Finite Group Databasebeta

← #13251211

Subgroups — #13251211

Order 1792.

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

normal subgroups only

This group has 570 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
1C7×#2975C211yesmaximal, characteristic
2#11739824C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic
4#11728534C212yesmaximal, complemented
5#11725552C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6487—22no—
8C2×#6432C2211yescharacteristic
9C7×#436—24no—
10C7×#431C2212yes—
11C7×#407—22no—
12#89993—21no—
13C2×#6407—21no—
14#89748—21no—
15#89721C414yes—
16C4×C4⋊Dic7C2211yescharacteristic
17#89696C2212yes—
18#89325—21no—
19#89277—24no—
20#9403—71nomaximal, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7D412yes—
23C22×Dic14—21no—
24C2×C7×M16—24no—
25C4×C7×D4—22no—
26C4×C7×D4—22no—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C7×C22⋊C4C2×C412yes—
29C2×C42×C7C2311yescharacteristic
30C2×C22⋊Dic7—24no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7C2×C412yes—
34#6487—48no—
35C2×Dic7⋊C4—24no—
36C2×C4×Dic7—21no—
37C2×#1136—22no—
38#6432—22no—
39#6432—22no—
40C7×C4⋊C8D414yes—
41C7×C22⋊C8—24no—
42C7×C22⋊C8—22no—
43C4×C7×C8—22no—
44#6407—42no—
45C7⋊#95C2×C412yes—
46C4⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—22no—
48#2975D711yescharacteristic
49#2542—71no—
50#1800—71no—
51#1549—72no—
52#1303—72no—
53C24×C7C22⋊C412yes—
54C2×C7×D4—22no—
55C2×C7×D4—21no—
56C2×C7×D4—22no—
57C2×C7×D4—21no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—44no—
61C22×C4×C7C22⋊C412yes—
62C22×C4×C7C22×C411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D411yescharacteristic
65C22×Dic7—22no—
66C22×Dic7—22no—
67C2×Dic14—44no—
68C2×Dic14—41no—
69C2×Dic14—41no—
70C7×M16—84no—
71C2×C7×C8—24no—
72C7×C4⋊C4—22no—
73C7×C22⋊C4—24no—
74C42×C7C2×D412yes—
75C42×C7#4112yes—
76C22⋊Dic7—48no—
77C4⋊Dic7—24no—
78C4⋊Dic7—24no—
79C4⋊Dic7—42no—
80Dic7⋊C4—48no—
81C4×Dic7—41no—
82C4×Dic7—41no—
83#1136—42no—
84#1136—42no—
85C2×C7⋊C8—42no—
86C2×C4⋊Q8—71no—
87C2×C22⋊Q8—142no—
88C2×C4×D4C2×D711yescharacteristic
89#436—24no—
90#431C2×D712yes—
91#423—72no—
92C2×#100—141no—
93#407—22no—
94#399—74no—
95#390—141no—
96#384—141no—
97C4×C4⋊C4—71no—
98#343—141no—
99#332—144no—
100C23×C7—22no—
101C23×C7—22no—
102C23×C7—22no—
103C23×C7—42no—
104C23×C7—42no—
105C23×C7C2×C22⋊C411yescharacteristic
106C7×D4—42no—
107C7×D4—42no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—41no—
111C2×C4×C7—44no—
112C2×C4×C7—42no—
113C2×C4×C7—82no—
114C2×C4×C7C4×D412yes—
115C2×C4×C7#12011yescharacteristic
116C2×C4×C7#12112yes—
117C2×C4×C7#12311yescharacteristic
118C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C7#13712yes—
120C2×C4×C7#13711yescharacteristic
121C2×Dic7—24no—
122C2×Dic7—42no—
123C2×Dic7—42no—
124C2×Dic7—44no—
125C2×Dic7—42no—
126Dic14—82no—
127Dic14—82no—
128C7×C8—44no—
129C7⋊C8—82no—
130C22×Q8—141no—
131C22×D4D2811yescharacteristic
132C23×C4C22⋊D712yes—
