Finite Group Databasebeta

← #13327991

Subgroups — #13327991

Order 1792.

950 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 183 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×#544C211yesmaximal, characteristic
2#11740187C212yesmaximal
3#11739626C211yesmaximal, characteristic
4#11729658C214yesmaximal, complemented
5#11729652C214yesmaximal, complemented
6#11729442C212yesmaximal
7#11729306C211yesmaximal, characteristic
8C7×#544—24no—
9C7×#544—22no—
10C7×#544C2214yes—
11C2×C7×C4⋊Q8—22no—
12C2×C7×#125—22no—
13C2×C7×C22⋊Q8—21no—
14C2×C7×C22⋊Q8C2212yes—
15C2×C7×#117—21no—
16C22×C7×C4⋊C4C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18#90000—22no—
19#89982—21no—
20#89982—21no—
21C4×C22⋊Dic7C2211yescharacteristic
22#89979C2212yes—
23#89824—24no—
24#89822C2214yes—
25#89815C2214yes—
26#89755C2212yes—
27#89751C2212yes—
28#89744—22no—
29#89743—22no—
30#89333C2218yes—
31#89323C2214yes—
32#18327—71nomaximal, Sylow
33C22×C7×Q8—21no—
34C22×C7×Q8D412yes—
35C23×C4×C7D411yescharacteristic
36C23×C4×C7D411yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C22×Dic14C2311yescharacteristic
39C7×C4⋊Q8—28no—
40C7×C4⋊Q8—28no—
41C7×#125—28no—
42C7×#125—28no—
43C7×C22⋊Q8—24no—
44C7×C22⋊Q8—24no—
45C7×C22⋊Q8—22no—
46C7×C22⋊Q8—22no—
47C7×C22⋊Q8D414yes—
48C7×C22⋊Q8C2314yes—
49C7×#117—24no—
50C7×#117—24no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4D412yes—
59C2×C7×C4⋊C4D412yes—
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C22⋊C4—22no—
62C2×C42×C7—22no—
63C2×C22⋊Dic7—21no—
64C2×C22⋊Dic7—21no—
65C2×#1144—24no—
66C2×C4⋊Dic7C2311yescharacteristic
67#6487—22no—
68#6487—22no—
69C2×Dic7⋊C4—21no—
70C2×Dic7⋊C4—22no—
71C2×Dic7⋊C4—22no—
72C2×C4×Dic7—21no—
73C2×#1136—22no—
74#6409—24no—
75C7⋊#95—22no—
76C7⋊#95—22no—
77C7⋊#95C2312yes—
78C22⋊(C7⋊C8)—22no—
79C22⋊(C7⋊C8)C2314yes—
80#6384—24no—
81#6382—24no—
82C2×#544D711yescharacteristic
83#3195—72no—
84#2583—71no—
85#1611—74no—
86#1605—74no—
87#1508—72no—
88#1463—71no—
89C24×C7C2×D411yescharacteristic
90C2×C7×Q8—24no—
91C2×C7×Q8—22no—
92C2×C7×Q8—82no—
93C2×C7×Q8—82no—
94C2×C7×Q8C2×D414yes—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—24no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—81no—
103C22×C4×C7—81no—
104C22×C4×C7C2×D412yes—
105C22×C4×C7C2×D412yes—
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D412yes—
108C22×C4×C7C2×D412yes—
109C22×C4×C7C2411yescharacteristic
110C22×Dic7—22no—
111C22×Dic7—21no—
112C22×Dic7—21no—
113C2×Dic14—22no—
114C2×Dic14—81no—
115C7×C4⋊C4—24no—
116C7×C4⋊C4—24no—
117C7×C4⋊C4—24no—
118C7×C4⋊C4—28no—
119C7×C4⋊C4—24no—
120C7×C4⋊C4—24no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—24no—
123C7×C4⋊C4—24no—
124C7×C4⋊C4C2×D414yes—
125C7×C22⋊C4—24no—
126C7×C22⋊C4—24no—
127C42×C7—28no—
128C22⋊Dic7—24no—
129C22⋊Dic7—41no—
130C22⋊Dic7—41no—
131#1144—84no—
132C4⋊Dic7—22no—
133Dic7⋊C4—42no—
134Dic7⋊C4—44no—
135Dic7⋊C4—82no—
136C4×Dic7—81no—
137#1136—82no—
138C2×C7⋊C8—24no—
139#544—24no—
140#544—22no—
141#544C2×D714yes—
142C2×C4⋊Q8—22no—
143C2×#125—22no—
144C2×C22⋊Q8—21no—
145C2×C22⋊Q8—71no—
146C2×C22⋊Q8C2×D712yes—
147C2×#117—21no—
148C22×C4⋊C4C2×D711yescharacteristic
149#483—74no—
150#483—142no—
151#470—144no—
152#451—74no—
153#451—142no—
154#407—72no—
155#398—72no—
156#395—72no—
157#389—142no—
158#385—141no—
159#385—141no—
160C4×C22⋊C4—71no—
161#328—78no—
162#324—74no—
163C23×C7—22no—
164C23×C7—22no—
165C23×C7—41no—
166C23×C7#12011yescharacteristic
167C23×C7#12011yescharacteristic
168C23×C7C22×D411yescharacteristic
169C7×Q8—44no—
170C7×Q8—82no—
171C7×Q8—84no—
172C2×C4×C7—24no—
173C2×C4×C7—24no—
174C2×C4×C7—22no—
