Finite Group Databasebeta

← #13264120

Subgroups — #13264120

Order 1792.

648 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2388C211yesmaximal, characteristic
2C4×#6408C211yesmaximal, characteristic, complemented
3#11729355C211yesmaximal, characteristic, complemented
4#11725452C212yesmaximal, complemented
5#11725451C212yesmaximal, complemented
6C22×C7×#41C411yescharacteristic, complemented
7C2×C4×C7×D4—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C7×#395C411yescharacteristic, complemented
10C7×#394—21no—
11C7×#394C411yescharacteristic, complemented
12C7×#393C2211yescharacteristic
13C7×#392—21no—
14C7×#392C2211yescharacteristic
15C4×C7×C22⋊C4—21no—
16C4×C7×C22⋊C4C411yescharacteristic, complemented
17C2×#6408—21no—
18C4×C22⋊Dic7—21no—
19#89697—21no—
20#89321C2212yes—
21#89320—21no—
22#89320C2212yes—
23#11725—71nomaximal, Sylow
24C2×C7×#41—44no—
25C2×C7×#41—41no—
26C2×C7×#41—41no—
27C22×C7×Q8C2×C411yescharacteristic
28C22×C7×D4—21no—
29C22×C7×D4—21no—
30C22×C7×D4C2×C411yescharacteristic
31C22×C7×D4D411yescharacteristic
32C23×C4×C7—21no—
33C23×C4×C7C2×C411yescharacteristic
34C23×C4×C7D411yescharacteristic
35C4×C7×D4—41no—
36C4×C7×D4—41no—
37C4×C7×D4—44no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C7×C22⋊C4C2311yescharacteristic
47C2×C42×C7—21no—
48C2×C42×C7D411yescharacteristic
49C2×C22⋊Dic7—22no—
50C2×C4×Dic7—41no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95C2×C411yescharacteristic
54C7×#95C2×C411yescharacteristic
55#6408—24no—
56C7⋊#95—41no—
57C22⋊(C7⋊C8)—22no—
58C4⋊(C7⋊C8)—41no—
59C4×C7⋊C8—41no—
60#2388D711yescharacteristic
61C4×#99—71no—
62#1460—71no—
63#1351—72no—
64#1343—72no—
65C24×C7—21no—
66C24×C7C22⋊C411yescharacteristic
67C24×C7C2×D411yescharacteristic
68C7×#41—44no—
69C7×#41—44no—
70C7×#41—44no—
71C7×#41—44no—
72C2×C7×Q8—42no—
73C2×C7×Q8—41no—
74C2×C7×D4—22no—
75C2×C7×D4—22no—
76C2×C7×D4—41no—
77C2×C7×D4—42no—
78C2×C7×D4—42no—
79C2×C7×D4—41no—
80C2×C7×D4—42no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C2×C7×D4—42no—
84C2×C7×D4—42no—
85C22×C4×C7—24no—
86C22×C4×C7—21no—
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—42no—
94C22×C4×C7—42no—
95C22×C4×C7—42no—
96C22×C4×C7—41no—
97C22×C4×C7—41no—
98C22×C4×C7C22⋊C411yescharacteristic
99C22×C4×C7C22⋊C411yescharacteristic
100C22×C4×C7C22⋊C411yescharacteristic
101C22×C4×C7C22×C411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×Dic7—42no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—42no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—42no—
109C7×C22⋊C4—42no—
110C7×C22⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C42×C7—21no—
114C42×C7—21no—
115C42×C7—42no—
116C22⋊Dic7—24no—
117C22⋊Dic7—42no—
118C4×Dic7—82no—
119C2×C7⋊C8—42no—
120C22×#41Dic711yescharacteristic, complemented
121C2×C4×D4—21no—
122C2×C4×D4C2×D711yescharacteristic
123#431—141no—
124C2×#99—141no—
125#395Dic711yescharacteristic, complemented
126#394—21no—
127#394Dic711yescharacteristic, complemented
128#393C2×D711yescharacteristic
129#392—21no—
130#392C2×D711yescharacteristic
131C4×C22⋊C4—21no—
132C4×C22⋊C4—141no—
133C4×C22⋊C4Dic711yescharacteristic, complemented
134#342—72no—
135#323—72no—
136#323—141no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—22no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7C2×C22⋊C411yescharacteristic
154C23×C7#13512yes—
155C7×Q8—42no—
156C7×Q8—42no—
157C7×D4—44no—
158C7×D4—42no—
159C7×D4—48no—
160C7×D4—48no—
161C7×D4—42no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—22no—
165C2×C4×C7—22no—
166C2×C4×C7—22no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—44no—
171C2×C4×C7—42no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—42no—
177C2×C4×C7—44no—
178C2×C4×C7—42no—
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—81no—