133C2×M16—24no—
134C2×M16—142no—
135C4⋊Q8—142no—
136C4⋊Q8—142no—
137C22⋊Q8—288no—
138C4×D4—22no—
139C4×D4—22no—
140C2×C4⋊C4—72no—
141C2×C4⋊C4—141no—
142C2×C4⋊C4—144no—
143C2×C4⋊C4—141no—
144C2×C4⋊C4D2811yescharacteristic
145C2×C22⋊C4—144no—
146C2×C22⋊C4C4×D712yes—
147C2×C42—141no—
148C2×C42C22×D711yescharacteristic
149C4⋊C8—142no—
150C4⋊C8—142no—
151C4⋊C8C22⋊D714yes—
152#100—282no—
153C22⋊C8—24no—
154C22⋊C8—22no—
155C4×C8—22no—
156#95—72no—
157C22×C7—21no—
158C22×C7—21no—
159C22×C7—44no—
160C22×C7—42no—
161C22×C7—42no—
162C22×C7—42no—
163C22×C7—82no—
164C22×C7#38611yescharacteristic
165C22×C7#41711yescharacteristic
166C22×C7#41911yescharacteristic
167C4×C7—21no—
168C4×C7—22no—
169C4×C7—42no—
170C4×C7#45312yes—
171C4×C7#46812yes—
172C4×C7#47012yes—
173Dic7—44no—
174Dic7—82no—
175C24(C2×D7)⋊C412yes—
176C2×Q8—284no—
177C2×Q8—281no—
178C2×Q8—281no—
179C2×D4—21no—
180C2×D4—22no—
181C2×D4—21no—
182C2×D4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—44no—
186C22×C4—142no—
187C22×C4—142no—
188C22×C4(C2×D7)⋊C412yes—
189C22×C4C2×C4×D711yescharacteristic
190C22×C4C2×D2811yescharacteristic
191C22×C4C2×C22⋊D711yescharacteristic
192M16—84no—
193M16—282no—
194M16—282no—
195C2×C8—24no—
196C2×C8—282no—
197C4⋊C4—22no—
198C4⋊C4—144no—
199C4⋊C4—144no—
200C4⋊C4—288no—
201C4⋊C4—282no—
202C22⋊C4—24no—
203C22⋊C4—288no—
204C42—281no—
205C42—281no—
206C42#115712yes—
207C42C2×C22⋊D712yes—
208C2×C7—21no—
209C2×C7—21no—
210C2×C7—82no—
211C2×C7#188711yescharacteristic
212C2×C7#189411yescharacteristic
213C2×C7#191511yescharacteristic
214C23—22no—
215C23—22no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23C2×(C2×D7)⋊C411yescharacteristic
220Q8—562no—
221Q8—562no—
222D4—42no—
223D4—42no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4—144no—
231C2×C4—282no—
232C2×C4—282no—
233C2×C4—284no—
234C2×C4—282no—
235C2×C4#647111yescharacteristic
236C2×C4#647212yes—
237C2×C4#647211yescharacteristic
238C2×C4C4×C22⋊D712yes—
239C2×C4#649211yescharacteristic
240C2×C4#649312yes—
241C2×C4#651611yescharacteristic
242C8—44no—
243C8—562no—
244C7#940311yescharacteristic, complemented, Sylow
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22#8989211yescharacteristic
253C22#8990511yescharacteristic
254C22#8998511yescharacteristic
255C4—22no—
256C4—21no—
257C4—42no—
258C4—284no—
259C4—562no—
260C4#8990612yes—
261C4#8990912yes—
262C4#9001312yes—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#1172940411yescharacteristic
267C2#1172974311yescharacteristic
268C2#1172978511yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2975C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#2975D7
hypocenterC7#9403
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
4. #11728534C2semidirect
5. #11725552C2semidirect
244. C7#9403semidirect

Maximal subgroups

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

subgroupindex
C7×#29752
#117398242
#117293712
#117285342
#117255522
#94037