Finite Group Databasebeta

← #13335083

Subgroups — #13335083

Order 1792.

960 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C2×C7×#562C211yesmaximal, characteristic, complemented
2#11740457C211yesmaximal, characteristic, complemented
3#11740392C212yesmaximal, complemented
4#11729708C214yesmaximal, complemented
5#11729706C214yesmaximal, complemented
6#11729490C211yesmaximal, characteristic, complemented
7#11729472C212yesmaximal, complemented
8C7×#562—22no—
9C7×#562—24no—
10C7×#562C2214yes—
11C2×C7×#125C2211yescharacteristic
12C2×C7×#124C2212yes—
13C2×C7×#123—22no—
14C2×C7×C22⋊Q8—22no—
15C2×C7×#121—22no—
16#90168C2211yescharacteristic
17#89999—22no—
18#89989—22no—
19#89841C2212yescomplemented
20#89840—22no—
21#89839C2212yescomplemented
22#89830C2214yescomplemented
23#89829—24no—
24#89827C2214yescomplemented
25#89738—22no—
26#89698C2212yes—
27C4×#1136C2211yescharacteristic
28#89338C2214yescomplemented
29#89335C2218yescomplemented
30#19001—71nomaximal, Sylow
31C22×C7×Q8D412yes—
32C22×C7×D4—22no—
33C22×C7×D4D412yes—
34C23×C4×C7—22no—
35C2×#1157—21no—
36C7×#125—22no—
37C7×#125D412yes—
38C7×#125C2312yes—
39C7×#124—22no—
40C7×#124—22no—
41C7×#124D414yes—
42C7×#124C2314yes—
43C7×#123—28no—
44C7×#123—28no—
45C7×C22⋊Q8—28no—
46C7×C22⋊Q8—28no—
47C7×#121—28no—
48C7×#121—28no—
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4D412yes—
52C2×C7×C22⋊C4—22no—
53C2×C7×C22⋊C4—22no—
54C2×C7×C22⋊C4—22no—
55C2×C42×C7C2311yescharacteristic
56C2×#1143—24no—
57C2×#1141—24no—
58#6493—22no—
59C2×#1136—21no—
60#6438—21no—
61#6437C2311yescharacteristic
62C4×D28—22no—
63#6432C2311yescharacteristic
64#6409—24no—
65#6406—24no—
66C22⋊(C7⋊C8)—22no—
67#6384—24no—
68#6383—24no—
69C4⋊(C7⋊C8)—22no—
70#6377C2312yes—
71#6377C2314yes—
72C4×C7⋊C8—21no—
73C2×#562D711yescharacteristic, complemented
74#3213—72no—
75#3142—71no—
76#1662—74no—
77#1656—74no—
78#1534—71no—
79#1527—72no—
80C24×C7—22no—
81C2×C7×Q8—22no—
82C2×C7×Q8—82no—
83C2×C7×Q8C2×D414yes—
84C2×C7×D4—28no—
85C2×C7×D4—22no—
86C2×C7×D4—82no—
87C2×C7×D4—84no—
88C2×C7×D4C2×D414yes—
89C22×C4×C7—22no—
90C22×C4×C7—22no—
91C22×C4×C7—28no—
92C22×C4×C7—84no—
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D412yes—
95C2×C22⋊D7—41no—
96#1157—44no—
97C2×D28—22no—
98C2×D28—21no—
99C2×C4×D7—41no—
100C2×Dic14—21no—
101C7×C4⋊C4—28no—
102C7×C4⋊C4—24no—
103C7×C4⋊C4—22no—
104C7×C4⋊C4—24no—
105C7×C4⋊C4C2×D414yes—
106C7×C22⋊C4—28no—
107C7×C22⋊C4—24no—
108C7×C22⋊C4—24no—
109C7×C22⋊C4—24no—
110C7×C22⋊C4—24no—
111C42×C7C2×D412yes—
112C42×C7C2×D411yescharacteristic
113C42×C7C2411yescharacteristic
114#1143—84no—
115#1141—84no—
116(C2×D7)⋊C4—42no—
117C4⋊Dic7—22no—
118#1136—42no—
119C2×C7⋊C8—24no—
120C2×C7⋊C8—22no—
121#562—22no—
122#562—24no—
123#562C2×D714yes—
124#532—71no—
125C2×#125C2×D711yescharacteristic
126C2×#124C2×D712yes—
127C2×#123—22no—
128C2×C22⋊Q8—22no—
129C2×#121—22no—
130#496—74no—
131#491—72no—
132#490—72no—
133#489—74no—
134#488—142no—
135#463—144no—
136#448—142no—
137#447—142no—
138#432—72no—
139#417—142no—
140C4×M16—71no—
141#330—74no—
142#329—78no—
143C23×C7—24no—
144C23×C7—44no—
145C23×C7—44no—
146C23×C7—44no—
147C23×C7#12011yescharacteristic
148C22×D7—41no—
149C7×Q8—44no—
150C7×Q8—82no—
151C7×D4—44no—
152C7×D4—82no—
153C7×D4—88no—
154C2×C4×C7—22no—
155C2×C4×C7—24no—
156C2×C4×C7—24no—
157C2×C4×C7—42no—
158C2×C4×C7—44no—
159C2×C4×C7—44no—
160C2×C4×C7—88no—
161C2×C4×C7#12012yes—
162C2×C4×C7#12011yescharacteristic
163C2×C4×C7C22×D412yes—
