Finite Group Databasebeta

← #13328543

Subgroups — #13328543

Order 1792.

1170 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1170 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×#3501C211yesmaximal, characteristic, complemented
2#11740800C212yesmaximal, complemented
3#11739823C211yesmaximal, characteristic
4#11729646C218yesmaximal, complemented
5#11729589C212yesmaximal, complemented
6#11729355C211yesmaximal, characteristic, complemented
7C7×#554—22no—
8C7×#554—24no—
9C7×D4×Q8—22no—
10C7×#544—21no—
11C7×#543C2214yes—
12C7×#533C2212yes—
13C2×C7×C22⋊Q8—22no—
14C7×#518—21no—
15C2×C4×C7×D4C2211yescharacteristic
16C2×#6503—21no—
17C2×#6493—22no—
18C2×#6437C2211yescharacteristic
19#89860—22no—
20#89857—22no—
21#89818C2218yes—
22#89799—24no—
23#89760—21no—
24#89732C2214yes—
25C4×C4⋊Dic7C2211yescharacteristic
26#89697C2212yes—
27#89333C2218yes—
28#89320C2214yes—
29#18389—71nomaximal, Sylow
30C22×C7×Q8—22no—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7D412yes—
33C22×C22⋊D7—24no—
34C22×D28—21no—
35C22×D28—21no—
36C22×D28C2312yes—
37C7×C4⋊Q8—24no—
38C7×C4⋊Q8D412yescomplemented
39C7×C4⋊Q8D412yescomplemented
40C7×#125—24no—
41C7×#125—22no—
42C7×C22⋊Q8—24no—
43C7×C22⋊Q8—24no—
44C7×C22⋊Q8—28no—
45C7×C22⋊Q8—22no—
46C7×C22⋊Q8—24no—
47C7×C22⋊Q8—28no—
48C7×C22⋊Q8C2314yes—
49C4×C7×Q8—22no—
50C4×C7×Q8D414yescomplemented
51C4×C7×D4—24no—
52C4×C7×D4—24no—
53C4×C7×D4—22no—
54C4×C7×D4—22no—
55C7×#117—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4C2312yes—
61C2×C42×C7C2311yescharacteristic
62C2×#1143—28no—
63#6503—44no—
64#6493—48no—
65C2×(C2×D7)⋊C4—24no—
66C2×C4⋊Dic7—21no—
67C2×C4×Dic7—21no—
68#6437—22no—
69#6437—22no—
70C7⋊#95C2312yes—
71C22⋊(C7⋊C8)C2314yes—
72#6383—28no—
73C4⋊(C7⋊C8)—22no—
74C4×C7⋊C8—22no—
75#3501D711yescharacteristic, complemented
76#3208—72no—
77#2541—71no—
78#1596—78no—
79#1503—72no—
80#1460—71no—
81C24×C7C2×D412yes—
82C23×D7—22no—
83C2×C7×Q8—24no—
84C2×C7×Q8—22no—
85C2×C7×Q8—84no—
86C2×C7×Q8C2×D414yes—
87C2×C7×D4—22no—
88C2×C7×D4—22no—
89C2×C7×D4SD1614yescomplemented
90C22×C4×C7—24no—
91C22×C4×C7—24no—
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—44no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D412yes—
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2411yescharacteristic
99C2×C22⋊D7—48no—
100C2×C22⋊D7—48no—
101C2×C22⋊D7—84no—
102C22×Dic7—22no—
103C2×D28—24no—
104C2×D28—44no—
105C2×D28—41no—
106C2×D28—42no—
107C2×D28—41no—
108C2×D28—42no—
109C2×D28—42no—
110C2×D28—82no—
111C7×C4⋊C4—24no—
112C7×C4⋊C4—28no—
113C7×C4⋊C4—22no—
114C7×C4⋊C4—24no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—24no—
119C7×C4⋊C4C2×D414yes—
120C7×C22⋊C4—28no—
121C7×C22⋊C4—24no—
122C7×C22⋊C4—24no—
123C42×C7—24no—
