Finite Group Databasebeta

← #13257000

Subgroups — #13257000

Order 1792.

446 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 75 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
1#11731403C211yesmaximal, characteristic, complemented
2C7×#1964C211yesmaximal, characteristic
3#11729428C211yesmaximal, characteristic
4#11729427C211yesmaximal, characteristic
5#11725432C211yesmaximal, characteristic, complemented
6#11725424C211yesmaximal, characteristic, complemented
7#11725422C211yesmaximal, characteristic, complemented
8C2×C7×#121C2211yescharacteristic
9C2×C7×#99—21no—
10C2×C7×#99—21no—
11C7×#380C2211yescharacteristic
12C4×C7×C22⋊C4C2211yescharacteristic
13#89983C2211yescharacteristic
14C4×C22⋊Dic7C2211yescharacteristic
15#89979C2211yescharacteristic
16#89979C2211yescharacteristic
17#89758—21no—
18#89751—21no—
19#89720—21no—
20#89719—21no—
21#89427—21no—
22#89418—21no—
23#89332—21no—
24#89332—21no—
25#89332—21no—
26#89332—21no—
27#10382—71nomaximal, Sylow
28C22×C7×D4—21no—
29C22×C7×D4Q811yescharacteristic, complemented
30C23×C4×C7C2311yescharacteristic
31C22×Dic14—21no—
32C7×#121—21no—
33C7×#121—21no—
34C7×#121—21no—
35C7×#121—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4Q811yescharacteristic, complemented
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C42×C7—21no—
43C2×C22⋊Dic7—21no—
44C2×C22⋊Dic7—21no—
45C2×C4⋊Dic7—21no—
46C2×Dic7⋊C4—21no—
47C2×Dic7⋊C4—21no—
48C2×C4×Dic7—21no—
49C2×#1136—21no—
50C2×#1136—21no—
51C7×#99—42no—
52C7×#99—42no—
53C7×#95—21no—
54C7×#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95D411yescharacteristic
61C7⋊#95D411yescharacteristic
62C7⋊#95D411yescharacteristic
63C7⋊#95D411yescharacteristic
64C22⋊(C7⋊C8)—21no—
65C22⋊(C7⋊C8)—21no—
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)—21no—
68#2665—71no—
69#1964D711yescharacteristic
70#1510—71no—
71#1507—71no—
72#1305—71no—
73#1297—71no—
74#1284—71no—
75C24×C7—21no—
76C24×C7C2×D411yescharacteristic
77C2×C7×D4—21no—
78C2×C7×D4—21no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C2×C7×D4—41no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—81no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—41no—
90C22×C4×C7—81no—
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×Q811yescharacteristic
94C22×C4×C7#4111yescharacteristic
95C22×C4×C7#4111yescharacteristic
96C22×C4×C7#4111yescharacteristic
97C22×Dic7—21no—
98C22×Dic7—21no—
99C22×Dic7—21no—
100C22×Dic7—21no—
101C2×Dic14—42no—
102C2×Dic14—81no—
103C2×Dic14—81no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—81no—
107C7×C22⋊C4—22no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C42×C7—81no—
114C22⋊Dic7—42no—
115C22⋊Dic7—81no—
116C4⋊Dic7—81no—
117Dic7⋊C4—42no—
118Dic7⋊C4—81no—
119C4×Dic7—81no—
120#1136—81no—
121#1136—81no—
122C2×C7⋊C8—41no—
123C2×C7⋊C8—41no—
124C2×#121C2×D711yescharacteristic
125C2×#99—21no—
126C2×#99—21no—
127#407—71no—
128#407—71no—
129#400—141no—
130#396—141no—
131#395—141no—
132#393—141no—
133#391—141no—
134#387—141no—
135#381—71no—
136#380C2×D711yescharacteristic
137C4×C22⋊C4—71no—
138C4×C22⋊C4C2×D711yescharacteristic
139#327—141no—
140#327—141no—
141#327—141no—
142#327—141no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—81no—
151C23×C7—81no—
152C23×C7#12011yescharacteristic
153C23×C7C22⋊Q811yescharacteristic
154C23×C7#12411yescharacteristic
155C7×D4—81no—
156C7×D4—81no—
157C7×D4—81no—
158C7×D4—81no—
159C7×D4—81no—
160C7×D4—81no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—42no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—81no—
178C2×C4×C7—81no—
179C2×C4×C7C22⋊Q811yescharacteristic
180C2×C4×C7C22⋊Q811yescharacteristic
181C2×C4×C7#12411yescharacteristic
182C2×C4×C7#12411yescharacteristic
183C2×Dic7—22no—
184C2×Dic7—41no—
185C2×Dic7—41no—
186C2×Dic7—41no—
187C2×Dic7—41no—
188C2×Dic7—41no—
189C2×Dic7—41no—
190C2×Dic7—41no—
191C2×Dic7—41no—
192C2×Dic7—41no—
193C2×Dic7—41no—
