Finite Group Databasebeta

← #11731386

Subgroups — #11731386

Order 896.

532 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D7C211yesmaximal, characteristic
2C2×#6453C211yesmaximal, characteristic, complemented
3C7×#380C211yesmaximal, characteristic, complemented
4#89984C211yesmaximal, characteristic, complemented
5#89759C211yesmaximal, characteristic, complemented
6#89759C211yesmaximal, characteristic, complemented
7C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
8#89733C212yesmaximal
9#89716C212yesmaximal, complemented
10#89441C212yesmaximal, complemented
11#89432C212yesmaximal, complemented
12C23×C4×C7C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2211yescharacteristic
19C2×C22⋊Dic7C2212yes—
20C4×C22⋊D7—24no—
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×C4⋊Dic7C2212yes—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×Dic7⋊C4C2212yes—
28C2×Dic7⋊C4C2211yescharacteristic
29C2×C4×Dic7C2212yes—
30C2×C4×Dic7C2211yescharacteristic
31#6453—24no—
32C7×#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2562—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C22×C4×C7—22no—
41C22×C4×C7—22no—
42C22×C4×C7—22no—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2312yes—
45C22×C4×C7C2312yes—
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—22no—
48C2×C22⋊D7—22no—
49C2×C22⋊D7—22no—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2312yes—
52C22×Dic7C2312yes—
53C22×Dic7C2311yescharacteristic
54C22×Dic7C2311yescharacteristic
55C2×C4×D7—22no—
56C2×C4×D7—22no—
57C2×C4×D7—22no—
58C7×C4⋊C4—22no—
59C7×C4⋊C4—22no—
60C7×C22⋊C4—42no—
61C22⋊Dic7—24no—
62C22⋊Dic7—22no—
63(C2×D7)⋊C4—24no—
64(C2×D7)⋊C4—22no—
65(C2×D7)⋊C4—42no—
66C4⋊Dic7—22no—
67C4⋊Dic7—22no—
68C4⋊Dic7—24no—
69Dic7⋊C4—24no—
70Dic7⋊C4—22no—
71C4×Dic7—42no—
72C4×Dic7D414yes—
73C2×#125—71no—
74C2×C4×D4—71no—
75#400—72no—
76#394—72no—
77#388—72no—
78#388—72no—
79#380—71no—
80#380—71no—
81#380—71no—
82#380D711yescharacteristic, complemented
83C4×C4⋊C4—71no—
84C23×C7—21no—
85C23×C7—21no—
86C23×C7—21no—
87C23×C7—21no—
88C23×C7—21no—
89C23×C7—21no—
90C23×C7#4112yes—
91C23×C7C2411yescharacteristic
92C22×D7—21no—
93C22×D7—21no—
94C22×D7—21no—
95C22×D7—21no—
96C22×D7—21no—
97C22×D7—21no—
98C22×D7#4112yes—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—44no—
109C2×C4×C7C2×D412yes—
110C2×C4×C7#4112yes—
111C2×C4×C7#4112yes—
112C22⋊D7—28no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—22no—
118C2×Dic7—21no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—21no—
124C2×Dic7—21no—
125C2×Dic7—22no—
126C2×Dic7C2×D412yes—
127C2×Dic7C2×D412yes—
128C4×D7—44no—
129C22×D4—71no—
130C23×C4—71no—
131C23×C4C2×D711yescharacteristic
132#125—144no—
133C4×D4—144no—
134C2×C4⋊C4—71no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4—72no—
138C2×C4⋊C4—71no—
139C2×C4⋊C4C2×D711yescharacteristic
140C2×C4⋊C4C2×D711yescharacteristic
141C2×C22⋊C4—71no—
142C2×C22⋊C4—72no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4C2×D712yes—
147C2×C42—71no—
148C2×C42—72no—
149#95—72no—
150#95—72no—
151#95—72no—
152#95—72no—
153#95C2×D712yes—
154C22×C7—22no—
155C22×C7—22no—
156C22×C7—22no—
157C22×C7—41no—
158C22×C7—41no—
159C22×C7—41no—
160C22×C7—41no—
161C22×C7C22×D411yescharacteristic
