Finite Group Databasebeta

← #13335051

Subgroups — #13335051

Order 1792.

1066 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1066 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×H(3,𝔽4)C211yesmaximal, characteristic
2#11740394C212yesmaximal, complemented
3#11740384C211yesmaximal, characteristic, complemented
4#11729703C218yesmaximal, complemented
5#11729473C211yesmaximal, characteristic, complemented
6#11729472C212yesmaximal, complemented
7C7×H(3,𝔽4)—24no—
8C7×H(3,𝔽4)—22no—
9C7×H(3,𝔽4)C2214yes—
10C2×C7×#124—21no—
11C2×C7×#124—22no—
12C2×C7×#124C2212yes—
13C2×C7×#124C2211yescharacteristic
14C2×C7×#120—21no—
15C2×C7×#120—22no—
16#90161C2211yescharacteristic, complemented
17#90000—22no—
18#89997—21no—
19#89990—22no—
20#89987—21no—
21#89832C2214yescomplemented
22#89828—22no—
23#89827C2214yescomplemented
24#89827C2214yescomplemented
25#89826—24no—
26#89698C2212yescomplemented
27C4×#1136C2211yescharacteristic, complemented
28#89335C2218yescomplemented
29#89335C2214yescomplemented
30#18997—71nomaximal, Sylow
31C25×C7—21no—
32C22×C7×Q8D411yescharacteristic
33C22×C7×Q8D412yes—
34C22×C7×D4—22no—
35C22×C7×D4—21no—
36C22×C7×D4D411yescharacteristic
37C22×C7×D4D412yes—
38C7×#124—24no—
39C7×#124—28no—
40C7×#124—24no—
41C7×#124—28no—
42C7×#124—22no—
43C7×#124—22no—
44C7×#124—22no—
45C7×#124D414yes—
46C7×#124D412yes—
47C7×#124C2312yes—
48C7×#124C2314yes—
49C7×#120—24no—
50C7×#120—28no—
51C7×#120—24no—
52C7×#120—28no—
53C2×C7×C22⋊C4—21no—
54C2×C7×C22⋊C4—22no—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4—22no—
57C2×C7×C22⋊C4—22no—
58C2×C42×C7—21no—
59C2×C42×C7C2311yescharacteristic
60C2×#1144—24no—
61C2×#1142—24no—
62#6489—21no—
63#6487—22no—
64C2×#1136—21no—
65#6433—21no—
66#6432C2312yescomplemented
67C4×Dic14—22no—
68#6409—24no—
69#6409—24no—
70#6406—24no—
71#6406—24no—
72C22⋊(C7⋊C8)—22no—
73C4⋊(C7⋊C8)—22no—
74#6377C2312yescomplemented
75#6377C2314yescomplemented
76C4×C7⋊C8—21no—
77C2×H(3,𝔽4)D711yescharacteristic
78#3215—72no—
79#3140—71no—
80#1659—78no—
81#1533—71no—
82#1527—72no—
83C24×C7—24no—
84C24×C7—21no—
85C24×C7—22no—
86C24×C7—84no—
87C24×C7—82no—
88C2×C7×Q8—21no—
89C2×C7×Q8—22no—
90C2×C7×Q8—42no—
91C2×C7×Q8—82no—
92C2×C7×Q8C2×D412yes—
93C2×C7×Q8C2×D414yes—
94C2×C7×D4—24no—
95C2×C7×D4—28no—
96C2×C7×D4—21no—
97C2×C7×D4—22no—
98C2×C7×D4—42no—
99C2×C7×D4—82no—
100C2×C7×D4—84no—
101C2×C7×D4—82no—
102C2×C7×D4C2×D412yes—
103C2×C7×D4C2×D414yes—
104C22×C4×C7—22no—
105C22×C4×C7—21no—
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D412yes—
108C2×Dic14—22no—
109C7×C22⋊C4—28no—
110C7×C22⋊C4—24no—
111C7×C22⋊C4—24no—
112C7×C22⋊C4—24no—
113C7×C22⋊C4—24no—
114C7×C22⋊C4—24no—
115C7×C22⋊C4—24no—
116C7×C22⋊C4—24no—
117C42×C7—24no—
118C42×C7C2×D412yes—
119C42×C7C2×D411yescharacteristic
120C42×C7C2411yescharacteristic
121C22⋊Dic7—41no—
122#1144—84no—
123#1142—84no—
124C4⋊Dic7—22no—
125C4⋊Dic7—22no—
126Dic7⋊C4—42no—
127C4×Dic7—41no—
128#1136—42no—
129C2×C7⋊C8—24no—
130C2×C7⋊C8—22no—
131H(3,𝔽4)—24no—
132H(3,𝔽4)—22no—
133H(3,𝔽4)C2×D714yes—
134#533—71no—
135C2×#124—21no—
136C2×#124—22no—
137C2×#124C2×D711yescharacteristic
138C2×#124C2×D712yes—
139C2×#120—21no—
140C2×#120—22no—
141#497—74no—
142#489—74no—
143#489—74no—
144#488—142no—
145#464—144no—
146#451—142no—
147#450—142no—
148#432—72no—
149#419—141no—
150#418—141no—
151C4×M16—71no—
152#329—78no—
153#329—74no—
154C23×C7—22no—
155C23×C7—24no—
156C23×C7—44no—
157C23×C7—44no—
158C23×C7—44no—
159C23×C7—42no—
160C23×C7—44no—
161C23×C7—84no—
162C23×C7—88no—
163C23×C7—816no—
164C23×C7#12011yescharacteristic
165C7×Q8—44no—
166C7×Q8—42no—
167C7×Q8—81no—
168C7×Q8—82no—
169C7×D4—44no—
170C7×D4—42no—
171C7×D4—81no—
172C7×D4—82no—
173C7×D4—84no—
174C7×D4—88no—
175C2×C4×C7—22no—
176C2×C4×C7—22no—
177C2×C4×C7—24no—
178C2×C4×C7—42no—
179C2×C4×C7—44no—
180C2×C4×C7—42no—
