Finite Group Databasebeta

← #11734694

Subgroups — #11734694

Order 896.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#1161C211yesmaximal, characteristic
2C2×#6457C212yesmaximal, complemented
3C2×D7×C4⋊C4C211yesmaximal, characteristic, complemented
4C7×#395C211yesmaximal, characteristic, complemented
5#90001C211yesmaximal, characteristic, complemented
6#89763C211yesmaximal, characteristic, complemented
7#89761C212yesmaximal, complemented
8#89444C211yesmaximal, characteristic, complemented
9#89440C212yesmaximal, complemented
10#89437C212yesmaximal, complemented
11#89436C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2211yescharacteristic
13C2×#1161—28no—
14C2×#1161—24no—
15C22×D28C2211yescharacteristic
16C22×D28C2212yes—
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2212yes—
20C22×C4×D7C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C4⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2212yes—
32#6457—28no—
33D7×C4⋊C4—24no—
34C7×#95C2211yescharacteristic
35C7×#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38C7⋊#95C2212yes—
39#2406—71nomaximal, Sylow
40C23×D7C2312yes—
41C23×D7C2311yescharacteristic
42C2×C7×Q8—24no—
43C2×C7×Q8—22no—
44C22×C4×C7C2312yes—
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2312yes—
47C22×C4×C7C2311yescharacteristic
48C22×C4×C7C2311yescharacteristic
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2311yescharacteristic
53#1161—232no—
54C2×D28—24no—
55C2×D28—24no—
56C2×D28—24no—
57C2×D28—22no—
58C2×D28D418yes—
59C2×C4×D7—22no—
60C2×C4×D7—24no—
61C2×C4×D7—24no—
62C2×C4×D7—24no—
63C2×C4×D7—22no—
64C2×C4×D7—24no—
65C2×C4×D7—22no—
66C2×C4×D7—22no—
67C2×C4×D7—24no—
68C2×C4×D7D414yes—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—24no—
71(C2×D7)⋊C4—24no—
72(C2×D7)⋊C4—24no—
73(C2×D7)⋊C4—24no—
74(C2×D7)⋊C4—24no—
75(C2×D7)⋊C4—24no—
76C4⋊Dic7—22no—
77Dic7⋊C4—22no—
78Dic7⋊C4—24no—
79Dic7⋊C4—22no—
80C22×#41—71no—
81C2×#123—72no—
82C22×C4⋊C4—71no—
83#395—71no—
84#395D711yescharacteristic, complemented
85#394—71no—
86#394—71no—
87#386—72no—
88#385—72no—
89#385—72no—
90#381—71no—
91C23×C7C2411yescharacteristic
92C22×D7—21no—
93C22×D7—21no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7—22no—
97C22×D7—22no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7C2×D414yes—
102C22×D7C2×D414yes—
103C22×D7C2×D412yes—
104C7×Q8—28no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7C2×D414yes—
116C2×C4×C7C2×D412yes—
117C2×C4×C7#4112yescomplemented
118C2×Dic7—21no—
119C2×Dic7—22no—
120C2×Dic7—21no—
121C2×Dic7—21no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7C2×D412yes—
127C2×Dic7#4112yescomplemented
128D28—28no—
129D28—216no—
130C4×D7—28no—
131C4×D7—216no—
132C4×D7—28no—
133C4×D7—44no—
134C2×#41—144no—
135C2×#41—148no—
136C22×Q8C2×D711yescharacteristic
137C22×D4—72no—
138C22×D4—71no—
139C23×C4—71no—
140C23×C4—71no—
141C23×C4—71no—
142C23×C4—72no—
143#123—148no—
144C2×C4⋊C4—71no—
145C2×C4⋊C4—71no—
146C2×C4⋊C4—71no—
147C2×C4⋊C4—72no—
148C2×C4⋊C4—144no—
149C2×C4⋊C4C2×D711yescharacteristic
150C2×C4⋊C4C2×D712yes—
151C2×C22⋊C4—72no—
152C2×C22⋊C4—72no—
153C2×C22⋊C4—72no—
154C2×C22⋊C4—72no—
155C2×C22⋊C4—72no—
156#95—72no—
157#95—71no—
158#95—72no—
159#95C2×D711yescharacteristic
