Finite Group Databasebeta

← #13250826

Subgroups — #13250826

Order 1792.

478 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3195C211yesmaximal, characteristic
2#11740195C211yesmaximal, characteristic
3#11729455C211yesmaximal, characteristic
4#11729443C211yesmaximal, characteristic
5#11729442C211yesmaximal, characteristic
6#11725491C211yesmaximal, characteristic
7#11725488C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×C22⋊Q8C2211yescharacteristic
10C7×#483—21no—
11C7×#483C412yescomplemented
12C7×#470—22no—
13C7×#451—21no—
14C7×#451C412yescomplemented
15C7×#407C2211yescharacteristic
16#90002—21no—
17#89984C2211yescharacteristic
18#89983C2211yescharacteristic
19C4×C22⋊Dic7C2211yescharacteristic
20#89979C2211yescharacteristic
21#89757—21no—
22#89753—21no—
23#89753—21no—
24#89723—21no—
25#89723—21no—
26#89333—22no—
27#89333—21no—
28#89333—21no—
29#89321—21no—
30#89321—21no—
31#9342—71nomaximal, Sylow
32C22×C7×Q8C2×C411yescharacteristic
33C22×C7×Q8D411yescharacteristic
34C23×C4×C7C2311yescharacteristic
35C2×C7×Q16—22no—
36C2×C7×M16—21no—
37C7×C22⋊Q8—21no—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8C2×C412yes—
44C7×C22⋊Q8D412yes—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4C2×C411yescharacteristic
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C22⋊Dic7—21no—
52C2×C22⋊Dic7—21no—
53C2×C22⋊Dic7D411yescharacteristic
54C2×C22⋊Dic7D411yescharacteristic
55C2×C4⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—21no—
58C2×C4×Dic7—21no—
59C2×C4×Dic7—21no—
60C2×C4×Dic7—21no—
61C2×#1136—21no—
62C7×#106—22no—
63C7×#103—22no—
64C7×#103—22no—
65C7×C22⋊C8—21no—
66C7×C22⋊C8C2×C412yes—
67C7⋊#95—21no—
68C7⋊#95—21no—
69C7⋊#95—21no—
70C7⋊#95—21no—
71C7⋊#95—21no—
72C7⋊#95—21no—
73C7⋊#95—21no—
74C7⋊#95D411yescharacteristic
75C7⋊#95D411yescharacteristic
76C22⋊(C7⋊C8)—21no—
77C22⋊(C7⋊C8)—21no—
78#3195D711yescharacteristic
79#2378—71no—
80#1996—71no—
81#1511—71no—
82#1508—71no—
83#1305—71no—
84#1284—71no—
85C24×C7C2×D411yescharacteristic
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—21no—
89C2×C7×Q8—81no—
90C2×C7×Q8—81no—
91C2×C7×Q8C22⋊C412yes—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—81no—
97C22×C4×C7C22×C411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7#4111yescharacteristic
102C22×C4×C7#4111yescharacteristic
103C22×Dic7—21no—
104C22×Dic7—21no—
105C22×Dic7—21no—
106C22×Dic7—21no—
107C22×Dic7—41no—
108C7×Q16—82no—
109C7×M16—81no—
110C2×C7×C8—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4C22⋊C412yes—
119C7×C22⋊C4—22no—
120C7×C22⋊C4—22no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—41no—
123C22⋊Dic7—22no—
124C22⋊Dic7—41no—
125C22⋊Dic7—41no—
126C22⋊Dic7—41no—
127C22⋊Dic7—81no—
128C4⋊Dic7—81no—
129C4⋊Dic7—81no—
130C4⋊Dic7—81no—
131C4×Dic7—81no—
132C4×Dic7—81no—
133C4×Dic7—81no—
134#1136—81no—
135C2×C7⋊C8—41no—
136C2×C22⋊Q8C2×D711yescharacteristic
137C2×C22⋊Q8C2×D711yescharacteristic
138#483—21no—
139#483Dic712yescomplemented
140#470—22no—
141#451—21no—
142#451Dic712yescomplemented
143#407—71no—
144#407C2×D711yescharacteristic
145#391—141no—
146#389—141no—
147#388—141no—
148#388—141no—
149#387—141no—
150#387—141no—
151#381—71no—
152#380—71no—
153C4×C22⋊C4—71no—
154#342—141no—
155#342—141no—
156#328—141no—
157#328—141no—
158#328—142no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7C4×D411yescharacteristic
164C23×C7#12011yescharacteristic
165C23×C7#12111yescharacteristic
166C7×Q8—42no—
167C7×Q8—42no—
168C7×Q8—81no—
169C7×Q8—81no—
170C2×C4×C7—22no—
171C2×C4×C7—22no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—81no—
185C2×C4×C7—81no—
186C2×C4×C7C2×C22⋊C411yescharacteristic
187C2×C4×C7C4×D411yescharacteristic
188C2×C4×C7#12111yescharacteristic
189C2×C4×C7#12311yescharacteristic
190C2×Dic7—41no—
191C2×Dic7—41no—
192C2×Dic7—41no—
193C2×Dic7—41no—
194C2×Dic7—42no—
195C2×Dic7—41no—
196C2×Dic7—41no—
197C2×Dic7—41no—
198C2×Dic7—41no—
199C2×Dic7—41no—
200C2×Dic7—41no—
