Finite Group Databasebeta

← #13264297

Subgroups — #13264297

Order 1792.

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

normal subgroups only

This group has 622 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×#2406C211yesmaximal, characteristic
2C2×#89325C211yesmaximal, characteristic, complemented
3#11729306C211yesmaximal, characteristic, complemented
4#11725454C212yesmaximal, complemented
5#11725453C212yesmaximal, complemented
6C22×C7×#41C411yescharacteristic
7C2×C7×#123—21no—
8C22×C7×C4⋊C4C2211yescharacteristic
9C22×#1136—21no—
10C7×#395C411yescharacteristic
11C7×#395C2211yescharacteristic
12C7×#394C411yescharacteristic
13C7×#394C2211yescharacteristic
14C7×#386—21no—
15C7×#385—21no—
16C7×#385—21no—
17C7×#381C411yescharacteristic
18#89979—21no—
19#89325—21no—
20#89325C2212yes—
21#89323—21no—
22#89323C2212yes—
23#11762—71nomaximal, Sylow
24C2×C7×#41—41no—
25C2×C7×#41—41no—
26C2×C7×#41—44no—
27C22×C7×Q8C2×C411yescharacteristic
28C22×C7×D4—21no—
29C22×C7×D4C2×C411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7C2×C411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C23×C4×C7D411yescharacteristic
34C7×#123—44no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—41no—
38C2×C7×C4⋊C4—41no—
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C4⋊C4C2311yescharacteristic
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×#1136—24no—
49C2×#1136—22no—
50C2×#1136—41no—
51C2×#1136—42no—
52C2×#1136—41no—
53C22×C7⋊C8—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95C2×C411yescharacteristic
58C7×#95C2×C411yescharacteristic
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—41no—
61#2406D711yescharacteristic
62C2×#343—71no—
63#1463—71no—
64#1354—72no—
65#1345—72no—
66C24×C7—21no—
67C24×C7C22⋊C411yescharacteristic
68C7×#41—44no—
69C7×#41—44no—
70C7×#41—44no—
71C7×#41—44no—
72C2×C7×Q8—42no—
73C2×C7×Q8—41no—
74C2×C7×D4—24no—
75C2×C7×D4—41no—
76C2×C7×D4—42no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—42no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—22no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
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—42no—
93C22×C4×C7—42no—
94C22×C4×C7—42no—
95C22×C4×C7—42no—
96C22×C4×C7—42no—
97C22×C4×C7C22⋊C411yescharacteristic
98C22×C4×C7C22⋊C411yescharacteristic
99C22×C4×C7C22⋊C411yescharacteristic
100C22×C4×C7C22×C411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—42no—
108C7×C4⋊C4—42no—
109C7×C22⋊C4—42no—
110C7×C22⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—41no—
115#1136—24no—
116#1136—44no—
117#1136—42no—
118#1136—82no—
119C2×C7⋊C8—44no—
120C2×C7⋊C8—42no—
121C2×C7⋊C8—42no—
122C22×#41Dic711yescharacteristic
123C22×M16—141no—
124C2×#123—21no—
125C22×C4⋊C4C2×D711yescharacteristic
126#407—141no—
127#395Dic711yescharacteristic
128#395C2×D711yescharacteristic
129#394Dic711yescharacteristic
130#394C2×D711yescharacteristic
131#386—21no—
132#385—21no—
133#385—21no—
134#381Dic711yescharacteristic
135#343—72no—
136#343—141no—
137#324—72no—
138#324—141no—
139C23×C7—22no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—22no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7#9911yescharacteristic
151C23×C7#9911yescharacteristic
152C23×C7C2×C22⋊C411yescharacteristic
153C7×Q8—42no—
154C7×Q8—42no—
155C7×D4—44no—
156C7×D4—42no—
157C7×D4—42no—
158C7×D4—44no—
159C2×C4×C7—22no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—22no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—48no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
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—81no—
