Finite Group Databasebeta

← #13254321

Subgroups — #13254321

Order 1792.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#2503C211yesmaximal, characteristic
2#11740186C211yesmaximal, characteristic
3C2×#89743C211yesmaximal, characteristic
4#11729444C211yesmaximal, characteristic
5#11729307C211yesmaximal, characteristic
6#11727211C211yesmaximal, characteristic
7#11725423C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C22×C7×C4⋊C4C2211yescharacteristic
10C2×#6487C2211yescharacteristic
11C7×#400—22no—
12C7×#391—21no—
13C7×#382—21no—
14C7×#381—22no—
15C4×C7×C22⋊C4C2211yescharacteristic
16#90000—21no—
17#89984C2211yescharacteristic
18C2×C22⋊(C7⋊C8)C2211yescharacteristic
19C2×C22⋊(C7⋊C8)C2211yescharacteristic
20#89823—22no—
21#89822—21no—
22#89816—21no—
23#89743—22no—
24#89743—22no—
25#89742—21no—
26C2×#6384—21no—
27C2×#6382—21no—
28#89333—22no—
29#89324—22no—
30#89322—22no—
31#9961—71nomaximal, Sylow
32C22×C7×Q8D411yescharacteristic
33C23×C4×C7—21no—
34C23×C4×C7C2311yescharacteristic
35C22×Dic14D411yescharacteristic
36C7×C22⋊Q8—21no—
37C7×C22⋊Q8—21no—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4D411yescharacteristic
47C2×C7×C4⋊C4D412yes—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4D412yes—
50C2×C42×C7—21no—
51C2×C22⋊Dic7—21no—
52C2×C22⋊Dic7—21no—
53C2×#1144—22no—
54C2×C4⋊Dic7—21no—
55C2×C4⋊Dic7D411yescharacteristic
56#6487—21no—
57#6487—21no—
58#6487—21no—
59#6487—21no—
60C2×Dic7⋊C4—21no—
61C22×C7⋊C8—21no—
62C22×C7⋊C8D412yes—
63C7×#95—24no—
64C7×#95—24no—
65C7×#95—21no—
66#6409—22no—
67C7⋊#95—21no—
68C22⋊(C7⋊C8)—21no—
69C22⋊(C7⋊C8)—21no—
70C22⋊(C7⋊C8)—21no—
71C22⋊(C7⋊C8)D412yes—
72#6384—24no—
73#6384—22no—
74#6382—24no—
75#6381—22no—
76#3189—71no—
77C2×#483—71no—
78#2503D711yescharacteristic
79#1996—71no—
80#1823—71no—
81#1520—71no—
82#1282—71no—
83C24×C7C2×D411yescharacteristic
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—42no—
87C22×C4×C7—22no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7—42no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—42no—
95C22×C4×C7—42no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7#4111yescharacteristic
102C22×Dic7—21no—
103C22×Dic7—21no—
104C22×Dic7—41no—
105C2×Dic14—21no—
106C2×Dic14—21no—
107C2×Dic14—42no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C22⋊C4—22no—
117C7×C22⋊C4—22no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120C42×C7—42no—
121C22⋊Dic7—41no—
122C22⋊Dic7—41no—
123C22⋊Dic7—42no—
124#1144—82no—
125C4⋊Dic7—21no—
126C4⋊Dic7—21no—
127C4⋊Dic7—42no—
128Dic7⋊C4—41no—
129Dic7⋊C4—41no—
130C2×C7⋊C8—22no—
131C2×C7⋊C8—22no—
132C2×C7⋊C8—42no—
133C2×C7⋊C8—42no—
134C2×C22⋊Q8—71no—
135C2×C22⋊Q8C2×D711yescharacteristic
136C22×C4⋊C4C2×D711yescharacteristic
137#484—141no—
138#484—141no—
139#483—142no—
140#483—142no—
141#467—142no—
142#451—141no—
143#451—141no—
144C2×#106—141no—
145C2×#103—141no—
146C2×C22⋊C8—71no—
147C2×C22⋊C8—71no—
148#400—22no—
149#391—21no—
150#382—21no—
151#381—22no—
152#380—71no—
153C4×C22⋊C4C2×D711yescharacteristic
154#342—142no—
155#340—142no—
156#328—142no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7#12011yescharacteristic
160C23×C7#12111yescharacteristic
161C23×C7#12711yescharacteristic
162C23×C7#13512yes—
163C23×C7#13712yes—
164C7×Q8—42no—
165C7×Q8—81no—
166C2×C4×C7—22no—
167C2×C4×C7—22no—
168C2×C4×C7—22no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—42no—
179C2×C4×C7—42no—
180C2×C4×C7—44no—
181C2×C4×C7—82no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C2×C4×C7#12011yescharacteristic
185C2×C4×C7#12011yescharacteristic
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7#12111yescharacteristic
