Finite Group Databasebeta

← #11735597

Subgroups — #11735597

Order 896.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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×#6459C211yesmaximal, characteristic
2C2×D7×C4⋊C4C211yesmaximal, characteristic
3C7×#400C211yesmaximal, characteristic, complemented
4#89758C211yesmaximal, characteristic, complemented
5#89754C211yesmaximal, characteristic, complemented
6#89438C211yesmaximal, characteristic
7#89436C212yesmaximal
8#89436C212yesmaximal
9#89431C211yesmaximal, characteristic, complemented
10#89422C212yesmaximal, complemented
11#89418C212yesmaximal, complemented
12C22×C4×D7C2212yes—
13C22×C4×D7C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×(C2×D7)⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2211yescharacteristic
24C2×C4×Dic7C2212yes—
25#6459—24no—
26D7×C4⋊C4—24no—
27C7×#95C2212yes—
28C7×#95C2212yes—
29C7×#95C2211yescharacteristic
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2503—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×Dic7C2312yes—
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×C4×D7—24no—
50C2×C4×D7—24no—
51C2×C4×D7—28no—
52C2×C4×D7D414yes—
53C2×Dic14—22no—
54C2×Dic14—24no—
55C7×C4⋊C4—22no—
56C7×C4⋊C4—22no—
57(C2×D7)⋊C4—22no—
58(C2×D7)⋊C4—22no—
59(C2×D7)⋊C4—24no—
60C4⋊Dic7—22no—
61Dic7⋊C4—22no—
62Dic7⋊C4—24no—
63Dic7⋊C4—22no—
64C4×Dic7—24no—
65C2×C22⋊Q8—71no—
66C22×C4⋊C4—71no—
67#400—71no—
68#400—71no—
69#400—71no—
70#400D711yescharacteristic, complemented
71#391—72no—
72#382—72no—
73#381—72no—
74#381—72no—
75C4×C22⋊C4—71no—
76C23×C7C2411yescharacteristic
77C22×D7—22no—
78C22×D7—21no—
79C22×D7—21no—
80C22×D7C2×D412yes—
81C22×D7#4114yes—
82C2×C4×C7—22no—
83C2×C4×C7—21no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—24no—
90C2×C4×C7—21no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7C2×D412yes—
94C2×Dic7—21no—
95C2×Dic7—22no—
96C2×Dic7—21no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—24no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7—21no—
104C2×Dic7C2×D412yes—
105C2×Dic7#4114yes—
106C4×D7—28no—
107C4×D7—48no—
108Dic14—44no—
109C22×Q8—71no—
110C23×C4—71no—
111C23×C4—72no—
112C22⋊Q8—144no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—144no—
118C2×C4⋊C4C2×D711yescharacteristic
119C2×C4⋊C4C2×D711yescharacteristic
120C2×C22⋊C4—71no—
121C2×C22⋊C4—71no—
122C2×C22⋊C4—72no—
123C2×C42—72no—
124#95—71no—
125#95—72no—
126#95—72no—
127#95—72no—
128#95—72no—
129#95—72no—
130#95—72no—
131#95C2×D711yescharacteristic
132#95C2×D712yes—
133#95C2×D712yes—
134C22×C7C22×D411yescharacteristic
135C22×C7C2×#4112yes—
136C22×C7C2×#4112yes—
137C22×C72−1+411yescharacteristic
138C22×C72−1+411yescharacteristic
139C2×D7—22no—
140C2×D7—22no—
141C2×D7—24no—
142C2×D7—24no—
143C2×D7#12414yes—
144C4×C7—22no—
145C4×C7—41no—
146C4×C7—42no—
147C4×C7—41no—
148C4×C7—42no—
149Dic7—41no—
150Dic7—42no—
151Dic7—41no—
152Dic7—42no—
153Dic7#12314yes—
154C24—71no—
155C2×Q8—142no—
156C2×Q8—144no—
157C22×C4—71no—
158C22×C4—74no—
159C22×C4—72no—
160C22×C4—71no—
161C22×C4—71no—
162C22×C4—72no—
163C22×C4—144no—
164C22×C4—144no—
165C22×C4—148no—
166C22×C4C22×D712yes—
167C22×C4C22×D711yescharacteristic
168C22×C4C22×D711yescharacteristic
169C22×C4C22×D712yes—
170C22×C4C22×D711yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—144no—
174C4⋊C4—142no—
175C4⋊C4—142no—
176C4⋊C4—142no—
177C22⋊C4—144no—
178C22⋊C4—142no—
179C22⋊C4—142no—
180C42—144no—
181C2×C7C2×#12311yescharacteristic
182C2×C7C2×#12411yescharacteristic
183C2×C7#53611yescharacteristic
184C2×C7#55212yes—
185C2×C7#55212yes—
186D7—24no—
187C23—72no—
188C23—74no—
189C23—141no—
190C23—142no—
191C23—141no—
192C23C23×D711yescharacteristic
193Q8—284no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—72no—
206C2×C4—74no—
207C2×C4—148no—
208C2×C4—141no—
209C2×C4—141no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—141no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—144no—
216C2×C4—142no—
217C2×C4—141no—
218C2×C4—288no—
219C2×C4D4×D712yes—
220C7#250311yescharacteristic, complemented, Sylow
221C22—74no—
222C22—144no—
223C22—144no—
224C22—142no—
225C22—142no—
226C22C2×D4×D711yescharacteristic
227C22#655111yescharacteristic
228C22D7×#4112yes—
229C22D7×#4112yes—
230C22#655411yescharacteristic
231C4—22no—
232C4—42no—
233C4—41no—
234C4—41no—
235C4—42no—
236C4—74no—
237C4—282no—
238C4—281no—
239C4—281no—
240C4—282no—
241C2—144no—
242C2#9031911yescharacteristic
243C2D7×#12311yescharacteristic
244C2D7×#12411yescharacteristic
245C2#9038412yes—
246C2#9039912yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC23×D7C23
hypercenter#400D7
hypocenterC7#2503
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C7×#400C2semidirect
4. #89758C2semidirect
5. #89754C2semidirect
9. #89431C2semidirect
10. #89422C2semidirect
11. #89418C2semidirect
70. #400D7semidirect
220. C7#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×D7×C4⋊C42
C7×#4002
#897582
#897542
#894382
#894362
#894312
#894222
#894182
#25037