189C2×C4×C7#10412yes—
190C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C7#13711yescharacteristic
192C7⋊C8—44no—
193C7⋊C8—82no—
194C2×#41—41no—
195C2×#41—41no—
196C2×#41—44no—
197C22×Q8C2×Dic711yescharacteristic
198C22×D4—21no—
199C22×D4C2×Dic711yescharacteristic
200C23×C4—21no—
201C23×C4C2×Dic711yescharacteristic
202C23×C4C22⋊D711yescharacteristic
203C23×C4C22⋊D711yescharacteristic
204C2×M16—144no—
205C2×M16—142no—
206C2×M16—281no—
207C2×M16—282no—
208C2×M16—281no—
209C22×C8—281no—
210#123—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C2×Dic711yescharacteristic
216C2×C4⋊C4C22⋊D711yescharacteristic
217C2×C4⋊C4C22⋊D711yescharacteristic
218C2×C4⋊C4C22×D711yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C22⋊C8—142no—
225C22⋊C8—281no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95C2×Dic711yescharacteristic
230#95C2×Dic711yescharacteristic
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—42no—
239C22×C7—42no—
240C22×C7—42no—
241C22×C7—42no—
242C22×C7—42no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7AΣL(1,𝔽16)12yes—
248C22×C7C2×#9911yescharacteristic
249C22×C7#41711yescharacteristic
250C22×C7C2×#10411yescharacteristic
251C4×C7—21no—
252C4×C7—22no—
253C4×C7—21no—
254C4×C7—41no—
255C4×C7—41no—
256C4×C7—42no—
257C4×C7—41no—
258C4×C7—41no—
259C4×C7—81no—
260C4×C7—81no—
261C24—21no—
262C24C22⋊Dic711yescharacteristic
263#41—44no—
264#41—44no—
265#41—44no—
266#41—44no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×D4—24no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4C22⋊Dic711yescharacteristic
293C22×C4C22⋊Dic711yescharacteristic
294C22×C4C22⋊Dic711yescharacteristic
295C22×C4C22×Dic711yescharacteristic
296C22×C4C2×C22⋊D711yescharacteristic
297C22×C4C2×C22⋊D711yescharacteristic
298M16—144no—
299M16—284no—
300M16—282no—
301M16—562no—
302C2×C8—284no—
303C2×C8—282no—
304C2×C8—282no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C2×C7—21no—
318C2×C7—21no—
319C2×C7—22no—
320C2×C7—41no—
321C2×C7—82no—
322C2×C7#150411yescharacteristic
323C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
324C2×C7#213311yescharacteristic
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#640811yescharacteristic
337C23#640811yescharacteristic
338C23C2×C22⋊Dic711yescharacteristic
339Q8—42no—
340Q8—42no—
341D4—44no—
342D4—42no—
343D4—42no—
344D4—44no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—48no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
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—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4#641112yes—
376C2×C4#651211yescharacteristic
377C2×C4#651411yescharacteristic
378C8—284no—
379C8—562no—
380C7#1176211yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22#8933112yes—
398C22C2×#640811yescharacteristic
399C22#9000411yescharacteristic
400C22C2×#641111yescharacteristic
401C4—21no—
402C4—22no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4—81no—
410C4—81no—
411C2—21no—
412C2—21no—
413C2—22no—
414C2—41no—
415C2—82no—
416C2C2×#8933111yescharacteristic
417C2#1172946211yescharacteristic
418C2#1172962611yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2406C2
socleC22×C7C2×#99
prime radicalC22×C7×#41C4
cosocleC2×C7×C4⋊C4C23
hypercenter#2406D7
hypocenterC7#11762
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. C2×#89325C2semidirect
3. #11729306C2semidirect
4. #11725454C2semidirect
5. #11725453C2semidirect
380. C7#11762semidirect

Maximal subgroups

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

subgroupindex
C7×#24062
C2×#893252
#117293062
#117254542
#117254532
#117627