Finite Group Databasebeta

← #11735482

Subgroups — #11735482

Order 896.

676 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 171 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×#6459C212yesmaximal, complemented
2C2×#6451C211yesmaximal, characteristic, complemented
3C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
4C2×#6441C212yesmaximal, complemented
5C7×#399C211yesmaximal, characteristic, complemented
6#89764C211yesmaximal, characteristic, complemented
7#89748C212yesmaximal, complemented
8#89729C212yesmaximal
9#89721C211yesmaximal, characteristic, complemented
10#89449C211yesmaximal, characteristic, complemented
11#89421C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C23×Dic7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C22×Dic14C2212yes—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2212yes—
18C2×C7×C22⋊C4C2212yes—
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C22⋊Dic7C2212yes—
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×C4⋊Dic7C2212yes—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2212yes—
29C2×Dic7⋊C4C2212yes—
30C2×Dic7⋊C4C2212yes—
31C2×C4×Dic7C2212yes—
32C2×C4×Dic7C2211yescharacteristic
33#6459—28no—
34#6451—24no—
35(C22⋊D7)⋊C4—24no—
36#6441—28no—
37C7×#95C2211yescharacteristic
38C7⋊#95C2211yescharacteristic
39#2542—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C23×D7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2312yes—
46C2×C22⋊D7—22no—
47C2×C22⋊D7—22no—
48C2×C22⋊D7Q814yescomplemented
49C22×Dic7—24no—
50C22×Dic7—22no—
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2312yes—
53C22×Dic7C2312yes—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C2×C4×D7—22no—
57C2×C4×D7—24no—
58C2×Dic14—24no—
59C2×Dic14—24no—
60C2×Dic14—24no—
61C7×C4⋊C4—22no—
62C7×C4⋊C4—24no—
63C7×C22⋊C4—24no—
64C7×C22⋊C4—22no—
65C22⋊Dic7—24no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—24no—
69C4⋊Dic7—24no—
70C4⋊Dic7—22no—
71Dic7⋊C4—24no—
72Dic7⋊C4—22no—
73Dic7⋊C4—24no—
74Dic7⋊C4—24no—
75Dic7⋊C4D418yescomplemented
76C4×Dic7—24no—
77C4×Dic7D414yes—
78C2×C4⋊Q8—71no—
79C2×C22⋊Q8—72no—
80C2×C22⋊Q8—72no—
81C2×C4×D4—71no—
82#399—71no—
83#399—71no—
84#399—71no—
85#399D711yescharacteristic, complemented
86#390—72no—
87#384—72no—
88C4×C4⋊C4—71no—
89C23×C7—22no—
90C23×C7—21no—
91C23×C7—21no—
92C23×C7—22no—
93C23×C7C2×Q812yes—
94C23×C7C2411yescharacteristic
95C22×D7—22no—
96C22×D7—21no—
97C22×D7—21no—
98C22×D7—22no—
99C22×D7C2×Q812yes—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—21no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7C2×D412yes—
109C2×C4×C7C2×D414yes—
110C22⋊D7—28no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—21no—
117C2×Dic7—22no—
118C2×Dic7—44no—
119C2×Dic7C2×D414yes—
120C2×Dic7C2×D412yes—
121C2×Dic7C2×D412yes—
122C2×Dic7C2×D414yes—
123C2×Dic7C2×Q812yes—
124C2×Dic7#4112yes—
125C4×D7—44no—
126Dic14—48no—
127C22×Q8—72no—
128C22×D4—71no—
129C23×C4—71no—
130C23×C4—71no—
131C4⋊Q8—144no—
132C22⋊Q8—148no—
133C22⋊Q8—148no—
134C4×D4—144no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—71no—
137C2×C4⋊C4—72no—
138C2×C4⋊C4—72no—
139C2×C4⋊C4—72no—
140C2×C4⋊C4—72no—
141C2×C4⋊C4—71no—
142C2×C4⋊C4C2×D712yes—
143C2×C4⋊C4C2×D711yescharacteristic
144C2×C22⋊C4—72no—
