Finite Group Databasebeta

← #11740533

Subgroups — #11740533

Order 896.

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

normal subgroups only

This group has 470 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×C7×#126C211yesmaximal, characteristic
2C2×C4×Dic14C211yesmaximal, characteristic, complemented
3#89758C211yesmaximal, characteristic, complemented
4#89751C211yesmaximal, characteristic, complemented
5#89750C212yesmaximal, complemented
6#89744C212yesmaximal, complemented
7#89721C211yesmaximal, characteristic
8#89719C211yesmaximal, characteristic
9#89717C212yesmaximal
10#89716C212yesmaximal
11#89704C211yesmaximal, characteristic, complemented
12C22×Dic14C2211yescharacteristic
13C7×#126—24no—
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C42×C7C2211yescharacteristic
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2212yes—
23C2×C4⋊Dic7C2211yescharacteristic
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2212yes—
29C4×Dic14—24no—
30C7⋊#95C2212yes—
31C7⋊#95C2211yescharacteristic
32C7⋊#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2604—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C2×Dic14—22no—
50C2×Dic14—24no—
51C7×C4⋊C4—22no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4—22no—
54C7×C22⋊C4—22no—
55C7×C22⋊C4—22no—
56C7×C22⋊C4—22no—
57C42×C7D414yes—
58C22⋊Dic7—24no—
59C22⋊Dic7—24no—
60C4⋊Dic7—22no—
61Dic7⋊C4—24no—
62Dic7⋊C4—24no—
63Dic7⋊C4—24no—
64C4×Dic7—24no—
65C4×Dic7—24no—
66C2×#126D711yescharacteristic
67C2×C4×Q8—71no—
68#400—71no—
69#399—71no—
70#395—71no—
71#393—71no—
72#389—72no—
73#388—72no—
74#388—72no—
75#387—72no—
76#376—71no—
77C23×C7—21no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7—21no—
81C23×C7—21no—
82C23×C7C2411yescharacteristic
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7C2×D412yes—
95C2×C4×C7C2×D412yes—
96C2×C4×C7C2×D412yes—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7#4114yes—
108C2×Dic7#4114yes—
109Dic14—28no—
110C22×Q8—71no—
111#126—24no—
112C4×Q8—144no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—72no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4C2×D711yescharacteristic
118C2×C4⋊C4C2×D711yescharacteristic
119C2×C4⋊C4C2×D711yescharacteristic
120C2×C22⋊C4—72no—
121C2×C22⋊C4—72no—
122C2×C22⋊C4C2×D711yescharacteristic
123C2×C22⋊C4C2×D711yescharacteristic
124C2×C22⋊C4C2×D711yescharacteristic
125C2×C42—72no—
126C2×C42—72no—
127C2×C42C2×D711yescharacteristic
128#95—72no—
129#95—71no—
130#95—71no—
131#95—72no—
132#95—72no—
133#95—72no—
134#95—72no—
135C22×C7—41no—
136C22×C7—41no—
137C22×C7—41no—
138C22×C7—42no—
139C22×C7—42no—
140C22×C7C22×D411yescharacteristic
141C22×C7C2×#4112yes—
142C22×C7C2×#4112yes—
143C22×C72−1+411yescharacteristic
144C22×C72−1+411yescharacteristic
145C4×C7—22no—
146C4×C7—22no—
147C4×C7—22no—
148C4×C7—41no—
149C4×C7—41no—
150C4×C7—41no—
151Dic7—24no—
152Dic7—24no—
153Dic7—42no—
154Dic7—42no—
155C24C22×D711yescharacteristic
156C2×Q8—144no—
157C2×Q8—142no—
158C22×C4—72no—
159C22×C4—72no—
160C22×C4—72no—
161C22×C4—72no—
162C22×C4C22×D711yescharacteristic
163C22×C4C22×D711yescharacteristic
164C22×C4C22×D711yescharacteristic
165C22×C4C22×D711yescharacteristic
166C22×C4C22×D711yescharacteristic
167C22×C4C22×D711yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—144no—
172C4⋊C4—144no—
173C4⋊C4—144no—
174C4⋊C4—142no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—144no—
179C22⋊C4—144no—
180C42—144no—
181C42—144no—
182C42C22⋊D714yes—
183C2×C7—81no—
184C2×C7#52511yescharacteristic
185C2×C7#53211yescharacteristic
186C2×C7#53611yescharacteristic
187C2×C7#55212yes—
188C2×C7#55612yes—
189C23—22no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23C23×D711yescharacteristic
195Q8—148no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—74no—
208C2×C4—74no—
209C2×C4—142no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4C2×C22⋊D712yes—
220C2×C4C2×C22⋊D712yes—
221C2×C4C2×C22⋊D712yes—
222C7#260411yescharacteristic, complemented, Sylow
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22D7×#4112yes—
229C22D7×#4112yes—
230C22#655411yescharacteristic
231C22#655411yescharacteristic
232C22C22×C22⋊D711yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—144no—
240C4—144no—
241C4—282no—
242C4—282no—
243C2—81no—
244C2#9039312yes—
245C2#9039912yes—
246C2#9051611yescharacteristic
247C2#9051911yescharacteristic
248C2#9052311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#126C2
prime radicalC24×C7C23
hypercenterC2×#126D7
hypocenterC7#2604
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic14C2semidirect
3. #89758C2semidirect
4. #89751C2semidirect
5. #89750C2semidirect
6. #89744C2semidirect
11. #89704C2semidirect
222. C7#2604semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1262
C2×C4×Dic142
#897582
#897512
#897502
#897442
#897212
#897192
#897172
#897162
#897042
#26047