q8—22no—
53C2×C7×Q8—21no—
54C2×C7×Q8—21no—
55C2×C7×Q8—22no—
56C22×C4×C7C22⋊C412yes—
57C22×C4×C7C22⋊C412yes—
58C22×C4×C7C22×C411yescharacteristic
59C22×C4×C7C2×D411yescharacteristic
60C22×C4×C7C2×D411yescharacteristic
61C22×Dic7—22no—
62C22×Dic7—22no—
63C2×Dic14—24no—
64C2×Dic14—82no—
65C7×M16—42no—
66C7×M16—42no—
67C2×C7×C8—42no—
68C7×C4⋊C4—24no—
69C7×C4⋊C4—22no—
70C42×C7—22no—
71C42×C7—22no—
72C42×C7C2×D412yes—
73C42×C7#4112yes—
74C4⋊Dic7—24no—
75C4⋊Dic7—41no—
76C4⋊Dic7—42no—
77C4⋊Dic7—41no—
78Dic7⋊C4—84no—
79C4×Dic7—41no—
80C4×Dic7—41no—
81C2×C7⋊C8—24no—
82C2×C4⋊Q8—71no—
83C2×C4×Q8C2×D711yescharacteristic
84#446—144no—
85#432—142no—
86#431—72no—
87#423C2×D712yes—
88C2×#101—21no—
89#395—74no—
90#391—141no—
91#389—142no—
92#384—141no—
93C4×C4⋊C4—71no—
94#343—21no—
95#333—144no—
96C23×C7C2×C22⋊C411yescharacteristic
97C7×Q8—42no—
98C7×Q8—42no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—41no—
105C2×C4×C7—42no—
106C2×C4×C7C4×D412yes—
107C2×C4×C7#12011yescharacteristic
108C2×C4×C7#12112yes—
109C2×C4×C7#12311yescharacteristic
110C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C7#13711yescharacteristic
112C2×C4×C7#13712yes—
113C2×Dic7—24no—
114C2×Dic7—42no—
115C2×Dic7—42no—
116C2×Dic7—44no—
117C2×Dic7—42no—
118Dic14—84no—
119C7×C8—82no—
120C7⋊C8—44no—
121C22×Q8—72no—
122C22×Q8C22⋊D711yescharacteristic
123C2×M16—22no—
124C4⋊Q8—142no—
125C4⋊Q8—142no—
126C4×Q8—22no—
127C4×Q8—22no—
128C2×C4⋊C4—141no—
129C2×C4⋊C4—141no—
130C2×C4⋊C4—141no—
131C2×C4⋊C4—144no—
132C2×C4⋊C4C4×D712yes—
133C2×C4⋊C4C22⋊D711yescharacteristic
134C2×C42—141no—
135C2×C42D2812yes—
136C2×C42C22×D711yescharacteristic
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—74no—
140C4⋊C8—144no—
141#101—42no—
142C22⋊C8—142no—
143#97—144no—
144C4×C8—142no—
145#95—72no—
146#95—144no—
147C22×C7—21no—
148C22×C7—21no—
149C22×C7#38611yescharacteristic
150C22×C7#41711yescharacteristic
151C22×C7#41911yescharacteristic
152C4×C7—21no—
153C4×C7—22no—
154C4×C7—42no—
155C4×C7—42no—
156C4×C7#45612yes—
157C4×C7#46812yes—
158C4×C7#47012yes—
159Dic7—44no—
160Dic7—82no—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—21no—
164C2×Q8—22no—
165C2×Q8—144no—
166C2×Q8—562no—
167C22×C4—142no—
168C22×C4—142no—
169C22×C4(C2×D7)⋊C412yes—
170C22×C4(C2×D7)⋊C412yes—
171C22×C4C2×C4×D711yescharacteristic
172C22×C4C2×D2811yescharacteristic
173C22×C4C2×C22⋊D711yescharacteristic
174M16—42no—
175M16—42no—
176C2×C8—42no—
177C2×C8—144no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—144no—
181C4⋊C4—281no—
182C4⋊C4—281no—
183C4⋊C4—282no—
184C4⋊C4—564no—
185C42—22no—
186C42—22no—
187C42—281no—
188C42—281no—
189C42D4×D712yes—
190C42#115912yes—
191C2×C7—21no—
192C2×C7—21no—
193C2×C7#188611yescharacteristic
194C2×C7#189411yescharacteristic
195C2×C7#191411yescharacteristic
196C23C2×(C2×D7)⋊C411yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—564no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—144no—
208C2×C4—282no—
209C2×C4—282no—
210C2×C4—284no—
211C2×C4—282no—
212C2×C4(C22⋊D7)⋊C412yes—
213C2×C4#644511yescharacteristic
214C2×C4#644911yescharacteristic
215C2×C4#649511yescharacteristic
216C2×C4#650212yes—
217C2×C4#650511yescharacteristic
218C2×C4#651412yes—
219C8—82no—
220C8—284no—
221C7#940011yescharacteristic, complemented, Sylow
222C22—21no—
223C22—21no—
224C22#8973011yescharacteristic
225C22#8973811yescharacteristic
226C22#8977611yescharacteristic
227C4—22no—
228C4—21no—
229C4—42no—
230C4—42no—
231C4—284no—
232C4—562no—
233C4#8952412yes—
234C4#8981412yes—
235C4#8982412yes—
236C2—21no—
237C2—21no—
238C2#1172727011yescharacteristic
239C2#1172785711yescharacteristic
240C2#1172789111yescharacteristic
subgroupG/H
centerC22#89730
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1801C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1801D7
hypocenterC7#9400
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
221. C7#9400semidirect

Maximal subgroups

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

subgroupindex
#117399232
#117398812
C7×#18012
#117282612
#117255212
#94007