Finite Group Databasebeta

← #13254637

Subgroups — #13254637

Order 1792.

734 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2397C211yesmaximal, characteristic, complemented
2#11739753C212yesmaximal, complemented
3#11729352C212yesmaximal, complemented
4#11727194C211yesmaximal, characteristic, complemented
5#11725420C211yesmaximal, characteristic, complemented
6C2×C4×C7×D4—22no—
7C2×C4×C7×D4C2212yes—
8#90168C2211yescharacteristic
9C7×#399—22no—
10C7×#392—22no—
11C7×#376C2211yescharacteristic
12#89989—22no—
13#89787—22no—
14C4×#1141—24no—
15#89779—22no—
16#89739—22no—
17C2×#6379—22no—
18#89697C2212yes—
19#89696C2211yescharacteristic
20D4⋊(C7⋊C8)—24no—
21#10012—71nomaximal, Sylow
22C22×C7×D4—22no—
23C22×C7×D4D412yes—
24C23×C4×C7—22no—
25C2×#1157—21no—
26C4×C7×D4—22no—
27C4×C7×D4—22no—
28C4×C7×D4—22no—
29C4×C7×D4—22no—
30C4×C7×D4—44no—
31C4×C7×D4—44no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4D412yes—
34C2×C7×C22⋊C4—24no—
35C2×C7×C22⋊C4—22no—
36C2×C42×C7D412yes—
37C2×C42×C7C2311yescharacteristic
38C2×#1141—24no—
39#6493—22no—
40C2×#1136—21no—
41#6438—21no—
42#6437D411yescharacteristic, complemented
43C4×D28—22no—
44#6432D411yescharacteristic, complemented
45C7×#95—22no—
46#6406—24no—
47C22⋊(C7⋊C8)—22no—
48#6383—24no—
49#6381—24no—
50#6379—44no—
51C4⋊(C7⋊C8)—22no—
52C4⋊(C7⋊C8)—22no—
53C4×C7⋊C8D414yescomplemented
54#3100—72no—
55#2397D711yescharacteristic, complemented
56#1963—71no—
57#1486—72no—
58#1274—71no—
59C24×C7—22no—
60C2×C7×D4—22no—
61C2×C7×D4—22no—
62C2×C7×D4—44no—
63C2×C7×D4—44no—
64C2×C7×D4—48no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—44no—
70C22×C4×C7—44no—
71C22×C4×C7C2×D412yes—
72C22×C4×C7C2×D411yescharacteristic
73C2×C22⋊D7—41no—
74#1157—42no—
75#1157—42no—
76C2×D28—22no—
77C2×D28—21no—
78C2×C4×D7—41no—
79C2×Dic14—21no—
80C7×C4⋊C4—22no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—44no—
83C7×C22⋊C4—48no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—42no—
86C42×C7—22no—
87C42×C7D814yescomplemented
88C42×C7C2×D411yescharacteristic
89C42×C7C2×D412yes—
90C42×C7#4111yescharacteristic
91#1141—48no—
92(C2×D7)⋊C4—42no—
93C4⋊Dic7—22no—
94#1136—42no—
95C2×C7⋊C8—24no—
96C2×C7⋊C8—41no—
97#532—71no—
98C2×C4×D4—22no—
99C2×C4×D4C2×D712yes—
100#480—142no—
101#474—142no—
102#459—142no—
103#447—142no—
104C4×D8—144no—
105#431—72no—
106#423—71no—
107C2×#104—142no—
108#399—22no—
109#392—22no—
110#376C2×D711yescharacteristic
111D4⋊C8—144no—
112C23×C7—24no—
113C23×C7—44no—
114C23×C7—42no—
115C23×C7—42no—
116C23×C7—44no—
117C23×C7#12011yescharacteristic
118C22×D7—41no—
119C7×D4—24no—
120C7×D4—24no—
121C7×D4—42no—
122C7×D4—42no—
123C7×D4—416no—
124C2×C4×C7—24no—
125C2×C4×C7—42no—
126C2×C4×C7—44no—
127C2×C4×C7—42no—
128C2×C4×C7—44no—
129C2×C4×C7—84no—
130C2×C4×C7—82no—
131C2×C4×C7—82no—
132C2×C4×C7#12012yes—
133C2×C4×C7#12111yescharacteristic
134C2×C4×C7#12112yes—
135C2×C4×C7#12711yescharacteristic
136C2×C4×C7C2×D812yes—
137C2×C4×C7AGL(1,ℤ/8)12yes—
138C22⋊D7—82no—
139C2×Dic7—41no—
140D28—44no—
141D28—42no—
142C4×D7—81no—
143C4×D7—81no—
144Dic14—42no—
145C7⋊C8—44no—
146C7⋊C8—81no—
147C2×#41—141no—
148C22×D4—22no—
149C22×D4C22⋊D712yes—
