Finite Group Databasebeta

← #13254263

Subgroups — #13254263

Order 1792.

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

normal subgroups only

This group has 594 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×#3187C211yesmaximal, characteristic
2#11740221C211yesmaximal, characteristic, complemented
3C2×#89805C211yesmaximal, characteristic
4#11729444C211yesmaximal, characteristic, complemented
5#11729440C211yesmaximal, characteristic
6#11727678C211yesmaximal, characteristic, complemented
7#11725483C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×#121C2211yescharacteristic
10C22×C4⋊Dic7C2211yescharacteristic
11C7×#483—21no—
12C7×#481—21no—
13C7×#465—22no—
14C7×#450—21no—
15C7×#448—21no—
16C2×C7×C22⋊C8C2211yescharacteristic
17#90002—21no—
18C2×#6408—22no—
19C2×#6406—21no—
20#89984C2211yescharacteristic
21C4×C22⋊Dic7C2211yescharacteristic
22C2×C22⋊(C7⋊C8)C2211yescharacteristic
23#89805—22no—
24#89805—22no—
25#89758—22no—
26#89749—21no—
27C2×#6381—21no—
28#89715—22no—
29#89346—22no—
30#89333—22no—
31#9953—71nomaximal, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7C2311yescharacteristic
36C23×Dic7—21no—
37C2×C7×SD16—22no—
38C22×C7×C8D412yes—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×#121—21no—
44C7×#121—21no—
45C7×#121—21no—
46C7×#121—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C22⋊Dic7—21no—
53C2×C22⋊Dic7D412yes—
54C2×C4⋊Dic7—21no—
55C2×C4⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—21no—
58C2×C4⋊Dic7—21no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7D412yes—
61C2×C4×Dic7—21no—
62C22×C7⋊C8—21no—
63C7×#106—22no—
64C7×#103—22no—
65C7×#102—22no—
66C7×C22⋊C8—21no—
67C7×C22⋊C8—21no—
68#6408—44no—
69#6406—24no—
70C7⋊#95—24no—
71C7⋊#95—24no—
72C7⋊#95—21no—
73C7⋊#95—21no—
74C22⋊(C7⋊C8)—21no—
75C22⋊(C7⋊C8)D412yes—
76#6381—24no—
77#3187D711yescharacteristic
78C2×#480—71no—
79#2503—71no—
80#1964—71no—
81#1823—71no—
82#1520—71no—
83#1282—71no—
84C24×C7—21no—
85C24×C7C2×D411yescharacteristic
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—42no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—42no—
92C2×C7×D4—41no—
93C2×C7×D4—41no—
94C2×C7×D4—42no—
95C2×C7×D4—42no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—42no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7#4111yescharacteristic
106C22×Dic7—22no—
107C22×Dic7—22no—
108C22×Dic7—41no—
109C22×Dic7—41no—
110C22×Dic7—42no—
111C22×Dic7—42no—
112C22×Dic7—41no—
113C7×SD16—82no—
114C2×C7×C8—22no—
115C2×C7×C8—42no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C7×C22⋊C4—41no—
125C7×C22⋊C4—41no—
126C22⋊Dic7—22no—
127C22⋊Dic7—22no—
128C22⋊Dic7—42no—
129C4⋊Dic7—22no—
130C4⋊Dic7—22no—
131C4⋊Dic7—41no—
132C4⋊Dic7—41no—
133C4⋊Dic7—42no—
134C4×Dic7—42no—
135C2×C7⋊C8—22no—
136C2×C7⋊C8—42no—
137C2×C22⋊Q8C2×D711yescharacteristic
138C2×#121C2×D711yescharacteristic
139C22×C4⋊C4—71no—
140#483—21no—
141#481—21no—
142#480—142no—
143#480—142no—
144#465—22no—
145#450—21no—
146#448—21no—
147C2×#107—141no—
148C2×#102—141no—
149C2×#99—142no—
150C2×C22⋊C8—71no—
151C2×C22⋊C8C2×D711yescharacteristic
152#400—142no—
153#391—141no—
154#382—141no—
155#381—142no—
156#380—71no—
157C4×C22⋊C4—71no—
158#340—142no—
159#328—142no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—42no—
164C23×C7—81no—
165C23×C7—81no—
166C23×C7#12011yescharacteristic
167C23×C7#12111yescharacteristic
168C23×C7#12711yescharacteristic
169C23×C7#13512yes—
170C23×C7AGL(1,ℤ/8)12yes—
171C7×Q8—42no—
172C7×Q8—81no—
173C7×D4—42no—
174C7×D4—42no—
175C7×D4—82no—
176C7×D4—81no—
177C7×D4—81no—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—22no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—42no—
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×Dic7—42no—
195C2×Dic7—42no—
196C2×Dic7—42no—
197C2×Dic7—42no—
198C2×Dic7—44no—
199C2×Dic7—82no—
200C2×Dic7—81no—
201C2×Dic7—81no—
202C2×Dic7—81no—
203C2×Dic7—81no—
204C2×Dic7—81no—
205C7×C8—42no—
