Finite Group Databasebeta

← #13328588

Subgroups — #13328588

Order 1792.

1290 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 207 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×#3496C211yesmaximal, characteristic
2#11740716C212yesmaximal, complemented
3#11739824C211yesmaximal, characteristic, complemented
4#11729598C218yesmaximal, complemented
5#11729565C212yesmaximal, complemented
6#11729355C211yesmaximal, characteristic, complemented
7C2×C7×H(5,𝔽2)—21no—
8C7×#547—22no—
9C7×#546—24no—
10C7×D42—22no—
11C7×#537—21no—
12C7×#534C2214yes—
13C7×#532C2212yes—
14C2×C7×#121—22no—
15C2×C4×C7×D4C2211yescharacteristic
16C2×#6487—22no—
17C2×#6432C2211yescharacteristic, complemented
18#89993—21no—
19#89851—22no—
20#89842—22no—
21#89806C2218yescomplemented
22#89780—24no—
23#89748—21no—
24#89721C2214yescomplemented
25C4×C4⋊Dic7C2211yescharacteristic, complemented
26#89697C2212yescomplemented
27#89332C2218yescomplemented
28#89320C2214yescomplemented
29#18395—71nomaximal, Sylow
30C7×H(5,𝔽2)—44no—
31C7×H(5,𝔽2)—44no—
32C2×C7×#41—24no—
33C2×C7×#41—22no—
34C22×C7×D4—24no—
35C22×C7×D4—24no—
36C22×C7×D4—22no—
37C22×C7×D4D411yescharacteristic
38C23×C4×C7D412yes—
39C22×Dic14—21no—
40C7×C4⋊Q8D412yes—
41C7×#127D412yes—
42C7×#124—22no—
43C7×#123—28no—
44C7×C22⋊Q8—24no—
45C7×#121—24no—
46C7×#121—24no—
47C7×#121—24no—
48C7×#121—22no—
49C7×#121—24no—
50C7×#121C2314yes—
51C7×#120—28no—
52C4×C7×D4—22no—
53C4×C7×D4—22no—
54C4×C7×D4—22no—
55C4×C7×D4D414yes—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4C2312yes—
60C2×C42×C7C2311yescharacteristic
61C2×C22⋊Dic7—24no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7—21no—
64C2×C4⋊Dic7C2312yes—
65#6487—48no—
66C2×Dic7⋊C4—24no—
67C2×C4×Dic7—21no—
68#6432—22no—
69#6432—22no—
70#6406—28no—
71C7⋊#95C2312yes—
72C22⋊(C7⋊C8)C2314yes—
73#6382—28no—
74C4⋊(C7⋊C8)—22no—
75C4×C7⋊C8—22no—
76#3496D711yescharacteristic
77#3227—72no—
78#2542—71no—
79#1613—78no—
80#1502—72no—
81#1460—71no—
82C24×C7—24no—
83C24×C7C2×D412yes—
84C7×#41—416no—
85C7×#41—44no—
86C7×#41—44no—
87C2×C7×Q8—22no—
88C2×C7×D4—28no—
89C2×C7×D4—24no—
90C2×C7×D4—24no—
91C2×C7×D4—22no—
92C2×C7×D4—22no—
93C2×C7×D4—22no—
94C2×C7×D4—22no—
95C2×C7×D4—416no—
96C2×C7×D4—48no—
97C2×C7×D4—48no—
98C2×C7×D4—84no—
99C2×C7×D4SD1614yes—
100C2×C7×D4C2×D414yes—
101C22×C4×C7—24no—
102C22×C4×C7—22no—
103C22×C4×C7—24no—
104C22×C4×C7—22no—
105C22×C4×C7—44no—
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D412yes—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2411yescharacteristic
110C22×Dic7—22no—
111C22×Dic7—22no—
112C2×Dic14—44no—
113C2×Dic14—41no—
114C2×Dic14—41no—
115C7×C4⋊C4—24no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4C2×D414yes—
119C7×C22⋊C4—28no—
120C7×C22⋊C4—24no—
121C7×C22⋊C4—24no—
