Finite Group Databasebeta

← #13264582

Subgroups — #13264582

Order 1792.

476 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2631C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic, complemented
3#11729356C211yesmaximal, characteristic, complemented
4#11725452C212yesmaximal, complemented
5#11725451C212yesmaximal, complemented
6C2×C7×#124C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C7×#397—21no—
9C7×#393C411yescharacteristic, complemented
10C7×#393C2211yescharacteristic
11C7×#392—21no—
12C7×#392C411yescharacteristic, complemented
13C7×#392C2211yescharacteristic
14C7×#386—21no—
15C7×#386—21no—
16C7×#383C411yescharacteristic, complemented
17C2×#6408—21no—
18#89984—21no—
19#89979—21no—
20#89323—21no—
21#89321C2212yes—
22#89320C2212yes—
23#11825—71nomaximal, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4—21no—
26C22×C7×D4—21no—
27C22×C7×D4C2×C411yescharacteristic
28C22×C7×D4D411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C7×#124—41no—
32C7×#124—41no—
33C4×C7×D4—41no—
34C4×C7×D4—41no—
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4C2311yescharacteristic
44C2×C42×C7C2×C411yescharacteristic
45C2×C22⋊Dic7—22no—
46C2×C4⋊Dic7—41no—
47C2×#1136—41no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95C2×C411yescharacteristic
53C7×#95C2×C411yescharacteristic
54#6408—42no—
55C7⋊#95—41no—
56C22⋊(C7⋊C8)—22no—
57C22⋊(C7⋊C8)—41no—
58#2631D711yescharacteristic
59#1799—71no—
60#1465—71no—
61#1351—72no—
62#1343—72no—
63C24×C7—21no—
64C24×C7C22⋊C411yescharacteristic
65C24×C7C2×D411yescharacteristic
66C2×C7×Q8—41no—
67C2×C7×Q8—81no—
68C2×C7×D4—21no—
69C2×C7×D4—21no—
70C2×C7×D4—41no—
71C2×C7×D4—41no—
72C2×C7×D4—41no—
73C2×C7×D4—42no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C2×C7×D4—81no—
77C2×C7×D4—81no—
78C2×C7×D4—81no—
79C2×C7×D4—81no—
80C2×C7×D4—81no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—81no—
89C22×C4×C7—81no—
90C22×C4×C7C22⋊C411yescharacteristic
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22×C411yescharacteristic
94C22×C4×C7C2×D411yescharacteristic
95C22×Dic7—42no—
96C7×C4⋊C4—41no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—81no—
103C7×C22⋊C4—81no—
104C7×C22⋊C4—81no—
105C7×C22⋊C4—81no—
106C42×C7—21no—
107C42×C7—21no—
108C22⋊Dic7—42no—
109C22⋊Dic7—42no—
110C4⋊Dic7—82no—
111#1136—82no—
112C2×C7⋊C8—42no—
113C2×#124Dic711yescharacteristic, complemented
114C2×C4×D4C2×D711yescharacteristic
115C2×#99—141no—
116#407—141no—
117#397—21no—
118#393Dic711yescharacteristic, complemented
119#393C2×D711yescharacteristic
120#392—21no—
121#392Dic711yescharacteristic, complemented
122#392C2×D711yescharacteristic
123#386—21no—
124#386—21no—
125#383Dic711yescharacteristic, complemented
126#380—141no—
127#342—72no—
128#324—141no—
129#323—72no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7C2×C22⋊C411yescharacteristic
147C23×C7AGL(1,ℤ/8)11yescharacteristic
148C23×C7#13711yescharacteristic
149C7×Q8—82no—
150C7×D4—42no—
151C7×D4—42no—
152C7×D4—82no—
153C7×D4—82no—
154C7×D4—82no—
155C7×D4—84no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—41no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
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—81no—
178C2×C4×C7—81no—
179C2×C4×C7#9911yescharacteristic
180C2×C4×C7#9911yescharacteristic
181C2×Dic7—82no—
182C2×Dic7—82no—
183C2×Dic7—82no—
184C7⋊C8—82no—
185C22×Q8C2×Dic711yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C2×Dic711yescharacteristic
189C22×D4C22⋊D711yescharacteristic
190C23×C4C22⋊D711yescharacteristic
