Finite Group Databasebeta

← #11735761

Subgroups — #11735761

Order 896.

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

normal subgroups only

This group has 444 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×#6460C211yesmaximal, characteristic
2C2×#6459C211yesmaximal, characteristic
3C7×#400C211yesmaximal, characteristic, complemented
4#89754C211yesmaximal, characteristic, complemented
5#89751C211yesmaximal, characteristic, complemented
6#89447C212yesmaximal
7#89446C212yesmaximal
8#89436C211yesmaximal, characteristic
9#89432C212yesmaximal, complemented
10#89428C212yesmaximal, complemented
11#89428C211yesmaximal, characteristic, complemented
12C22×C4×D7C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×(C2×D7)⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×C4⋊Dic7C2212yes—
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26#6460—24no—
27#6459—24no—
28C7×#95C2211yescharacteristic
29C7×#95C2212yes—
30C7×#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2817—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×Dic7C2312yes—
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C2×C4×D7—42no—
50C2×C4×D7D414yes—
51C2×Dic14—22no—
52C2×Dic14—22no—
53C2×Dic14—42no—
54C2×Dic14—42no—
55C7×C4⋊C4—22no—
56C7×C4⋊C4—22no—
57(C2×D7)⋊C4—22no—
58(C2×D7)⋊C4—22no—
59(C2×D7)⋊C4—24no—
60(C2×D7)⋊C4—42no—
61C4⋊Dic7—24no—
62C4⋊Dic7—42no—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65Dic7⋊C4—42no—
66C2×C22⋊Q8—71no—
67C2×C22⋊Q8—71no—
68#400—71no—
69#400—72no—
70#400D711yescharacteristic, complemented
71#397—72no—
72#397—72no—
73#395—71no—
74#395—72no—
75#395—71no—
76#381—71no—
77C23×C7C2411yescharacteristic
78C22×D7—22no—
79C22×D7—21no—
80C22×D7—21no—
81C22×D7—22no—
82C22×D7C2×D412yes—
83C2×C4×C7—22no—
84C2×C4×C7—21no—
85C2×C4×C7—22no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7—24no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7C2×D412yes—
95C2×Dic7—22no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7—21no—
106C2×Dic7—22no—
107C2×Dic7C2×D412yes—
108C4×D7—44no—
109Dic14—44no—
110Dic14—44no—
111C22×Q8—71no—
112C22×Q8—71no—
113C23×C4—71no—
114C22⋊Q8—144no—
115C22⋊Q8—144no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4C2×D711yescharacteristic
122C2×C4⋊C4C2×D711yescharacteristic
123C2×C22⋊C4—71no—
124C2×C22⋊C4—71no—
125C2×C22⋊C4—72no—
126C2×C22⋊C4—72no—
127#95—72no—
128#95—71no—
129#95—72no—
130#95—72no—
131#95—72no—
132#95—72no—
133#95C2×D711yescharacteristic
134#95C2×D712yes—
135#95C2×D712yes—
136C22×C7C22×D411yescharacteristic
137C22×C7H(5,𝔽2)12yes—
138C22×C72−1+411yescharacteristic
139C22×C72−1+412yes—
140C22×C72−1+411yescharacteristic
141C2×D7—22no—
142C2×D7—22no—
143C2×D7—22no—
144C2×D7—42no—
145C2×D7—42no—
146C4×C7—22no—
147C4×C7—41no—
148C4×C7—41no—
149C4×C7—42no—
150C4×C7—42no—
151Dic7—22no—
152Dic7—41no—
153Dic7—42no—
154Dic7—41no—
155Dic7—42no—
156C24—71no—
157C2×Q8—142no—
158C2×Q8—142no—
159C2×Q8—282no—
160C2×Q8—282no—
161C22×C4—71no—
162C22×C4—74no—
163C22×C4—72no—
164C22×C4—71no—
165C22×C4—71no—
166C22×C4—72no—
167C22×C4—282no—
168C22×C4C22×D712yes—
169C22×C4C22×D711yescharacteristic
170C22×C4C22×D712yes—
171C22×C4C22×D711yescharacteristic
172C22×C4C22×D711yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—142no—
176C4⋊C4—144no—
177C4⋊C4—142no—
178C4⋊C4—282no—
179C4⋊C4—282no—
180C22⋊C4—142no—
181C22⋊C4—144no—
182C22⋊C4—142no—
183C22⋊C4—282no—
184C2×C7#53411yescharacteristic
185C2×C7#53611yescharacteristic
186C2×C7#53611yescharacteristic
187C2×C7#56312yes—
188C2×C7#56312yes—
189D7—42no—
190C23—72no—
191C23—141no—
192C23—142no—
193C23—141no—
194C23—142no—
195C23C23×D711yescharacteristic
196Q8—284no—
197Q8—284no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—72no—
210C2×C4—142no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—141no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—141no—
222C2×C4—284no—
223C2×C4D4×D712yes—
224C7#281711yescharacteristic, complemented, Sylow
225C22—142no—
226C22—142no—
227C22—142no—
228C22—282no—
229C22—282no—
230C22C2×D4×D711yescharacteristic
231C22#655111yescharacteristic
232C22#655312yes—
233C22#655412yes—
234C22#655411yescharacteristic
235C4—22no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—42no—
240C4—142no—
241C4—282no—
242C4—281no—
243C4—282no—
244C4—281no—
245C2—282no—
246C2#9031911yescharacteristic
247C2#9034411yescharacteristic
248C2#9036611yescharacteristic
249C2#9039012yes—
250C2#9040312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC23×D7C23
hypercenter#400D7
hypocenterC7#2817
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. #89754C2semidirect
5. #89751C2semidirect
9. #89432C2semidirect
10. #89428C2semidirect
11. #89428C2semidirect
70. #400D7semidirect
224. C7#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×#64592
C7×#4002
#897542
#897512
#894472
#894462
#894362
#894322
#894282
#28177