Finite Group Databasebeta

← #13271345

Subgroups — #13271345

Order 1792.

676 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C22×C7×#101C211yesmaximal, characteristic
2#11741510C211yesmaximal, characteristic
3#11729887C211yesmaximal, characteristic
4C2×#89273C212yesmaximal, complemented
5#11725505C212yesmaximal, complemented
6C23×C7×Q8C2211yescharacteristic
7C22×C7×M16—21no—
8C2×#6507C411yescharacteristic
9C2×#6507C2211yescharacteristic
10C2×#6487C411yescharacteristic
11C2×C7×#101—21no—
12C2×C7×#101—24no—
13C2×C7×#101C2212yes—
14#90002C411yescharacteristic
15#90002C2211yescharacteristic
16#90001—21no—
17#90001—21no—
18C2×#6410—21no—
19C2×#6410C2212yes—
20#89982—21no—
21#89982—21no—
22C4×C22⋊Dic7C411yescharacteristic
23#89979—21no—
24#89273—24no—
25#13094—71nomaximal, Sylow
26C22×C7×Q8—22no—
27C22×C7×Q8—22no—
28C22×C7×Q8—41no—
29C22×C7×Q8—41no—
30C22×C7×Q8—42no—
31C22×C7×Q8C2×C411yescharacteristic
32C22×C7×Q8D411yescharacteristic
33C22×C7×Q8D411yescharacteristic
34C22×C7×Q8C2311yescharacteristic
35C23×C4×C7C2×C411yescharacteristic
36C23×C4×C7D411yescharacteristic
37C23×C4×C7D411yescharacteristic
38C22×Dic14C2×C411yescharacteristic
39C2×C7×M16—24no—
40C2×C7×M16—22no—
41C2×C7×M16—41no—
42C2×C7×M16—42no—
43C22×C7×C8—41no—
44C2×C22⋊Dic7—21no—
45C2×C22⋊Dic7—21no—
46#6507—22no—
47#6507—22no—
48#6507—42no—
49C2×C4⋊Dic7C2×C411yescharacteristic
50#6487—41no—
51#6487—41no—
52C2×Dic7⋊C4—21no—
53C2×Dic7⋊C4—21no—
54C2×Dic7⋊C4—21no—
55C2×Dic7⋊C4—21no—
56C2×Dic7⋊C4—21no—
57C2×C4×Dic7C2×C411yescharacteristic
58C2×C4×Dic7C2×C411yescharacteristic
59C2×#1136—22no—
60C7×#101—24no—
61C7×#101—22no—
62C7×#101D414yes—
63#6410—22no—
64#6410—22no—
65#6410—42no—
66C7⋊#95—21no—
67C7⋊#95—21no—
68C7⋊#95—21no—
69C7⋊#95—21no—
70C7⋊#95—21no—
71C22⋊(C7⋊C8)—41no—
72C22⋊(C7⋊C8)—41no—
73C22×#101D711yescharacteristic
74#2434—71no—
75C2×#356—72no—
76#1784—71no—
77#1394—72no—
78C24×C7C22⋊C411yescharacteristic
79C2×C7×Q8—22no—
80C2×C7×Q8—22no—
81C2×C7×Q8—22no—
82C2×C7×Q8—21no—
83C2×C7×Q8—22no—
84C2×C7×Q8—21no—
85C2×C7×Q8—41no—
86C2×C7×Q8—41no—
87C2×C7×Q8—41no—
88C2×C7×Q8—41no—
89C2×C7×Q8—41no—
90C2×C7×Q8—41no—
91C2×C7×Q8—44no—
92C2×C7×Q8—44no—
93C2×C7×Q8—44no—
94C2×C7×Q8—44no—
95C2×C7×Q8C2×D412yes—
96C2×C7×Q8#4112yes—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—41no—
109C22×C4×C7—41no—
110C22×C4×C7C22⋊C411yescharacteristic
111C22×C4×C7C22⋊C411yescharacteristic
112C22×C4×C7C22⋊C411yescharacteristic
113C22×C4×C7C22×C411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×Dic7—21no—
117C22×Dic7—21no—
118C22×Dic7—21no—
119C22×Dic7—21no—
120C2×Dic14—22no—
121C2×Dic14—41no—
122C2×Dic14—41no—
123C7×M16—24no—
124C7×M16—44no—
125C7×M16—42no—
126C2×C7×C8—44no—