191C23×C4C22⋊D711yescharacteristic
192C2×M16—281no—
193#124—41no—
194#124—41no—
195C4×D4—41no—
196C4×D4—41no—
197C2×C4⋊C4—281no—
198C2×C4⋊C4C22⋊D711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—142no—
206C2×C22⋊C4C2×Dic711yescharacteristic
207C2×C22⋊C4C22×D711yescharacteristic
208C2×C42C2×Dic711yescharacteristic
209#99—282no—
210C22⋊C8—142no—
211C22⋊C8—281no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—281no—
217#95C2×Dic711yescharacteristic
218#95C2×Dic711yescharacteristic
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—42no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—81no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7—81no—
238C22×C7—81no—
239C22×C7—81no—
240C22×C7C2×#9911yescharacteristic
241C22×C7#41711yescharacteristic
242C22×C7#42111yescharacteristic
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—41no—
246C4×C7—41no—
247C4×C7—41no—
248C4×C7—41no—
249C4×C7—81no—
250C4×C7—81no—
251C4×C7—81no—
252Dic7—162no—
253C24—21no—
254C24C22⋊Dic711yescharacteristic
255C24C2×C22⋊D711yescharacteristic
256C2×Q8—41no—
257C2×Q8—81no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—81no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4—81no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—282no—
281C22×C4C22⋊Dic711yescharacteristic
282C22×C4C22⋊Dic711yescharacteristic
283C22×C4C22⋊Dic711yescharacteristic
284C22×C4C22×Dic711yescharacteristic
285C22×C4C2×C22⋊D711yescharacteristic
286M16—562no—
287C2×C8—282no—
288C4⋊C4—41no—
289C4⋊C4—562no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C22⋊C4—282no—
300C22⋊C4—282no—
301C42—21no—
302C42—21no—
303C2×C7—21no—
304C2×C7—21no—
305C2×C7—41no—
306C2×C7—41no—
307C2×C7—81no—
308C2×C7—161no—
309C2×C7#151311yescharacteristic
310C2×C7#212211yescharacteristic
311C2×C7#212711yescharacteristic
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23#651211yescharacteristic
329C23#651411yescharacteristic
330C23C2×C22⋊Dic711yescharacteristic
331Q8—82no—
332D4—42no—
333D4—42no—
334D4—82no—
335D4—82no—
336D4—82no—
337D4—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
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—81no—
360C2×C4—81no—
361C2×C4—562no—
362C2×C4—562no—
363C2×C4—562no—
364C2×C4#640811yescharacteristic
365C2×C4#640811yescharacteristic
366C8—562no—
367C7#1182511yescharacteristic, complemented, Sylow
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C2×#640811yescharacteristic
390C22#9000411yescharacteristic
391C22#9000811yescharacteristic
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—1122no—
402C2—21no—
403C2—21no—
404C2—41no—
405C2—41no—
406C2—81no—
407C2—161no—
408C2#1172946311yescharacteristic
409C2#1172961311yescharacteristic
410C2#1172961411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
Fitting, prime radicalC7×#2631C2
socleC22×C7C2×#99
cosocleC2×C7×C22⋊C4C23
hypercenter#2631D7
hypocenterC7#11825
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2631

Complements

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

NH ≅ G/Ntype
2. #11729373C2semidirect
3. #11729356C2semidirect
4. #11725452C2semidirect
5. #11725451C2semidirect
6. C2×C7×#124C4semidirect
9. C7×#393C4semidirect
12. C7×#392C4semidirect
16. C7×#383C4semidirect
113. C2×#124Dic7semidirect
118. #393Dic7semidirect
121. #392Dic7semidirect
125. #383Dic7semidirect
367. C7#11825semidirect

Maximal subgroups

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

subgroupindex
C7×#26312
#117293732
#117293562
#117254522
#117254512
#118257