Finite Group Databasebeta

← #11735268

Subgroups — #11735268

Order 896.

416 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 131 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
2C2×#6450C211yesmaximal, characteristic
3C7×#398C211yesmaximal, characteristic
4#89758C211yesmaximal, characteristic
5#89753C212yesmaximal
6#89752C211yesmaximal, characteristic
7#89751C212yesmaximal
8#89431C211yesmaximal, characteristic
9#89430C212yesmaximal
10#89427C212yesmaximal
11#89423C211yesmaximal, characteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2212yes—
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2211yescharacteristic
18C2×Dic7⋊C4C2211yescharacteristic
19C2×Dic7⋊C4C2211yescharacteristic
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×C4×Dic7C2211yescharacteristic
23C2×C4×Dic7C2211yescharacteristic
24C2×C4×Dic7C2211yescharacteristic
25#6453—24no—
26#6450—24no—
27C7×#95C2211yescharacteristic
28C7×#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2634—71nomaximal, Sylow
38C22×C4×C7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×Dic7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×Dic14—22no—
50C2×Dic14—22no—
51C2×Dic14—22no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4—22no—
54C7×C4⋊C4—24no—
55C4⋊Dic7—22no—
56C4⋊Dic7—24no—
57Dic7⋊C4—22no—
58Dic7⋊C4—22no—
59Dic7⋊C4—24no—
60Dic7⋊C4—24no—
61C4×Dic7—22no—
62C4×Dic7—22no—
63C4×Dic7D414yes—
64C2×#125—71no—
65C2×C4×Q8—71no—
66#400—71no—
67#400—72no—
68#400—71no—
69#398—71no—
70#398D711yescharacteristic
71#395—72no—
72#387—72no—
73#387—72no—
74#383—71no—
75C23×C7C2411yescharacteristic
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—22no—
79C2×C4×C7—21no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7C2×D412yes—
88C2×C4×C7#4112yes—
89C2×Dic7—21no—
90C2×Dic7—21no—
91C2×Dic7—21no—
92C2×Dic7—21no—
93C2×Dic7—22no—
94C2×Dic7—21no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—24no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7C2×D412yes—
102C2×Dic7C2×D412yes—
103C2×Dic7#4112yes—
104Dic14—28no—
105C22×Q8—71no—
106#125—144no—
107C4×Q8—144no—
108C2×C4⋊C4—71no—
109C2×C4⋊C4—71no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—71no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4C2×D711yescharacteristic
115C2×C4⋊C4C2×D711yescharacteristic
116C2×C4⋊C4C2×D712yes—
117C2×C42—71no—
118C2×C42—71no—
119C2×C42—71no—
120#95—72no—
121#95—72no—
122#95—71no—
123#95—72no—
124#95—72no—
125#95—72no—
126#95—72no—
127#95—72no—
128#95C2×D711yescharacteristic
129#95C2×D712yes—
130C22×C7C22×D411yescharacteristic
131C22×C7C2×#4111yescharacteristic
132C22×C7C2×#4111yescharacteristic
133C22×C72−1+412yes—
134C22×C72−1+412yes—
135C4×C7—22no—
136C4×C7—22no—
137C4×C7—42no—
138C4×C7—41no—
139C4×C7—42no—
140Dic7—22no—
141Dic7—22no—
142Dic7—22no—
143Dic7—42no—
144Dic7—41no—
145Dic7—42no—
146C2×Q8—142no—
147C2×Q8—142no—
148C2×Q8—142no—
149C22×C4—71no—
150C22×C4—71no—
151C22×C4—71no—
152C22×C4—72no—
153C22×C4—71no—
154C22×C4—72no—
155C22×C4C22×D712yes—
156C22×C4C22×D711yescharacteristic
157C22×C4C22×D712yes—
158C22×C4C22×D711yescharacteristic
159C22×C4C22×D711yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—24no—
163C4⋊C4—144no—
164C4⋊C4—142no—
165C4⋊C4—144no—
166C4⋊C4—144no—
167C4⋊C4—142no—
168C4⋊C4—142no—
169C42—74no—
170C42—142no—
171C42—142no—
172C2×C7#52511yescharacteristic
173C2×C7#53611yescharacteristic
174C2×C7#53611yescharacteristic
175C2×C7#54112yes—
176C2×C7#54112yes—
177C23C23×D711yescharacteristic
178Q8—148no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—72no—
191C2×C4—72no—
192C2×C4—72no—
193C2×C4—141no—
194C2×C4—141no—
195C2×C4—141no—
196C2×C4—142no—
197C2×C4—141no—
198C2×C4—141no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—144no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4D4×D712yes—
206C2×C4#115912yes—
207C7#263411yescharacteristic, complemented, Sylow
208C22C2×D4×D711yescharacteristic
209C22C2×#115911yescharacteristic
210C22#655112yes—
211C22D7×#4111yescharacteristic
212C22#655412yes—
213C4—22no—
214C4—22no—
215C4—42no—
216C4—42no—
217C4—41no—
218C4—142no—
219C4—142no—
220C4—142no—
221C4—282no—
222C4—282no—
223C4—281no—
224C2#9029711yescharacteristic
225C2#9031612yes—
226C2#9033911yescharacteristic
227C2#9038712yes—
228C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#398C2
hypercenter#398D7
hypocenterC7#2634
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
207. C7#2634semidirect

Maximal subgroups

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

subgroupindex
C2×#64532
C2×#64502
C7×#3982
#897582
#897532
#897522
#897512
#894312
#894302
#894272
#894232
#26347