122C7×C22⋊C4—24no—
123C42×C7C2×D412yes—
124C42×C7C2×D412yes—
125C22⋊Dic7—48no—
126C4⋊Dic7—24no—
127C4⋊Dic7—24no—
128C4⋊Dic7—42no—
129Dic7⋊C4—48no—
130C4×Dic7—42no—
131C2×C7⋊C8—24no—
132C2×H(5,𝔽2)—21no—
133#547—22no—
134#546—24no—
135D42—22no—
136#537—21no—
137#534C2×D714yes—
138#532C2×D712yes—
139C2×C4⋊Q8—71no—
140C2×C22⋊Q8—142no—
141C2×#121—22no—
142C2×C4×D4C2×D711yescharacteristic
143#498—142no—
144#486—142no—
145#481—78no—
146#476—144no—
147#431—72no—
148#399—74no—
149#390—141no—
150#384—141no—
151C4×C4⋊C4—71no—
152#327—78no—
153#323—74no—
154C23×C7—22no—
155C23×C7—22no—
156C23×C7—24no—
157C23×C7—42no—
158C23×C7—42no—
159C23×C7—48no—
160C23×C7—84no—
161C23×C7—84no—
162C23×C7C2×SD1614yes—
163C23×C7C22×D411yescharacteristic
164C7×Q8—44no—
165C7×D4—28no—
166C7×D4—44no—
167C7×D4—416no—
168C7×D4—44no—
169C7×D4—44no—
170C7×D4—44no—
171C7×D4—44no—
172C7×D4—84no—
173C2×C4×C7—22no—
174C2×C4×C7—22no—
175C2×C4×C7—24no—
176C2×C4×C7—41no—
177C2×C4×C7—48no—
178C2×C4×C7—48no—
179C2×C4×C7—44no—
180C2×C4×C7—42no—
181C2×C4×C7—82no—
182C2×C4×C7#12011yescharacteristic
183C2×C4×C7#12012yes—
184C2×C4×C7#12011yescharacteristic
185C2×C4×C7C2×SD1612yes—
186C2×C4×C7AGL(1,ℤ/8)12yes—
187C2×C4×C7C22×D412yes—
188C2×Dic7—24no—
189C2×Dic7—42no—
190C2×Dic7—42no—
191C2×Dic7—44no—
192C2×Dic7—42no—
193Dic14—82no—
194Dic14—82no—
195C7⋊C8—44no—
196H(5,𝔽2)—44no—
197H(5,𝔽2)—44no—
198C2×#41—24no—
199C2×#41—22no—
200C22×Q8—141no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4C22⋊D711yescharacteristic
205C23×C4C22⋊D712yes—
206C4⋊Q8—142no—
207C4⋊Q8—142no—
208C4⋊Q8C22⋊D712yes—
209#127C22⋊D712yes—
210#124—22no—
211#123—28no—
212C22⋊Q8—24no—
213C22⋊Q8—288no—
214#121—24no—
215#121—24no—
216#121—24no—
217#121—22no—
218#121—24no—
219#121C22×D714yes—
220#120—28no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4C22⋊D714yes—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—72no—
227C2×C4⋊C4—141no—
228C2×C4⋊C4—144no—
229C2×C4⋊C4—141no—
230C2×C4⋊C4C22⋊D711yescharacteristic
231C2×C22⋊C4—24no—
232C2×C22⋊C4—144no—
233C2×C22⋊C4C22×D712yes—
234C2×C42—141no—
235C2×C42C22×D711yescharacteristic
236#106—148no—
237C4⋊C8—142no—
238#102—148no—
239C22⋊C8—74no—
240C4×C8—142no—
241#95—72no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—24no—
245C22×C7—44no—
246C22×C7—42no—
247C22×C7—42no—
248C22×C7—44no—
249C22×C7—44no—
250C22×C7—44no—
251C22×C7—82no—
252C22×C7—168no—
253C22×C7C2×#12011yescharacteristic
254C22×C7C22×SD1611yescharacteristic
255C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
256C4×C7—21no—
257C4×C7—22no—
258C4×C7—44no—
259C4×C7—42no—
260C4×C7#45014yes—
261C4×C7#45312yes—
262Dic7—44no—
263Dic7—82no—
264C24—24no—
265C24C2×C22⋊D712yes—
266#41—416no—
267#41—44no—
268#41—44no—
269C2×Q8—22no—
270C2×Q8—284no—