175C2×C4×C7—22no—
176C2×C4×C7—24no—
177C2×C4×C7—24no—
178C2×C4×C7—22no—
179C2×C4×C7—42no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—44no—
185C2×C4×C7—44no—
186C2×C4×C7—42no—
187C2×C4×C7—81no—
188C2×C4×C7#12012yes—
189C2×C4×C7#13714yes—
190C2×C4×C7C22×D412yes—
191C2×Dic7—24no—
192C2×Dic7—42no—
193C2×Dic7—42no—
194C2×Dic7—42no—
195C2×Dic7—41no—
196C2×Dic7—41no—
197C2×Dic7—42no—
198Dic14—44no—
199C7⋊C8—44no—
200C22×Q8—21no—
201C22×Q8—71no—
202C22×Q8C22⋊D712yes—
203C23×C4C22⋊D711yescharacteristic
204C23×C4C22⋊D711yescharacteristic
205C23×C4C22×D711yescharacteristic
206C2×Q16—144no—
207C2×M16—142no—
208C4⋊Q8—28no—
209C4⋊Q8—28no—
210#125—28no—
211#125—28no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—142no—
217C22⋊Q8—142no—
218C22⋊Q8C22⋊D714yes—
219C22⋊Q8C22×D714yes—
220#117—24no—
221#117—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—71no—
230C2×C4⋊C4—141no—
231C2×C4⋊C4—142no—
232C2×C4⋊C4—142no—
233C2×C4⋊C4C22⋊D712yes—
234C2×C4⋊C4C22⋊D712yes—
235C2×C4⋊C4C22×D712yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—141no—
238C2×C22⋊C4—141no—
239C2×C42—22no—
240C2×C42—141no—
241#106—144no—
242#103—144no—
243#103—144no—
244C22⋊C8—74no—
245C22⋊C8—142no—
246#95—72no—
247#95—142no—
248#95—142no—
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—44no—
257C22×C7#45612yes—
258C22×C7C2×#12011yescharacteristic
259C22×C7C2×#13712yes—
260C4×C7—24no—
261C4×C7—22no—
262C4×C7—22no—
263C4×C7—44no—
264C4×C7—42no—
265C4×C7—44no—
266Dic7—44no—
267Dic7—81no—
268Dic7—81no—
269C24C2×C22⋊D711yescharacteristic
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—142no—
275C2×Q8—561no—
276C2×Q8C2×C22⋊D714yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—142no—
287C22×C4—141no—
288C22×C4—141no—
289C22×C4C2×C22⋊D712yes—
290C22×C4C2×C22⋊D712yes—
291C22×C4C2×C22⋊D711yescharacteristic
292C22×C4C2×C22⋊D712yes—
293C22×C4C2×C22⋊D712yes—
294C22×C4C23×D711yescharacteristic
295Q16—564no—
296M16—562no—
297C2×C8—144no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—142no—
308C4⋊C4—284no—
309C4⋊C4—282no—
310C4⋊C4—562no—
311C4⋊C4C2×C22⋊D714yes—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—144no—
315C22⋊C4—281no—
316C22⋊C4—281no—
317C42—28no—
318C42—561no—
319C2×C7—22no—
320C2×C7—22no—
321C2×C7—41no—
322C2×C7#300411yescharacteristic
323C2×C7C2×#45611yescharacteristic
324C2×C7#302611yescharacteristic
325C23—22no—
326C23—22no—
327C23—41no—
328C23#651611yescharacteristic
329C23#651611yescharacteristic
330C23C22×C22⋊D711yescharacteristic
331Q8—44no—
332Q8—82no—
333Q8—84no—
334Q8—284no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—144no—
352C2×C4—282no—
353C2×C4—282no—
354C2×C4—282no—
355C2×C4—282no—
356C2×C4—281no—
357C2×C4—281no—
358C2×C4#650514yes—
359C2×C4#651612yes—
360C2×C4C22×C22⋊D712yes—
361C8—284no—
362C7#1832711yescharacteristic, complemented, Sylow
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22#9001812yes—
372C22C2×#650512yes—
373C22C2×#651611yescharacteristic
374C4—24no—
375C4—22no—
376C4—22no—
377C4—42no—
378C4—44no—
379C4—44no—
380C4—284no—
381C4—561no—
382C4—561no—
383C2—22no—
384C2—22no—
385C2—41no—
386C2#1174150411yescharacteristic
387C2#1174155511yescharacteristic
388C2C2×#9001811yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#544C2
socleC22×C7C2×#120
prime radicalC24×C7C2×D4
hypercenterC2×#544D7
hypocenterC7#18327
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
4. #11729658C2semidirect
5. #11729652C2semidirect
362. C7#18327semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5442
#117401872
#117396262
#117296582
#117296522
#117294422
#117293062
#183277