Finite Group Databasebeta

← #13251745

Subgroups — #13251745

Order 1792.

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

normal subgroups only

This group has 684 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#11739794C211yesmaximal, characteristic
3#11729356C211yesmaximal, characteristic
4#11727744C212yesmaximal, complemented
5#11725490C212yesmaximal, complemented
6C23×Dic14C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C2×#6487—21no—
9C7×#435—24no—
10C7×#432—21no—
11C7×#407C2212yes—
12C2×C7×C22⋊C8—21no—
13#89992C412yes—
14#89984C2211yescharacteristic
15#89979C2212yes—
16C2×#6398—24no—
17#89751—21no—
18#89719—21no—
19#89718C412yes—
20#89701—21no—
21#89323—21no—
22#89320—21no—
23#9505—71nomaximal, Sylow
24C22×C7×D4C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C23×Dic7C2×C412yes—
28C22×Dic14—22no—
29C22×Dic14—21no—
30C22×Dic14—21no—
31C22×Dic14—22no—
32C22×Dic14—44no—
33C2×C7×M16D414yes—
34C22×C7×C8—22no—
35C4×C7×D4—22no—
36C4×C7×D4—22no—
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×C22⋊Dic7—22no—
42C2×C22⋊Dic7C2×C412yes—
43C2×C4⋊Dic7—21no—
44#6487—44no—
45C2×Dic7⋊C4—22no—
46C2×#1136—22no—
47C7×C4⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—22no—
51C7×#97—22no—
52C7⋊#95—22no—
53C7⋊#95—22no—
54C7⋊#95—21no—
55C22⋊(C7⋊C8)—22no—
56C22⋊(C7⋊C8)—22no—
57#6398—48no—
58#2973D711yescharacteristic
59#2409—71no—
60#1784—72no—
61#1465—71no—
62#1305—72no—
63C24×C7C22⋊C411yescharacteristic
64C24×C7C22×C411yescharacteristic
65C2×C7×D4—22no—
66C2×C7×D4—22no—
67C2×C7×D4—41no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C2×D412yes—
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7#4112yes—
79C22×Dic7—22no—
80C22×Dic7—24no—
81C22×Dic7—22no—
82C22×Dic7—44no—
83C2×Dic14—24no—
84C2×Dic14—28no—
85C2×Dic14—44no—
86C2×Dic14—48no—
87C2×Dic14—44no—
88C2×Dic14—42no—
89C2×Dic14—41no—
90C2×Dic14—42no—
91C2×Dic14—41no—
92C7×M16—44no—
93C2×C7×C8—24no—
94C2×C7×C8—82no—
95C7×C4⋊C4—22no—
96C7×C22⋊C4—22no—
97C7×C22⋊C4—22no—
98C42×C7—21no—
99C42×C7—21no—
100C22⋊Dic7—44no—
101C22⋊Dic7—42no—
102C4⋊Dic7—42no—
103Dic7⋊C4—44no—
104#1136—44no—
105C2×C7⋊C8—42no—
106C23×Q8—71no—
107C2×C22⋊Q8—141no—
108C2×C4×D4C2×D711yescharacteristic
109#435—24no—
110#432—21no—
111C2×#101—144no—
112#407—72no—
113#407C2×D712yes—
114C2×C22⋊C8—21no—
115#395—141no—
116#393—141no—
117#391—141no—
118#386—72no—
119#385—72no—
120#380—71no—
121#324—141no—
122#323—141no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7#10112yescomplemented
132C23×C7C2×C22⋊C411yescharacteristic
133C23×C7C4×D412yes—
134C7×D4—44no—
135C2×C4×C7—24no—
136C2×C4×C7—22no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—41no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—81no—
148C2×C4×C7#12011yescharacteristic
149C2×C4×C7#12112yes—
150C2×C4×C7#12311yescharacteristic
151C2×Dic7—24no—
152C2×Dic7—28no—
153C2×Dic7—42no—
154C2×Dic7—44no—
155C2×Dic7—42no—
156C2×Dic7—42no—
157C2×Dic7—42no—
158C2×Dic7—42no—
159Dic14—48no—
160Dic14—44no—
161Dic14—44no—
162C7×C8—44no—
163C7⋊C8—82no—
164C22×Q8—141no—
165C22×Q8—142no—
166C22×Q8—141no—
167C22×Q8—142no—
168C22×Q8—284no—
169C22×D4C4×D711yescharacteristic
170C23×C4—72no—
171C23×C4C22⋊D711yescharacteristic
172C23×C4C22×D711yescharacteristic
173C2×M16—142no—
174C2×M16C22⋊D714yes—
175C22×C8—22no—
