Finite Group Databasebeta

← #13264707

Subgroups — #13264707

Order 1792.

396 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 83 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#11739621C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic
3C2×#89323C211yesmaximal, characteristic
4#11725302C212yesmaximal, complemented
5#11725301C212yesmaximal, complemented
6C22×C7×C4⋊C4C2211yescharacteristic
7C2×#6489C411yescharacteristic
8C7×#407—21no—
9#89983—21no—
10#89983C411yescharacteristic
11#89982—21no—
12C2×C22⋊(C7⋊C8)—21no—
13#89758C411yescharacteristic
14#89758C2211yescharacteristic
15#89755C411yescharacteristic
16#89755C2211yescharacteristic
17#89750—21no—
18#89750—21no—
19C7×#343—21no—
20C7×#343C2212yes—
21#89323—21no—
22#89323C2212yes—
23#11850—71nomaximal, Sylow
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C2×C7×M16—22no—
28C2×C7×C4⋊C4—41no—
29C2×C7×C4⋊C4—41no—
30C2×C7×C4⋊C4C2×C411yescharacteristic
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C4⋊C4C2311yescharacteristic
34C2×C22⋊Dic7—21no—
35C2×C22⋊Dic7—21no—
36#6489—41no—
37#6489—41no—
38C2×C4⋊Dic7C2×C411yescharacteristic
39C2×C4⋊Dic7C2×C411yescharacteristic
40C2×Dic7⋊C4—21no—
41C2×Dic7⋊C4—21no—
42C2×C4×Dic7—21no—
43C22×C7⋊C8—41no—
44C7×C22⋊C8—41no—
45C7×C22⋊C8—41no—
46C7⋊#95—21no—
47C7⋊#95—21no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95C2×C411yescharacteristic
54C7⋊#95C2×C411yescharacteristic
55C22⋊(C7⋊C8)—22no—
56C22⋊(C7⋊C8)—41no—
57#2484—71no—
58#1821D711yescharacteristic
59C2×#324—71no—
60#1353—72no—
61#1346—72no—
62C24×C7C22⋊C411yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—41no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×Dic7—21no—
80C22×Dic7—21no—
81C22×Dic7—21no—
82C22×Dic7—21no—
83C7×M16—42no—
84C7×M16—42no—
85C2×C7×C8—42no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C22⋊Dic7—42no—
91C22⋊Dic7—41no—
92C22⋊Dic7—41no—
93C4⋊Dic7—41no—
94C4⋊Dic7—41no—
95Dic7⋊C4—42no—
96Dic7⋊C4—42no—
97C4×Dic7—42no—
98C2×C7⋊C8—42no—
99C2×C7⋊C8—82no—
100C2×#117—71no—
101C22×C4⋊C4C2×D711yescharacteristic
102#407—21no—
103C2×C22⋊C8—141no—
104#400—71no—
105#400—71no—
106#398—71no—
107#398—71no—
108#387—141no—
109#387—141no—
110#385—141no—
111#381—71no—
112#381—141no—
113#343—21no—
114#343C2×D712yes—
115#324—72no—
116#324—141no—
117C23×C7—22no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—41no—
121C23×C7#9911yescharacteristic
122C23×C7#9911yescharacteristic
123C23×C7C2×C22⋊C411yescharacteristic
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—81no—
143C2×C4×C7#13512yes—
144C2×Dic7—41no—
145C2×Dic7—41no—
146C2×Dic7—41no—
147C2×Dic7—41no—
148C2×Dic7—42no—
149C2×Dic7—41no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—41no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C7×C8—82no—
156C7⋊C8—82no—
157C23×C4C4×D711yescharacteristic
158C23×C4D2811yescharacteristic
159C23×C4C22⋊D711yescharacteristic
160C2×M16—22no—
161C22×C8—281no—
162#117—281no—
163#117—281no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—141no—
169C2×C4⋊C4—141no—
170C2×C4⋊C4C4×D711yescharacteristic
171C2×C4⋊C4D2811yescharacteristic
172C2×C4⋊C4C22⋊D711yescharacteristic
173C2×C4⋊C4C22×D711yescharacteristic
174C2×C22⋊C4—141no—
175C2×C22⋊C4—141no—
176C2×C42—141no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—142no—
180C22⋊C8—281no—
181#95—71no—
182#95—71no—
183#95—141no—
184#95—141no—
185#95—141no—
186#95—141no—
187#95—141no—
188#95—141no—
189#95—141no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—42no—
198C22×C7—42no—
199C22×C7#35212yes—
200C22×C7C2×#9911yescharacteristic
201C22×C7#41611yescharacteristic
202C22×C7#42111yescharacteristic
203C4×C7—22no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—41no—
209Dic7—81no—
210Dic7—81no—
211Dic7—81no—
212Dic7—81no—
213C24(C2×D7)⋊C411yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—141no—
225C22×C4—141no—
226C22×C4—141no—
227C22×C4—141no—
228C22×C4(C2×D7)⋊C411yescharacteristic
229C22×C4(C2×D7)⋊C411yescharacteristic
230C22×C4(C2×D7)⋊C411yescharacteristic
231C22×C4C2×C4×D711yescharacteristic
232C22×C4C2×D2811yescharacteristic
233C22×C4C2×C22⋊D711yescharacteristic
234M16—42no—
235M16—42no—
236C2×C8—42no—
237C2×C8—282no—
238C2×C8—562no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—281no—
244C4⋊C4—281no—
245C4⋊C4—282no—
246C4⋊C4—282no—
247C22⋊C4—282no—
248C22⋊C4—281no—
249C22⋊C4—281no—
250C42—282no—
251C2×C7—21no—
252C2×C7—21no—
253C2×C7—22no—
254C2×C7—41no—
255C2×C7#151611yescharacteristic
256C2×C7C2×#35211yescharacteristic
257C2×C7#213311yescharacteristic
258C23—21no—
259C23—21no—
260C23—22no—
261C23—41no—
262C23#638011yescharacteristic
263C23#638011yescharacteristic
264C23C2×(C2×D7)⋊C411yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—281no—
285C2×C4—281no—
286C2×C4—281no—
287C2×C4—281no—
288C2×C4—282no—
289C2×C4—281no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—281no—
293C2×C4—281no—
294C2×C4—281no—
295C2×C4#651312yes—
296C8—82no—
297C8—562no—
298C7#1185011yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#8927112yes—
309C22C2×#638011yescharacteristic
310C22#8977411yescharacteristic
311C22#8977711yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—561no—
319C4—561no—
320C4—561no—
321C4—561no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#1172728611yescharacteristic
327C2C2×#8927111yescharacteristic
328C2#1172788211yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1821C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1821D7
hypocenterC7#11850
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
4. #11725302C2semidirect
5. #11725301C2semidirect
298. C7#11850semidirect

Maximal subgroups

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

subgroupindex
#117396212
C7×#18212
C2×#893232
#117253022
#117253012
#118507