Finite Group Databasebeta

← #11734606

Subgroups — #11734606

Order 896.

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

normal subgroups only

This group has 578 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×#6457C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic, complemented
4#89995C211yesmaximal, characteristic, complemented
5#89758C211yesmaximal, characteristic, complemented
6#89733C212yesmaximal, complemented
7#89732C212yesmaximal, complemented
8#89731C212yesmaximal, complemented
9#89720C212yesmaximal, complemented
10#89449C211yesmaximal, characteristic, complemented
11#89439C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C22⋊D7C2212yes—
15C22×D28C2211yescharacteristic
16C22×C4×D7C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2212yes—
22#6501—24no—
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2212yes—
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2212yes—
32#6457—24no—
33C7×#95C2211yescharacteristic
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2813—71nomaximal, Sylow
38C24×C7C2312yes—
39C23×D7C2311yescharacteristic
40C23×D7C2311yescharacteristic
41C2×C7×D4—22no—
42C2×C7×D4—42no—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2312yes—
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—24no—
48C2×C22⋊D7—42no—
49C2×C22⋊D7—42no—
50C2×C22⋊D7—42no—
51C2×C22⋊D7—42no—
52C2×C22⋊D7—42no—
53C22×Dic7C2312yes—
54C22×Dic7C2312yes—
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C2×D28—22no—
58C2×D28—42no—
59C2×C4×D7—42no—
60C2×C4×D7D414yescomplemented
61C7×C4⋊C4—22no—
62C7×C22⋊C4—42no—
63C7×C22⋊C4—42no—
64C22⋊Dic7—24no—
65C22⋊Dic7—42no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—22no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4—42no—
70(C2×D7)⋊C4—42no—
71C4⋊Dic7—22no—
72C4⋊Dic7—42no—
73Dic7⋊C4—22no—
74Dic7⋊C4—42no—
75C2×#123—71no—
76C2×#121—71no—
77#400—71no—
78#399—71no—
79#396—72no—
80#394—72no—
81#394—72no—
82#394D711yescharacteristic, complemented
83#392—71no—
84#392—72no—
85#380—71no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—22no—
91C23×C7—22no—
92C23×C7—22no—
93C23×C7C2411yescharacteristic
94C22×D7—21no—
95C22×D7—22no—
96C22×D7—22no—
97C22×D7—21no—
98C22×D7—21no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—21no—
102C22×D7—21no—
103C22×D7C2×D412yes—
104C7×D4—44no—
105C2×C4×C7—22no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—22no—
111C2×C4×C7—21no—
112C2×C4×C7—22no—
113C2×C4×C7—21no—
114C2×C4×C7C2×D412yes—
115C22⋊D7—48no—
116C22⋊D7—48no—
117C2×Dic7—22no—
118C2×Dic7—21no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7C2×D412yes—
127D28—44no—
128C4×D7—44no—
129C22×D4—72no—
130C22×D4—72no—
131C22×D4—71no—
132C22×D4C2×D711yescharacteristic
133C23×C4—71no—
134#123—144no—
135#121—144no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4—71no—
138C2×C4⋊C4—72no—
139C2×C4⋊C4—72no—
140C2×C4⋊C4C2×D711yescharacteristic
141C2×C22⋊C4—72no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—71no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4—72no—
148C2×C22⋊C4C2×D712yes—
149C2×C22⋊C4C2×D712yes—
150#95—71no—
151#95—72no—
152#95—72no—
153#95C2×D711yescharacteristic
154C22×C7—42no—
155C22×C7—42no—
156C22×C7—42no—
157C22×C7—42no—
158C22×C7—42no—
159C22×C7—42no—
160C22×C7—42no—
161C22×C7C22×D411yescharacteristic
162C22×C7H(5,𝔽2)12yes—
163C22×C7H(5,𝔽2)11yescharacteristic
164C22×C7H(5,𝔽2)12yes—
165C22×C72−1+411yescharacteristic
166C2×D7—22no—
167C2×D7—22no—
168C2×D7—22no—
169C2×D7—42no—
170C2×D7—42no—
171C2×D7—41no—
172C2×D7—41no—
173C2×D7—41no—
174C2×D7—42no—
175C2×D7—42no—
176C4×C7—22no—
177C4×C7—41no—
178C4×C7—41no—
179C4×C7—42no—
180Dic7—22no—
181Dic7—41no—
182Dic7—42no—
183Dic7—42no—
184C24—71no—
185C24—71no—
186C24C22×D712yes—
187C2×D4—22no—
188C2×D4—42no—
189C2×D4—142no—
190C2×D4—144no—
191C2×D4—282no—
192C2×D4—282no—
193C2×D4—282no—
194C2×D4—282no—
195C2×D4—282no—
196C2×D4—282no—
197C22×C4—71no—
198C22×C4—74no—
199C22×C4—72no—
200C22×C4—72no—
201C22×C4—71no—
202C22×C4—282no—
203C22×C4C22×D711yescharacteristic
204C22×C4C22×D711yescharacteristic
205C22×C4C22×D712yes—
206C22×C4C22×D711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—142no—
209C4⋊C4—142no—
210C4⋊C4—282no—
211C4⋊C4—282no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—142no—
215C22⋊C4—142no—
216C22⋊C4—144no—
217C22⋊C4—142no—
218C22⋊C4—282no—
219C22⋊C4—282no—
220C22⋊C4—282no—
221C2×C7—82no—
222C2×C7#53411yescharacteristic
223C2×C7#53511yescharacteristic
224C2×C7#53711yescharacteristic
225C2×C7#56012yes—
226C2×C7#56212yes—
227D7—42no—
228D7—81no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—72no—
237C23—141no—
238C23—142no—
239C23—142no—
240C23—141no—
241C23—141no—
242C23—142no—
243C23—141no—
244C23—142no—
245C23—141no—
246C23C23×D711yescharacteristic
247D4—44no—
248D4—284no—
249D4—288no—
250D4—288no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—72no—
261C2×C4—142no—
262C2×C4—141no—
263C2×C4—142no—
264C2×C4—142no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—142no—
269C2×C4—142no—
270C2×C4—284no—
271C2×C4D4×D712yes—
272C7#281311yescharacteristic, complemented, Sylow
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—142no—
281C22—142no—
282C22—142no—
283C22—282no—
284C22—282no—
285C22—281no—
286C22—281no—
287C22—282no—
288C22—282no—
289C22—281no—
290C22C2×D4×D711yescharacteristic
291C22#654812yes—
292C22#654811yescharacteristic
293C22#655312yes—
294C22#655411yescharacteristic
295C4—22no—
296C4—41no—
297C4—41no—
298C4—42no—
299C4—142no—
300C4—282no—
301C4—282no—
302C4—281no—
303C2—82no—
304C2—282no—
305C2—561no—
306C2#9028011yescharacteristic
307C2#9029911yescharacteristic
308C2#9031112yes—
309C2#9035812yes—
310C2#9036511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
hypercenter#394D7
hypocenterC7#2813
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#6501C2semidirect
2. C2×#6457C2semidirect
3. C7×#394C2semidirect
4. #89995C2semidirect
5. #89758C2semidirect
6. #89733C2semidirect
7. #89732C2semidirect
8. #89731C2semidirect
9. #89720C2semidirect
10. #89449C2semidirect
11. #89439C2semidirect
60. C2×C4×D7D4semidirect
82. #394D7semidirect
272. C7#2813semidirect

Maximal subgroups

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

subgroupindex
C2×#65012
C2×#64572
C7×#3942
#899952
#897582
#897332
#897322
#897312
#897202
#894492
#894392
#28137