Finite Group Databasebeta

← #13251166

Subgroups — #13251166

Order 1792.

506 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C7×#2982C211yesmaximal, characteristic
2#11739920C211yesmaximal, characteristic
3#11729391C211yesmaximal, characteristic
4#11728532C212yesmaximal
5#11725552C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6507—21no—
8C2×#6432C2211yescharacteristic
9C7×C8×Q8—24no—
10C7×#431—22no—
11C7×#431C2212yes—
12C2×#6410—21no—
13#89755C414yes—
14#89746—21no—
15C4×C4⋊Dic7C2211yescharacteristic
16#89700—22no—
17#89696C2212yes—
18#89325—21no—
19#89281—24no—
20#9394—71nomaximal, Sylow
21C22×C7×Q8D411yescharacteristic
22C22×Dic14—21no—
23C4×C7×Q8—22no—
24C4×C7×Q8—22no—
25C2×C7×C4⋊C4C2×C412yes—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C42×C7D412yes—
28C2×C42×C7C2311yescharacteristic
29#6507—42no—
30#6507—42no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7C2×C412yes—
34C2×Dic7⋊C4—24no—
35C2×C4×Dic7—21no—
36C2×#1136—22no—
37#6432—22no—
38#6432—22no—
39C7×C4⋊C8—24no—
40C7×C4⋊C8D414yes—
41C7×C22⋊C8—22no—
42C4×C7×C8—24no—
43C4×C7×C8—22no—
44#6410—42no—
45C7⋊#95—24no—
46C7⋊#95C2×C412yes—
47C4⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)—22no—
49#2982D711yescharacteristic
50#2546—71no—
51#1801—71no—
52#1546—72no—
53#1303—72no—
54C2×C7×Q8—22no—
55C2×C7×Q8—22no—
56C2×C7×Q8D812yes—
57C2×C7×Q8SD1612yes—
58C22×C4×C7C22⋊C412yes—
59C22×C4×C7C22⋊C412yes—
60C22×C4×C7C22×C411yescharacteristic
61C22×C4×C7C2×D411yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C22×Dic7—22no—
64C22×Dic7—22no—
65C2×Dic14—42no—
66C2×Dic14—41no—
67C2×Dic14—41no—
68C2×Dic14—42no—
69C2×C7×C8—24no—
70C7×C4⋊C4—24no—
71C7×C4⋊C4—22no—
72C42×C7—22no—
73C42×C7—22no—
74C42×C7C2×D412yes—
75C42×C7#4112yes—
76C4⋊Dic7—24no—
77C4⋊Dic7—24no—
78C4⋊Dic7—41no—
79C4⋊Dic7—41no—
80Dic7⋊C4—48no—
81C4×Dic7—41no—
82C4×Dic7—41no—
83#1136—42no—
84#1136—42no—
85C2×C7⋊C8—42no—
86C2×C4⋊Q8—71no—
87C2×C4⋊Q8—141no—
88C2×C4×Q8C2×D711yescharacteristic
89C8×Q8—24no—
90#431—22no—
91#431C2×D712yes—
92#423—72no—
93C2×#101—141no—
94#398—74no—
95#391—141no—
96#389—142no—
97C4×C4⋊C4—71no—
98#343—141no—
99#332—144no—
100C23×C7C2×C22⋊C411yescharacteristic
101C7×Q8—24no—
102C7×Q8—24no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—41no—
108C2×C4×C7—42no—
109C2×C4×C7#10214yes—
110C2×C4×C7C4×D412yes—
111C2×C4×C7#12011yescharacteristic
112C2×C4×C7#12112yes—
113C2×C4×C7#12311yescharacteristic
114C2×C4×C7C2×D811yescharacteristic
115C2×C4×C7C2×SD1611yescharacteristic
116C2×C4×C7#13512yes—
117C2×Dic7—24no—
118C2×Dic7—42no—
119C2×Dic7—42no—
120C2×Dic7—42no—
121C2×Dic7—42no—
122C2×Dic7—42no—
123Dic14—82no—
124Dic14—82no—
125C7×C8—44no—
126C7⋊C8—82no—
127C22×Q8—141no—
128C22×Q8D2811yescharacteristic
129C2×M16—142no—