176C22⋊Q8—284no—
177C4×D4—22no—
178C4×D4—22no—
179C2×C4⋊C4—142no—
180C2×C4⋊C4—141no—
181C2×C4⋊C4D2811yescharacteristic
182C2×C22⋊C4—72no—
183C2×C22⋊C4—142no—
184C2×C22⋊C4C4×D711yescharacteristic
185C2×C22⋊C4D2811yescharacteristic
186C2×C42C22⋊D711yescharacteristic
187C4⋊C8—22no—
188#101—288no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—142no—
193C22⋊C8—142no—
194#97—22no—
195#95—142no—
196#95—142no—
197#95—141no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—22no—
201C22×C7—22no—
202C22×C7—22no—
203C22×C7—22no—
204C22×C7—42no—
205C22×C7—42no—
206C22×C7—42no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—81no—
210C22×C7#38611yescharacteristic
211C22×C7C2×#10111yescharacteristic
212C22×C7#41311yescharacteristic
213C4×C7—22no—
214C4×C7—21no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7B(4,𝔽2)12yes—
219Dic7—44no—
220Dic7—82no—
221C24(C2×D7)⋊C411yescharacteristic
222C24C2×C4×D711yescharacteristic
223C2×Q8—148no—
224C2×Q8—144no—
225C2×Q8—282no—
226C2×Q8—284no—
227C2×Q8—284no—
228C2×Q8—288no—
229C2×Q8—281no—
230C2×Q8—282no—
231C2×Q8—281no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—142no—
240C22×C4—144no—
241C22×C4—142no—
242C22×C4—284no—
243C22×C4(C2×D7)⋊C411yescharacteristic
244C22×C4(C2×D7)⋊C411yescharacteristic
245C22×C4(C2×D7)⋊C411yescharacteristic
246C22×C4C2×D2811yescharacteristic
247C22×C4#115712yes—
248C22×C4C2×C22⋊D711yescharacteristic
249C22×C4C2×C22⋊D712yes—
250M16—44no—
251M16—284no—
252C2×C8—24no—
253C2×C8—82no—
254C2×C8—282no—
255C4⋊C4—22no—
256C4⋊C4—282no—
257C4⋊C4—284no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—284no—
261C22⋊C4—282no—
262C42—21no—
263C42—21no—
264C2×C7—21no—
265C2×C7—21no—
266C2×C7—42no—
267C2×C7—42no—
268C2×C7#188911yescharacteristic
269C2×C7#190111yescharacteristic
270C2×C7#191511yescharacteristic
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23#639812yescomplemented
280C23C2×(C2×D7)⋊C411yescharacteristic
281C23C4×C22⋊D712yes—
282Q8—288no—
283Q8—284no—
284Q8—284no—
285D4—44no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—144no—
300C2×C4—148no—
301C2×C4—282no—
302C2×C4—284no—
303C2×C4—282no—
304C2×C4—282no—
305C2×C4—282no—
306C2×C4—282no—
307C2×C4#649211yescharacteristic
308C2×C4#649312yes—
309C2×C4#651611yescharacteristic
310C8—44no—
311C8—562no—
312C7#950511yescharacteristic, complemented, Sylow
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#8990411yescharacteristic
326C22C2×#639811yescharacteristic
327C22#8998511yescharacteristic
328C4—22no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—284no—
334C4—562no—
335C4#9001612yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—42no—
340C2#1172940811yescharacteristic
341C2#1172975711yescharacteristic
342C2#1172978511yescharacteristic
subgroupG/H
centerC22#89985
commutatorC24×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2973C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC23×C4×C7C23
hypercenter#2973D7
hypocenterC7#9505
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #11727744C2semidirect
5. #11725490C2semidirect
131. C23×C7#101semidirect
279. C23#6398semidirect
312. C7#9505semidirect

Maximal subgroups

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

subgroupindex
C7×#29732
#117397942
#117293562
#117277442
#117254902
#95057