Finite Group Databasebeta

← #13249522

Subgroups — #13249522

Order 1792.

694 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 143 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×#2855C211yesmaximal, characteristic
2C2×#89704C211yesmaximal, characteristic
3C2×#89333C212yesmaximal
4#11729285C211yesmaximal, characteristic
5#11725430C212yesmaximal
6C2×C7×C22⋊Q8C2211yescharacteristic
7C22×C42×C7C2211yescharacteristic
8C22×C4×Dic7—21no—
9C7×#391C412yescomplemented
10C7×#389C412yescomplemented
11C7×#387—22no—
12C7×#385—21no—
13C7×#381—21no—
14C7×#380C2211yescharacteristic
15C2×C7⋊#95—21no—
16C2×C7⋊#95C2212yes—
17C2×C22⋊(C7⋊C8)C2212yes—
18#89704—24no—
19#89333—24no—
20#89319—22no—
21#9145—71nomaximal, Sylow
22C22×C7×Q8C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C23×Dic7—22no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C22⋊C4—21no—
34C2×C7×C22⋊C4—21no—
35C2×C42×C7—22no—
36C2×C42×C7—22no—
37C2×C42×C7C2×C412yes—
38C2×C42×C7D412yes—
39C2×C4×Dic7—24no—
40C2×C4×Dic7—28no—
41C22×C7⋊C8—22no—
42C7×#95—22no—
43C7×#95—22no—
44C7×#95—22no—
45C7×#95—21no—
46C7⋊#95—24no—
47C7⋊#95—22no—
48C7⋊#95D414yes—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)—22no—
51#2855D711yescharacteristic
52C2×#376—71no—
53#1817—71no—
54C2×#328—72no—
55#1294—72no—
56C24×C7C2×D411yescharacteristic
57C2×C7×Q8—22no—
58C2×C7×Q8—42no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—24no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—42no—
70C22×C4×C7C22⋊C412yes—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D412yes—
75C22×C4×C7#4112yes—
76C22×Dic7—28no—
77C22×Dic7—24no—
78C22×Dic7—44no—
79C22×Dic7—42no—
80C22×Dic7—42no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—44no—
84C7×C4⋊C4—42no—
85C7×C22⋊C4—41no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—42no—
88C42×C7—24no—
89C42×C7—24no—
90C4×Dic7—216no—
91C2×C7⋊C8—44no—
92C2×C7⋊C8—42no—
93C2×C22⋊Q8C2×D711yescharacteristic
94C22×C42—141no—
95C22×C42C2×D711yescharacteristic
96C2×C22⋊C8—72no—
97#391Dic712yescomplemented
98#389Dic712yescomplemented
99#387—22no—
100#385—21no—
101#381—21no—
102#380C2×D711yescharacteristic
103#376—144no—
104C2×#95—72no—
105C2×#95—141no—
106#336—142no—
107#328—144no—
108C23×C7—22no—
109C23×C7—22no—
110C23×C7C2×C22⋊C411yescharacteristic
111C23×C7#12011yescharacteristic
112C23×C7#12311yescharacteristic
113C23×C7#13512yes—
114C23×C7#13712yes—
115C7×Q8—82no—
116C2×C4×C7—24no—
117C2×C4×C7—24no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—22no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7#10414yes—
137C2×C4×C7C4×D412yes—
138C2×C4×C7#12112yes—
139C2×Dic7—28no—
140C2×Dic7—44no—
141C2×Dic7—48no—
142C2×Dic7—48no—
143C2×Dic7—44no—
144C7⋊C8—82no—
145C22×Q8C2×Dic711yescharacteristic
146C23×C4—142no—
147C23×C4C22⋊D711yescharacteristic
148C23×C4C22⋊D711yescharacteristic
149C23×C4C22×D711yescharacteristic
150C22×C8—142no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4C2×Dic711yescharacteristic
156C2×C4⋊C4C2×Dic712yes—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—144no—
162C2×C42—148no—
163C2×C42C2×Dic712yes—
164C2×C42C22⋊D712yes—
165C22⋊C8—142no—
166C22⋊C8—142no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—21no—
171#95—74no—
172#95—144no—
173#95—142no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—24no—
179C22×C7—24no—
180C22×C7#38611yescharacteristic
181C22×C7#41012yes—
182C22×C7#41611yescharacteristic
183C22×C7#41911yescharacteristic
184C22×C7C2×#10412yes—
185C22×C7#45212yes—
186C22×C7#48512yes—
187C4×C7—24no—
188C4×C7—22no—
189C4×C7—22no—
190C4×C7—42no—
191C4×C7—81no—
192C4×C7—81no—
193C4×C7—82no—
194Dic7—48no—
195C24C2×C22⋊D711yescharacteristic
196C2×Q8—22no—
197C2×Q8—42no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—42no—
209C22×C4—148no—
210C22×C4—144no—
211C22×C4—284no—
212C22×C4—282no—
213C22×C4—282no—
214C22×C4C22⋊Dic712yes—
215C22×C4C22⋊Dic712yes—
216C22×C4D4×D712yes—
217C22×C4#115912yes—
218C22×C4C22×Dic711yescharacteristic
219C22×C4C2×C22⋊D711yescharacteristic
220C2×C8—284no—
221C2×C8—282no—
222C4⋊C4—22no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—42no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—42no—
229C42—24no—
230C42—24no—
231C42—1416no—
232C2×C7—22no—
233C2×C7—22no—
234C2×C7#150012yes—
235C2×C7#150612yes—
236C2×C7#187311yescharacteristic
237C2×C7#189611yescharacteristic
238C2×C7#189911yescharacteristic
239C23—22no—
240C23—22no—
241C23#649811yescharacteristic
242C23#650011yescharacteristic
243C23#650512yes—
244C23#651312yes—
245C23C2×C22⋊Dic711yescharacteristic
246Q8—82no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—148no—
268C2×C4—284no—
269C2×C4—288no—
270C2×C4—288no—
271C2×C4—284no—
272C2×C4#641114yes—
273C2×C4D4×Dic712yes—
274C2×C4#650212yes—
275C8—562no—
276C7#914511yescharacteristic, complemented, Sylow
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—24no—
283C22#8981712yes—
284C22#8981912yes—
285C22#8999211yescharacteristic
286C22#8999711yescharacteristic
287C22#9000511yescharacteristic
288C22C2×#641112yes—
289C22#9001212yes—
290C4—22no—
291C4—22no—
292C4—24no—
293C4—42no—
294C4—81no—
295C4—81no—
296C4—82no—
297C4—288no—
298C2—22no—
299C2—22no—
300C2#1172944711yescharacteristic
301C2#1172945911yescharacteristic
302C2#1172946511yescharacteristic
303C2#1172950812yes—
304C2#1172959012yes—
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2855C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2855D7
hypocenterC7#9145
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
9. C7×#391C4semidirect
10. C7×#389C4semidirect
97. #391Dic7semidirect
98. #389Dic7semidirect
276. C7#9145semidirect

Maximal subgroups

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

subgroupindex
C7×#28552
C2×#897042
C2×#893332
#117292852
#117254302
#91457