Finite Group Databasebeta

← #13251476

Subgroups — #13251476

Order 1792.

518 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2547C211yesmaximal, characteristic
2#11739747C211yesmaximal, characteristic
3#11729531C212yesmaximal, complemented
4#11729357C211yesmaximal, characteristic
5#11725427C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×#124C412yes—
8C2×C4×C7×D4C2211yescharacteristic
9C7×#397—21no—
10C7×#397—21no—
11C7×#391C412yes—
12C7×#380—21no—
13C7×#380—21no—
14C4×C7×C4⋊C4C2211yescharacteristic
15#89979—21no—
16C2×C22⋊(C7⋊C8)—21no—
17#89783—24no—
18#89698C2212yes—
19#89698C2212yes—
20#89341—24no—
21#89320—21no—
22#89320—21no—
23#9451—71nomaximal, Sylow
24C22×C7×Q8C2×C412yes—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C7×C4⋊Q8—22no—
29C7×C4⋊Q8—22no—
30C7×#124—44no—
31C4×C7×D4—22no—
32C4×C7×D4—22no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C42×C7C2×C412yes—
42C2×C42×C7C2311yescharacteristic
43C2×#1136—22no—
44C22×C7⋊C8—22no—
45C7×#95—22no—
46C7×#95—22no—
47C7×#95—21no—
48C22⋊(C7⋊C8)—22no—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)—22no—
51C22⋊(C7⋊C8)—22no—
52C4⋊(C7⋊C8)—22no—
53C4⋊(C7⋊C8)D414yes—
54#6377—22no—
55#6377—22no—
56#2973—71no—
57#2547D711yescharacteristic
58#1558—72no—
59#1466—71no—
60#1285—72no—
61C24×C7C22⋊C411yescharacteristic
62C24×C7C22⋊C411yescharacteristic
63C2×C7×Q8—24no—
64C2×C7×Q8—42no—
65C2×C7×Q8—42no—
66C2×C7×D4—22no—
67C2×C7×D4—22no—
68C2×C7×D4—41no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—42no—
76C22×C4×C7—42no—
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22×C411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C7×C4⋊C4—22no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—42no—
86C7×C4⋊C4—41no—
87C7×C22⋊C4—22no—
88C7×C22⋊C4—22no—
89C7×C22⋊C4—44no—
90C7×C22⋊C4—44no—
91C42×C7—42no—
92C42×C7C2×D412yes—
93C42×C7#4112yes—
94#1136—82no—
95C2×C7⋊C8—24no—
96C2×C7⋊C8—42no—
97C2×C7⋊C8—82no—
98C2×C4⋊Q8C2×D711yescharacteristic
99C2×#124Dic712yes—
100C2×C4×D4C2×D711yescharacteristic
101#435—144no—
102#432—72no—
103#432—72no—
104#407—141no—
105C2×C22⋊C8—141no—
106#397—21no—
107#397—21no—
108#391Dic712yes—
109#380—21no—
110#380—21no—
111C4×C4⋊C4C2×D711yescharacteristic
112#333—144no—
113#323—141no—
114#323—141no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7#10012yes—
125C23×C7C2×C22⋊C411yescharacteristic
126C7×Q8—84no—
127C7×D4—44no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—22no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—41no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
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—81no—
146C2×C4×C7C4×D412yes—
147C2×C4×C7#12011yescharacteristic
148C2×C4×C7#12112yes—
149C2×C4×C7#12311yescharacteristic
150C7⋊C8—44no—
151C7⋊C8—82no—
152C22×Q8C2×Dic712yes—
153C22×D4C2×Dic711yescharacteristic
154C23×C4C22⋊D711yescharacteristic
155C23×C4C22⋊D711yescharacteristic
156C2×M16—142no—
157C22×C8—142no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160#124—44no—
161C4×D4—22no—
162C4×D4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C2×Dic711yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4C22⋊D711yescharacteristic
170C2×C22⋊C4C22⋊D711yescharacteristic
171C2×C42C2×Dic712yes—
172C2×C42C22×D711yescharacteristic
173C4⋊C8—74no—
174C4⋊C8—142no—
175C22⋊C8—142no—
176C22⋊C8—142no—
177C22⋊C8—142no—
178C22⋊C8—142no—
179#97—142no—
180#97—142no—
181#95—22no—
182#95—22no—
183#95—21no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—22no—
187C22×C7—42no—
188C22×C7—42no—
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—81no—
195C22×C7—81no—
196C22×C7#38611yescharacteristic
197C22×C7C2×#10011yescharacteristic
198C22×C7#41311yescharacteristic
199C4×C7—21no—
200C4×C7—22no—
201C4×C7—22no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—82no—
205C4×C7—82no—
206C4×C7#45512yes—
207C24C22⋊Dic711yescharacteristic
208C24C22⋊Dic711yescharacteristic
209C2×Q8—24no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C22⋊Dic711yescharacteristic
224C22×C4C22⋊Dic711yescharacteristic
225C22×C4C22×Dic711yescharacteristic
226C22×C4C2×C22⋊D711yescharacteristic
227C22×C4C2×C22⋊D711yescharacteristic
228M16—562no—
229C2×C8—144no—
230C2×C8—282no—
231C2×C8—562no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C42—42no—
242C42D4×D712yes—
243C42#115912yes—
244C2×C7—21no—
245C2×C7—21no—
246C2×C7—42no—
247C2×C7—81no—
248C2×C7#188511yescharacteristic
249C2×C7#190011yescharacteristic
250C2×C7#190911yescharacteristic
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#640712yes—
261C23C2×C22⋊Dic711yescharacteristic
262Q8—84no—
263D4—44no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4D4×Dic712yes—
283C2×C4#649811yescharacteristic
284C2×C4#650011yescharacteristic
285C2×C4#650212yes—
286C8—284no—
287C8—562no—
288C7#945111yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22C2×#640711yescharacteristic
302C22#8999211yescharacteristic
303C22#9000911yescharacteristic
304C4—22no—
305C4—21no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4#8983312yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#1172943711yescharacteristic
317C2#1172947711yescharacteristic
318C2#1172961911yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2547C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#2547D7
hypocenterC7#9451
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #11729531C2semidirect
5. #11725427C2semidirect
288. C7#9451semidirect

Maximal subgroups

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

subgroupindex
C7×#25472
#117397472
#117295312
#117293572
#117254272
#94517