Finite Group Databasebeta

← #13251154

Subgroups — #13251154

Order 1792.

486 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 131 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#11739880C211yesmaximal, characteristic
3C7×#1801C211yesmaximal, characteristic
4#11728261C212yesmaximal
5#11725521C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6432C2211yescharacteristic
8C7×#423C2212yes—
9C2×C7×#101—21no—
10#90002—21no—
11Q8×C7⋊C8—24no—
12#89751C414yes—
13#89748—21no—
14C4×C4⋊Dic7C2211yescharacteristic
15#89700—22no—
16#89697—22no—
17#89697C2212yes—
18C7×#343—21no—
19#89276—24no—
20#9392—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—
45C4×C7⋊C8—24no—
46C4×C7⋊C8—22no—
47#2982—71no—
48#2567—71no—
49#1801D711yescharacteristic
50#1548—72no—
51#1303—72no—
52C2×C7×Q8—22no—
53C2×C7×Q8—22no—
54C2×C7×Q8SD1612yes—
55C2×C7×Q8Q1612yes—
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—42no—
80C2×C7⋊C8—24no—
81C2×C4⋊Q8—71no—
82C2×C4×Q8C2×D711yescharacteristic
83C8×Q8—144no—
84#431—72no—
85#431—142no—
86#423C2×D712yes—
87C2×#101—21no—
88#395—74no—
89#391—141no—
90#389—142no—
91#384—141no—
92C4×C4⋊C4—71no—
93#343—21no—
94#333—144no—
95C23×C7C2×C22⋊C411yescharacteristic
96C7×Q8—24no—
97C7×Q8—24no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—41no—
103C2×C4×C7—42no—
104C2×C4×C7#10314yes—
105C2×C4×C7C4×D412yes—
106C2×C4×C7#12011yescharacteristic
107C2×C4×C7#12112yes—
108C2×C4×C7#12311yescharacteristic
109C2×C4×C7C2×SD1611yescharacteristic
110C2×C4×C7C2×Q1611yescharacteristic
111C2×C4×C7#13512yes—
112C2×Dic7—24no—
113C2×Dic7—42no—
114C2×Dic7—42no—
115C2×Dic7—44no—
116C2×Dic7—42no—
117Dic14—84no—
118C7×C8—82no—
119C7⋊C8—44no—
120C22×Q8—72no—
121C22×Q8C22⋊D711yescharacteristic
122C2×M16—22no—
123C4⋊Q8—142no—
124C4⋊Q8—142no—
125C4×Q8—22no—
126C4×Q8—22no—
127C2×C4⋊C4—141no—
128C2×C4⋊C4—141no—
129C2×C4⋊C4—141no—
130C2×C4⋊C4—144no—
131C2×C4⋊C4C4×D712yes—
132C2×C4⋊C4C22⋊D711yescharacteristic
133C2×C42—141no—
134C2×C42D2812yes—
135C2×C42C22×D711yescharacteristic
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—74no—
139C4⋊C8—144no—
140#101—42no—
141C22⋊C8—142no—
142C4×C8—144no—
143C4×C8—142no—
144#95—72no—
145#95—144no—
146C22×C7—21no—
147C22×C7—21no—
148C22×C7#38611yescharacteristic
149C22×C7C2×#10311yescharacteristic
150C22×C7#41611yescharacteristic
151C4×C7—21no—
152C4×C7—24no—
153C4×C7—22no—
154C4×C7—42no—
155C4×C7#45612yes—
156C4×C7#46512yes—
157C4×C7#46712yes—
158Dic7—44no—
159Dic7—82no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—144no—
163C2×Q8—562no—
164C2×Q8#114212yes—
165C2×Q8#114412yes—
166C22×C4—142no—
167C22×C4—142no—
168C22×C4(C2×D7)⋊C412yes—
169C22×C4(C2×D7)⋊C412yes—
170C22×C4C2×C4×D711yescharacteristic
171C22×C4C2×D2811yescharacteristic
172C22×C4C2×C22⋊D711yescharacteristic
173M16—42no—
174M16—42no—
175C2×C8—42no—
176C2×C8—144no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—144no—
180C4⋊C4—281no—
181C4⋊C4—281no—
182C4⋊C4—282no—
183C4⋊C4—564no—
184C42—22no—
185C42—22no—
186C42—282no—
187C42D4×D712yes—
188C42#115912yes—
189C2×C7—21no—
190C2×C7—21no—
191C2×C7#188611yescharacteristic
192C2×C7#189211yescharacteristic
193C2×C7#191411yescharacteristic
194C23C2×(C2×D7)⋊C411yescharacteristic
195Q8—24no—
196Q8—24no—
197Q8—564no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—144no—
205C2×C4—282no—
206C2×C4—282no—
207C2×C4—284no—
208C2×C4—282no—
209C2×C4#638414yes—
210C2×C4(C22⋊D7)⋊C412yes—
211C2×C4#644511yescharacteristic
212C2×C4#644911yescharacteristic
213C2×C4C2×#114211yescharacteristic
214C2×C4#650212yes—
215C2×C4C2×#114411yescharacteristic
216C2×C4#651312yes—
217C8—82no—
218C8—284no—
219C7#939211yescharacteristic, complemented, Sylow
220C22—21no—
221C22—21no—
222C22#8973011yescharacteristic
223C22C2×#638411yescharacteristic
224C22#8977411yescharacteristic
225C4—22no—
226C4—24no—
227C4—21no—
228C4—42no—
229C4—284no—
230C4—562no—
231C4#8952412yes—
232C4#8981312yes—
233C4#8982312yes—
234C2—21no—
235C2—21no—
236C2#1172726811yescharacteristic
237C2#1172785711yescharacteristic
238C2#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#9392
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
219. C7#9392semidirect

Maximal subgroups

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

subgroupindex
#117399232
#117398802
C7×#18012
#117282612
#117255212
#93927