Finite Group Databasebeta

← #13251485

Subgroups — #13251485

Order 1792.

498 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2973C211yesmaximal, characteristic
2#11739829C211yesmaximal, characteristic
3#11729357C211yesmaximal, characteristic
4#11728576C212yesmaximal, complemented
5#11725556C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6432C2211yescharacteristic
8C7×#435—24no—
9C7×#432C2212yes—
10C7×#407—21no—
11C2×C7×C22⋊C8—21no—
12#89993C412yes—
13#89984—21no—
14#89984—21no—
15#89749C412yes—
16#89719—21no—
17#89719—21no—
18C4×C4⋊Dic7C2211yescharacteristic
19#89698C2212yes—
20#89320—21no—
21#89320—21no—
22#89281—24no—
23#9453—71nomaximal, Sylow
24C22×C7×D4C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C22×Dic14—21no—
28C2×C7×M16—22no—
29C22×C7×C8—22no—
30C4×C7×D4—22no—
31C4×C7×D4—22no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C22⋊C4D411yescharacteristic
34C2×C7×C22⋊C4D411yescharacteristic
35C2×C42×C7C2311yescharacteristic
36C2×C22⋊Dic7—22no—
37C2×C22⋊Dic7—22no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7—21no—
40C2×C4⋊Dic7C2×C412yes—
41C2×C4×Dic7C2×C412yes—
42#6432—22no—
43#6432—22no—
44C7×C4⋊C8D414yes—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—22no—
48C7×#97—22no—
49C7⋊#95—22no—
50C7⋊#95—22no—
51C7⋊#95—21no—
52C22⋊(C7⋊C8)—22no—
53C4⋊(C7⋊C8)—22no—
54#6377—22no—
55#2973D711yescharacteristic
56#2563—71no—
57#1556—72no—
58#1466—71no—
59#1285—72no—
60C24×C7C22⋊C411yescharacteristic
61C24×C7C22⋊C411yescharacteristic
62C2×C7×D4—22no—
63C2×C7×D4—22no—
64C2×C7×D4—41no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×Dic7—22no—
77C22×Dic7—22no—
78C2×Dic14—42no—
79C2×Dic14—82no—
80C7×M16—82no—
81C2×C7×C8—24no—
82C2×C7×C8—82no—
83C7×C4⋊C4—22no—
84C7×C22⋊C4—22no—
85C7×C22⋊C4—22no—
86C42×C7C2×D412yes—
87C42×C7#4112yes—
88C22⋊Dic7—82no—
89C22⋊Dic7—82no—
90C4⋊Dic7—24no—
91C4⋊Dic7—42no—
92C4⋊Dic7—41no—
93C4⋊Dic7—41no—
94C4×Dic7—42no—
95C2×C7⋊C8—42no—
96C2×C4⋊Q8—71no—
97C2×C4×D4C2×D711yescharacteristic
98#435—24no—
99#432—72no—
100#432C2×D712yes—
101#407—21no—
102C2×C22⋊C8—21no—
103#393—141no—
104#393—141no—
105#390—72no—
106#382—72no—
107#380—141no—
108#380—141no—
109C4×C4⋊C4—71no—
110#332—144no—
111#323—141no—
112#323—141no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7#10112yes—
123C23×C7C2×C22⋊C411yescharacteristic
124C7×D4—44no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—22no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—41no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—81no—
136C2×C4×C7—81no—
137C2×C4×C7C4×D412yes—
138C2×C4×C7#12011yescharacteristic
139C2×C4×C7#12112yes—
140C2×C4×C7#12311yescharacteristic
141C2×Dic7—42no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144C2×Dic7—42no—
145C2×Dic7—42no—
146C2×Dic7—42no—
147Dic14—84no—
148C7×C8—44no—
149C7⋊C8—82no—
150C22×Q8—141no—
151C22×D4C4×D711yescharacteristic
152C23×C4C22⋊D711yescharacteristic
153C23×C4C22⋊D711yescharacteristic
154C2×M16—22no—
155C22×C8—22no—
156C4⋊Q8—142no—
157C4⋊Q8—142no—
158C4×D4—22no—
159C4×D4—22no—
160C2×C4⋊C4—72no—
161C2×C4⋊C4—141no—
162C2×C4⋊C4—141no—
163C2×C4⋊C4C4×D711yescharacteristic
164C2×C22⋊C4—142no—
165C2×C22⋊C4—142no—
166C2×C22⋊C4D2811yescharacteristic
167C2×C22⋊C4D2811yescharacteristic
168C2×C42—72no—
169C2×C42C22×D711yescharacteristic
170C4⋊C8—142no—
171C4⋊C8C22⋊D714yes—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—142no—
176#97—22no—
177#97—142no—
178#95—142no—
179#95—142no—
180#95—141no—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—22no—
184C22×C7—42no—
185C22×C7—42no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—41no—
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—81no—
192C22×C7—81no—
193C22×C7#38611yescharacteristic
194C22×C7C2×#10111yescharacteristic
195C22×C7#41311yescharacteristic
196C4×C7—21no—
197C4×C7—22no—
198C4×C7—22no—
199C4×C7—41no—
200C4×C7—41no—
201C4×C7#45512yes—
202Dic7—82no—
203Dic7—82no—
204C24(C2×D7)⋊C411yescharacteristic
205C24(C2×D7)⋊C411yescharacteristic
206C2×Q8—282no—
207C2×Q8—562no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—142no—
218C22×C4—142no—
219C22×C4(C2×D7)⋊C411yescharacteristic
220C22×C4(C2×D7)⋊C411yescharacteristic
221C22×C4C2×C4×D711yescharacteristic
222C22×C4C2×D2811yescharacteristic
223C22×C4C2×C22⋊D711yescharacteristic
224M16—82no—
225C2×C8—24no—
226C2×C8—82no—
227C2×C8—282no—
228C4⋊C4—22no—
229C4⋊C4—144no—
230C4⋊C4—282no—
231C4⋊C4—281no—
232C4⋊C4—281no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—562no—
236C22⋊C4—562no—
237C42—282no—
238C42#115712yes—
239C42C2×C22⋊D712yes—
240C2×C7—21no—
241C2×C7—21no—
242C2×C7—42no—
243C2×C7—81no—
244C2×C7#188511yescharacteristic
245C2×C7#190111yescharacteristic
246C2×C7#191011yescharacteristic
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#639812yes—
257C23C2×(C2×D7)⋊C411yescharacteristic
258Q8—564no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—282no—
273C2×C4—282no—
274C2×C4—282no—
275C2×C4—282no—
276C2×C4—282no—
277C2×C4—282no—
278C2×C4C4×C22⋊D712yes—
279C2×C4#649211yescharacteristic
280C2×C4#649312yes—
281C2×C4#651611yescharacteristic
282C8—44no—
283C8—562no—
284C7#945311yescharacteristic, complemented, Sylow
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#8990411yescharacteristic
298C22C2×#639811yescharacteristic
299C22#8998511yescharacteristic
300C4—22no—
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—562no—
306C4—562no—
307C4#9001412yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#1172940311yescharacteristic
313C2#1172975711yescharacteristic
314C2#1172977911yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2973C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#2973D7
hypocenterC7#9453
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #11728576C2semidirect
5. #11725556C2semidirect
284. C7#9453semidirect

Maximal subgroups

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

subgroupindex
C7×#29732
#117398292
#117293572
#117285762
#117255562
#94537