124C42×C7C2×D412yes—
125C42×C7C2×D412yes—
126#1143—88no—
127(C2×D7)⋊C4—48no—
128C4⋊Dic7—24no—
129C4×Dic7—42no—
130C2×C7⋊C8—24no—
131#554—22no—
132#554—24no—
133D4×Q8—22no—
134#544—21no—
135#543C2×D714yes—
136#533C2×D712yes—
137C2×#127—71no—
138C2×#127—141no—
139C2×C22⋊Q8—22no—
140C2×#121—142no—
141#518—21no—
142C2×C4×D4C2×D711yescharacteristic
143#492—142no—
144#486—142no—
145#460—144no—
146#450—78no—
147#431—72no—
148#392—74no—
149#390—141no—
150C4×C4⋊C4—71no—
151#328—78no—
152#323—74no—
153C23×C7—22no—
154C23×C7—22no—
155C23×C7—42no—
156C23×C7—42no—
157C23×C7C2×SD1614yes—
158C23×C7C22×D411yescharacteristic
159C22×D7—44no—
160C22×D7—44no—
161C22×D7—42no—
162C22×D7—42no—
163C22×D7—42no—
164C7×Q8—44no—
165C7×Q8—44no—
166C7×Q8—84no—
167C7×D4—28no—
168C2×C4×C7—22no—
169C2×C4×C7—24no—
170C2×C4×C7—24no—
171C2×C4×C7—22no—
172C2×C4×C7—41no—
173C2×C4×C7—44no—
174C2×C4×C7—42no—
175C2×C4×C7—82no—
176C2×C4×C7—84no—
177C2×C4×C7—84no—
178C2×C4×C7#12011yescharacteristic
179C2×C4×C7#12012yes—
180C2×C4×C7#12011yescharacteristic
181C2×C4×C7C2×SD1612yes—
182C2×C4×C7#13712yes—
183C2×C4×C7C22×D412yes—
184C22⋊D7—816no—
185C2×Dic7—24no—
186C2×Dic7—42no—
187C2×Dic7—42no—
188D28—48no—
189D28—82no—
190D28—84no—
191D28—82no—
192C7⋊C8—44no—
193C22×Q8—22no—
194C22×D4—72no—
195C22×D4—144no—
196C22×D4—141no—
197C22×D4—141no—
198C22×D4C22⋊D711yescharacteristic
199C23×C4C22⋊D712yes—
200C2×SD16—148no—
201C4⋊Q8—24no—
202C4⋊Q8C22⋊D712yescomplemented
203C4⋊Q8C22⋊D712yescomplemented
204#127—142no—
205#127—142no—
206#127—284no—
207#125—24no—
208#125—22no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8—28no—
212C22⋊Q8—22no—
213C22⋊Q8—24no—
214C22⋊Q8—28no—
215C22⋊Q8C22×D714yes—
216#121—288no—
217C4×Q8—22no—
218C4×Q8C22⋊D714yescomplemented
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—22no—
222C4×D4—22no—
223#117—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—141no—
227C2×C4⋊C4C22⋊D711yescharacteristic
228C2×C22⋊C4—24no—
229C2×C22⋊C4—144no—
230C2×C22⋊C4C22×D712yes—
231C2×C42—141no—
232C2×C42C22×D711yescharacteristic
233C4⋊C8—142no—
234#102—148no—
235C22⋊C8—74no—
236C4×C8—142no—
237#95—72no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—24no—
241C22×C7—44no—
242C22×C7—42no—
243C22×C7—42no—
244C22×C7—82no—
245C22×C7C2×#12011yescharacteristic
246C22×C7C22×SD1611yescharacteristic
247C22×C7C2×#13711yescharacteristic
248C2×D7—84no—
249C2×D7—84no—
250C2×D7—82no—
251C2×D7—82no—
252C2×D7—82no—
253C4×C7—21no—
254C4×C7—22no—
255C4×C7—42no—
256C4×C7—44no—
257C4×C7—44no—
258C4×C7#44814yes—
259C4×C7#45312yes—
260Dic7—44no—
261C24—142no—
262C24C2×C22⋊D712yes—
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—84no—
266C2×Q8C2×C22⋊D714yes—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—144no—
