Finite Group Databasebeta

← #13250824

Subgroups — #13250824

Order 1792.

478 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2378C211yesmaximal, characteristic
2#11740187C211yesmaximal, characteristic
3#11729442C211yesmaximal, characteristic
4#11727287C211yesmaximal, characteristic
5#11727283C211yesmaximal, characteristic
6#11725434C211yesmaximal, characteristic
7#11725432C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×#6487C2211yescharacteristic
10C7×#391—21no—
11C7×#389—21no—
12C7×#388—21no—
13C7×#388—21no—
14C7×#387—21no—
15C7×#387—21no—
16C7×#381C2211yescharacteristic
17C7×#380C2211yescharacteristic
18#90000—21no—
19C4×C22⋊Dic7C2211yescharacteristic
20#89979C2211yescharacteristic
21#89979C2211yescharacteristic
22#89824—22no—
23#89822C412yescomplemented
24#89815C412yescomplemented
25#89743—21no—
26#89333—22no—
27#89322—21no—
28#89322—21no—
29#89247—21no—
30#89247—21no—
31#9342—71nomaximal, Sylow
32C22×C7×Q8D411yescharacteristic
33C23×C4×C7C2311yescharacteristic
34C22×Dic14C2×C411yescharacteristic
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—21no—
37C7×C22⋊Q8C2×C412yes—
38C7×C22⋊Q8D412yes—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4D411yescharacteristic
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C42×C7—21no—
49C2×C42×C7—21no—
50C2×C22⋊Dic7—21no—
51C2×C22⋊Dic7—21no—
52C2×#1144—22no—
53C2×C4⋊Dic7C2×C411yescharacteristic
54#6487—21no—
55#6487—21no—
56#6487—21no—
57#6487—21no—
58C2×Dic7⋊C4—21no—
59C2×C4×Dic7—21no—
60C2×#1136—21no—
61C2×#1136—21no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95—21no—
65C7×#95—21no—
66C7×#95—21no—
67C7×#95—21no—
68C7×#95D411yescharacteristic
69C7×#95D411yescharacteristic
70#6409—22no—
71C7⋊#95—21no—
72C22⋊(C7⋊C8)—21no—
73C22⋊(C7⋊C8)—21no—
74C22⋊(C7⋊C8)—21no—
75C22⋊(C7⋊C8)C2×C412yes—
76#6384—22no—
77#6382—22no—
78#3195—71no—
79#2378D711yescharacteristic
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—81no—
88C2×C7×Q8C22⋊C412yes—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—81no—
96C22×C4×C7C22×C411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7#4111yescharacteristic
101C22×C4×C7#4111yescharacteristic
102C22×Dic7—21no—
103C22×Dic7—21no—
104C22×Dic7—41no—
105C2×Dic14—21no—
106C2×Dic14—21no—
107C2×Dic14—81no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—81no—
112C7×C4⋊C4—81no—
113C7×C4⋊C4—81no—
114C7×C4⋊C4C22⋊C412yes—
115C7×C22⋊C4—22no—
116C7×C22⋊C4—22no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—81no—
120C42×C7—81no—
121C42×C7—81no—
122C22⋊Dic7—22no—
123C22⋊Dic7—41no—
124C22⋊Dic7—41no—
125C22⋊Dic7—41no—
126#1144—82no—
127C4⋊Dic7—21no—
128C4⋊Dic7—21no—
129Dic7⋊C4—41no—
130Dic7⋊C4—41no—
131C4×Dic7—81no—
132#1136—81no—
133#1136—81no—
134C2×C7⋊C8—22no—
135C2×C7⋊C8—41no—
136C2×C22⋊Q8—71no—
137C2×C22⋊Q8C2×D711yescharacteristic
138#483—72no—
139#483—141no—
140#470—142no—
141#451—72no—
142#451—141no—
143#407—71no—
144#407—71no—
145#391—21no—
146#389—21no—
147#388—21no—
148#388—21no—
149#387—21no—
150#387—21no—
151#381C2×D711yescharacteristic
152#380C2×D711yescharacteristic
153C4×C22⋊C4—71no—
154#342—141no—
155#342—141no—
156#328—142no—
157#328—141no—
158#328—141no—
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—81no—
168C2×C4×C7—22no—
169C2×C4×C7—22no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
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—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×C7C2×C22⋊C411yescharacteristic
193C2×C4×C7C4×D411yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12311yescharacteristic
