Finite Group Databasebeta

← #13272115

Subgroups — #13272115

Order 1792.

602 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2544C211yesmaximal, characteristic
2C2×#89340C212yesmaximal
3C2×#89336C212yesmaximal, complemented
4#11729373C211yesmaximal, characteristic, complemented
5C4×#6408C211yesmaximal, characteristic, complemented
6C2×C7×#127C2211yescharacteristic
7C2×C7×#124C411yescharacteristic, complemented
8C2×C7×#124C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C7×#394—21no—
11C7×#394—21no—
12C7×#391C411yescharacteristic, complemented
13C7×#390C411yescharacteristic, complemented
14C7×#386—21no—
15C7×#386—21no—
16C4×C7×C4⋊C4C411yescharacteristic, complemented
17C2×#6408C2212yes—
18C2×#6408C2212yes—
19#89984—21no—
20C4×C22⋊Dic7—21no—
21#89340—24no—
22#89336—24no—
23#89321—21no—
24#89321—21no—
25#13225—71nomaximal, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C22×C7×D4—21no—
28C22×C7×D4—21no—
29C22×C7×D4C2×C411yescharacteristic
30C22×C7×D4C2311yescharacteristic
31C23×C4×C7D411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C7×#127—22no—
34C7×#127—22no—
35C7×#124—22no—
36C7×#124—22no—
37C7×#124—41no—
38C7×#124—41no—
39C4×C7×D4—41no—
40C4×C7×D4—41no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4D411yescharacteristic
50C2×C7×C22⋊C4D411yescharacteristic
51C2×C42×C7C2×C411yescharacteristic
52C2×C42×C7C2×C411yescharacteristic
53C2×C42×C7C2×C411yescharacteristic
54C2×C22⋊Dic7—22no—
55C2×C22⋊Dic7—22no—
56C2×C4⋊Dic7—41no—
57C2×C4×Dic7—41no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—21no—
61#6408—22no—
62#6408—22no—
63#6408—22no—
64#6408D414yes—
65C7⋊#95—41no—
66C7⋊#95—41no—
67#2544D711yescharacteristic
68C2×#354—72no—
69C2×#353—72no—
70#1799—71no—
71C4×#99—71no—
72C24×C7—21no—
73C24×C7C2×D411yescharacteristic
74C24×C7C2×D411yescharacteristic
75C2×C7×Q8—22no—
76C2×C7×Q8—41no—
77C2×C7×Q8—41no—
78C2×C7×D4—22no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—42no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—81no—
89C2×C7×D4—81no—
90C2×C7×D4C2×D412yes—
91C2×C7×D4#4112yes—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—81no—
100C22×C4×C7—81no—
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22⋊C411yescharacteristic
103C22×C4×C7C22⋊C411yescharacteristic
104C22×C4×C7C22⋊C411yescharacteristic
105C22×C4×C7C22×C411yescharacteristic
106C22×Dic7—42no—
107C22×Dic7—42no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—42no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—81no—
119C7×C22⋊C4—81no—
120C42×C7—21no—
121C42×C7—21no—
122C42×C7—22no—
123C42×C7—22no—
124C22⋊Dic7—24no—
125C22⋊Dic7—42no—
126C22⋊Dic7—42no—
127C22⋊Dic7—42no—
128C4⋊Dic7—82no—
129C4×Dic7—82no—
130C2×#127C2×D711yescharacteristic
131C2×#124Dic711yescharacteristic, complemented
132C2×#124C2×D711yescharacteristic
133C2×C4×D4C2×D711yescharacteristic
134C2×#99—72no—
135C2×#99—72no—
136#394—21no—
137#394—21no—
138#391Dic711yescharacteristic, complemented
139#390Dic711yescharacteristic, complemented
140#386—21no—
141#386—21no—
142#380—141no—
143C4×C4⋊C4Dic711yescharacteristic, complemented
144C4×C22⋊C4—141no—
145#354—144no—
146#353—144no—
147#342—141no—
148#342—141no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—21no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7C2×C22⋊C411yescharacteristic
165C23×C7#12112yes—
166C23×C7#12411yescharacteristic
167C23×C7#12711yescharacteristic
168C7×Q8—82no—
169C7×D4—42no—
170C7×D4—42no—
171C7×D4—82no—
172C7×D4—84no—
173C7×D4—82no—
174C7×D4—82no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—22no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—81no—
196C2×C4×C7—81no—
197C2×C4×C7#9911yescharacteristic
198C2×C4×C7#9911yescharacteristic
199C2×C4×C7C4×D412yes—
200C2×Dic7—44no—
201C2×Dic7—82no—
202C2×Dic7—82no—
203C2×Dic7—82no—
204C2×Dic7—82no—
205C2×Dic7—82no—
206C22×Q8C2×Dic711yescharacteristic
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C2×Dic711yescharacteristic
210C22×D4C22×D711yescharacteristic
