Finite Group Databasebeta

← #11734187

Subgroups — #11734187

Order 896.

644 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#6501C211yesmaximal, characteristic, complemented
2C2×#6447C211yesmaximal, characteristic, complemented
3C7×#392C211yesmaximal, characteristic, complemented
4#89995C212yesmaximal, complemented
5#89995C211yesmaximal, characteristic, complemented
6#89995C212yesmaximal, complemented
7#89732C212yesmaximal, complemented
8#89725C211yesmaximal, characteristic, complemented
9#89724C212yesmaximal, complemented
10#89446C211yesmaximal, characteristic, complemented
11#89436C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C7×D4C2212yes—
14C22×C22⋊D7C2211yescharacteristic
15C22×C22⋊D7C2211yescharacteristic
16C22×C22⋊D7C2212yes—
17C22×C22⋊D7C2212yes—
18C22×D28C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2212yes—
22C2×C22⋊Dic7C2212yes—
23C2×C22⋊Dic7C2212yes—
24C2×C22⋊Dic7C2212yes—
25#6501—24no—
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2212yes—
28C2×(C2×D7)⋊C4C2212yes—
29C2×C4⋊Dic7C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31#6447—24no—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2815—71nomaximal, Sylow
38C24×C7C2312yes—
39C24×C7C2311yescharacteristic
40C23×D7C2311yescharacteristic
41C23×D7C2311yescharacteristic
42C2×C7×D4—22no—
43C2×C7×D4—42no—
44C2×C7×D4—42no—
45C2×C7×D4—42no—
46C2×C7×D4—42no—
47C22×C4×C7C2312yes—
48C22×C4×C7C2311yescharacteristic
49C22×C4×C7C2311yescharacteristic
50C2×C22⋊D7—22no—
51C2×C22⋊D7—24no—
52C2×C22⋊D7—22no—
53C2×C22⋊D7—42no—
54C2×C22⋊D7—42no—
55C2×C22⋊D7—42no—
56C2×C22⋊D7—42no—
57C2×C22⋊D7—42no—
58C2×C22⋊D7—42no—
59C2×C22⋊D7—42no—
60C22×Dic7C2312yes—
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2312yes—
63C22×Dic7C2311yescharacteristic
64C2×D28—22no—
65C2×D28—42no—
66C2×C4×D7—42no—
67C2×C4×D7D414yescomplemented
68C7×C22⋊C4—22no—
69C7×C22⋊C4—42no—
70C22⋊Dic7—24no—
71C22⋊Dic7—42no—
72C22⋊Dic7—42no—
73(C2×D7)⋊C4—22no—
74(C2×D7)⋊C4—42no—
75(C2×D7)⋊C4—42no—
76C4⋊Dic7—22no—
77Dic7⋊C4—22no—
78C2×#121—71no—
79C2×#121—71no—
80#397—72no—
81#397—71no—
82#397—71no—
83#392—72no—
84#392—71no—
85#392—72no—
86#392—72no—
87#392D711yescharacteristic, complemented
88#381—71no—
89C23×C7—22no—
90C23×C7—21no—
91C23×C7—21no—
92C23×C7—22no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—22no—
97C23×C7—22no—
98C23×C7—22no—
99C23×C7—22no—
100C23×C7—21no—
101C23×C7C2411yescharacteristic
102C22×D7—22no—
103C22×D7—22no—
104C22×D7—21no—
105C22×D7—21no—
106C22×D7—21no—
107C22×D7—21no—
108C22×D7—22no—
109C22×D7—22no—
110C22×D7—21no—
111C22×D7C2×D412yes—
112C7×D4—44no—
113C7×D4—48no—
114C2×C4×C7—22no—
115C2×C4×C7—21no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7C2×D412yes—
121C22⋊D7—44no—
122C22⋊D7—44no—
123C22⋊D7—48no—
124C22⋊D7—48no—
125C2×Dic7—22no—
126C2×Dic7—21no—
127C2×Dic7—22no—
128C2×Dic7—22no—
129C2×Dic7—22no—
130C2×Dic7—22no—
131C2×Dic7—22no—
132C2×Dic7—22no—
133C2×Dic7—22no—
134C2×Dic7C2×D412yes—
135D28—44no—
136C4×D7—44no—
137C22×D4—71no—
138C22×D4—72no—
139C22×D4—71no—
140C22×D4—72no—
141C22×D4—71no—
142C22×D4C2×D711yescharacteristic
143C22×D4C2×D712yes—
144C23×C4—71no—
145#121—144no—
146#121—144no—
147C2×C4⋊C4—71no—
148C2×C4⋊C4—71no—
149C2×C22⋊C4—72no—
150C2×C22⋊C4—71no—
151C2×C22⋊C4—72no—
152C2×C22⋊C4—72no—
153C2×C22⋊C4—72no—
154C2×C22⋊C4—72no—
155C2×C22⋊C4C2×D711yescharacteristic
