Finite Group Databasebeta

← #13251531

Subgroups — #13251531

Order 1792.

700 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2434C211yesmaximal, characteristic
2#11741507C211yesmaximal, characteristic
3#11729887C211yesmaximal, characteristic
4#11729443C212yesmaximal
5#11725422C212yesmaximal
6C23×C7×Q8C2211yescharacteristic
7C2×C7×C4⋊Q8C412yescomplemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×#6489—21no—
10C7×#395—22no—
11C7×#391C412yescomplemented
12C7×#385—21no—
13C7×#385—21no—
14C4×C7×C22⋊C4C2211yescharacteristic
15#90002—24no—
16C2×#6410—22no—
17#89983—21no—
18#89983C2212yes—
19#89979C2212yes—
20#89333—24no—
21#9462—71nomaximal, Sylow
22C22×C7×Q8—22no—
23C22×C7×Q8—22no—
24C22×C7×Q8—44no—
25C22×C7×Q8C2×C411yescharacteristic
26C22×C7×Q8C2×C412yes—
27C22×C7×Q8D412yes—
28C23×C4×C7D411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7C2311yescharacteristic
31C7×C4⋊Q8—44no—
32C7×C22⋊Q8—22no—
33C7×C22⋊Q8—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C42×C7C2×C412yes—
41C2×C22⋊Dic7—22no—
42#6489—44no—
43C2×C4⋊Dic7—22no—
44C2×C4×Dic7—22no—
45C2×#1136—22no—
46C7×#95—22no—
47C7×#95—22no—
48C7×#95—21no—
49#6410—44no—
50C7⋊#95—24no—
51C7⋊#95—22no—
52C7⋊#95D414yes—
53C22⋊(C7⋊C8)—22no—
54C22⋊(C7⋊C8)—22no—
55#2434D711yescharacteristic
56#2310—71no—
57#1784—71no—
58#1511—72no—
59#1284—72no—
60C24×C7C2×D411yescharacteristic
61C2×C7×Q8—24no—
62C2×C7×Q8—24no—
63C2×C7×Q8—24no—
64C2×C7×Q8—24no—
65C2×C7×Q8—22no—
66C2×C7×Q8—42no—
67C2×C7×Q8—42no—
68C2×C7×Q8—44no—
69C2×C7×Q8—44no—
70C2×C7×Q8—48no—
71C2×C7×Q8—42no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—24no—
75C22×C4×C7—22no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—42no—
81C22×C4×C7—42no—
82C22×C4×C7C22⋊C412yes—
83C22×C4×C7C22⋊C412yes—
84C22×C4×C7C22×C411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D412yes—
87C22×C4×C7#4112yes—
88C22×Dic7—24no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—44no—
91C7×C4⋊C4—44no—
92C7×C4⋊C4—42no—
93C7×C22⋊C4—24no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C42×C7—42no—
97C22⋊Dic7—44no—
98C4⋊Dic7—44no—
99C4×Dic7—44no—
100#1136—44no—
101C2×C7⋊C8—42no—
102C23×Q8C2×D711yescharacteristic
103C2×C4⋊Q8Dic712yescomplemented
104C2×C22⋊Q8C2×D711yescharacteristic
105C2×#117—141no—
106C2×#101—142no—
107#407—72no—
108#395—22no—
109#391—144no—
110#391Dic712yescomplemented
111#385—21no—
112#385—21no—
113#381—72no—
114#381—141no—
115C4×C22⋊C4C2×D711yescharacteristic
116#328—144no—
117C23×C7—22no—
118C23×C7—22no—
119C23×C7—41no—
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7#12011yescharacteristic
122C23×C7#12311yescharacteristic
123C7×Q8—48no—
124C7×Q8—48no—
125C7×Q8—82no—
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—24no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—24no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—42no—
147C2×C4×C7#9914yes—
148C2×C4×C7C4×D412yes—
149C2×C4×C7#12112yes—
150C2×Dic7—44no—
151C2×Dic7—44no—
152C2×Dic7—44no—
153C7⋊C8—82no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—44no—
157C22×Q8C2×Dic711yescharacteristic
158C22×Q8C2×Dic712yes—
159C22×Q8C22⋊D712yes—
160C23×C4C22⋊D711yescharacteristic
161C23×C4C22⋊D711yescharacteristic
162C23×C4C22×D711yescharacteristic
163C2×M16—142no—
164C4⋊Q8—44no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167#117—284no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—142no—
172C2×C4⋊C4C2×Dic711yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—142no—
176C2×C42—142no—
177C2×C42C2×Dic712yes—
178#101—284no—
179C22⋊C8—142no—
180C22⋊C8—142no—
181#95—22no—
182#95—22no—
183#95—21no—
184#95—74no—
185#95—144no—
186#95—142no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—44no—
195C22×C7#38611yescharacteristic
196C22×C7C2×#9912yes—
197C22×C7#45612yes—
198C4×C7—22no—
199C4×C7—24no—
200C4×C7—22no—
201C4×C7—44no—
202C4×C7—42no—
203C4×C7—81no—
204C4×C7—81no—
205Dic7—84no—
206C24C2×C22⋊D711yescharacteristic
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8—48no—
217C2×Q8—42no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—144no—
229C22×C4C22⋊Dic712yes—
230C22×C4C22⋊Dic712yes—
231C22×C4D4×D712yes—
232C22×C4#115912yes—
233C22×C4C22×Dic711yescharacteristic
234C22×C4C2×C22⋊D711yescharacteristic
235M16—284no—
236C2×C8—282no—
237C4⋊C4—22no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—284no—
242C22⋊C4—24no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—284no—
246C42—42no—
247C42—284no—
248C2×C7—22no—
249C2×C7—22no—
250C2×C7—41no—
251C2×C7#188611yescharacteristic
252C2×C7#189811yescharacteristic
253C2×C7#191211yescharacteristic
254C23—22no—
255C23—22no—
256C23—41no—
257C23#649811yescharacteristic
258C23#650011yescharacteristic
259C23C2×C22⋊Dic711yescharacteristic
260Q8—48no—
261Q8—48no—
262Q8—82no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—284no—
285C2×C4—284no—
286C2×C4—284no—
287C2×C4#640814yes—
288C2×C4D4×Dic712yes—
289C2×C4#650212yes—
290C8—562no—
291C7#946211yescharacteristic, complemented, Sylow
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#8986712yes—
301C22C2×#640812yes—
302C22#8999211yescharacteristic
303C4—22no—
304C4—24no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—564no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#1172942411yescharacteristic
315C2#1172953811yescharacteristic
316C2#1172961811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2434C2
socleC22×C7#386
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2434D7
hypocenterC7#9462
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
7. C2×C7×C4⋊Q8C4semidirect
11. C7×#391C4semidirect
103. C2×C4⋊Q8Dic7semidirect
110. #391Dic7semidirect
291. C7#9462semidirect

Maximal subgroups

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

subgroupindex
C7×#24342
#117415072
#117298872
#117294432
#117254222
#94627