Finite Group Databasebeta

← #13264924

Subgroups — #13264924

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×#2487C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic
3#11729355C211yesmaximal, characteristic
4#11725457C214yesmaximal, complemented
5C2×C4×C7×D4C2211yescharacteristic
6C22×C7×C4⋊C4C411yescharacteristic, complemented
7C7×#400C412yescomplemented
8C7×#399C2212yes—
9C7×#389—21no—
10C7×#386—21no—
11C7×#385—21no—
12C7×#380—21no—
13C7×#380C411yescharacteristic, complemented
14C2×#6408—21no—
15#89984—21no—
16#89697—21no—
17#89321C2212yes—
18#89320—21no—
19#89320C2212yes—
20#11899—71nomaximal, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7—21no—
23C23×C4×C7C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C4×C7×D4—41no—
26C4×C7×D4—41no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—41no—
31C2×C7×C4⋊C4—41no—
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C7×C22⋊C4C2311yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×C22⋊Dic7—22no—
40C2×C4⋊Dic7—41no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95C2×C412yes—
44#6408—42no—
45C7⋊#95—41no—
46C22⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—41no—
48C4×C7⋊C8—41no—
49#2487D711yescharacteristic
50#1799—71no—
51#1460—71no—
52#1352—74no—
53C24×C7C22⋊C411yescharacteristic
54C24×C7C2×D411yescharacteristic
55C2×C7×D4—41no—
56C2×C7×D4—41no—
57C2×C7×D4SD1612yes—
58C2×C7×D4Q1612yes—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7—82no—
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×C4⋊C4—82no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—41no—
83C7×C22⋊C4—42no—
84C7×C22⋊C4—82no—
85C42×C7—21no—
86C42×C7—21no—
87C22⋊Dic7—42no—
88C22⋊Dic7—42no—
89C4⋊Dic7—82no—
90C2×C7⋊C8—42no—
91C2×C4×D4C2×D711yescharacteristic
92C22×C4⋊C4Dic711yescharacteristic, complemented
93#431—141no—
94C2×#99—141no—
95#400Dic712yescomplemented
96#399C2×D712yes—
97#389—21no—
98#386—21no—
99#385—21no—
100#380—21no—
101#380—141no—
102#380Dic711yescharacteristic, complemented
103#342—72no—
104#323—72no—
105#323—141no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7#10312yes—
115C23×C7C2×C22⋊C411yescharacteristic
116C23×C7C2×SD1611yescharacteristic
117C23×C7C2×Q1611yescharacteristic
118C7×D4—42no—
119C7×D4—42no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—22no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—81no—
139C2×C4×C7—84no—
140C2×C4×C7#9911yescharacteristic
141C2×C4×C7#9911yescharacteristic
142C2×C4×C7#10312yes—
143C2×Dic7—82no—
144C2×Dic7—82no—
145C2×Dic7—82no—
146C7⋊C8—82no—
147C22×D4C22⋊D711yescharacteristic
148C23×C4—21no—
149C23×C4C2×Dic711yescharacteristic
150C23×C4C22⋊D711yescharacteristic
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—281no—
159C2×C4⋊C4C2×Dic712yes—
160C2×C4⋊C4C2×Dic711yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—142no—
164C2×C22⋊C4C22⋊D711yescharacteristic
165C2×C22⋊C4C22×D711yescharacteristic
166C2×C42C22⋊D711yescharacteristic
167C4⋊C8—281no—
168#99—282no—
169C22⋊C8—142no—
170C4×C8—281no—
171#95—22no—
172#95—22no—
173#95—281no—
174#95C2×Dic712yes—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—22no—
179C22×C7—21no—
180C22×C7—42no—
181C22×C7—41no—
182C22×C7—41no—
183C22×C7—42no—
184C22×C7—41no—
185C22×C7—41no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—81no—
189C22×C7C2×#9911yescharacteristic
190C22×C7C2×#10311yescharacteristic
191C22×C7#42111yescharacteristic
192C4×C7—21no—
193C4×C7—22no—
194C4×C7—41no—
195C4×C7—41no—
196C4×C7—41no—
197C4×C7—82no—
198C4×C7—82no—
199C4×C7#35212yes—
200Dic7—162no—
201C24C22⋊Dic711yescharacteristic
202C24C2×C22⋊D711yescharacteristic
203C2×D4—41no—
204C2×D4—41no—
205C2×D4#114312yes—
206C2×D4#114412yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—82no—
217C22×C4—282no—
218C22×C4C22⋊Dic711yescharacteristic
219C22×C4C22⋊Dic711yescharacteristic
220C22×C4C22⋊Dic711yescharacteristic
221C22×C4C22×Dic711yescharacteristic
222C22×C4C2×C22⋊D711yescharacteristic
223C2×C8—282no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—82no—
230C4⋊C4—562no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—82no—
235C22⋊C4—282no—
236C22⋊C4—282no—
237C42—21no—
238C42—21no—
239C2×C7—21no—
240C2×C7—21no—
241C2×C7—41no—
242C2×C7—42no—
243C2×C7—41no—
244C2×C7#152011yescharacteristic
245C2×C7C2×#35211yescharacteristic
246C2×C7#212811yescharacteristic
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23#640912yes—
256C23C2×#114311yescharacteristic
257C23C2×#114411yescharacteristic
258C23C2×C22⋊Dic711yescharacteristic
259D4—42no—
260D4—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4—562no—
282C2×C4—562no—
283C2×C4—562no—
284C2×C4#640811yescharacteristic
285C2×C4#640811yescharacteristic
286C2×C4#640912yes—
287C8—562no—
288C7#1189911yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#640811yescharacteristic
304C22C2×#640911yescharacteristic
305C22#9000811yescharacteristic
306C4—22no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4—1122no—
314C4#8933412yes—
315C2—21no—
316C2—21no—
317C2—41no—
318C2—42no—
319C2—41no—
320C2#1172944411yescharacteristic
321C2C2×#8933411yescharacteristic
322C2#1172961511yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2487C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C22⋊C4C23
hypercenter#2487D7
hypocenterC7#11899
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
4. #11725457C2semidirect
6. C22×C7×C4⋊C4C4semidirect
7. C7×#400C4semidirect
13. C7×#380C4semidirect
92. C22×C4⋊C4Dic7semidirect
95. #400Dic7semidirect
102. #380Dic7semidirect
288. C7#11899semidirect

Maximal subgroups

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

subgroupindex
C7×#24872
#117293732
#117293552
#117254572
#118997