150C23×C4—22no—
151C2×D8—144no—
152C2×M16—141no—
153C4⋊Q8—71no—
154#127—71no—
155#124—141no—
156#121—142no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4—142no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊D712yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—22no—
168C2×C42C22⋊D712yes—
169C2×C42C22×D711yescharacteristic
170#107—144no—
171C4⋊C8—142no—
172C4⋊C8—142no—
173#104—284no—
174#102—144no—
175#102—144no—
176C22⋊C8—142no—
177C4×C8—74no—
178#95—22no—
179C22×C7—22no—
180C22×C7—48no—
181C22×C7—44no—
182C22×C7—84no—
183C22×C7—82no—
184C22×C7—82no—
185C22×C7#39211yescharacteristic
186C22×C7#44712yes—
187C2×D7—82no—
188C2×D7—81no—
189C4×C7—22no—
190C4×C7—22no—
191C4×C7—44no—
192C4×C7—82no—
193C4×C7#45912yes—
194C4×C7#49311yescharacteristic
195C4×C7#49611yescharacteristic
196Dic7—81no—
197C24—22no—
198#41—282no—
199#41—282no—
200C2×Q8—141no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—48no—
206C2×D4—142no—
207C2×D4—141no—
208C2×D4—281no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—281no—
216C22×C4C2×C22⋊D712yes—
217C22×C4C2×C22⋊D711yescharacteristic
218D8—288no—
219M16—282no—
220C2×C8—144no—
221C2×C8—281no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—44no—
225C4⋊C4—142no—
226C22⋊C4—48no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—282no—
230C42—22no—
231C42#114114yescomplemented
232C42D4×D711yescharacteristic
233C42D4×D712yes—
234C42#115911yescharacteristic
235C2×C7—22no—
236C2×C7—84no—
237C2×C7#201012yes—
238C2×C7#201111yescharacteristic
239D7—161no—
240C23—24no—
241C23—44no—
242C23—42no—
243C23—44no—
244C23—42no—
245C23—281no—
246C23#651611yescharacteristic
247Q8—282no—
248D4—24no—
249D4—24no—
250D4—42no—
251D4—42no—
252D4—416no—
253D4—284no—
254D4—282no—
255D4—562no—
256C2×C4—24no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—281no—
265C2×C4—561no—
266C2×C4—561no—
267C2×C4C2×#114112yes—
268C2×C4#649512yes—
269C2×C4#650012yes—
270C2×C4#650111yescharacteristic
271C2×C4#650212yes—
272C2×C4#650311yescharacteristic
273C8—284no—
274C8—561no—
275C7#1001211yescharacteristic, complemented, Sylow
276C22—22no—
277C22—48no—
278C22—44no—
279C22—84no—
280C22—82no—
281C22—82no—
282C22—562no—
283C22—561no—
284C22#8998912yes—
285C22#8999511yescharacteristic
286C4—22no—
287C4—22no—
288C4—44no—
289C4—82no—
290C4—561no—
291C4#8984412yes—
292C4#8984511yescharacteristic
293C4#8984611yescharacteristic
294C2—22no—
295C2—84no—
296C2—1121no—
297C2#1172946912yes—
298C2#1172952111yescharacteristic
subgroupG/H
centerC2×C4#6503
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2397C2
socleC22×C7#392
hypercenter#2397D7
hypocenterC7#10012
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. C7×#2397C2semidirect
2. #11739753C2semidirect
3. #11729352C2semidirect
4. #11727194C2semidirect
5. #11725420C2semidirect
42. #6437D4semidirect
44. #6432D4semidirect
53. C4×C7⋊C8D4semidirect
55. #2397D7semidirect
87. C42×C7D8semidirect
231. C42#1141semidirect
275. C7#10012semidirect

Maximal subgroups

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

subgroupindex
C7×#23972
#117397532
#117293522
#117271942
#117254202
#100127