Finite Group Databasebeta

← #13264928

Subgroups — #13264928

Order 1792.

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

normal subgroups only

This group has 452 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×#2563C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic
3#11729354C211yesmaximal, characteristic
4#11725452C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C7×#393C412yescomplemented
8C7×#393C2212yes—
9C7×#390—21no—
10C7×#382—21no—
11C7×#380—21no—
12C7×#380—21no—
13C4×C7×C4⋊C4C411yescharacteristic, complemented
14C2×#6408—21no—
15#89984—21no—
16C2×C22⋊(C7⋊C8)—21no—
17#89323—21no—
18#89321C2212yes—
19#89320C2212yes—
20#11900—71nomaximal, Sylow
21C22×C7×Q8C2×C412yes—
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×C4⋊Q8—41no—
26C7×C4⋊Q8—41no—
27C4×C7×D4—41no—
28C4×C7×D4—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4C2311yescharacteristic
37C2×C42×C7—21no—
38C2×C42×C7C2×C411yescharacteristic
39C2×C22⋊Dic7—22no—
40C2×C4⋊Dic7—41no—
41C22×C7⋊C8—41no—
42C7×#95—22no—
43C7×#95—22no—
44C7×#95C2×C412yes—
45#6408—42no—
46C7⋊#95—41no—
47C22⋊(C7⋊C8)—22no—
48C22⋊(C7⋊C8)—41no—
49#2563D711yescharacteristic
50#1799—71no—
51#1459—71no—
52#1351—74no—
53C24×C7C22⋊C411yescharacteristic
54C24×C7C2×D411yescharacteristic
55C2×C7×Q8—42no—
56C2×C7×Q8—82no—
57C2×C7×D4—41no—
58C2×C7×D4—41no—
59C2×C7×D4SD1612yes—
60C2×C7×D4Q1612yes—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×Dic7—42no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—82no—
83C7×C22⋊C4—82no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—42no—
87C22⋊Dic7—42no—
88C22⋊Dic7—42no—
89C4⋊Dic7—82no—
90C2×C7⋊C8—42no—
91C2×C7⋊C8—82no—
92C2×C4⋊Q8Dic711yescharacteristic, complemented
93C2×C4×D4C2×D711yescharacteristic
94C2×#99—141no—
95C2×C22⋊C8—141no—
96#393Dic712yescomplemented
97#393C2×D712yes—
98#390—21no—
99#382—21no—
100#380—21no—
101#380—21no—
102#380—141no—
103C4×C4⋊C4Dic711yescharacteristic, complemented
104#342—72no—
105#324—141no—
106#323—72no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7—41no—
115C23×C7#10312yes—
116C23×C7C2×C22⋊C411yescharacteristic
117C23×C7C2×SD1611yescharacteristic
118C23×C7C2×Q1611yescharacteristic
119C7×Q8—84no—
120C7×D4—42no—
121C7×D4—42no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—22no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—41no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7#9911yescharacteristic
142C2×C4×C7#9911yescharacteristic
143C2×C4×C7#10312yes—
144C2×Dic7—82no—
145C2×Dic7—82no—
146C2×Dic7—82no—
147C7⋊C8—82no—
148C22×Q8C2×Dic712yes—
149C22×D4C22⋊D711yescharacteristic
150C23×C4C22⋊D711yescharacteristic
151C23×C4C22⋊D711yescharacteristic
152C22×C8—281no—
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4×D4—41no—
156C4×D4—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—281no—
161C2×C4⋊C4C22⋊D711yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—142no—
165C2×C22⋊C4C2×Dic711yescharacteristic
166C2×C22⋊C4C22×D711yescharacteristic
167C2×C42—21no—
168C2×C42C2×Dic711yescharacteristic
169#99—282no—
170C22⋊C8—142no—
171C22⋊C8—281no—
172#95—22no—
173#95—22no—
174#95—281no—
175#95C2×Dic712yes—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—21no—
179C22×C7—22no—
180C22×C7—21no—
181C22×C7—42no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—41no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—81no—
189C22×C7—81no—
190C22×C7C2×#9911yescharacteristic
191C22×C7C2×#10311yescharacteristic
192C22×C7#42111yescharacteristic
193C4×C7—21no—
194C4×C7—22no—
195C4×C7—41no—
196C4×C7—41no—
197C4×C7—41no—
198C4×C7—82no—
199C4×C7—82no—
200C4×C7#35412yes—
201Dic7—162no—
202C24C22⋊Dic711yescharacteristic
203C24C2×C22⋊D711yescharacteristic
204C2×Q8—42no—
205C2×Q8—82no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4#114312yes—
209C2×D4#114412yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—282no—
219C22×C4C22⋊Dic711yescharacteristic
220C22×C4C22⋊Dic711yescharacteristic
221C22×C4C22⋊Dic711yescharacteristic
222C22×C4C22×Dic711yescharacteristic
223C22×C4C2×C22⋊D711yescharacteristic
224C2×C8—282no—
225C2×C8—562no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—562no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C22⋊C4—282no—
237C22⋊C4—282no—
238C42—21no—
239C42—21no—
240C42—42no—
241C2×C7—21no—
242C2×C7—21no—
243C2×C7—41no—
244C2×C7—42no—
245C2×C7—41no—
246C2×C7#152011yescharacteristic
247C2×C7#212211yescharacteristic
248C2×C7C2×#35411yescharacteristic
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#640912yes—
258C23C2×#114311yescharacteristic
259C23C2×#114411yescharacteristic
260C23C2×C22⋊Dic711yescharacteristic
261Q8—84no—
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—562no—
284C2×C4—562no—
285C2×C4—562no—
286C2×C4#640811yescharacteristic
287C2×C4#640811yescharacteristic
288C2×C4#640912yes—
289C8—562no—
290C7#1190011yescharacteristic, complemented, Sylow
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#640811yescharacteristic
306C22C2×#640911yescharacteristic
307C22#9000811yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C4—82no—
315C4—1122no—
316C4#8933612yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2#1172944411yescharacteristic
323C2C2×#8933611yescharacteristic
324C2#1172961311yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2563C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C22⋊C4C23
hypercenter#2563D7
hypocenterC7#11900
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
4. #11725452C2semidirect
5. C2×C7×C4⋊Q8C4semidirect
7. C7×#393C4semidirect
13. C4×C7×C4⋊C4C4semidirect
92. C2×C4⋊Q8Dic7semidirect
96. #393Dic7semidirect
103. C4×C4⋊C4Dic7semidirect
290. C7#11900semidirect

Maximal subgroups

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

subgroupindex
C7×#25632
#117293732
#117293542
#117254522
#119007