271C2×Q8—281no—
272C2×Q8—281no—
273C2×D4—28no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—416no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—84no—
284C2×D4#114214yes—
285C2×D4C2×C22⋊D714yes—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—44no—
291C22×C4—142no—
292C22×C4—142no—
293C22×C4C2×C22⋊D711yescharacteristic
294C22×C4C2×C22⋊D712yes—
295C22×C4C2×C22⋊D711yescharacteristic
296C22×C4C23×D711yescharacteristic
297C2×C8—144no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—144no—
302C4⋊C4—144no—
303C4⋊C4—288no—
304C4⋊C4—282no—
305C4⋊C4C2×C22⋊D714yes—
306C22⋊C4—28no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—288no—
311C42—282no—
312C42C2×C22⋊D712yes—
313C42C2×C22⋊D712yes—
314C2×C7—21no—
315C2×C7—21no—
316C2×C7—44no—
317C2×C7—84no—
318C2×C7C2×#45011yescharacteristic
319C2×C7C2×#45311yescharacteristic
320C2×C7#302811yescharacteristic
321C23—22no—
322C23—22no—
323C23—24no—
324C23—42no—
325C23—42no—
326C23—48no—
327C23—84no—
328C23—84no—
329C23C2×#114214yes—
330C23C22×C22⋊D711yescharacteristic
331Q8—44no—
332Q8—562no—
333Q8—562no—
334D4—28no—
335D4—44no—
336D4—416no—
337D4—44no—
338D4—44no—
339D4—44no—
340D4—44no—
341D4—84no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—41no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—144no—
352C2×C4—282no—
353C2×C4—282no—
354C2×C4—284no—
355C2×C4—282no—
356C2×C4#649512yes—
357C2×C4C2×#114212yes—
358C2×C4#651611yescharacteristic
359C2×C4#651612yes—
360C2×C4#651611yescharacteristic
361C2×C4C22×C22⋊D712yes—
362C8—284no—
363C7#1839511yescharacteristic, complemented, Sylow
364C22—21no—
365C22—21no—
366C22—24no—
367C22—44no—
368C22—42no—
369C22—42no—
370C22—44no—
371C22—44no—
372C22—44no—
373C22—82no—
374C22—168no—
375C22C2×#649511yescharacteristic
376C22C22×#114211yescharacteristic
377C22C2×#651611yescharacteristic
378C4—21no—
379C4—22no—
380C4—44no—
381C4—42no—
382C4—284no—
383C4—562no—
384C4#8999014yes—
385C4#9001312yes—
386C2—21no—
387C2—21no—
388C2—44no—
389C2—84no—
390C2C2×#8999011yescharacteristic
391C2C2×#9001311yescharacteristic
392C2#1174156911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#3496C2
socleC22×C7C2×#120
hypercenter#3496D7
hypocenterC7#18395
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
2. #11740716C2semidirect
3. #11739824C2semidirect
4. #11729598C2semidirect
5. #11729565C2semidirect
6. #11729355C2semidirect
17. C2×#6432C22semidirect
21. #89806C22semidirect
24. #89721C22semidirect
25. C4×C4⋊Dic7C22semidirect
26. #89697C22semidirect
27. #89332C22semidirect
28. #89320C22semidirect
363. C7#18395semidirect

Maximal subgroups

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

subgroupindex
C7×#34962
#117407162
#117398242
#117295982
#117295652
#117293552
#183957