188C2×Dic7—41no—
189C2×Dic7—41no—
190C2×Dic7—81no—
191C2×Dic7—81no—
192C2×Dic7—81no—
193C2×Dic7—81no—
194C2×Dic7—81no—
195Dic14—42no—
196Dic14—42no—
197C7⋊C8—42no—
198C7⋊C8—42no—
199C22×Q8—71no—
200C22×Q8C22⋊D711yescharacteristic
201C23×C4—21no—
202C23×C4C22×D711yescharacteristic
203C2×Q16—142no—
204C22×C8—72no—
205C22×C8—141no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—141no—
211C22⋊Q8—141no—
212C22⋊Q8—141no—
213C22⋊Q8—141no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—71no—
221C2×C4⋊C4—141no—
222C2×C4⋊C4—141no—
223C2×C4⋊C4C22⋊D712yes—
224C2×C4⋊C4C22⋊D711yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—141no—
227C2×C22⋊C4—141no—
228C2×C22⋊C4C22⋊D712yes—
229C2×C42—21no—
230#107—142no—
231#106—144no—
232#103—144no—
233#103—142no—
234#103—142no—
235C22⋊C8—72no—
236C22⋊C8—141no—
237C22⋊C8—141no—
238C22⋊C8—141no—
239#95—24no—
240#95—24no—
241#95—21no—
242#95—141no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—42no—
250C22×C7—42no—
251C22×C7#39211yescharacteristic
252C22×C7#45212yes—
253C22×C7#46412yes—
254C22×C7#49511yescharacteristic
255C22×C7#49711yescharacteristic
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—82no—
260C4×C7—81no—
261C4×C7—82no—
262C4×C7—81no—
263C4×C7#45812yes—
264Dic7—81no—
265Dic7—81no—
266Dic7—161no—
267C24C2×C22⋊D711yescharacteristic
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—42no—
271C2×Q8—141no—
272C2×Q8—141no—
273C2×Q8—282no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—141no—
284C22×C4—141no—
285C22×C4—281no—
286C22×C4D4×D711yescharacteristic
287C22×C4D4×D711yescharacteristic
288C22×C4D4×D711yescharacteristic
289C22×C4#115911yescharacteristic
290C22×C4C2×C22⋊D711yescharacteristic
291C22×C4C2×C22⋊D711yescharacteristic
292Q16—562no—
293C2×C8—142no—
294C2×C8—142no—
295C2×C8—282no—
296C2×C8—282no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—141no—
306C4⋊C4—141no—
307C4⋊C4—282no—
308C4⋊C4—281no—
309C4⋊C4—281no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—281no—
315C22⋊C4—281no—
316C22⋊C4—282no—
317C42—42no—
318C2×C7—21no—
319C2×C7—22no—
320C2×C7—22no—
321C2×C7—21no—
322C2×C7#200911yescharacteristic
323C2×C7#201511yescharacteristic
324C2×C7#202211yescharacteristic
325C23—41no—
326C23—41no—
327C23#650111yescharacteristic
328C23#650311yescharacteristic
329C23#651312yes—
330C23#651412yes—
331C23#651611yescharacteristic
332Q8—42no—
333Q8—81no—
334Q8—282no—
335Q8—282no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—281no—
355C2×C4—281no—
356C2×C4—561no—
357C2×C4—561no—
358C2×C4—561no—
359C2×C4—561no—
360C2×C4—561no—
361C2×C4#650011yescharacteristic
362C2×C4#650011yescharacteristic
363C2×C4#650211yescharacteristic
364C2×C4#650211yescharacteristic
365C8—282no—
366C8—282no—
367C7#996111yescharacteristic, complemented, Sylow
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#8982612yes—
377C22#8983111yescharacteristic
378C22#8983211yescharacteristic
379C22#8999511yescharacteristic
380C22#9001112yes—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—561no—
389C4—561no—
390C4—1121no—
391C4#8982512yes—
392C2—22no—
393C2—21no—
394C2—22no—
395C2—21no—
396C2#1172946011yescharacteristic
397C2#1172948311yescharacteristic
398C2#1172962311yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#2503C2
socleC22×C7#392
prime radicalC24×C7C2×D4
hypercenter#2503D7
hypocenterC7#9961
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
367. C7#9961semidirect

Maximal subgroups

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

subgroupindex
C7×#25032
#117401862
C2×#897432
#117294442
#117293072
#117272112
#117254232
#99617