130C4⋊Q8—142no—
131C4⋊Q8—142no—
132C4⋊Q8—282no—
133C4⋊Q8—282no—
134C4×Q8—22no—
135C4×Q8—22no—
136C2×C4⋊C4—72no—
137C2×C4⋊C4—141no—
138C2×C4⋊C4—144no—
139C2×C4⋊C4—141no—
140C2×C4⋊C4C4×D712yes—
141C2×C4⋊C4D2811yescharacteristic
142C2×C42—141no—
143C2×C42C22⋊D712yes—
144C2×C42C22×D711yescharacteristic
145C4⋊C8—24no—
146C4⋊C8—142no—
147C4⋊C8—142no—
148C4⋊C8C22⋊D714yes—
149#101—282no—
150C22⋊C8—22no—
151C4×C8—24no—
152C4×C8—22no—
153#95—72no—
154#95—144no—
155C22×C7—21no—
156C22×C7—21no—
157C22×C7#38611yescharacteristic
158C22×C7C2×#10211yescharacteristic
159C22×C7#41611yescharacteristic
160C4×C7—21no—
161C4×C7—24no—
162C4×C7—22no—
163C4×C7—42no—
164C4×C7#45612yes—
165C4×C7#46512yes—
166C4×C7#46612yes—
167Dic7—44no—
168Dic7—82no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—281no—
172C2×Q8—281no—
173C2×Q8—282no—
174C2×Q8—282no—
175C2×Q8#113212yes—
176C2×Q8D5612yes—
177C22×C4—142no—
178C22×C4—142no—
179C22×C4(C2×D7)⋊C412yes—
180C22×C4(C2×D7)⋊C412yes—
181C22×C4C2×C4×D711yescharacteristic
182C22×C4C2×D2811yescharacteristic
183C22×C4C2×C22⋊D711yescharacteristic
184M16—282no—
185M16—282no—
186C2×C8—24no—
187C2×C8—282no—
188C4⋊C4—24no—
189C4⋊C4—22no—
190C4⋊C4—144no—
191C4⋊C4—144no—
192C4⋊C4—281no—
193C4⋊C4—281no—
194C4⋊C4—288no—
195C42—22no—
196C42—22no—
197C42—281no—
198C42—281no—
199C42#115712yes—
200C42C2×C22⋊D712yes—
201C2×C7—21no—
202C2×C7—21no—
203C2×C7#188611yescharacteristic
204C2×C7#189311yescharacteristic
205C2×C7#191511yescharacteristic
206C23C2×(C2×D7)⋊C411yescharacteristic
207Q8—24no—
208Q8—24no—
209Q8—562no—
210Q8—562no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—144no—
218C2×C4—282no—
219C2×C4—282no—
220C2×C4—282no—
221C2×C4—282no—
222C2×C4—282no—
223C2×C4#639514yes—
224C2×C4C2×#113211yescharacteristic
225C2×C4C2×D5611yescharacteristic
226C2×C4#646712yes—
227C2×C4C4×C22⋊D712yes—
228C2×C4#649211yescharacteristic
229C2×C4#649312yes—
230C2×C4#651611yescharacteristic
231C8—44no—
232C8—562no—
233C7#939411yescharacteristic, complemented, Sylow
234C22—21no—
235C22—21no—
236C22C2×#639511yescharacteristic
237C22#8988511yescharacteristic
238C22#8998511yescharacteristic
239C4—21no—
240C4—24no—
241C4—22no—
242C4—42no—
243C4—284no—
244C4—562no—
245C4#8988612yes—
246C4#8988712yes—
247C4#9001812yes—
248C2—21no—
249C2—21no—
250C2#1172941411yescharacteristic
251C2#1172972711yescharacteristic
252C2#1172978511yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2982C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2982D7
hypocenterC7#9394
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
233. C7#9394semidirect

Maximal subgroups

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

subgroupindex
C7×#29822
#117399202
#117293912
#117285322
#117255522
#93947