162C22×C7C2×#4111yescharacteristic
163C22×C7C2×#4111yescharacteristic
164C22×C7C2×#4111yescharacteristic
165C22×C7C2×#4111yescharacteristic
166C22×C7H(5,𝔽2)11yescharacteristic
167C22×C72−1+411yescharacteristic
168C2×D7—22no—
169C2×D7—22no—
170C2×D7—22no—
171C2×D7—41no—
172C2×D7—41no—
173C2×D7—41no—
174C2×D7—41no—
175C4×C7—41no—
176C4×C7—42no—
177C4×C7—42no—
178C4×C7#12414yes—
179Dic7—22no—
180Dic7—22no—
181Dic7—42no—
182Dic7—41no—
183Dic7—42no—
184C24—71no—
185C24C22×D711yescharacteristic
186C2×D4—142no—
187C2×D4—142no—
188C2×D4—142no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—71no—
193C22×C4—71no—
194C22×C4—71no—
195C22×C4—72no—
196C22×C4—72no—
197C22×C4—142no—
198C22×C4—142no—
199C22×C4—142no—
200C22×C4C22×D711yescharacteristic
201C22×C4C22×D712yes—
202C22×C4C22×D712yes—
203C22×C4C22×D711yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—142no—
207C4⋊C4—144no—
208C4⋊C4—144no—
209C4⋊C4—142no—
210C4⋊C4—142no—
211C22⋊C4—42no—
212C22⋊C4—142no—
213C22⋊C4—144no—
214C22⋊C4—142no—
215C22⋊C4—144no—
216C22⋊C4—282no—
217C42—74no—
218C42—282no—
219C2×C7—42no—
220C2×C7#52511yescharacteristic
221C2×C7C2×#12411yescharacteristic
222C2×C7#53711yescharacteristic
223C2×C7#55211yescharacteristic
224C2×C7#55211yescharacteristic
225C2×C7#55311yescharacteristic
226C2×C7#55311yescharacteristic
227D7—42no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—72no—
235C23—141no—
236C23—141no—
237C23—141no—
238C23—141no—
239C23—141no—
240C23—141no—
241C23#116112yes—
242C23C23×D711yescharacteristic
243D4—148no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—72no—
255C2×C4—72no—
256C2×C4—141no—
257C2×C4—141no—
258C2×C4—141no—
259C2×C4—141no—
260C2×C4—142no—
261C2×C4—141no—
262C2×C4—141no—
263C2×C4—142no—
264C2×C4—141no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—142no—
269C2×C4—284no—
270C2×C4#115712yes—
271C2×C4D4×D712yes—
272C2×C4#115912yes—
273C7#256211yescharacteristic, complemented, Sylow
274C22—22no—
275C22—22no—
276C22—22no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—142no—
282C22—142no—
283C22—142no—
284C22—281no—
285C22—281no—
286C22—281no—
287C22—281no—
288C22C2×#115711yescharacteristic
289C22C2×D4×D711yescharacteristic
290C22C2×#115911yescharacteristic
291C22#654811yescharacteristic
292C22C2×#116111yescharacteristic
293C22D7×#4111yescharacteristic
294C22#655411yescharacteristic
295C4—41no—
296C4—42no—
297C4—42no—
298C4—142no—
299C4—142no—
300C4—281no—
301C4—282no—
302C4—282no—
303C4#644814yes—
304C2—42no—
305C2—282no—
306C2C2×#644811yescharacteristic
307C2#9020511yescharacteristic
308C2#9022011yescharacteristic
309C2#9029911yescharacteristic
310C2#9030511yescharacteristic
311C2#9031811yescharacteristic
312C2#9033311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#380C2
prime radicalC22×C22⋊D7C22
hypercenter#380D7
hypocenterC7#2562
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#6453C2semidirect
3. C7×#380C2semidirect
4. #89984C2semidirect
5. #89759C2semidirect
6. #89759C2semidirect
7. C4⋊C4×Dic7C2semidirect
9. #89716C2semidirect
10. #89441C2semidirect
11. #89432C2semidirect
82. #380D7semidirect
273. C7#2562semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D72
C2×#64532
C7×#3802
#899842
#897592
C4⋊C4×Dic72
#897332
#897162
#894412
#894322
#25627