Finite Group Databasebeta

← #13272102

Subgroups — #13272102

Order 1792.

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

normal subgroups only

This group has 628 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×#2534C211yesmaximal, characteristic
2C2×#89334C212yesmaximal, complemented
3C2×#89331C212yesmaximal
4C4×#6408C211yesmaximal, characteristic, complemented
5#11729371C211yesmaximal, characteristic, complemented
6C2×C7×#123C2211yescharacteristic
7C2×C7×#120C2211yescharacteristic
8C2×C4×C7×D4C2211yescharacteristic
9C22×C7×C22⋊C4C411yescharacteristic, complemented
10C7×#396—21no—
11C7×#394—21no—
12C7×#388—21no—
13C7×#386—21no—
14C7×#381C411yescharacteristic, complemented
15C7×#380C411yescharacteristic, complemented
16C7×#379C411yescharacteristic, complemented
17C2×#6408C2212yes—
18C2×#6407C2212yes—
19C4×C22⋊Dic7—21no—
20#89696—21no—
21#89334—24no—
22#89331—24no—
23#89325—21no—
24#89321—21no—
25#13222—71nomaximal, Sylow
26C25×C7C2×C411yescharacteristic
27C22×C7×D4—21no—
28C22×C7×D4C2311yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7C2×C411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C7×#123—22no—
33C7×#123—22no—
34C7×#120—22no—
35C7×#120—22no—
36C4×C7×D4—41no—
37C4×C7×D4—41no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—41no—
48C2×C7×C22⋊C4—41no—
49C2×C7×C22⋊C4C2×C411yescharacteristic
50C2×C7×C22⋊C4C2×C411yescharacteristic
51C2×C7×C22⋊C4C2×C411yescharacteristic
52C2×C7×C22⋊C4D411yescharacteristic
53C2×C42×C7D411yescharacteristic
54C2×C22⋊Dic7—22no—
55C2×C4×Dic7—41no—
56C2×#1136—22no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60#6408—22no—
61#6408D414yes—
62#6407—22no—
63#6407—22no—
64C7⋊#95—41no—
65C4⋊(C7⋊C8)—41no—
66C4⋊(C7⋊C8)—41no—
67#2534D711yescharacteristic
68C2×#352—72no—
69C2×AΣL(1,𝔽16)—72no—
70#1800—71no—
71C4×#99—71no—
72C24×C7—21no—
73C24×C7—22no—
74C24×C7—41no—
75C24×C7—42no—
76C24×C7—41no—
77C24×C7—81no—
78C24×C7C22×C411yescharacteristic
79C24×C7C2×D411yescharacteristic
80C2×C7×D4—41no—
81C2×C7×D4—41no—
82C2×C7×D4—42no—
83C2×C7×D4—42no—
84C2×C7×D4—81no—
85C2×C7×D4C2×D412yes—
86C2×C7×D4#4112yes—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—22no—
95C22×C4×C7—41no—
96C22×C4×C7—41no—
97C22×C4×C7—81no—
98C22×C4×C7—81no—
99C22×C4×C7C22⋊C411yescharacteristic
100C22×C4×C7C22⋊C411yescharacteristic
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22⋊C411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×Dic7—42no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—81no—
109C7×C22⋊C4—22no—
110C7×C22⋊C4—22no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—81no—
121C42×C7—21no—
122C42×C7—21no—
123C22⋊Dic7—24no—
124C22⋊Dic7—42no—
125C4×Dic7—82no—
126#1136—42no—
127#1136—42no—
128C2×C7⋊C8—42no—
129C2×#123C2×D711yescharacteristic
130C2×#120C2×D711yescharacteristic
131C2×C4×D4C2×D711yescharacteristic
132C22×C22⋊C4Dic711yescharacteristic, complemented
133#423—141no—
134C2×#100—72no—
135C2×#99—72no—
136#396—21no—
137#394—21no—
138#388—21no—
139#386—21no—
140#381Dic711yescharacteristic, complemented
141#380Dic711yescharacteristic, complemented
142#379Dic711yescharacteristic, complemented
143C4×C22⋊C4—141no—
144#352—144no—
145AΣL(1,𝔽16)—144no—
146#343—141no—
147#342—141no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—22no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—42no—
161C23×C7—42no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7—82no—
166C23×C7—84no—
167C23×C7—84no—
168C23×C7—82no—
169C23×C7C2×C22⋊C411yescharacteristic
170C23×C7C4×D412yes—
171C23×C7#12112yes—
172C7×D4—42no—
173C7×D4—42no—
174C7×D4—84no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—41no—
181C2×C4×C7—42no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
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—81no—
195C2×C4×C7—81no—
196C2×C4×C7—81no—
197C2×C4×C7—82no—
198C2×C4×C7#9911yescharacteristic
199C2×C4×C7#9911yescharacteristic
200C2×C4×C7#12411yescharacteristic
201C2×C4×C7#12711yescharacteristic
202C2×Dic7—44no—
203C2×Dic7—82no—
204C2×Dic7—82no—
205C7⋊C8—82no—
206C25C2×Dic711yescharacteristic
