Finite Group Databasebeta

← #13264692

Subgroups — #13264692

Order 1792.

432 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#11739729C211yesmaximal, characteristic
2C7×#1822C211yesmaximal, characteristic, complemented
3#11729325C211yesmaximal, characteristic, complemented
4#11725302C212yesmaximal, complemented
5#11725299C212yesmaximal, complemented
6C2×C7×#117C2211yescharacteristic
7C2×C4×D28C411yescharacteristic
8C7×#407—21no—
9#89985—21no—
10#89763C2211yescharacteristic
11#89761—21no—
12#89758C2211yescharacteristic
13#89732C411yescharacteristic
14#89728—21no—
15#89721C411yescharacteristic
16#89712—21no—
17#89709C411yescharacteristic
18#89697—21no—
19C7×#343—21no—
20C7×#343C2212yes—
21#89323C2212yes—
22#89320—21no—
23#11847—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C22×C22⋊D7—21no—
26C22×D28C2×C411yescharacteristic
27C22×C4×D7—21no—
28C2×C7×M16—22no—
29C7×#117—41no—
30C7×#117—41no—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C4⋊C4C2311yescharacteristic
33C2×C7×C22⋊C4C2×C411yescharacteristic
34C2×C7×C22⋊C4D411yescharacteristic
35C2×C42×C7C2×C411yescharacteristic
36C2×C42×C7D411yescharacteristic
37C2×C22⋊Dic7—21no—
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—21no—
40C2×(C2×D7)⋊C4—21no—
41C2×(C2×D7)⋊C4—21no—
42C2×C4⋊Dic7C2×C411yescharacteristic
43C2×Dic7⋊C4—21no—
44C4×D28—41no—
45C4×D28—41no—
46C7×C22⋊C8—41no—
47C7×C22⋊C8—41no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95C2×C411yescharacteristic
53C7⋊#95C2×C411yescharacteristic
54C22⋊(C7⋊C8)—22no—
55C4⋊(C7⋊C8)—41no—
56C4×C7⋊C8—41no—
57#2598—71no—
58#1822D711yescharacteristic, complemented
59#1458—71no—
60#1353—72no—
61#1345—72no—
62C24×C7C22⋊C411yescharacteristic
63C23×D7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—22no—
67C22×C4×C7—41no—
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C2×C22⋊D7—42no—
75C2×C22⋊D7—42no—
76C2×C22⋊D7—81no—
77C22×Dic7—21no—
78C22×Dic7—21no—
79C22×Dic7—21no—
80C2×D28—41no—
81C2×D28—42no—
82C2×C4×D7—42no—
83C2×C4×D7—81no—
84C2×C4×D7—81no—
85C7×M16—42no—
86C7×M16—42no—
87C2×C7×C8—42no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—21no—
95C42×C7—21no—
96C22⋊Dic7—81no—
97(C2×D7)⋊C4—42no—
98(C2×D7)⋊C4—41no—
99(C2×D7)⋊C4—41no—
100(C2×D7)⋊C4—42no—
101(C2×D7)⋊C4—81no—
102C4⋊Dic7—41no—
103Dic7⋊C4—81no—
104C2×C7⋊C8—42no—
105C2×C4×D4—71no—
106C2×#117C2×D711yescharacteristic
107#431—141no—
108#407—21no—
109#400—71no—
110#399—71no—
111#394—71no—
112#392—71no—
113#388—141no—
114#386—141no—
115#386—141no—
116#385—141no—
117#383—71no—
118#343—21no—
119#343C2×D712yes—
120#324—72no—
121#323—141no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7C2×C22⋊C411yescharacteristic
128C22×D7—22no—
129C22×D7—41no—
130C22×D7—41no—
131C22×D7—41no—
132C22×D7—41no—
133C22×D7—41no—
134C22×D7—41no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7#9911yescharacteristic
150C2×C4×C7#9911yescharacteristic
151C2×C4×C7#13512yes—
152C22⋊D7—84no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C2×Dic7—41no—
156C2×Dic7—41no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162D28—42no—
163D28—42no—
164C4×D7—84no—
165C7×C8—82no—
166C7⋊C8—82no—
167C22×D4—71no—
168C22×D4—141no—
169C23×C4—141no—
170C23×C4C22⋊D711yescharacteristic
171C2×M16—22no—
172C4×D4—281no—
173C4×D4—281no—
174#117—41no—
175#117—41no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—141no—
178C2×C4⋊C4C22⋊D711yescharacteristic
179C2×C4⋊C4C22×D711yescharacteristic