145C2×C22⋊C4—71no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4—72no—
148C2×C22⋊C4C2×D711yescharacteristic
149C2×C22⋊C4C2×D712yes—
150C2×C42—71no—
151C2×C42—72no—
152#95—71no—
153#95C2×D711yescharacteristic
154C22×C7—22no—
155C22×C7—24no—
156C22×C7—41no—
157C22×C7—41no—
158C22×C7—42no—
159C22×C7C22×D411yescharacteristic
160C22×C7C22×D412yes—
161C22×C7C22×Q811yescharacteristic
162C22×C7C2×#4111yescharacteristic
163C22×C72−1+412yes—
164C2×D7—22no—
165C2×D7—24no—
166C2×D7—41no—
167C2×D7—41no—
168C2×D7—42no—
169C4×C7—22no—
170C4×C7—24no—
171C4×C7—42no—
172C4×C7—41no—
173Dic7—24no—
174Dic7—42no—
175Dic7—41no—
176Dic7C22⋊Q814yescomplemented
177Dic7#12714yes—
178C24—71no—
179C24C22×D711yescharacteristic
180C2×Q8—144no—
181C2×Q8—144no—
182C2×Q8—144no—
183C2×D4—74no—
184C2×D4—142no—
185C2×D4—142no—
186C22×C4—71no—
187C22×C4—71no—
188C22×C4—72no—
189C22×C4—72no—
190C22×C4—71no—
191C22×C4—142no—
192C22×C4—144no—
193C22×C4—142no—
194C22×C4—144no—
195C22×C4C22×D712yes—
196C22×C4C22×D711yescharacteristic
197C22×C4C22×D711yescharacteristic
198C22×C4C22×D712yes—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—78no—
202C4⋊C4—142no—
203C4⋊C4—144no—
204C4⋊C4—144no—
205C4⋊C4—144no—
206C4⋊C4—144no—
207C4⋊C4—142no—
208C22⋊C4—22no—
209C22⋊C4—24no—
210C22⋊C4—144no—
211C22⋊C4—144no—
212C22⋊C4—142no—
213C22⋊C4—144no—
214C42—74no—
215C42—144no—
216C2×C7—42no—
217C2×C7C2×C22⋊Q811yescharacteristic
218C2×C7C2×#12711yescharacteristic
219C2×C7#53611yescharacteristic
220C2×C7#54712yes—
221C2×C7D4×Q812yes—
222D7—42no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—72no—
228C23—142no—
229C23—141no—
230C23—141no—
231C23—142no—
232C23Q8×D712yes—
233C23C23×D711yescharacteristic
234Q8—288no—
235D4—148no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—72no—
245C2×C4—72no—
246C2×C4—74no—
247C2×C4—72no—
248C2×C4—72no—
249C2×C4—74no—
250C2×C4—142no—
251C2×C4—142no—
252C2×C4—141no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—142no—
256C2×C4—142no—
257C2×C4—284no—
258C2×C4—284no—
259C2×C4D4×D712yes—
260C2×C4D4×D714yes—
261C7#254211yescharacteristic, complemented, Sylow
262C22—22no—
263C22—24no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—142no—
268C22—144no—
269C22—281no—
270C22—281no—
271C22—282no—
272C22C2×D4×D711yescharacteristic
273C22C2×D4×D712yes—
274C22C2×Q8×D711yescharacteristic
275C22D7×#4111yescharacteristic
276C22#655412yes—
277C4—22no—
278C4—24no—
279C4—41no—
280C4—42no—
281C4—74no—
282C4—74no—
283C4—144no—
284C4—281no—
285C4—282no—
286C2—42no—
287C2—282no—
288C2D7×C22⋊Q811yescharacteristic
289C2#9032312yes—
290C2#9033911yescharacteristic
291C2#9034612yes—
292C2D7×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC22×C22⋊D7C22
hypercenter#399D7
hypocenterC7#2542
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6459C2semidirect
2. C2×#6451C2semidirect
4. C2×#6441C2semidirect
5. C7×#399C2semidirect
6. #89764C2semidirect
7. #89748C2semidirect
9. #89721C2semidirect
10. #89449C2semidirect
11. #89421C2semidirect
48. C2×C22⋊D7Q8semidirect
75. Dic7⋊C4D4semidirect
85. #399D7semidirect
176. Dic7C22⋊Q8semidirect
261. C7#2542semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×#64512
C2×(C22⋊D7)⋊C42
C2×#64412
C7×#3992
#897642
#897482
#897292
#897212
#894492
#894212
#25427