Finite Group Databasebeta

← #11740535

Subgroups — #11740535

Order 896.

434 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 137 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×#6433C211yesmaximal, characteristic, complemented
3#89758C211yesmaximal, characteristic, complemented
4#89756C212yesmaximal, complemented
5#89755C211yesmaximal, characteristic, complemented
6#89744C212yesmaximal, complemented
7#89725C211yesmaximal, characteristic
8#89721C211yesmaximal, characteristic
9#89720C212yesmaximal
10#89717C212yesmaximal
11C4×C4⋊Dic7C211yesmaximal, characteristic, complemented
12C7×#126—24no—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C42×C7C2211yescharacteristic
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2212yes—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2211yescharacteristic
24C2×C4⋊Dic7C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2212yes—
28C2×Dic7⋊C4C2212yes—
29C2×C4×Dic7C2212yes—
30#6433—24no—
31C7⋊#95C2211yescharacteristic
32C7⋊#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2664—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—
49C7×C4⋊C4—22no—
50C7×C4⋊C4—22no—
51C7×C4⋊C4—22no—
52C7×C22⋊C4—22no—
53C7×C22⋊C4—22no—
54C7×C22⋊C4—22no—
55C42×C7D414yes—
56C22⋊Dic7—24no—
57C22⋊Dic7—42no—
58C4⋊Dic7—22no—
59C4⋊Dic7—22no—
60C4⋊Dic7—24no—
61Dic7⋊C4—24no—
62Dic7⋊C4—24no—
63Dic7⋊C4—24no—
64Dic7⋊C4—42no—
65C4×Dic7—42no—
66C2×#126D711yescharacteristic
67C2×#125—71no—
68#400—71no—
69#399—71no—
70#398—71no—
71#398—72no—
72#397—71no—
73#396—72no—
74#389—72no—
75#388—72no—
76C4×C4⋊C4—71no—
77C23×C7—21no—
78C23×C7—22no—
79C23×C7—21no—
80C23×C7—22no—
81C23×C7—21no—
82C23×C7C2411yescharacteristic
83C2×C4×C7—22no—
84C2×C4×C7—21no—
85C2×C4×C7—22no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
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—22no—
108C2×Dic7#4114yes—
109#126—24no—
110#125—144no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—71no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—72no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4C2×D711yescharacteristic
119C2×C4⋊C4C2×D711yescharacteristic
120C2×C4⋊C4C2×D711yescharacteristic
121C2×C22⋊C4—72no—
122C2×C22⋊C4—72no—
123C2×C22⋊C4C2×D711yescharacteristic
124C2×C22⋊C4C2×D711yescharacteristic
125C2×C22⋊C4C2×D711yescharacteristic
126C2×C42—72no—
127C2×C42C2×D711yescharacteristic
128#95—71no—
129#95—71no—
130#95—72no—
131#95—72no—
132#95—72no—
133#95—72no—
134C22×C7—41no—
135C22×C7—41no—
136C22×C7—41no—
137C22×C7—42no—
138C22×C7—42no—
139C22×C7C22×D411yescharacteristic
140C22×C7C2×#4112yes—
141C22×C7H(5,𝔽2)11yescharacteristic
142C22×C72−1+412yes—
143C22×C72−1+411yescharacteristic
144C4×C7—22no—
145C4×C7—22no—
146C4×C7—22no—
147C4×C7—41no—
148C4×C7—41no—
149C4×C7—41no—
150Dic7—24no—
151Dic7—42no—
152Dic7—42no—
153Dic7—42no—
154C24C22×D711yescharacteristic
155C22×C4—72no—
156C22×C4—72no—
157C22×C4—72no—
158C22×C4—72no—
159C22×C4C22×D711yescharacteristic
160C22×C4C22×D711yescharacteristic
161C22×C4C22×D711yescharacteristic
162C22×C4C22×D711yescharacteristic
163C22×C4C22×D711yescharacteristic
164C22×C4C22×D711yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—144no—
169C4⋊C4—144no—
170C4⋊C4—144no—
171C4⋊C4—144no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—282no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—144no—
179C22⋊C4—282no—
180C42—282no—
181C42C22⋊D714yes—
182C2×C7—81no—
183C2×C7#53211yescharacteristic
184C2×C7#53611yescharacteristic
185C2×C7#53711yescharacteristic
186C2×C7#53912yes—
187C2×C7#54112yes—
188C23—22no—
189C23—21no—
190C23—21no—
191C23—22no—
192C23—21no—
193C23C23×D711yescharacteristic
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—74no—
206C2×C4—142no—
207C2×C4—142no—
208C2×C4—142no—
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×C4C2×C22⋊D712yes—
218C2×C4C2×C22⋊D712yes—
219C2×C4C2×C22⋊D712yes—
220C7#266411yescharacteristic, complemented, Sylow
221C22—42no—
222C22—42no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22D7×#4112yes—
227C22#655311yescharacteristic
228C22#655411yescharacteristic
229C22#655412yes—
230C22C22×C22⋊D711yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—144no—
238C4—282no—
239C4—282no—
240C4—282no—
241C2—81no—
242C2#9039212yes—
243C2#9039812yes—
244C2#9051611yescharacteristic
245C2#9052311yescharacteristic
246C2#9052411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#126C2
prime radicalC24×C7C23
hypercenterC2×#126D7
hypocenterC7#2664
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
2. C2×#6433C2semidirect
3. #89758C2semidirect
4. #89756C2semidirect
5. #89755C2semidirect
6. #89744C2semidirect
11. C4×C4⋊Dic7C2semidirect
220. C7#2664semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1262
C2×#64332
#897582
#897562
#897552
#897442
#897252
#897212
#897202
#897172
C4×C4⋊Dic72
#26647