127C2×C7×C8—42no—
128C22⋊Dic7—24no—
129C22⋊Dic7—41no—
130C22⋊Dic7—41no—
131C4⋊Dic7—41no—
132Dic7⋊C4—42no—
133Dic7⋊C4—42no—
134Dic7⋊C4—42no—
135Dic7⋊C4—41no—
136Dic7⋊C4—42no—
137Dic7⋊C4—41no—
138C4×Dic7—22no—
139C4×Dic7—41no—
140#1136—42no—
141#1136—42no—
142C2×C7⋊C8—42no—
143C23×Q8C2×D711yescharacteristic
144C22×M16—21no—
145C2×C4⋊Q8—71no—
146C2×C4⋊Q8—71no—
147C2×C22⋊Q8—71no—
148C2×#101—21no—
149C2×#101—24no—
150C2×#101—72no—
151C2×#101—141no—
152C2×#101C2×D712yes—
153#407—141no—
154#395—141no—
155#395—141no—
156#391—71no—
157#391—71no—
158#385—141no—
159#385—141no—
160C4×C22⋊C4—71no—
161#356—144no—
162C23×C7—22no—
163C23×C7—21no—
164C23×C7—21no—
165C23×C7—41no—
166C23×C7#9911yescharacteristic
167C23×C7#9911yescharacteristic
168C23×C7C2×C22⋊C411yescharacteristic
169C7×Q8—42no—
170C7×Q8—42no—
171C7×Q8—44no—
172C7×Q8—44no—
173C7×Q8—42no—
174C7×Q8—42no—
175C2×C4×C7—22no—
176C2×C4×C7—22no—
177C2×C4×C7—22no—
178C2×C4×C7—22no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—42no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7—42no—
197C2×C4×C7#9912yes—
198C2×C4×C7C4×D412yes—
199C2×C4×C7#12011yescharacteristic
200C2×C4×C7#12112yes—
201C2×C4×C7#12311yescharacteristic
202C2×Dic7—22no—
203C2×Dic7—22no—
204C2×Dic7—41no—
205C2×Dic7—41no—
206C2×Dic7—41no—
207C2×Dic7—41no—
208C2×Dic7—42no—
209C2×Dic7—41no—
210C2×Dic7—41no—
211C2×Dic7—41no—
212C2×Dic7—41no—
213Dic14—82no—
214C7×C8—44no—
215C7⋊C8—82no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—41no—
219C22×Q8—41no—
220C22×Q8—42no—
221C22×Q8—71no—
222C22×Q8C4×D711yescharacteristic
223C22×Q8D2811yescharacteristic
224C22×Q8C22⋊D711yescharacteristic
225C22×Q8C22×D711yescharacteristic
226C23×C4C4×D711yescharacteristic
227C23×C4D2811yescharacteristic
228C23×C4C22⋊D711yescharacteristic
229C2×M16—24no—
230C2×M16—22no—
231C2×M16—41no—
232C2×M16—42no—
233C2×M16—142no—
234C22×C8—41no—
235C4⋊Q8—142no—
236C4⋊Q8—142no—
237C4⋊Q8—282no—
238C22⋊Q8—281no—
239C22⋊Q8—281no—
240C2×C4⋊C4—71no—
241C2×C4⋊C4—141no—
242C2×C4⋊C4—141no—
243C2×C4⋊C4—141no—
244C2×C4⋊C4—141no—
245C2×C4⋊C4—141no—
246C2×C22⋊C4—141no—
247C2×C22⋊C4—141no—
248C2×C42—71no—
249C2×C42—71no—
250#101—24no—
251#101—22no—
252#101—142no—
253#101—142no—
254#101—282no—
255#101C22⋊D714yes—
256C22⋊C8—281no—
257C22⋊C8—281no—
258#95—141no—
259#95—141no—
260#95—141no—
261#95—141no—
262#95—141no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—42no—
271C22×C7—42no—
272C22×C7#35612yescomplemented
273C22×C7#38611yescharacteristic
274C22×C7C2×#9911yescharacteristic
275C22×C7C2×#9911yescharacteristic
276C4×C7—22no—
277C4×C7—21no—
278C4×C7—22no—
279C4×C7—22no—
280C4×C7—21no—
281C4×C7—41no—
282C4×C7—41no—
283Dic7—42no—
284Dic7—42no—
285Dic7—81no—
