Finite Group Databasebeta

← #13248841

Subgroups — #13248841

Order 1792.

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

normal subgroups only

This group has 740 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
1C2×C7×#399C211yesmaximal, characteristic
2#11739538C211yesmaximal, characteristic
3#11729308C212yesmaximal, complemented
4C2×#89320C211yesmaximal, characteristic
5#11725418C212yesmaximal, complemented
6C22×C7×C4⋊C4—21no—
7C22×C7×C4⋊C4C2211yescharacteristic
8C22×C7×C22⋊C4—21no—
9C22×C7×C22⋊C4C2211yescharacteristic
10C7×#399—22no—
11C7×#399C414yescomplemented
12C2×C7×#95C2211yescharacteristic
13#90019—21no—
14#89984C2212yes—
15C4×C22⋊Dic7—21no—
16C2×C22⋊(C7⋊C8)C2212yes—
17#89723—24no—
18#89324—24no—
19#89320—22no—
20#9024—71nomaximal, Sylow
21C25×C7D411yescharacteristic
22C23×C4×C7—21no—
23C23×C4×C7—21no—
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C2×C7×C4⋊C4—22no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—24no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—44no—
31C2×C7×C4⋊C4C2×C412yes—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—24no—
35C2×C7×C22⋊C4—44no—
36C2×C7×C22⋊C4C2×C412yes—
37C2×C7×C22⋊C4D412yes—
38C2×C22⋊Dic7—22no—
39C2×C22⋊Dic7—22no—
40C2×C22⋊Dic7—22no—
41C2×C4⋊Dic7D414yes—
42C2×C4×Dic7—22no—
43C22×C7⋊C8—22no—
44C7×#95—21no—
45C7×#95C2×C412yes—
46C7⋊#95—22no—
47C7⋊#95—22no—
48C22⋊(C7⋊C8)—22no—
49C22⋊(C7⋊C8)—22no—
50C2×#399D711yescharacteristic
51#2312—71no—
52#1824—72no—
53C2×#323—71no—
54#1294—72no—
55C24×C7—21no—
56C24×C7—22no—
57C24×C7—21no—
58C24×C7—22no—
59C24×C7—42no—
60C24×C7—42no—
61C24×C7C22⋊C412yes—
62C24×C7C2×D411yescharacteristic
63C22×C4×C7—24no—
64C22×C4×C7—22no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—22no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7—42no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C22×C4×C7C22⋊C412yes—
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D412yes—
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7#4112yes—
81C22×Dic7—24no—
82C7×C4⋊C4—24no—
83C7×C4⋊C4—48no—
84C7×C4⋊C4—42no—
85C7×C22⋊C4—24no—
86C7×C22⋊C4—24no—
87C7×C22⋊C4—48no—
88C22⋊Dic7—42no—
89C22⋊Dic7—42no—
90C22⋊Dic7—82no—
91C22⋊Dic7—82no—
92C4⋊Dic7—44no—
93C4×Dic7—82no—
94C2×C7⋊C8—44no—
95C2×C7⋊C8—42no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4C2×D711yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4C2×D711yescharacteristic
100C2×C22⋊C8—72no—
101#399—22no—
102#399Dic714yescomplemented
103#388—144no—
104#380—72no—
105#379—141no—
106C4×C22⋊C4—141no—
107C2×#95C2×D711yescharacteristic
108#340—144no—
109#323—142no—
110C23×C7—22no—
111C23×C7—22no—
112C23×C7—22no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7—42no—
116C23×C7—42no—
117C23×C7—48no—
118C23×C7—42no—
119C23×C7—42no—
120C23×C7—42no—
121C23×C7—42no—
122C23×C7—81no—
123C23×C7—81no—
124C23×C7—84no—
125C23×C7C2×C22⋊C411yescharacteristic
126C23×C7#12011yescharacteristic
127C23×C7#12311yescharacteristic
128C23×C7#13512yes—
129C23×C7#13712yes—
130C2×C4×C7—24no—
131C2×C4×C7—24no—
132C2×C4×C7—22no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—42no—
136C2×C4×C7—48no—
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—81no—
145C2×C4×C7—82no—
146C2×C4×C7—82no—
147C2×C4×C7C4×D412yes—
148C2×C4×C7#12112yes—
149C2×Dic7—44no—
150C2×Dic7—44no—
151C2×Dic7—44no—
