Finite Group Databasebeta

← #13249765

Subgroups — #13249765

Order 1792.

530 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2638C211yesmaximal, characteristic
2#11739749C211yesmaximal, characteristic
3#11729442C212yesmaximal, complemented
4C4×#6408C211yesmaximal, characteristic
5#11725434C212yesmaximal, complemented
6C2×C7×C22⋊Q8C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#398C411yescharacteristic, complemented
9C7×#397—21no—
10C7×#397C411yescharacteristic, complemented
11C7×#395C411yescharacteristic, complemented
12C7×#394—21no—
13C7×#393C411yescharacteristic, complemented
14C7×#387—21no—
15C7×#385—21no—
16C7×#381C2211yescharacteristic
17C2×#6408—21no—
18C4×C22⋊Dic7C2212yes—
19#89979—21no—
20#89979C2212yes—
21#89784—24no—
22#89697—21no—
23#89333—24no—
24#89321—21no—
25#9184—71nomaximal, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C23×C4×C7C2311yescharacteristic
30C7×C22⋊Q8—22no—
31C7×C22⋊Q8—22no—
32C4×C7×D4—22no—
33C4×C7×D4—22no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C2×C22⋊Dic7—22no—
47C2×C4×Dic7—22no—
48C2×#1136—22no—
49C2×#1136—22no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#6408—24no—
58C7⋊#95—22no—
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—22no—
61C22⋊(C7⋊C8)D414yes—
62C4⋊(C7⋊C8)—22no—
63C4×C7⋊C8—22no—
64#2975—71no—
65#2638D711yescharacteristic
66C4×#99—71no—
67#1508—72no—
68#1284—72no—
69C24×C7C22⋊C411yescharacteristic
70C24×C7C2×D411yescharacteristic
71C2×C7×Q8—22no—
72C2×C7×Q8—81no—
73C2×C7×D4—22no—
74C2×C7×D4—22no—
75C2×C7×D4—41no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—81no—
83C22×C4×C7—81no—
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7#4112yes—
91C22×Dic7—42no—
92C7×C4⋊C4—22no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—42no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—81no—
98C7×C22⋊C4—22no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—81no—
103C7×C22⋊C4—81no—
104C42×C7—21no—
105C42×C7—21no—
106C22⋊Dic7—24no—
107C22⋊Dic7—42no—
108C4×Dic7—82no—
109#1136—82no—
110#1136—82no—
111C2×C7⋊C8—24no—
112C2×C22⋊Q8C2×D711yescharacteristic
113C2×C4×D4C2×D711yescharacteristic
114#436—144no—
115#431—141no—
116C2×#99—141no—
117#407—72no—
118#407—141no—
119#398Dic711yescharacteristic, complemented
120#397—21no—
121#397Dic711yescharacteristic, complemented
122#395Dic711yescharacteristic, complemented
123#394—21no—
124#393Dic711yescharacteristic, complemented
125#387—21no—
126#385—21no—
127#381C2×D711yescharacteristic
128C4×C22⋊C4—72no—
129#342—141no—
130#328—144no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7C2×C22⋊C411yescharacteristic
141C23×C7#12112yes—
142C7×Q8—82no—
143C7×D4—44no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—42no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—81no—
168C2×C4×C7#10412yes—
169C2×C4×C7C4×D412yes—
170C2×C4×C7#12011yescharacteristic
171C2×C4×C7#12311yescharacteristic
172C2×Dic7—44no—
173C2×Dic7—82no—
174C2×Dic7—82no—
175C7⋊C8—44no—
176C22×Q8C2×Dic711yescharacteristic
177C22×D4C22⋊D711yescharacteristic
178C23×C4C22⋊D711yescharacteristic
179C23×C4C22×D711yescharacteristic
180C2×M16—142no—
181C2×M16—142no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C4×D4—22no—
185C4×D4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C2×Dic711yescharacteristic
190C2×C4⋊C4C2×Dic711yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—142no—
196C2×C22⋊C4C2×Dic711yescharacteristic
197C2×C22⋊C4C22⋊D711yescharacteristic
198C2×C42—142no—
199C2×C42C22⋊D711yescharacteristic
200C4⋊C8—142no—
201#99—144no—
202C22⋊C8—74no—
203C22⋊C8—142no—
204C22⋊C8—142no—
205C4×C8—142no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—142no—
212#95C2×Dic711yescharacteristic
213#95C2×Dic711yescharacteristic
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—22no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—42no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7#38611yescharacteristic
227C22×C7C2×#10411yescharacteristic
228C22×C7#42111yescharacteristic
229C4×C7—22no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—22no—
233C4×C7—41no—
234C4×C7—41no—
235C4×C7—81no—
236C4×C7—81no—
237C4×C7—81no—
238C4×C7—81no—
239Dic7—84no—
240C24C22⋊Dic711yescharacteristic
241C24C2×C22⋊D711yescharacteristic
242C2×Q8—22no—
243C2×Q8—81no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—282no—
256C22×C4C22⋊Dic711yescharacteristic
257C22×C4C22⋊Dic711yescharacteristic
258C22×C4C22⋊Dic711yescharacteristic
259C22×C4D4×D712yes—
260C22×C4#115912yes—
261C22×C4C22×Dic711yescharacteristic
262C22×C4C2×C22⋊D711yescharacteristic
263M16—562no—
264M16—562no—
265C2×C8—144no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—81no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C22⋊C4—144no—
279C22⋊C4—282no—
280C42—21no—
281C42—21no—
282C42—562no—
283C2×C7—21no—
284C2×C7—21no—
285C2×C7—42no—
286C2×C7—81no—
287C2×C7#187111yescharacteristic
288C2×C7#190911yescharacteristic
289C2×C7#191311yescharacteristic
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23#650212yes—
300C23C2×C22⋊Dic711yescharacteristic
301Q8—82no—
302D4—44no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—284no—
328C2×C4—562no—
329C2×C4—562no—
330C2×C4#641112yes—
331C2×C4D4×Dic712yes—
332C2×C4#649811yescharacteristic
333C2×C4#650011yescharacteristic
334C8—284no—
335C7#918411yescharacteristic, complemented, Sylow
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#8999211yescharacteristic
349C22C2×#641111yescharacteristic
350C22#9000811yescharacteristic
351C4—22no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—564no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—81no—
366C2#1172946711yescharacteristic
367C2#1172960211yescharacteristic
368C2#1172961911yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2638C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenter#2638D7
hypocenterC7#9184
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #11729442C2semidirect
5. #11725434C2semidirect
8. C7×#398C4semidirect
10. C7×#397C4semidirect
11. C7×#395C4semidirect
13. C7×#393C4semidirect
119. #398Dic7semidirect
121. #397Dic7semidirect
122. #395Dic7semidirect
124. #393Dic7semidirect
335. C7#9184semidirect

Maximal subgroups

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

subgroupindex
C7×#26382
#117397492
#117294422
C4×#64082
#117254342
#91847