Finite Group Databasebeta

← #13251651

Subgroups — #13251651

Order 1792.

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

normal subgroups only

This group has 542 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×#2536C211yesmaximal, characteristic
2#11739880C211yesmaximal, characteristic, complemented
3#11729391C211yesmaximal, characteristic, complemented
4#11729352C212yesmaximal
5#11725437C212yesmaximal, complemented
6C2×C7×C22⋊Q8—21no—
7C2×C4×C7×Q8C2211yescharacteristic
8C2×C4×C7×D4C2211yescharacteristic
9C7×#398C412yes—
10C7×#394C412yes—
11C7×#388—22no—
12C7×#382C2211yescharacteristic
13C7×#380—21no—
14C2×#6410—21no—
15Q8×C7⋊C8—24no—
16#89697—22no—
17#89697C2212yes—
18#89696C2212yes—
19D4⋊(C7⋊C8)—24no—
20#89325—21no—
21#9486—71nomaximal, Sylow
22C22×C7×Q8D411yescharacteristic
23C22×C7×D4C2×C411yescharacteristic
24C23×C4×C7—21no—
25C7×C22⋊Q8—42no—
26C7×C22⋊Q8—42no—
27C4×C7×Q8—22no—
28C4×C7×Q8—22no—
29C4×C7×D4—22no—
30C4×C7×D4—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C412yes—
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—21no—
39C2×C42×C7D412yes—
40C2×C42×C7C2311yescharacteristic
41C2×#1136—22no—
42C7×#95—22no—
43C7×#95—21no—
44C7×#95C2×C412yes—
45#6410—42no—
46C22⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—24no—
48C4⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—22no—
51C4×C7⋊C8—24no—
52C4×C7⋊C8D414yes—
53#2982—71no—
54#2536D711yescharacteristic
55#1801—71no—
56#1486—72no—
57#1286—72no—
58C24×C7—21no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8D812yes—
62C2×C7×Q8SD1612yes—
63C2×C7×D4—22no—
64C2×C7×D4—42no—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—42no—
68C22×C4×C7—41no—
69C22×C4×C7—41no—
70C22×C4×C7—42no—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C7×C4⋊C4—24no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—44no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C22⋊C4—44no—
83C7×C22⋊C4—44no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—41no—
86C42×C7—22no—
87C42×C7—22no—
88C42×C7C2×D412yes—
89C42×C7#4112yes—
90#1136—42no—
91#1136—42no—
92C2×C7⋊C8—24no—
93C2×C7⋊C8—42no—
94C2×C22⋊Q8—21no—
95C2×C4×Q8C2×D711yescharacteristic
96C2×C4×D4C2×D711yescharacteristic
97C8×Q8—144no—
98#431—72no—
99#431—142no—
100#423—72no—
101C2×#101—141no—
102#398Dic712yes—
103#394Dic712yes—
104#388—22no—
105#382C2×D711yescharacteristic
106#380—21no—
107#343—141no—
108D4⋊C8—144no—
109C23×C7—22no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7—41no—
115C23×C7—41no—
116C23×C7C2×C22⋊C411yescharacteristic
117C7×Q8—24no—
118C7×Q8—24no—
119C7×D4—24no—
120C7×D4—42no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—41no—
125C2×C4×C7—41no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—41no—
130C2×C4×C7—42no—
131C2×C4×C7—41no—
132C2×C4×C7—41no—
133C2×C4×C7—82no—
134C2×C4×C7—82no—
135C2×C4×C7#10214yes—
136C2×C4×C7C4×D412yes—
137C2×C4×C7#12011yescharacteristic
138C2×C4×C7#12112yes—
139C2×C4×C7#12311yescharacteristic
140C2×C4×C7C2×D811yescharacteristic
141C2×C4×C7C2×SD1611yescharacteristic
142C2×C4×C7#13512yes—
143C7⋊C8—44no—
144C7⋊C8—82no—
145C22×Q8C22⋊D711yescharacteristic
146C22×D4C2×Dic711yescharacteristic
147C23×C4—21no—
148C2×M16—142no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×D4—22no—
154C4×D4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×Dic711yescharacteristic
158C2×C4⋊C4C2×Dic712yes—
159C2×C4⋊C4C22⋊D711yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C42C22⋊D712yes—
164C2×C42C22×D711yescharacteristic
165C4⋊C8—144no—
166C4⋊C8—142no—
167C4⋊C8—142no—
168C4⋊C8—142no—
169#101—282no—
170C22⋊C8—142no—
171C4×C8—74no—
172C4×C8—144no—
173#95—22no—
174#95—21no—
175#95C2×Dic712yes—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—42no—
179C22×C7—42no—
180C22×C7—42no—
181C22×C7—81no—
182C22×C7—81no—
183C22×C7—81no—
184C22×C7—81no—
185C22×C7#38611yescharacteristic
186C22×C7C2×#10211yescharacteristic
187C22×C7#41611yescharacteristic
188C4×C7—22no—
189C4×C7—24no—
190C4×C7—21no—
191C4×C7—21no—
192C4×C7—42no—
193C4×C7—82no—
194C4×C7—81no—
195C4×C7#46512yes—
196C4×C7#46612yes—
197C24—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8#114112yes—
201C2×Q8#114212yes—
202C2×D4—22no—
203C2×D4—42no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4C22⋊Dic712yes—
211C22×C4C22⋊Dic712yes—
212C22×C4C22×Dic711yescharacteristic
213C22×C4C2×C22⋊D711yescharacteristic
214C22×C4C2×C22⋊D711yescharacteristic
215M16—282no—
216M16—282no—
217C2×C8—144no—
218C2×C8—282no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—44no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C42—22no—
230C42—22no—
231C42D4×D712yes—
232C42#115912yes—
233C2×C7—21no—
234C2×C7—21no—
235C2×C7—82no—
236C2×C7#189111yescharacteristic
237C2×C7#189311yescharacteristic
238C2×C7#191211yescharacteristic
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23C2×C22⋊Dic711yescharacteristic
247Q8—24no—
248Q8—24no—
249D4—24no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4#640614yes—
266C2×C4C2×#114111yescharacteristic
267C2×C4C2×#114211yescharacteristic
268C2×C4D4×Dic712yes—
269C2×C4#649811yescharacteristic
270C2×C4#650011yescharacteristic
271C2×C4#650212yes—
272C2×C4#651312yes—
273C8—284no—
274C8—562no—
275C7#948611yescharacteristic, complemented, Sylow
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22C2×#640611yescharacteristic
286C22#8999211yescharacteristic
287C22#9000511yescharacteristic
288C4—24no—
289C4—22no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—82no—
294C4—81no—
295C4#8981012yes—
296C4#8981312yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#1172943511yescharacteristic
301C2#1172946811yescharacteristic
302C2#1172961811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2536C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#2536D7
hypocenterC7#9486
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. #11739880C2semidirect
3. #11729391C2semidirect
5. #11725437C2semidirect
275. C7#9486semidirect

Maximal subgroups

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

subgroupindex
C7×#25362
#117398802
#117293912
#117293522
#117254372
#94867