286Dic7—81no—
287C24(C2×D7)⋊C411yescharacteristic
288C2×Q8—22no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—142no—
305C2×Q8—281no—
306C2×Q8—281no—
307C2×Q8#115712yes—
308C2×Q8C2×C22⋊D712yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—141no—
323C22×C4—141no—
324C22×C4—141no—
325C22×C4—141no—
326C22×C4(C2×D7)⋊C411yescharacteristic
327C22×C4(C2×D7)⋊C411yescharacteristic
328C22×C4(C2×D7)⋊C411yescharacteristic
329C22×C4C2×C4×D711yescharacteristic
330C22×C4C2×D2811yescharacteristic
331C22×C4C2×C22⋊D711yescharacteristic
332M16—24no—
333M16—44no—
334M16—42no—
335M16—282no—
336M16—282no—
337C2×C8—44no—
338C2×C8—42no—
339C2×C8—282no—
340C4⋊C4—281no—
341C4⋊C4—282no—
342C4⋊C4—282no—
343C4⋊C4—281no—
344C4⋊C4—282no—
345C4⋊C4—281no—
346C4⋊C4—282no—
347C22⋊C4—144no—
348C22⋊C4—281no—
349C22⋊C4—281no—
350C42—142no—
351C42—281no—
352C2×C7—21no—
353C2×C7—21no—
354C2×C7—22no—
355C2×C7—41no—
356C2×C7#189811yescharacteristic
357C2×C7C2×#35611yescharacteristic
358C2×C7#213311yescharacteristic
359C23—21no—
360C23—21no—
361C23—22no—
362C23—41no—
363C23#638011yescharacteristic
364C23#638011yescharacteristic
365C23C2×(C2×D7)⋊C411yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—44no—
369Q8—44no—
370Q8—42no—
371Q8—42no—
372Q8—562no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—142no—
396C2×C4—142no—
397C2×C4—281no—
398C2×C4—281no—
399C2×C4—281no—
400C2×C4—281no—
401C2×C4—282no—
402C2×C4—281no—
403C2×C4—281no—
404C2×C4—281no—
405C2×C4—281no—
406C2×C4#638012yes—
407C2×C4C4×C22⋊D712yes—
408C2×C4#649211yescharacteristic
409C2×C4#649312yes—
410C2×C4#651611yescharacteristic
411C8—44no—
412C8—562no—
413C7#1309411yescharacteristic, complemented, Sylow
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22#8927312yescomplemented
424C22C2×#638011yescharacteristic
425C22C2×#638011yescharacteristic
426C22#8998511yescharacteristic
427C4—21no—
428C4—22no—
429C4—21no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—282no—
435C4—282no—
436C4—561no—
437C4—561no—
438C2—21no—
439C2—21no—
440C2—22no—
441C2—41no—
442C2#1172788211yescharacteristic
443C2C2×#8927311yescharacteristic
444C2#1172929711yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC22×Q8C22×D7
FittingC22×C7×#101C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC22×C7×Q8C23
hypercenterC22×#101D7
hypocenterC7#13094
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C22×#101

Complements

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

NH ≅ G/Ntype
4. C2×#89273C2semidirect
5. #11725505C2semidirect
272. C22×C7#356semidirect
413. C7#13094semidirect
423. C22#89273semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1012
#117415102
#117298872
C2×#892732
#117255052
#130947