Finite Group Databasebeta

← #11734552

Subgroups — #11734552

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×#1159C211yesmaximal, characteristic
2C2×D7×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#6446C212yesmaximal, complemented
4C7×#394C211yesmaximal, characteristic, complemented
5#89994C211yesmaximal, characteristic, complemented
6#89751C211yesmaximal, characteristic, complemented
7#89728C212yesmaximal, complemented
8#89722C212yesmaximal, complemented
9#89718C212yesmaximal, complemented
10#89436C211yesmaximal, characteristic, complemented
11#89428C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C23×Dic7C2212yes—
15C2×#1159—28no—
16C2×#1159—24no—
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×Dic14C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2212yes—
23C2×C7×C22⋊C4C2212yes—
24C2×C22⋊Dic7C2212yes—
25C2×C22⋊Dic7C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×C4⋊Dic7C2212yes—
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2212yes—
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32D7×C4⋊C4—24no—
33#6446—28no—
34C7×#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38C7⋊#95C2212yes—
39#2406—71nomaximal, Sylow
40C24×C7C2312yes—
41C23×D7C2311yescharacteristic
42C2×C7×D4—24no—
43C2×C7×D4—22no—
44C22×C4×C7C2312yes—
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2311yescharacteristic
48C2×C22⋊D7—24no—
49C2×C22⋊D7—24no—
50C2×C22⋊D7D418yes—
51C22×Dic7—24no—
52C22×Dic7—24no—
53C22×Dic7—24no—
54C22×Dic7C2312yes—
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2312yes—
57C22×Dic7C2311yescharacteristic
58C22×Dic7C2311yescharacteristic
59#1159—232no—
60C2×C4×D7—22no—
61C2×C4×D7—24no—
62C2×C4×D7—22no—
63C2×C4×D7—22no—
64C2×C4×D7—24no—
65C2×C4×D7—22no—
66C2×C4×D7D414yes—
67C2×Dic14—22no—
68C2×Dic14—24no—
69C7×C4⋊C4—22no—
70C7×C22⋊C4—24no—
71C7×C22⋊C4—24no—
72C22⋊Dic7—24no—
73C22⋊Dic7—24no—
74(C2×D7)⋊C4—24no—
75C4⋊Dic7—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#395—71no—
85#394—71no—
86#394D711yescharacteristic, complemented
87#386—72no—
88#385—72no—
89#385—72no—
90#381—71no—
91C23×C7—22no—
92C23×C7—22no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—22no—
97C23×C7C2×D414yes—
98C23×C7C2411yescharacteristic
99C22×D7—21no—
100C22×D7—21no—
101C22×D7—22no—
102C22×D7C2×D414yes—
103C22×D7C2×D412yes—
104C7×D4—28no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7—21no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7C2×D412yes—
114C2×C4×C7#4112yescomplemented
115C22⋊D7—216no—
116C2×Dic7—216no—
117C2×Dic7—21no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—21no—
122C2×Dic7—21no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7C2×D414yes—
128C2×Dic7C2×D412yes—
129C2×Dic7#4112yescomplemented
130C4×D7—28no—
131C4×D7—28no—
132C4×D7—44no—
133Dic14—28no—
134C2×#41—144no—
135C2×#41—148no—
136C22×Q8—71no—
137C22×D4—72no—
138C22×D4C2×D711yescharacteristic
139C23×C4—71no—
140C23×C4—72no—
141C23×C4—71no—
142C23×C4—71no—
143#123—148no—
144C2×C4⋊C4—71no—
145C2×C4⋊C4—71no—
146C2×C4⋊C4—71no—
147C2×C4⋊C4—72no—
148C2×C4⋊C4—72no—
149C2×C4⋊C4—144no—
150C2×C4⋊C4C2×D711yescharacteristic
151C2×C22⋊C4—72no—
152C2×C22⋊C4—72no—
153C2×C22⋊C4—72no—
154C2×C22⋊C4C2×D712yes—
155C2×C22⋊C4C2×D712yes—