194C2×Dic7—41no—
195Dic14—84no—
196C7⋊C8—81no—
197C7⋊C8—81no—
198C22×Q8—141no—
199C22×D4—21no—
200C22×D4Dic1411yescharacteristic, complemented
201C23×C4C22×D711yescharacteristic
202C2×M16—141no—
203C2×M16—141no—
204#121—21no—
205#121—21no—
206#121—21no—
207#121—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—141no—
210C2×C4⋊C4—141no—
211C2×C4⋊C4—141no—
212C2×C4⋊C4Dic1411yescharacteristic, complemented
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—141no—
216C2×C22⋊C4—141no—
217C2×C22⋊C4C22⋊D711yescharacteristic
218C2×C22⋊C4C22⋊D711yescharacteristic
219C2×C42—21no—
220C2×C42—141no—
221#99—42no—
222#99—42no—
223C22⋊C8—141no—
224C22⋊C8—141no—
225C22⋊C8—141no—
226C22⋊C8—141no—
227#95—21no—
228#95—21no—
229#95—71no—
230#95—71no—
231#95—71no—
232#95—71no—
233#95—141no—
234#95—141no—
235#95—141no—
236#95—141no—
237#95—141no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7—81no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7—161no—
252C22×C7—161no—
253C22×C7#39311yescharacteristic
254C22×C7#45311yescharacteristic
255C22×C7#45411yescharacteristic
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—21no—
260C4×C7—81no—
261C4×C7—81no—
262C4×C7—82no—
263C4×C7—81no—
264Dic7—42no—
265Dic7—81no—
266Dic7—81no—
267Dic7—81no—
268C24—21no—
269C24C2×C22⋊D711yescharacteristic
270C2×Q8—282no—
271C2×Q8—561no—
272C2×Q8—561no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—81no—
287C22×C4—141no—
288C22×C4—141no—
289C22×C4—141no—
290C22×C4—141no—
291C22×C4C2×Dic1411yescharacteristic
292C22×C4#115711yescharacteristic
293C22×C4D4×D711yescharacteristic
294C22×C4D4×D711yescharacteristic
295C22×C4#115911yescharacteristic
296C22×C4#115911yescharacteristic
297M16—561no—
298M16—561no—
299C2×C8—281no—
300C2×C8—281no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—81no—
304C4⋊C4—282no—
305C4⋊C4—561no—
306C4⋊C4—561no—
307C22⋊C4—22no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—282no—
314C22⋊C4—561no—
315C42—81no—
316C42—561no—
317C2×C7—21no—
318C2×C7—21no—
319C2×C7—41no—
320C2×C7—41no—
321C2×C7—161no—
322C2×C7#202711yescharacteristic
323C2×C7#203111yescharacteristic
324C2×C7#203211yescharacteristic
325C23—21no—
326C23—21no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—81no—
333C23—81no—
334C23#648711yescharacteristic
335C23#649911yescharacteristic
336C23#650011yescharacteristic
337Q8—564no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—142no—
363C2×C4—281no—
364C2×C4—281no—
365C2×C4—281no—
366C2×C4—281no—
367C2×C4—281no—
368C2×C4—281no—
369C2×C4—281no—
370C2×C4—281no—
371C2×C4—281no—
372C2×C4—281no—
373C2×C4—281no—
374C2×C4#644111yescharacteristic
375C2×C4#644111yescharacteristic
376C2×C4#644811yescharacteristic
377C2×C4#644811yescharacteristic
378C8—561no—
379C8—561no—
380C7#1038211yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—161no—
395C22—161no—
396C22#8971911yescharacteristic
397C22#8983411yescharacteristic
398C22#8984911yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—81no—
404C4—81no—
405C4—82no—
406C4—81no—
407C4—282no—
408C4—561no—
409C4—561no—
410C4—561no—
411C2—21no—
412C2—21no—
413C2—41no—
414C2—41no—
415C2—161no—
416C2#1172739711yescharacteristic
417C2#1172777911yescharacteristic
418C2#1172779211yescharacteristic
subgroupG/H
centerC22#89719
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#1964C2
socleC22×C7#393
prime radicalC2×C7×#121C22
hypercenter#1964D7
hypocenterC7#10382
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
1. #11731403C2semidirect
5. #11725432C2semidirect
6. #11725424C2semidirect
7. #11725422C2semidirect
29. C22×C7×D4Q8semidirect
37. C2×C7×C4⋊C4Q8semidirect
200. C22×D4Dic14semidirect
212. C2×C4⋊C4Dic14semidirect
380. C7#10382semidirect

Maximal subgroups

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

subgroupindex
#117314032
C7×#19642
#117294282
#117294272
#117254322
#117254242
#117254222
#103827