181C2×C4×C7#12011yescharacteristic
182C2×C4×C7#12012yes—
183C2×C4×C7C22×D411yescharacteristic
184C2×C4×C7C22×D412yes—
185C2×Dic7—42no—
186Dic14—44no—
187C7⋊C8—42no—
188C7⋊C8—44no—
189C25—21no—
190C22×Q8C22⋊D711yescharacteristic
191C22×Q8C22⋊D712yes—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4C22⋊D711yescharacteristic
195C22×D4C22⋊D712yes—
196C2×Q16—144no—
197C2×SD16—144no—
198C2×M16—141no—
199C4⋊Q8—72no—
200#125—141no—
201#124—24no—
202#124—28no—
203#124—24no—
204#124—28no—
205#124—22no—
206#124—22no—
207#124—22no—
208#124C22⋊D714yes—
209#124C22⋊D712yes—
210#124C22×D712yes—
211#124C22×D714yes—
212C22⋊Q8—142no—
213#120—24no—
214#120—28no—
215#120—24no—
216#120—28no—
217C4×Q8—142no—
218#117—141no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—21no—
225C2×C42C22×D711yescharacteristic
226C4⋊C8—142no—
227#103—144no—
228#103—144no—
229#102—144no—
230#102—144no—
231C22⋊C8—142no—
232#97—72no—
233#97—74no—
234C4×C8—141no—
235C22×C7—22no—
236C22×C7—42no—
237C22×C7—44no—
238C22×C7—44no—
239C22×C7—44no—
240C22×C7—44no—
241C22×C7—84no—
242C22×C7—88no—
243C22×C7—816no—
244C22×C7#45512yes—
245C22×C7C2×#12011yescharacteristic
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—22no—
249C4×C7—22no—
250C4×C7—42no—
251C4×C7—44no—
252Dic7—82no—
253C24—24no—
254C24—21no—
255C24—22no—
256C24—84no—
257C24—82no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×Q8—82no—
262C2×Q8—142no—
263C2×Q8C2×C22⋊D712yes—
264C2×Q8C2×C22⋊D714yes—
265C2×D4—28no—
266C2×D4—24no—
267C2×D4—21no—
268C2×D4—22no—
269C2×D4—42no—
270C2×D4—82no—
271C2×D4—84no—
272C2×D4—82no—
273C2×D4C2×C22⋊D712yes—
274C2×D4C2×C22⋊D714yes—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4C2×C22⋊D711yescharacteristic
278C22×C4C2×C22⋊D712yes—
279Q16—564no—
280SD16—564no—
281M16—282no—
282C2×C8—142no—
283C2×C8—144no—
284C4⋊C4—142no—
285C4⋊C4—142no—
286C4⋊C4—282no—
287C22⋊C4—24no—
288C22⋊C4—28no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—281no—
296C42—24no—
297C42—281no—
298C42C2×C22⋊D711yescharacteristic
299C42C2×C22⋊D712yes—
300C42C23×D711yescharacteristic
301C2×C7—22no—
302C2×C7—82no—
303C2×C7—84no—
304C2×C7C2×#45511yescharacteristic
305C2×C7#301912yes—
306C23—24no—
307C23—22no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—44no—
313C23—88no—
314C23—84no—
315C23—816no—
316C23#651611yescharacteristic
317Q8—44no—
318Q8—42no—
319Q8—81no—
320Q8—82no—
321Q8—284no—
322D4—44no—
323D4—42no—
324D4—81no—
325D4—82no—
326D4—88no—
327D4—84no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—282no—
335C2×C4#651611yescharacteristic
336C2×C4#651612yes—
337C2×C4C22×C22⋊D711yescharacteristic
338C2×C4C22×C22⋊D712yes—
339C8—282no—
340C8—284no—
341C7#1899711yescharacteristic, complemented, Sylow
342C22—22no—
343C22—44no—
344C22—44no—
345C22—44no—
346C22—42no—
347C22—44no—
348C22—84no—
349C22—88no—
350C22—816no—
351C22#9001412yes—
352C22C2×#651611yescharacteristic
353C4—22no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—44no—
358C4—42no—
359C4—562no—
360C2—22no—
361C2—84no—
362C2—82no—
363C2#1174154912yes—
364C2C2×#9001411yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC2×C7×H(3,𝔽4)C2
socleC22×C7C2×#120
hypercenterC2×H(3,𝔽4)D7
hypocenterC7#18997
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
2. #11740394C2semidirect
3. #11740384C2semidirect
4. #11729703C2semidirect
5. #11729473C2semidirect
6. #11729472C2semidirect
16. #90161C22semidirect
21. #89832C22semidirect
23. #89827C22semidirect
24. #89827C22semidirect
26. #89698C22semidirect
27. C4×#1136C22semidirect
28. #89335C22semidirect
29. #89335C22semidirect
66. #6432C23semidirect
74. #6377C23semidirect
75. #6377C23semidirect
341. C7#18997semidirect

Maximal subgroups

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

subgroupindex
C2×C7×H(3,𝔽4)2
#117403942
#117403842
#117297032
#117294732
#117294722
#189977