164C2×C4×C7C22×D411yescharacteristic
165C22⋊D7—82no—
166C2×Dic7—41no—
167D28—44no—
168D28—42no—
169C4×D7—82no—
170Dic14—42no—
171C7⋊C8—42no—
172C7⋊C8—44no—
173C2×#41—141no—
174C22×Q8C22⋊D712yes—
175C22×D4—22no—
176C22×D4C22⋊D712yes—
177C23×C4—22no—
178C2×SD16—144no—
179C2×D8—144no—
180C2×M16—141no—
181C4⋊Q8—71no—
182#127—71no—
183#125—22no—
184#125C22⋊D712yes—
185#125C22×D712yes—
186#124—22no—
187#124—22no—
188#124—141no—
189#124C22⋊D714yes—
190#124C22×D714yes—
191#123—28no—
192#123—28no—
193C22⋊Q8—28no—
194C22⋊Q8—28no—
195#121—28no—
196#121—28no—
197#121—142no—
198C4×D4—142no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22⋊D712yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42C22×D711yescharacteristic
206C4⋊C8—142no—
207#103—144no—
208#103—144no—
209#102—144no—
210#102—144no—
211C22⋊C8—142no—
212#97—72no—
213#97—74no—
214C4×C8—141no—
215C22×C7—22no—
216C22×C7—44no—
217C22×C7—44no—
218C22×C7—44no—
219C22×C7—88no—
220C22×C7#45412yes—
221C22×C7C2×#12011yescharacteristic
222C2×D7—81no—
223C2×D7—82no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—22no—
227C4×C7—22no—
228C4×C7—44no—
229C4×C7—44no—
230Dic7—81no—
231C24—22no—
232#41—284no—
233C2×Q8—22no—
234C2×Q8—82no—
235C2×Q8—141no—
236C2×Q8C2×C22⋊D714yes—
237C2×D4—28no—
238C2×D4—22no—
239C2×D4—82no—
240C2×D4—84no—
241C2×D4—141no—
242C2×D4—142no—
243C2×D4—281no—
244C2×D4C2×C22⋊D714yes—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—28no—
248C22×C4—84no—
249C22×C4—281no—
250C22×C4C2×C22⋊D711yescharacteristic
251C22×C4C2×C22⋊D712yes—
252SD16—564no—
253D8—564no—
254M16—282no—
255C2×C8—142no—
256C2×C8—144no—
257C4⋊C4—28no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—142no—
262C4⋊C4C2×C22⋊D714yes—
263C22⋊C4—28no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—282no—
269C42C2×C22⋊D711yescharacteristic
270C42C2×C22⋊D712yes—
271C42C23×D711yescharacteristic
272C2×C7—22no—
273C2×C7—84no—
274C2×C7C2×#45411yescharacteristic
275C2×C7#301912yes—
276D7—161no—
277C23—24no—
278C23—44no—
279C23—44no—
280C23—44no—
281C23—281no—
282C23#651611yescharacteristic
283Q8—44no—
284Q8—82no—
285Q8—282no—
286D4—44no—
287D4—82no—
288D4—88no—
289D4—284no—
290D4—282no—
291D4—562no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—88no—
299C2×C4—281no—
300C2×C4—562no—
301C2×C4#651611yescharacteristic
302C2×C4#651612yes—
303C2×C4C22×C22⋊D711yescharacteristic
304C2×C4C22×C22⋊D712yes—
305C8—282no—
306C8—284no—
307C7#1900111yescharacteristic, complemented, Sylow
308C22—22no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—88no—
313C22—562no—
314C22—561no—
315C22#9001712yes—
316C22C2×#651611yescharacteristic
317C4—22no—
318C4—22no—
319C4—21no—
320C4—21no—
321C4—44no—
322C4—44no—
323C4—561no—
324C2—22no—
325C2—84no—
326C2—1121no—
327C2#1174154912yes—
328C2C2×#9001711yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#562C2
socleC22×C7C2×#120
hypercenterC2×#562D7
hypocenterC7#19001
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. C2×C7×#562C2semidirect
2. #11740457C2semidirect
3. #11740392C2semidirect
4. #11729708C2semidirect
5. #11729706C2semidirect
6. #11729490C2semidirect
7. #11729472C2semidirect
19. #89841C22semidirect
21. #89839C22semidirect
22. #89830C22semidirect
24. #89827C22semidirect
28. #89338C22semidirect
29. #89335C22semidirect
73. C2×#562D7semidirect
307. C7#19001semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5622
#117404572
#117403922
#117297082
#117297062
#117294902
#117294722
#190017