152C7⋊C8—82no—
153C25C22⋊D711yescharacteristic
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C22⋊D711yescharacteristic
157C23×C4C22×D711yescharacteristic
158C22×C8—142no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—74no—
165C2×C4⋊C4C2×Dic712yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—142no—
171C2×C22⋊C4—142no—
172C2×C22⋊C4—142no—
173C2×C22⋊C4C2×Dic712yes—
174C2×C22⋊C4C22⋊D712yes—
175C2×C42—142no—
176C22⋊C8—142no—
177C22⋊C8—142no—
178#95—21no—
179#95—142no—
180#95—142no—
181#95C2×Dic712yes—
182C22×C7—24no—
183C22×C7—24no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—44no—
189C22×C7—42no—
190C22×C7—42no—
191C22×C7—42no—
192C22×C7—82no—
193C22×C7—84no—
194C22×C7—84no—
195C22×C7#38611yescharacteristic
196C22×C7#41611yescharacteristic
197C22×C7#41911yescharacteristic
198C22×C7#45212yes—
199C22×C7#45312yes—
200C22×C7#45812yes—
201C22×C7#48512yes—
202C4×C7—22no—
203C4×C7—22no—
204C4×C7—44no—
205C4×C7—42no—
206C4×C7—82no—
207Dic7—84no—
208C24—21no—
209C24—22no—
210C24—21no—
211C24—22no—
212C24—42no—
213C24—42no—
214C24C22⋊Dic712yes—
215C24C2×C22⋊D711yescharacteristic
216C22×C4—24no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—144no—
230C22×C4C22⋊Dic712yes—
231C22×C4D4×D712yes—
232C22×C4#115912yes—
233C22×C4C22×Dic711yescharacteristic
234C22×C4C2×C22⋊D711yescharacteristic
235C2×C8—284no—
236C2×C8—282no—
237C4⋊C4—24no—
238C4⋊C4—48no—
239C4⋊C4—42no—
240C4⋊C4—284no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—48no—
244C22⋊C4—282no—
245C22⋊C4—282no—
246C22⋊C4—562no—
247C22⋊C4—562no—
248C42—562no—
249C2×C7—21no—
250C2×C7—21no—
251C2×C7—22no—
252C2×C7—82no—
253C2×C7#161814yes—
254C2×C7#187311yescharacteristic
255C2×C7#188711yescharacteristic
256C2×C7#189011yescharacteristic
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—42no—
263C23—42no—
264C23—48no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—81no—
270C23—81no—
271C23—84no—
272C23#649811yescharacteristic
273C23#650011yescharacteristic
274C23#650512yes—
275C23#651312yes—
276C23C2×C22⋊Dic711yescharacteristic
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—48no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—284no—
295C2×C4—284no—
296C2×C4—284no—
297C2×C4D4×Dic712yes—
298C2×C4#650212yes—
299C8—562no—
300C7#902411yescharacteristic, complemented, Sylow
301C22—21no—
302C22—21no—
303C22—24no—
304C22—24no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—84no—
313C22—84no—
314C22#8981712yes—
315C22#8981912yes—
316C22#8982512yes—
317C22#8984912yes—
318C22#8999211yescharacteristic
319C22#8999711yescharacteristic
320C22#9000511yescharacteristic
321C4—22no—
322C4—22no—
323C4—44no—
324C4—42no—
325C4—82no—
326C4—564no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2#1172746914yes—
332C2#1172944711yescharacteristic
333C2#1172945811yescharacteristic
334C2#1172951711yescharacteristic
subgroupG/H
centerC23#6500
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#399C2
socleC23×C7#120
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenterC2×#399D7
hypocenterC7#9024
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
3. #11729308C2semidirect
5. #11725418C2semidirect
11. C7×#399C4semidirect
102. #399Dic7semidirect
300. C7#9024semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3992
#117395382
#117293082
C2×#893202
#117254182
#90247