196C2×Dic7—42no—
197C2×Dic7—41no—
198C2×Dic7—41no—
199C2×Dic7—81no—
200C2×Dic7—81no—
201C2×Dic7—81no—
202C2×Dic7—81no—
203Dic14—42no—
204Dic14—81no—
205C7⋊C8—42no—
206C7⋊C8—81no—
207C22×Q8—71no—
208C22×Q8C22⋊D711yescharacteristic
209C23×C4C22×D711yescharacteristic
210C2×Q16—142no—
211C2×M16—141no—
212C2×M16—141no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—141no—
216C22⋊Q8—141no—
217C22⋊Q8—141no—
218C22⋊Q8—141no—
219C22⋊Q8C4×D712yes—
220C22⋊Q8D2812yes—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—71no—
226C2×C4⋊C4—141no—
227C2×C4⋊C4C22⋊D711yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—141no—
231C2×C22⋊C4—141no—
232C2×C22⋊C4C22⋊D711yescharacteristic
233C2×C22⋊C4C22⋊D711yescharacteristic
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—141no—
237#106—142no—
238#103—142no—
239#103—142no—
240C22⋊C8—72no—
241C22⋊C8—141no—
242C22⋊C8—141no—
243C22⋊C8—141no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—141no—
251#95C22⋊D711yescharacteristic
252#95C22⋊D711yescharacteristic
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—
274C4×C7—81no—
275C4×C7—81no—
276Dic7—82no—
277Dic7—81no—
278Dic7—81no—
279C24C2×C22⋊D711yescharacteristic
280C2×Q8—21no—
281C2×Q8—81no—
282C2×Q8—141no—
283C2×Q8—141no—
284C2×Q8—561no—
285C2×Q8(C2×D7)⋊C412yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—81no—
293C22×C4—141no—
294C22×C4—141no—
295C22×C4—281no—
296C22×C4C2×C4×D711yescharacteristic
297C22×C4C2×D2811yescharacteristic
298C22×C4#115711yescharacteristic
299C22×C4#115711yescharacteristic
300C22×C4C2×C22⋊D711yescharacteristic
301C22×C4C2×C22⋊D711yescharacteristic
302Q16—562no—
303M16—561no—
304M16—561no—
305C2×C8—142no—
306C2×C8—281no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C4⋊C4—81no—
313C4⋊C4—141no—
314C4⋊C4—141no—
315C4⋊C4—281no—
316C4⋊C4—281no—
317C4⋊C4(C2×D7)⋊C412yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—81no—
323C22⋊C4—142no—
324C22⋊C4—281no—
325C22⋊C4—281no—
326C22⋊C4—281no—
327C42—81no—
328C42—81no—
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—
341C23C4×C22⋊D711yescharacteristic
342C23#649311yescharacteristic
343C23#651611yescharacteristic
344Q8—42no—
345Q8—81no—
346Q8—282no—
347Q8—561no—
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—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—282no—
373C2×C4—281no—
374C2×C4—281no—
375C2×C4—561no—
376C2×C4—561no—
377C2×C4—561no—
378C2×C4—561no—
379C2×C4C2×(C2×D7)⋊C411yescharacteristic
380C2×C4C4×C22⋊D711yescharacteristic
381C2×C4#649211yescharacteristic
382C2×C4#649311yescharacteristic
383C8—282no—
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#8998511yescharacteristic
397C22#9001411yescharacteristic
398C22#9001811yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—562no—
410C4—561no—
411C4—561no—
412C2—21no—
413C2—21no—
414C2—41no—
415C2—41no—
416C2#1172941011yescharacteristic
417C2#1172941611yescharacteristic
418C2#1172942011yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2378C2
socleC22×C7#386
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2378D7
hypocenterC7#9342
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2378

Complements

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

NH ≅ G/Ntype
23. #89822C4semidirect
24. #89815C4semidirect
385. C7#9342semidirect

Maximal subgroups

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

subgroupindex
C7×#23782
#117401872
#117294422
#117272872
#117272832
#117254342
#117254322
#93427