186C2×C4×C7#9911yescharacteristic
187C2×C4×C7#9911yescharacteristic
188C2×C4×C7#10412yes—
189C2×Dic7—44no—
190C2×Dic7—82no—
191C2×Dic7—82no—
192C7⋊C8—82no—
193C2×#41—44no—
194C2×#41—41no—
195C2×#41—41no—
196C22×Q8C2×Dic711yescharacteristic
197C22×D4—21no—
198C22×D4—21no—
199C22×D4C2×Dic711yescharacteristic
200C22×D4C22⋊D711yescharacteristic
201C23×C4—21no—
202C23×C4C2×Dic711yescharacteristic
203C23×C4C22⋊D711yescharacteristic
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—44no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—142no—
215C2×C22⋊C4C22⋊D711yescharacteristic
216C2×C22⋊C4C22×D711yescharacteristic
217C2×C42—21no—
218C2×C42—281no—
219C2×C42C22⋊D711yescharacteristic
220C4⋊C8—281no—
221#99—144no—
222C22⋊C8—142no—
223C4×C8—281no—
224#95—21no—
225#95—21no—
226#95—281no—
227#95C2×Dic711yescharacteristic
228#95C2×Dic711yescharacteristic
229C22×C7—21no—
230C22×C7—21no—
231C22×C7—22no—
232C22×C7—42no—
233C22×C7—42no—
234C22×C7—42no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—41no—
238C22×C7—42no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7C2×#9911yescharacteristic
249C22×C7#41611yescharacteristic
250C22×C7C2×#10411yescharacteristic
251C4×C7—21no—
252C4×C7—22no—
253C4×C7—42no—
254C4×C7—42no—
255C4×C7—41no—
256C4×C7—41no—
257C4×C7—41no—
258C4×C7—81no—
259C4×C7AΣL(1,𝔽16)12yes—
260Dic7—84no—
261C24—21no—
262C24C22⋊Dic711yescharacteristic
263C24C2×C22⋊D711yescharacteristic
264#41—44no—
265#41—44no—
266#41—44no—
267#41—44no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—282no—
295C22×C4C22⋊Dic711yescharacteristic
296C22×C4C22⋊Dic711yescharacteristic
297C22×C4C22⋊Dic711yescharacteristic
298C22×C4C22×Dic711yescharacteristic
299C22×C4C2×C22⋊D711yescharacteristic
300C2×C8—282no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—144no—
311C22⋊C4—282no—
312C42—21no—
313C42—21no—
314C42—42no—
315C42—562no—
316C2×C7—21no—
317C2×C7—21no—
318C2×C7—42no—
319C2×C7—81no—
320C2×C7—82no—
321C2×C7#150111yescharacteristic
322C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
323C2×C7#212611yescharacteristic
324C23—21no—
325C23—21no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23#651312yes—
341C23C2×C22⋊Dic711yescharacteristic
342Q8—42no—
343Q8—42no—
344D4—44no—
345D4—42no—
346D4—48no—
347D4—48no—
348D4—42no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—284no—
374C2×C4—562no—
375C2×C4—562no—
376C2×C4#640811yescharacteristic
377C2×C4#640811yescharacteristic
378C2×C4#641112yes—
379C8—562no—
380C7#1172511yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C2×#640811yescharacteristic
401C22#9000511yescharacteristic
402C22C2×#641111yescharacteristic
403C4—21no—
404C4—22no—
405C4—42no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—564no—
412C4#8933112yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—82no—
417C2—81no—
418C2C2×#8933111yescharacteristic
419C2#1172946111yescharacteristic
420C2#1172957311yescharacteristic
subgroupG/H
centerC2×C4#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
Fitting, prime radicalC7×#2388C2
socleC22×C7C2×#99
cosocleC2×C7×C22⋊C4C23
hypercenter#2388D7
hypocenterC7#11725
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2388

Complements

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

NH ≅ G/Ntype
2. C4×#6408C2semidirect
3. #11729355C2semidirect
4. #11725452C2semidirect
5. #11725451C2semidirect
6. C22×C7×#41C4semidirect
9. C7×#395C4semidirect
11. C7×#394C4semidirect
16. C4×C7×C22⋊C4C4semidirect
120. C22×#41Dic7semidirect
125. #395Dic7semidirect
127. #394Dic7semidirect
133. C4×C22⋊C4Dic7semidirect
380. C7#11725semidirect

Maximal subgroups

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

subgroupindex
C7×#23882
C4×#64082
#117293552
#117254522
#117254512
#117257