207C22×D4—21no—
208C22×D4C22×D711yescharacteristic
209C23×C4C2×Dic711yescharacteristic
210C23×C4C2×Dic711yescharacteristic
211C23×C4C22⋊D711yescharacteristic
212C2×M16—142no—
213#123—22no—
214#123—22no—
215#120—22no—
216#120—22no—
217C4×D4—41no—
218C4×D4—41no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22⋊D711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—142no—
231C2×C22⋊C4C2×Dic711yescharacteristic
232C2×C22⋊C4C2×Dic711yescharacteristic
233C2×C22⋊C4C2×Dic711yescharacteristic
234C2×C22⋊C4C22⋊D711yescharacteristic
235C2×C42—281no—
236C2×C42C22⋊D711yescharacteristic
237C4⋊C8—281no—
238C4⋊C8—281no—
239#100—142no—
240#100—142no—
241#99—74no—
242#99—142no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—281no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—22no—
250C22×C7—22no—
251C22×C7—42no—
252C22×C7—42no—
253C22×C7—42no—
254C22×C7—42no—
255C22×C7—42no—
256C22×C7—41no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—42no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7—81no—
264C22×C7—84no—
265C22×C7—84no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7#39011yescharacteristic
269C22×C7C2×#9911yescharacteristic
270C22×C7#41011yescharacteristic
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274C4×C7—21no—
275C4×C7—41no—
276C4×C7—41no—
277C4×C7—41no—
278C4×C7—81no—
279C4×C7—81no—
280C4×C7—81no—
281Dic7—84no—
282C24—21no—
283C24—22no—
284C24—41no—
285C24—42no—
286C24—41no—
287C24—81no—
288C24C22×Dic711yescharacteristic
289C24C2×C22⋊D711yescharacteristic
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—81no—
295C2×D4D4×D712yes—
296C2×D4#115912yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—282no—
310C22×C4C22⋊Dic711yescharacteristic
311C22×C4C22⋊Dic711yescharacteristic
312C22×C4C22⋊Dic711yescharacteristic
313C22×C4C22⋊Dic711yescharacteristic
314C22×C4C2×C22⋊D711yescharacteristic
315M16—282no—
316M16—282no—
317C2×C8—282no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—81no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—81no—
334C22⋊C4—144no—
335C22⋊C4—282no—
336C42—21no—
337C42—21no—
338C42—562no—
339C2×C7—21no—
340C2×C7—21no—
341C2×C7—42no—
342C2×C7—42no—
343C2×C7—82no—
344C2×C7#196411yescharacteristic
345C2×C7#212011yescharacteristic
346C2×C7#212111yescharacteristic
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—22no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—82no—
365C23—84no—
366C23—84no—
367C23—82no—
368C23D4×Dic712yes—
369C23#650112yes—
370C23C2×C22⋊Dic711yescharacteristic
371D4—42no—
372D4—42no—
373D4—84no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
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—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—284no—
398C2×C4—562no—
399C2×C4—562no—
400C2×C4#640811yescharacteristic
401C2×C4#640811yescharacteristic
402C2×C4#649911yescharacteristic
403C2×C4#650311yescharacteristic
404C8—562no—
405C7#1322211yescharacteristic, complemented, Sylow
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—84no—
424C22—84no—
425C22—81no—
426C22—81no—
427C22C2×#640811yescharacteristic
428C22#8999311yescharacteristic
429C22#9001211yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—564no—
441C2—21no—
442C2—21no—
443C2—42no—
444C2—42no—
445C2—82no—
446C2#1172944011yescharacteristic
447C2#1172947011yescharacteristic
448C2#1172960111yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC22×D4C22×D7
FittingC7×#2534C2
socleC22×C7C2×#99
prime radicalC2×C7×#120C22
cosocleC22×C7×D4C23
hypercenter#2534D7
hypocenterC7#13222
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
2. C2×#89334C2semidirect
4. C4×#6408C2semidirect
5. #11729371C2semidirect
9. C22×C7×C22⋊C4C4semidirect
14. C7×#381C4semidirect
15. C7×#380C4semidirect
16. C7×#379C4semidirect
132. C22×C22⋊C4Dic7semidirect
140. #381Dic7semidirect
141. #380Dic7semidirect
142. #379Dic7semidirect
405. C7#13222semidirect

Maximal subgroups

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

subgroupindex
C7×#25342
C2×#893342
C2×#893312
C4×#64082
#117293712
#132227