Finite Group Databasebeta

← #11735504

Subgroups — #11735504

Order 896.

434 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 133 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×#6453C211yesmaximal, characteristic, complemented
2C2×#6442C211yesmaximal, characteristic
3C7×#399C211yesmaximal, characteristic
4#89757C212yesmaximal, complemented
5#89754C211yesmaximal, characteristic, complemented
6#89752C212yesmaximal, complemented
7#89725C211yesmaximal, characteristic
8#89723C212yesmaximal
9#89720C212yesmaximal
10#89433C211yesmaximal, characteristic, complemented
11#89421C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2212yes—
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2212yes—
16C2×C22⋊Dic7C2211yescharacteristic
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2211yescharacteristic
21C2×C4⋊Dic7C2212yes—
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29#6453—24no—
30#6442—24no—
31C7×#95C2211yescharacteristic
32C7⋊#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2664—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C7×C4⋊C4—22no—
50C7×C4⋊C4—24no—
51C7×C22⋊C4—22no—
52C7×C22⋊C4—24no—
53C22⋊Dic7—22no—
54C22⋊Dic7—22no—
55C22⋊Dic7—42no—
56C4⋊Dic7—22no—
57C4⋊Dic7—22no—
58C4⋊Dic7—24no—
59C4⋊Dic7—42no—
60Dic7⋊C4—22no—
61Dic7⋊C4—24no—
62Dic7⋊C4—22no—
63Dic7⋊C4—24no—
64C4×Dic7—42no—
65C4×Dic7D414yes—
66C2×#126—71no—
67C2×#125—71no—
68#400—71no—
69#399D711yescharacteristic
70#398—72no—
71#398—71no—
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—21no—
81C23×C7—22no—
82C23×C7C2411yescharacteristic
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7C2×D412yes—
92C2×C4×C7#4114yes—
93C2×Dic7—22no—
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—21no—
97C2×Dic7—22no—
98C2×Dic7—21no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7C2×D412yes—
108C2×Dic7C2×D412yes—
109#126—144no—
110#125—144no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—71no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4C2×D711yescharacteristic
120C2×C4⋊C4C2×D712yes—
121C2×C22⋊C4—71no—
122C2×C22⋊C4—71no—
123C2×C22⋊C4—72no—
124C2×C22⋊C4C2×D711yescharacteristic
125C2×C22⋊C4C2×D712yes—
126C2×C42—71no—
127C2×C42—72no—
128#95—71no—
129#95—72no—
130#95—72no—
131#95—72no—
132#95—72no—
133#95C2×D711yescharacteristic
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+411yescharacteristic
143C22×C72−1+412yes—
144C4×C7—22no—
145C4×C7—24no—
146C4×C7—41no—
147C4×C7—42no—
148Dic7—22no—
149Dic7—22no—
150Dic7—41no—
151Dic7—42no—
152Dic7—41no—
153Dic7—42no—
154C24C22×D711yescharacteristic
155C22×C4—71no—
156C22×C4—71no—
157C22×C4—72no—
158C22×C4—71no—
159C22×C4—71no—
160C22×C4—72no—
161C22×C4C22×D711yescharacteristic
162C22×C4C22×D712yes—
163C22×C4C22×D711yescharacteristic
164C22×C4C22×D712yes—
165C4⋊C4—22no—
166C4⋊C4—24no—
167C4⋊C4—144no—
168C4⋊C4—142no—
169C4⋊C4—144no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—142no—
173C4⋊C4—144no—
174C4⋊C4—282no—
175C22⋊C4—22no—
176C22⋊C4—24no—
177C22⋊C4—142no—
178C22⋊C4—142no—
179C22⋊C4—282no—
180C42—74no—
181C42—282no—
182C2×C7—81no—
183C2×C7#53211yescharacteristic
184C2×C7#53611yescharacteristic
185C2×C7#53711yescharacteristic
186C2×C7#53912yes—
187C2×C7#54112yes—
188C23—21no—
189C23—22no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23C23×D711yescharacteristic
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—72no—
203C2×C4—72no—
204C2×C4—142no—
205C2×C4—141no—
206C2×C4—141no—
207C2×C4—141no—
208C2×C4—141no—
209C2×C4—142no—
210C2×C4—142no—
211C2×C4—141no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—141no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4D4×D712yes—
219C2×C4#115914yes—
220C7#266411yescharacteristic, complemented, Sylow
221C22—41no—
222C22—42no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22C2×D4×D711yescharacteristic
227C22C2×#115912yes—
228C22#655111yescharacteristic
229C22#655311yescharacteristic
230C22#655412yes—
231C4—22no—
232C4—24no—
233C4—41no—
234C4—42no—
235C4—142no—
236C4—142no—
237C4—281no—
238C4—282no—
239C4—281no—
240C4—282no—
241C2—81no—
242C2#9031612yes—
243C2#9032011yescharacteristic
244C2#9033911yescharacteristic
245C2#9035612yes—
246C2#9040711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC24×C7C23
hypercenter#399D7
hypocenterC7#2664
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6453C2semidirect
4. #89757C2semidirect
5. #89754C2semidirect
6. #89752C2semidirect
10. #89433C2semidirect
11. #89421C2semidirect
220. C7#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#64532
C2×#64422
C7×#3992
#897572
#897542
#897522
#897252
#897232
#897202
#894332
#894212
#26647