201C2×Dic7—41no—
202C2×Dic7—41no—
203C2×Dic7—81no—
204C2×Dic7—81no—
205C7×C8—42no—
206C7⋊C8—81no—
207C22×Q8C2×Dic711yescharacteristic
208C22×Q8C22⋊D711yescharacteristic
209C23×C4C22×D711yescharacteristic
210C2×Q16—22no—
211C2×M16—21no—
212C2×M16—141no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8C2×Dic712yes—
220C22⋊Q8C22⋊D712yes—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—141no—
224C2×C4⋊C4—141no—
225C2×C4⋊C4—141no—
226C2×C4⋊C4C2×Dic711yescharacteristic
227C2×C4⋊C4C22⋊D711yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—71no—
231C2×C22⋊C4—71no—
232C2×C22⋊C4—141no—
233C2×C22⋊C4—141no—
234C2×C42—141no—
235C2×C42—141no—
236C2×C42—141no—
237#106—22no—
238#103—22no—
239#103—22no—
240C22⋊C8—21no—
241C22⋊C8—141no—
242C22⋊C8—141no—
243C22⋊C8C2×Dic712yes—
244#95—71no—
245#95—71no—
246#95—141no—
247#95—141no—
248#95—141no—
249#95—141no—
250#95—141no—
251#95—141no—
252#95—141no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7#38611yescharacteristic
264C22×C7#45511yescharacteristic
265C22×C7#45611yescharacteristic
266C4×C7—21no—
267C4×C7—21no—
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—42no—
271C4×C7—42no—
272C4×C7—81no—
273C4×C7—81no—
274Dic7—81no—
275Dic7—81no—
276Dic7—81no—
277Dic7—81no—
278Dic7—82no—
279C24D4×D711yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—81no—
284C2×Q8—81no—
285C2×Q8C22⋊Dic712yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—81no—
291C22×C4—141no—
292C22×C4—141no—
293C22×C4—141no—
294C22×C4—141no—
295C22×C4—281no—
296C22×C4D4×D711yescharacteristic
297C22×C4#115911yescharacteristic
298C22×C4#115911yescharacteristic
299C22×C4C22×Dic711yescharacteristic
300C22×C4C2×C22⋊D711yescharacteristic
301C22×C4C2×C22⋊D711yescharacteristic
302Q16—82no—
303M16—81no—
304M16—561no—
305C2×C8—22no—
306C2×C8—281no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—561no—
315C4⋊C4—561no—
316C4⋊C4—561no—
317C4⋊C4C22⋊Dic712yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—142no—
323C22⋊C4—281no—
324C22⋊C4—281no—
325C22⋊C4—281no—
326C22⋊C4—561no—
327C42—561no—
328C42—561no—
329C42—561no—
330C2×C7—21no—
331C2×C7—21no—
332C2×C7—41no—
333C2×C7—41no—
334C2×C7#188211yescharacteristic
335C2×C7#190511yescharacteristic
336C2×C7#191311yescharacteristic
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23D4×Dic711yescharacteristic
342C23#650011yescharacteristic
343C23#650211yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—81no—
347Q8—81no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
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—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—281no—
365C2×C4—281no—
366C2×C4—282no—
367C2×C4—281no—
368C2×C4—281no—
369C2×C4—281no—
370C2×C4—281no—
371C2×C4—281no—
372C2×C4—281no—
373C2×C4—281no—
374C2×C4—281no—
375C2×C4—281no—
376C2×C4—281no—
377C2×C4—561no—
378C2×C4—561no—
379C2×C4D4×Dic711yescharacteristic
380C2×C4#649811yescharacteristic
381C2×C4#650211yescharacteristic
382C2×C4C2×C22⋊Dic711yescharacteristic
383C8—42no—
384C8—561no—
385C7#934211yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—81no—
395C22—81no—
396C22#8997411yescharacteristic
397C22#8997711yescharacteristic
398C22#8999211yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—561no—
408C4—561no—
409C4—561no—
410C4—561no—
411C4—562no—
412C2—21no—
413C2—21no—
414C2—41no—
415C2—41no—
416C2#1172960211yescharacteristic
417C2#1172985411yescharacteristic
418C2#1172985511yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#3195C2
socleC22×C7#386
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#3195D7
hypocenterC7#9342
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
11. C7×#483C4semidirect
14. C7×#451C4semidirect
139. #483Dic7semidirect
142. #451Dic7semidirect
385. C7#9342semidirect

Maximal subgroups

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

subgroupindex
C7×#31952
#117401952
#117294552
#117294432
#117294422
#117254912
#117254882
#93427