160#95C2×D712yes—
161C22×C7C22×D411yescharacteristic
162C22×C7C22×D412yes—
163C22×C7C2×#4111yescharacteristic
164C22×C7C2×#4111yescharacteristic
165C22×C72−1+412yes—
166C2×D7—22no—
167C2×D7—22no—
168C2×D7—24no—
169C2×D7—24no—
170C2×D7—24no—
171C2×D7—24no—
172C2×D7—24no—
173C2×D7—42no—
174C2×D7—42no—
175C2×D7—42no—
176C2×D7—42no—
177C2×D7#12014yes—
178C4×C7—24no—
179C4×C7—22no—
180C4×C7—22no—
181C4×C7—41no—
182C4×C7—42no—
183Dic7—22no—
184Dic7—22no—
185Dic7—41no—
186Dic7—42no—
187C24—71no—
188C24—72no—
189#41—1432no—
190C2×Q8—22no—
191C2×Q8—24no—
192C2×D4—78no—
193C2×D4—144no—
194C2×D4—144no—
195C2×D4—144no—
196C2×D4—142no—
197C22×C4—71no—
198C22×C4—71no—
199C22×C4—74no—
200C22×C4—72no—
201C22×C4—71no—
202C22×C4—142no—
203C22×C4—144no—
204C22×C4—142no—
205C22×C4—144no—
206C22×C4—144no—
207C22×C4—142no—
208C22×C4—144no—
209C22×C4—144no—
210C22×C4—142no—
211C22×C4C22×D712yes—
212C22×C4C22×D711yescharacteristic
213C22×C4C22×D711yescharacteristic
214C22×C4C22×D712yes—
215C22×C4C22×D711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—142no—
219C4⋊C4—142no—
220C4⋊C4—142no—
221C4⋊C4—144no—
222C22⋊C4—144no—
223C22⋊C4—144no—
224C22⋊C4—144no—
225C22⋊C4—144no—
226C22⋊C4—144no—
227C2×C7C2×#12011yescharacteristic
228C2×C7#52511yescharacteristic
229C2×C7#53611yescharacteristic
230C2×C7#54712yes—
231C2×C7#54712yes—
232D7—24no—
233D7—44no—
234C23—74no—
235C23—72no—
236C23—74no—
237C23—141no—
238C23—142no—
239C23—141no—
240C23—142no—
241C23—142no—
242C23—142no—
243C23—142no—
244C23—142no—
245C23—142no—
246C23C23×D711yescharacteristic
247Q8—28no—
248D4—1416no—
249D4—148no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—72no—
261C2×C4—72no—
262C2×C4—148no—
263C2×C4—148no—
264C2×C4—1416no—
265C2×C4—141no—
266C2×C4—142no—
267C2×C4—141no—
268C2×C4—142no—
269C2×C4—142no—
270C2×C4—141no—
271C2×C4—142no—
272C2×C4—142no—
273C2×C4—284no—
274C2×C4D4×D712yes—
275C2×C4D4×D714yes—
276C2×C4#116112yescomplemented
277C7#240611yescharacteristic, complemented, Sylow
278C22—74no—
279C22—142no—
280C22—144no—
281C22—142no—
282C22—144no—
283C22—144no—
284C22—144no—
285C22—144no—
286C22—282no—
287C22—282no—
288C22—282no—
289C22—282no—
290C22C2×D4×D711yescharacteristic
291C22C2×D4×D712yes—
292C22C2×#116111yescharacteristic
293C22#655112yes—
294C22D7×#4111yescharacteristic
295C4—22no—
296C4—24no—
297C4—22no—
298C4—41no—
299C4—42no—
300C4—142no—
301C4—142no—
302C4—282no—
303C4—281no—
304C2—144no—
305C2—284no—
306C2D7×#12011yescharacteristic
307C2#9031811yescharacteristic
308C2#9032212yes—
309C2#9041511yescharacteristic
310C2#9041712yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#395C2
prime radicalC22×#1161C2
hypercenter#395D7
hypocenterC7#2406
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#6457C2semidirect
3. C2×D7×C4⋊C4C2semidirect
4. C7×#395C2semidirect
5. #90001C2semidirect
6. #89763C2semidirect
7. #89761C2semidirect
8. #89444C2semidirect
9. #89440C2semidirect
10. #89437C2semidirect
11. #89436C2semidirect
84. #395D7semidirect
117. C2×C4×C7#41semidirect
127. C2×Dic7#41semidirect
276. C2×C4#1161semidirect
277. C7#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#11612
C2×#64572
C2×D7×C4⋊C42
C7×#3952
#900012
#897632
#897612
#894442
#894402
#894372
#894362
#24067