211C23×C4C22⋊D711yescharacteristic
212C23×C4C22⋊D711yescharacteristic
213#127—22no—
214#127—22no—
215#124—22no—
216#124—22no—
217#124—41no—
218#124—41no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—281no—
223C2×C4⋊C4C2×Dic711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—142no—
231C2×C22⋊C4—142no—
232C2×C22⋊C4C22⋊D711yescharacteristic
233C2×C22⋊C4C22⋊D711yescharacteristic
234C2×C42—281no—
235C2×C42C2×Dic711yescharacteristic
236C2×C42C2×Dic711yescharacteristic
237C2×C42C2×Dic711yescharacteristic
238#99—74no—
239#99—142no—
240#99—142no—
241#99—142no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—281no—
246#95—281no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—22no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—42no—
253C22×C7—41no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7—81no—
260C22×C7—81no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7—81no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7#39011yescharacteristic
269C22×C7C2×#9911yescharacteristic
270C22×C7#41011yescharacteristic
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—22no—
274C4×C7—42no—
275C4×C7—41no—
276C4×C7—41no—
277C4×C7—41no—
278C4×C7—81no—
279C4×C7—81no—
280Dic7—84no—
281Dic7—162no—
282C24—21no—
283C24C2×C22⋊D711yescharacteristic
284C24C2×C22⋊D711yescharacteristic
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4D4×D712yes—
301C2×D4#115912yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—282no—
312C22×C4—282no—
313C22×C4C22⋊Dic711yescharacteristic
314C22×C4C22⋊Dic711yescharacteristic
315C22×C4C22⋊Dic711yescharacteristic
316C22×C4C22⋊Dic711yescharacteristic
317C22×C4C22×Dic711yescharacteristic
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—562no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—81no—
330C22⋊C4—81no—
331C22⋊C4—144no—
332C22⋊C4—282no—
333C22⋊C4—282no—
334C22⋊C4—282no—
335C42—21no—
336C42—21no—
337C42—22no—
338C42—22no—
339C42—562no—
340C2×C7—21no—
341C2×C7—21no—
342C2×C7—42no—
343C2×C7—41no—
344C2×C7—41no—
345C2×C7—161no—
346C2×C7#196411yescharacteristic
347C2×C7#212611yescharacteristic
348C2×C7#212711yescharacteristic
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23#649911yescharacteristic
365C23#650112yes—
366C23#650311yescharacteristic
367C23C2×C22⋊Dic711yescharacteristic
368Q8—82no—
369D4—42no—
370D4—42no—
371D4—82no—
372D4—82no—
373D4—84no—
374D4—82no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—284no—
398C2×C4—562no—
399C2×C4—562no—
400C2×C4—562no—
401C2×C4—562no—
402C2×C4—562no—
403C2×C4#640811yescharacteristic
404C2×C4#640811yescharacteristic
405C2×C4D4×Dic712yes—
406C7#1322511yescharacteristic, complemented, Sylow
407C22—21no—
408C22—21no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—41no—
413C22—42no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22C2×#640811yescharacteristic
429C22#8999311yescharacteristic
430C22#9001211yescharacteristic
431C4—22no—
432C4—21no—
433C4—21no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—81no—
439C4—81no—
440C4—564no—
441C4—1122no—
442C2—21no—
443C2—21no—
444C2—42no—
445C2—41no—
446C2—41no—
447C2—161no—
448C2#1172944011yescharacteristic
449C2#1172957311yescharacteristic
450C2#1172961411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC22×D4C22×D7
FittingC7×#2544C2
socleC22×C7C2×#99
prime radicalC2×C7×#127C22
cosocleC22×C7×D4C23
hypercenter#2544D7
hypocenterC7#13225
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2544

Complements

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

NH ≅ G/Ntype
3. C2×#89336C2semidirect
4. #11729373C2semidirect
5. C4×#6408C2semidirect
7. C2×C7×#124C4semidirect
12. C7×#391C4semidirect
13. C7×#390C4semidirect
16. C4×C7×C4⋊C4C4semidirect
131. C2×#124Dic7semidirect
138. #391Dic7semidirect
139. #390Dic7semidirect
143. C4×C4⋊C4Dic7semidirect
406. C7#13225semidirect

Maximal subgroups

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

subgroupindex
C7×#25442
C2×#893402
C2×#893362
#117293732
C4×#64082
#132257