156C2×C22⋊C4C2×D712yes—
157#95—72no—
158#95—71no—
159#95—72no—
160#95—72no—
161#95C2×D711yescharacteristic
162C22×C7—41no—
163C22×C7—42no—
164C22×C7—41no—
165C22×C7—42no—
166C22×C7—42no—
167C22×C7—41no—
168C22×C7—42no—
169C22×C7—42no—
170C22×C7—42no—
171C22×C7—42no—
172C22×C7—42no—
173C22×C7—42no—
174C22×C7C22×D411yescharacteristic
175C22×C7H(5,𝔽2)12yes—
176C22×C7H(5,𝔽2)12yes—
177C22×C7H(5,𝔽2)11yescharacteristic
178C22×C7H(5,𝔽2)11yescharacteristic
179C2×D7—22no—
180C2×D7—22no—
181C2×D7—22no—
182C2×D7—42no—
183C2×D7—42no—
184C2×D7—41no—
185C2×D7—41no—
186C2×D7—41no—
187C2×D7—42no—
188C2×D7—42no—
189C4×C7—22no—
190C4×C7—41no—
191C4×C7—42no—
192Dic7—22no—
193Dic7—41no—
194Dic7—42no—
195Dic7—42no—
196C24—71no—
197C24—71no—
198C24C22×D711yescharacteristic
199C24C22×D712yes—
200C2×D4—22no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—142no—
206C2×D4—142no—
207C2×D4—142no—
208C2×D4—144no—
209C2×D4—282no—
210C2×D4—282no—
211C2×D4—282no—
212C2×D4—282no—
213C2×D4—282no—
214C2×D4—282no—
215C2×D4—282no—
216C2×D4—282no—
217C22×C4—71no—
218C22×C4—74no—
219C22×C4—72no—
220C22×C4—71no—
221C22×C4—72no—
222C22×C4—282no—
223C22×C4C22×D712yes—
224C22×C4C22×D711yescharacteristic
225C22×C4C22×D711yescharacteristic
226C4⋊C4—142no—
227C4⋊C4—142no—
228C22⋊C4—22no—
229C22⋊C4—42no—
230C22⋊C4—144no—
231C22⋊C4—142no—
232C22⋊C4—282no—
233C22⋊C4—282no—
234C22⋊C4—282no—
235C22⋊C4—282no—
236C2×C7—81no—
237C2×C7—82no—
238C2×C7#53411yescharacteristic
239C2×C7#53511yescharacteristic
240C2×C7#53511yescharacteristic
241C2×C7#56012yes—
242C2×C7#56012yes—
243D7—42no—
244D7—81no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—22no—
257C23—72no—
258C23—142no—
259C23—142no—
260C23—141no—
261C23—141no—
262C23—141no—
263C23—141no—
264C23—142no—
265C23—142no—
266C23—141no—
267C23C23×D711yescharacteristic
268D4—44no—
269D4—48no—
270D4—284no—
271D4—284no—
272D4—284no—
273D4—288no—
274D4—288no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—72no—
282C2×C4—142no—
283C2×C4—141no—
284C2×C4—142no—
285C2×C4—142no—
286C2×C4—142no—
287C2×C4—142no—
288C2×C4—142no—
289C2×C4—142no—
290C2×C4—142no—
291C2×C4—284no—
292C2×C4D4×D712yes—
293C7#281511yescharacteristic, complemented, Sylow
294C22—41no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—142no—
307C22—142no—
308C22—142no—
309C22—282no—
310C22—282no—
311C22—281no—
312C22—281no—
313C22—282no—
314C22—281no—
315C22—282no—
316C22C2×D4×D711yescharacteristic
317C22#654812yes—
318C22#654812yes—
319C22#654811yescharacteristic
320C22#655311yescharacteristic
321C4—22no—
322C4—41no—
323C4—42no—
324C4—142no—
325C4—281no—
326C4—282no—
327C4—282no—
328C2—81no—
329C2—82no—
330C2—282no—
331C2—561no—
332C2#9028011yescharacteristic
333C2#9028612yes—
334C2#9029811yescharacteristic
335C2#9031112yes—
336C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2815
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#6501C2semidirect
2. C2×#6447C2semidirect
3. C7×#392C2semidirect
4. #89995C2semidirect
5. #89995C2semidirect
6. #89995C2semidirect
7. #89732C2semidirect
8. #89725C2semidirect
9. #89724C2semidirect
10. #89446C2semidirect
11. #89436C2semidirect
67. C2×C4×D7D4semidirect
87. #392D7semidirect
293. C7#2815semidirect

Maximal subgroups

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

subgroupindex
C2×#65012
C2×#64472
C7×#3922
#899952
#897322
#897252
#897242
#894462
#894362
#28157