156#95—72no—
157#95—71no—
158#95—72no—
159#95—72no—
160#95C2×D711yescharacteristic
161C22×C7—24no—
162C22×C7—24no—
163C22×C7—24no—
164C22×C7—42no—
165C22×C7—42no—
166C22×C7—42no—
167C22×C7—42no—
168C22×C7C22×D411yescharacteristic
169C22×C7C22×D412yes—
170C22×C7C2×#4111yescharacteristic
171C22×C7C2×#4111yescharacteristic
172C22×C72−1+412yes—
173C2×D7—22no—
174C2×D7—22no—
175C2×D7—24no—
176C2×D7—24no—
177C2×D7#12014yes—
178C4×C7—22no—
179C4×C7—22no—
180C4×C7—41no—
181C4×C7—42no—
182Dic7—24no—
183Dic7—22no—
184Dic7—22no—
185Dic7—41no—
186Dic7—42no—
187C24—71no—
188C24C22×D712yes—
189#41—1432no—
190C2×Q8—144no—
191C2×Q8—142no—
192C2×D4—24no—
193C2×D4—22no—
194C2×D4—78no—
195C2×D4—144no—
196C2×D4—144no—
197C22×C4—71no—
198C22×C4—71no—
199C22×C4—72no—
200C22×C4—74no—
201C22×C4—72no—
202C22×C4—71no—
203C22×C4—142no—
204C22×C4—142no—
205C22×C4—144no—
206C22×C4—142no—
207C22×C4—144no—
208C22×C4—144no—
209C22×C4—144no—
210C22×C4—144no—
211C22×C4—142no—
212C22×C4C22×D712yes—
213C22×C4C22×D711yescharacteristic
214C22×C4C22×D711yescharacteristic
215C22×C4C22×D711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—142no—
218C4⋊C4—144no—
219C4⋊C4—142no—
220C4⋊C4—144no—
221C4⋊C4—142no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—144no—
225C22⋊C4—144no—
226C22⋊C4—144no—
227C2×C7—44no—
228C2×C7C2×#12011yescharacteristic
229C2×C7#52511yescharacteristic
230C2×C7#53611yescharacteristic
231C2×C7#54712yes—
232C2×C7#54712yes—
233D7—24no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—74no—
241C23—72no—
242C23—141no—
243C23—141no—
244C23—142no—
245C23D4×D714yes—
246C23C23×D711yescharacteristic
247Q8—148no—
248D4—28no—
249D4—1416no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—72no—
259C2×C4—74no—
260C2×C4—72no—
261C2×C4—148no—
262C2×C4—148no—
263C2×C4—1416no—
264C2×C4—141no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—141no—
269C2×C4—142no—
270C2×C4—142no—
271C2×C4—141no—
272C2×C4—142no—
273C2×C4—142no—
274C2×C4—284no—
275C2×C4D4×D712yes—
276C2×C4#115912yescomplemented
277C7#240611yescharacteristic, complemented, Sylow
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—74no—
286C22—142no—
287C22—144no—
288C22—144no—
289C22—142no—
290C22C2×D4×D711yescharacteristic
291C22C2×D4×D712yes—
292C22C2×#115911yescharacteristic
293C22D7×#4111yescharacteristic
294C22#655412yes—
295C4—22no—
296C4—22no—
297C4—41no—
298C4—42no—
299C4—142no—
300C4—144no—
301C4—142no—
302C4—281no—
303C4—282no—
304C2—44no—
305C2—144no—
306C2D7×#12011yescharacteristic
307C2#9029711yescharacteristic
308C2#9030212yes—
309C2#9034612yes—
310C2#9036611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC22×#1159C2
hypercenter#394D7
hypocenterC7#2406
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×D7×C4⋊C4C2semidirect
3. C2×#6446C2semidirect
4. C7×#394C2semidirect
5. #89994C2semidirect
6. #89751C2semidirect
7. #89728C2semidirect
8. #89722C2semidirect
9. #89718C2semidirect
10. #89436C2semidirect
11. #89428C2semidirect
86. #394D7semidirect
114. C2×C4×C7#41semidirect
129. C2×Dic7#41semidirect
276. C2×C4#1159semidirect
277. C7#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#11592
C2×D7×C4⋊C42
C2×#64462
C7×#3942
#899942
#897512
#897282
#897222
#897182
#894362
#894282
#24067