270C2×D4—284no—
271C2×D4—288no—
272C2×D4—288no—
273C2×D4—282no—
274C2×D4—282no—
275C2×D4—281no—
276C2×D4—282no—
277C2×D4—281no—
278C2×D4—562no—
279C2×D4—564no—
280C2×D4#114314yescomplemented
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—44no—
286C22×C4—142no—
287C22×C4C2×C22⋊D711yescharacteristic
288C22×C4C2×C22⋊D712yes—
289C22×C4C2×C22⋊D711yescharacteristic
290C22×C4C23×D711yescharacteristic
291SD16—568no—
292C2×C8—144no—
293C4⋊C4—24no—
294C4⋊C4—28no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—144no—
302C4⋊C4C2×C22⋊D714yes—
303C22⋊C4—28no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—288no—
307C42—24no—
308C42—282no—
309C42C2×C22⋊D712yes—
310C42C2×C22⋊D712yes—
311C2×C7—21no—
312C2×C7—21no—
313C2×C7—44no—
314C2×C7C2×#44811yescharacteristic
315C2×C7C2×#45311yescharacteristic
316C2×C7#302711yescharacteristic
317D7—162no—
318C23—22no—
319C23—22no—
320C23—42no—
321C23—42no—
322C23—284no—
323C23—284no—
324C23—282no—
325C23—282no—
326C23—282no—
327C23C2×#114314yes—
328C23C22×C22⋊D711yescharacteristic
329Q8—44no—
330Q8—44no—
331Q8—84no—
332D4—28no—
333D4—288no—
334D4—562no—
335D4—5616no—
336D4—564no—
337D4—562no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—144no—
349C2×C4—282no—
350C2×C4—282no—
351C2×C4C2×#114312yes—
352C2×C4#650512yes—
353C2×C4#651611yescharacteristic
354C2×C4#651612yes—
355C2×C4#651611yescharacteristic
356C2×C4C22×C22⋊D712yes—
357C8—284no—
358C7#1838911yescharacteristic, complemented, Sylow
359C22—21no—
360C22—21no—
361C22—24no—
362C22—44no—
363C22—42no—
364C22—42no—
365C22—82no—
366C22—564no—
367C22—564no—
368C22—562no—
369C22—562no—
370C22—562no—
371C22C22×#114311yescharacteristic
372C22C2×#650511yescharacteristic
373C22C2×#651611yescharacteristic
374C4—21no—
375C4—22no—
376C4—42no—
377C4—44no—
378C4—44no—
379C4—284no—
380C4#8999914yes—
381C4#9001312yes—
382C2—21no—
383C2—21no—
384C2—44no—
385C2—1122no—
386C2C2×#8999911yescharacteristic
387C2C2×#9001311yescharacteristic
388C2#1174157111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3501C2
socleC22×C7C2×#120
prime radical#11739823C2
hypercenter#3501D7
hypocenterC7#18389
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. C7×#3501C2semidirect
2. #11740800C2semidirect
4. #11729646C2semidirect
5. #11729589C2semidirect
6. #11729355C2semidirect
38. C7×C4⋊Q8D4semidirect
39. C7×C4⋊Q8D4semidirect
50. C4×C7×Q8D4semidirect
75. #3501D7semidirect
89. C2×C7×D4SD16semidirect
202. C4⋊Q8C22⋊D7semidirect
203. C4⋊Q8C22⋊D7semidirect
218. C4×Q8C22⋊D7semidirect
280. C2×D4#1143semidirect
358. C7#18389semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
#117408002
#117398232
#117296462
#117295892
#117293552
#183897