Finite Group Databasebeta

← #13249562

Subgroups — #13249562

Order 1792.

586 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 135 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×#89709C211yesmaximal, characteristic
3C2×#89333C212yesmaximal
4#11729285C211yesmaximal, characteristic
5#11725430C212yesmaximal
6C2×C7×C22⋊Q8C2211yescharacteristic
7C22×C42×C7C2211yescharacteristic
8C7×#391C412yescomplemented
9C7×#389C412yescomplemented
10C7×#387—22no—
11C7×#385—21no—
12C7×#381—21no—
13C7×#380C2211yescharacteristic
14C2×C7⋊#95—21no—
15C2×C7⋊#95—21no—
16C2×C7⋊#95C2212yes—
17C2×C22⋊(C7⋊C8)C2212yes—
18#89709—24no—
19#89333—24no—
20#89319—22no—
21#9153—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—
39C22×C7⋊C8—22no—
40C7×#95—22no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95—21no—
44C7⋊#95—24no—
45C7⋊#95—24no—
46C7⋊#95—22no—
47C7⋊#95D414yes—
48C22⋊(C7⋊C8)—22no—
49C22⋊(C7⋊C8)—22no—
50#2855D711yescharacteristic
51C2×#383—71no—
52#1817—71no—
53C2×#328—72no—
54#1294—72no—
55C24×C7C2×D411yescharacteristic
56C2×C7×Q8—22no—
57C2×C7×Q8—42no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7C22⋊C412yes—
70C22×C4×C7C22⋊C412yes—
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D412yes—
74C22×C4×C7#4112yes—
75C22×Dic7—24no—
76C22×Dic7—44no—
77C22×Dic7—44no—
78C22×Dic7—42no—
79C22×Dic7—42no—
80C7×C4⋊C4—22no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—44no—
83C7×C4⋊C4—42no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—41no—
86C7×C22⋊C4—42no—
87C42×C7—24no—
88C42×C7—24no—
89C2×C7⋊C8—44no—
90C2×C7⋊C8—42no—
91C2×C22⋊Q8C2×D711yescharacteristic
92C22×C42C2×D711yescharacteristic
93C2×C22⋊C8—72no—
94#391Dic712yescomplemented
95#389Dic712yescomplemented
96#387—22no—
97#385—21no—
98#383—144no—
99#381—21no—
100#380C2×D711yescharacteristic
101C2×#95—72no—
102C2×#95—141no—
103C2×#95—141no—
104#336—142no—
105#328—144no—
106C23×C7—22no—
107C23×C7—22no—
108C23×C7C2×C22⋊C411yescharacteristic
109C23×C7#12011yescharacteristic
110C23×C7#12311yescharacteristic
111C23×C7#13512yes—
112C23×C7#13712yes—
113C7×Q8—82no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7C4×D412yes—
136C2×C4×C7#12112yes—
137C2×Dic7—44no—
138C2×Dic7—44no—
139C2×Dic7—44no—
140C2×Dic7—84no—
141C2×Dic7—84no—
142C7⋊C8—82no—
143C22×Q8C2×Dic711yescharacteristic
144C23×C4—142no—
145C23×C4C22⋊D711yescharacteristic
146C23×C4C22⋊D711yescharacteristic
147C23×C4C22×D711yescharacteristic
148C22×C8—142no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C2×Dic711yescharacteristic
154C2×C4⋊C4C2×Dic712yes—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42C2×Dic712yes—
160C2×C42C22⋊D712yes—
161C22⋊C8—142no—
162C22⋊C8—142no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—21no—
167#95—74no—
168#95—144no—
169#95—144no—
170#95—142no—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—24no—
176C22×C7—24no—
177C22×C7#38611yescharacteristic
178C22×C7#41012yes—
179C22×C7#41611yescharacteristic
180C22×C7#41911yescharacteristic
181C22×C7#42112yes—
182C22×C7#45212yes—
183C22×C7#48512yes—
184C4×C7—22no—
185C4×C7—22no—
186C4×C7—42no—
187C4×C7—42no—
188C4×C7—81no—
189C4×C7—81no—
190C4×C7—82no—
191Dic7—84no—
192C24C2×C22⋊D711yescharacteristic
193C2×Q8—22no—
194C2×Q8—42no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—144no—
207C22×C4—284no—
208C22×C4—284no—
209C22×C4—282no—
210C22×C4—282no—
211C22×C4C22⋊Dic712yes—
212C22×C4C22⋊Dic712yes—
213C22×C4D4×D712yes—
214C22×C4#115912yes—
215C22×C4C22×Dic711yescharacteristic
216C22×C4C2×C22⋊D711yescharacteristic
217C2×C8—284no—
218C2×C8—282no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—44no—
222C4⋊C4—42no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C42—24no—
227C42—24no—
228C2×C7—22no—
229C2×C7—22no—
230C2×C7#151512yes—
231C2×C7#152112yes—
232C2×C7#187311yescharacteristic
233C2×C7#189711yescharacteristic
234C2×C7#189911yescharacteristic
235C23—22no—
236C23—22no—
237C23#649811yescharacteristic
238C23#650011yescharacteristic
239C23#650512yes—
240C23#651312yes—
241C23C2×C22⋊Dic711yescharacteristic
242Q8—82no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—284no—
265C2×C4—284no—
266C2×C4—284no—
267C2×C4—564no—
268C2×C4—564no—
269C2×C4D4×Dic712yes—
270C2×C4#650212yes—
271C8—562no—
272C7#915311yescharacteristic, complemented, Sylow
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—24no—
278C22—24no—
279C22#8981712yes—
280C22#8981912yes—
281C22#8999211yescharacteristic
282C22#8999711yescharacteristic
283C22#9000511yescharacteristic
284C22#9000812yes—
285C22#9001212yes—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—81no—
291C4—81no—
292C4—82no—
293C4—564no—
294C2—22no—
295C2—22no—
296C2#1172944711yescharacteristic
297C2#1172945911yescharacteristic
298C2#1172946611yescharacteristic
299C2#1172951112yes—
300C2#1172959112yes—
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#9153
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
8. C7×#391C4semidirect
9. C7×#389C4semidirect
94. #391Dic7semidirect
95. #389Dic7semidirect
272. C7#9153semidirect

Maximal subgroups

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

subgroupindex
C7×#28552
C2×#897092
C2×#893332
#117292852
#117254302
#91537