206C7⋊C8—42no—
207C22×Q8C22⋊D711yescharacteristic
208C22×D4—21no—
209C22×D4C22⋊D711yescharacteristic
210C23×C4—141no—
211C23×C4C22×D711yescharacteristic
212C2×SD16—22no—
213C22×C8—141no—
214C22×C8C22⋊D712yes—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219#121—21no—
220#121—21no—
221#121—21no—
222#121—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—72no—
225C2×C4⋊C4—141no—
226C2×C4⋊C4—141no—
227C2×C4⋊C4—141no—
228C2×C4⋊C4—141no—
229C2×C4⋊C4—141no—
230C2×C4⋊C4—141no—
231C2×C4⋊C4C22⋊D711yescharacteristic
232C2×C4⋊C4C22⋊D711yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—72no—
236C2×C22⋊C4—141no—
237C2×C42—141no—
238#107—144no—
239#106—22no—
240#103—22no—
241#102—22no—
242#102—144no—
243#99—284no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—72no—
247C22⋊C8—141no—
248#95—144no—
249#95—144no—
250#95—141no—
251#95—141no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—42no—
259C22×C7—42no—
260C22×C7—81no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7—162no—
264C22×C7#39211yescharacteristic
265C22×C7#44912yes—
266C22×C7#46312yes—
267C22×C7#49511yescharacteristic
268C22×C7#49611yescharacteristic
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—81no—
273C4×C7—81no—
274C4×C7—81no—
275C4×C7#45812yes—
276Dic7—82no—
277Dic7—82no—
278Dic7—161no—
279C24—21no—
280C24D4×D711yescharacteristic
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—42no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—142no—
296C22×C4—142no—
297C22×C4—282no—
298C22×C4—282no—
299C22×C4—281no—
300C22×C4—281no—
301C22×C4—281no—
302C22×C4D4×D711yescharacteristic
303C22×C4D4×D711yescharacteristic
304C22×C4#115911yescharacteristic
305C22×C4C2×C22⋊D711yescharacteristic
306C22×C4C2×C22⋊D711yescharacteristic
307C22×C4C2×C22⋊D711yescharacteristic
308SD16—82no—
309C2×C8—22no—
310C2×C8—42no—
311C2×C8—142no—
312C2×C8—282no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—142no—
320C4⋊C4—142no—
321C4⋊C4—282no—
322C4⋊C4—281no—
323C4⋊C4—281no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—142no—
329C22⋊C4—142no—
330C22⋊C4—282no—
331C42—282no—
332C2×C7—22no—
333C2×C7—21no—
334C2×C7—22no—
335C2×C7—21no—
336C2×C7—161no—
337C2×C7#200911yescharacteristic
338C2×C7#201211yescharacteristic
339C2×C7#202211yescharacteristic
340C23—41no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23#647712yes—
347C23#647912yes—
348C23#650011yescharacteristic
349C23#650211yescharacteristic
350C23#650311yescharacteristic
351Q8—42no—
352Q8—81no—
353D4—42no—
354D4—42no—
355D4—81no—
356D4—82no—
357D4—81no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—282no—
371C2×C4—282no—
372C2×C4—282no—
373C2×C4—284no—
374C2×C4—282no—
375C2×C4—562no—
376C2×C4—561no—
377C2×C4—561no—
378C2×C4—561no—
379C2×C4—561no—
380C2×C4—561no—
381C2×C4#650011yescharacteristic
382C2×C4#650111yescharacteristic
383C2×C4#650211yescharacteristic
384C2×C4#651611yescharacteristic
385C8—42no—
386C8—282no—
387C7#995311yescharacteristic, complemented, Sylow
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—162no—
400C22#8993712yes—
401C22#8994011yescharacteristic
402C22#8994412yes—
403C22#8994811yescharacteristic
404C22#8999511yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—562no—
412C4—562no—
413C4—1121no—
414C4#9001512yes—
415C2—22no—
416C2—21no—
417C2—22no—
418C2—21no—
419C2—161no—
420C2#1172960311yescharacteristic
421C2#1172980911yescharacteristic
422C2#1172987411yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#3187C2
socleC22×C7#392
prime radicalC2×C7×#121C22
hypercenter#3187D7
hypocenterC7#9953
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. #11740221C2semidirect
4. #11729444C2semidirect
6. #11727678C2semidirect
7. #11725483C2semidirect
387. C7#9953semidirect

Maximal subgroups

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

subgroupindex
C7×#31872
#117402212
C2×#898052
#117294442
#117294402
#117276782
#117254832
#99537