180C2×C22⋊C4—141no—
181C2×C22⋊C4—141no—
182C2×C22⋊C4—141no—
183C2×C22⋊C4—141no—
184C2×C22⋊C4—141no—
185C2×C22⋊C4C4×D711yescharacteristic
186C2×C22⋊C4D2811yescharacteristic
187C2×C42C4×D711yescharacteristic
188C2×C42D2811yescharacteristic
189C4⋊C8—281no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—142no—
193C4×C8—281no—
194#95—71no—
195#95—71no—
196#95—141no—
197#95—141no—
198#95—141no—
199#95—141no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—42no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—81no—
207C22×C7C2×#9911yescharacteristic
208C22×C7#41611yescharacteristic
209C22×C7#42111yescharacteristic
210C2×D7—42no—
211C2×D7—42no—
212C2×D7—42no—
213C2×D7—81no—
214C2×D7—81no—
215C2×D7—81no—
216C2×D7—81no—
217C4×C7—22no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224Dic7—81no—
225Dic7—81no—
226Dic7—81no—
227C24—141no—
228C24(C2×D7)⋊C411yescharacteristic
229C2×D4—281no—
230C2×D4—282no—
231C2×D4—282no—
232C2×D4—282no—
233C2×D4—561no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—141no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4—282no—
242C22×C4—561no—
243C22×C4—561no—
244C22×C4(C2×D7)⋊C411yescharacteristic
245C22×C4(C2×D7)⋊C411yescharacteristic
246C22×C4(C2×D7)⋊C411yescharacteristic
247C22×C4C2×C4×D711yescharacteristic
248C22×C4C2×D2811yescharacteristic
249C22×C4C2×C22⋊D711yescharacteristic
250M16—42no—
251M16—42no—
252C2×C8—42no—
253C2×C8—282no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—281no—
257C4⋊C4—561no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—282no—
261C22⋊C4—282no—
262C22⋊C4—281no—
263C22⋊C4—281no—
264C22⋊C4—561no—
265C22⋊C4—561no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C2×C7—21no—
271C2×C7—21no—
272C2×C7—81no—
273C2×C7#151611yescharacteristic
274C2×C7#212011yescharacteristic
275C2×C7#213411yescharacteristic
276D7—82no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—142no—
283C23—281no—
284C23—281no—
285C23—281no—
286C23—281no—
287C23—281no—
288C23—281no—
289C23C2×(C2×D7)⋊C411yescharacteristic
290D4—282no—
291D4—282no—
292D4—564no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—281no—
308C2×C4—281no—
309C2×C4—281no—
310C2×C4—281no—
311C2×C4—281no—
312C2×C4—281no—
313C2×C4—281no—
314C2×C4—281no—
315C2×C4—281no—
316C2×C4—564no—
317C2×C4#638011yescharacteristic
318C2×C4#638011yescharacteristic
319C2×C4#651312yes—
320C8—82no—
321C8—562no—
322C7#1184711yescharacteristic, complemented, Sylow
323C22—21no—
324C22—21no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—282no—
331C22—282no—
332C22—282no—
333C22—561no—
334C22—561no—
335C22—561no—
336C22—561no—
337C22C2×#638011yescharacteristic
338C22#8977411yescharacteristic
339C22#8977711yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—561no—
348C4—561no—
349C4—561no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—562no—
354C2#1172728611yescharacteristic
355C2#1172789911yescharacteristic
356C2#1172790311yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1822C2
socleC22×C7C2×#99
prime radical#89732C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1822D7
hypocenterC7#11847
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. C7×#1822C2semidirect
3. #11729325C2semidirect
4. #11725302C2semidirect
5. #11725299C2semidirect
58. #1822D7semidirect
322. C7#11847semidirect

Maximal subgroups

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

subgroupindex
#117397292
C7×#18222
#117293252
#117253022
#117252992
#118477